Distributions of unramified extensions of global fields

DISTRIBUTIONS OF UNRAMIFIED EXTENSIONS OF GLOBAL FIELDS WILL SA WIN AND MELANIE MA TCHETT W OOD Abstra ct. Giv en a ﬁnite group Γ , w e pro v e results on the distribution of the prime to q | Γ | -part of fundamen tal groups of Γ -co v ers of the pro jective line P 1 F q o v er a ﬁnite ﬁeld F q as q → ∞ . Equiv alently , this is a result on the distribution of the Galois groups of maximal unramiﬁed extension of Γ -extensions of F q ( t ) , and thereby motiv ates a new conjecture on the distribution of Galois groups of maximal unramiﬁed extension of Γ -extensions of a n um ber ﬁeld. In particular, this allows us to see and predict the eﬀect of ro ots of unity in the base ﬁeld on such distributions. W e in troduce the idea to study these groups along with the class in their 3rd homology group that arises from Artin-V erdier Duality . This inv ariant reﬁnes the lifting inv ariant that, in the function ﬁeld setting, corresp onds to stable comp onents of Hurwitz space. One ma jor input into our function ﬁeld results is an application of our recen tly dev eloped metho ds to determine a distribution of groups (or more general algebraic structures) from its moments. W e prov e non-existence res ults in the num ber ﬁeld case that supp ort our conjectures in the case where our conjectures predict certain kinds of groups o ccur with probability zero. 1. Intr oduction This pap er is devoted to understanding the distribution of the Galois group of the max- imal unramiﬁed extension of a random num b er ﬁeld, and sp eciﬁcally to understanding the inﬂuence of the ro ots of unity on the distribution. W e mak e conjectural predictions, which are motiv ated b y function ﬁeld evidence in the q → ∞ limit, and also c heck ed against some n umerical evidence. It is useful to ﬁrst consider the ab elianization of the Galois group of the maximal unram- iﬁed extension. By class ﬁeld theory , this is the class group. Since work of Ac h ter [A c h06] and Malle [Mal08], it has b een understo o d that the ro ots of unity of a ﬁeld aﬀect the dis- tribution of the class groups of random extensions of a global ﬁeld. It w as not immediately clear ho w to give a formula for the probabilit y of obtaining a giv en class group that correctly accoun ts for ro ots of unit y , but a series of w orks ha v e made predictions for this in sp ecial cases compatible with numerical and function-ﬁeld evidence. Recently , the authors [SW23] ga ve a prediction for the distribution of the Sylo w p -subgroup of the class group of Galois extensions with Galois group Γ of a ﬁxed num b er ﬁeld, incorp orating ro ots of unity , as long as p is prime to the order of Γ . Predictions a voiding the inﬂuence of ro ots of unity for the distribution of the Galois group of the maximal unramiﬁed extension of a random extension of Q w ere recen tly made by Liu, W o o d, and Zurieck-Bro wn [L WZ24] in the case when the extension is split at ∞ and Liu and Willy ard [L W25] in general. Incorp orating ro ots of unity requires introducing several new ideas, whic h we will discuss shortly . W e no w giv e an example of a consequence of our main theorem in the function ﬁeld case, whic h concerns the maximal 2 -group quotient of the Galois group of the maximal unramiﬁed 1 extension, or, equiv alently , the Galois group of the composition of all unramiﬁed Galois extensions whose degree is a p o w er of 2 . Let q b e a prime p o wer, not divisible b y 2 or 3 , and let F q b e the ﬁnite ﬁeld of q elements. Let ∞ b e the unique place of F q ( t ) which do es not arise from a prime ideal of F q [ t ] , i.e. the place corresp onding to the v aluation v ( f ) = − deg f . F or K an extension of F q ( t ) , w e let rDisc K b e the pro duct of the norms of the places of F q ( t ) that ramify in K (the norm of a place v of F q ( t ) b eing q deg v ). Let E Z / 3 ( q m , F q ( t )) b e the set of Galois extensions of degree 3 of F q ( t ) (whic h necessarily ha v e Galois group Z / 3) , split at ∞ , such that rDisc K = q m . F or K an extension of F q ( t ) , let K un , 2 b e the comp osition of all ﬁnite Galois extensions of K which are everywhere unramiﬁed, split at all places lying ab ov e ∞ , and ha v e degree a p ow er of 2 . Th us Gal( K un , 2 /K ) is either a ﬁnite 2 -group or an inﬁnite pro- 2 group. The follo wing describ es the probabilit y of obtaining certain ﬁnite 2 -groups, and is a sample of a kind of corollary one can obtain from our main result. Theorem 1.1. L et H b e a ﬁnite 2 -gr oup. Then ther e exists p H ∈ [0 , 1] such that lim q →∞ q ≡ 3 mod 4 3 ∤ q lim sup b →∞ P m ≤ b   { K ∈ E Z / 3 ( q m , F q ( t )) | Gal( K un , 2 /K ) ∼ = H }   P m ≤ b   E Z / 3 ( q m , F q ( t ))   = p H and lim q →∞ q ≡ 3 mod 4 3 ∤ q lim inf b →∞ P m ≤ b   { K ∈ E Z / 3 ( q m , F q ( t )) | Gal( K un , 2 /K ) ∼ = H }   P m ≤ b   E Z / 3 ( q m , F q ( t ))   = p H and if | H | ≤ 8 then p H = ∞ Y k =0 (1 + 4 − k − 3 2 ) − 1 ·          1 if H is trivial 1 12 if H is the Klein four gr oup 7 2 5 · 3 if H is the eight-element quaternion gr oup 0 for al l other H with | H | ≤ 8 . Based on this, w e conjecture that the probabilit y that for K a random cyclic cubic exten- sion of Q (necessarily split at ∞ ), the Galois group of the comp osition of all ﬁnite Galois extension of K whic h are ev erywhere unramiﬁed (and th us split at all places lying ab o ve ∞ ), and hav e degree a p o wer of 2 , is giv en the same form ula p H . F or H the trivial group, ha ving the Galois group of K un , 2 isomorphic to H is equiv alent to the 2 -rank of the class group of K v anishing, and so in that case the conjecture agrees with an earlier conjecture of Malle [Mal08, Equation (1) on p. 2827]. F or H the Klein four group or eigh t-element quaternion group, this agrees with an earlier conjecture of Boston and Bush [BBH21, T able 7]. W e discuss the exact relationship to their w ork in more detail later. The conjecture of Boston and Bush was based on matc hing n umerical data (after guessing based on Malle’s conjecture [Mal10] that the probabilities are lik ely rational num b ers with small denominators times 1 6 Q ∞ k =0 (1 + 4 − k − 3 2 ) − 1 ) rather than an y theoretical mo del. Hence our conjecture also matc hes Boston and Bush’s data, as w ell as pro viding a conceptual explanation for their conjecture. 2 The condition 3 ∤ q in Theorem 1.1 is necessary to use comparison of fundamen tal groups b et ween zero and p ositiv e c haracteristic. The condition q ≡ 3 mo d 4 ensures that F q ( t ) has exactly 2 ro ots of unity of order a p o wer of 2 , the same n um b er that Q has, which is the case that mo dels the inﬂuence of the ro ots of unit y of Q on the statistics. How ev er, our main result in the function ﬁeld case allows for arbitrary groups of ro ots of unit y in the base ﬁeld. The restriction that | H | ≤ 8 is only there to give some sp eciﬁc examples. Our results giv e a metho d to enable the computation of p H for an y 2 -group H . An interesting consequence of Theorem 1.1 is that for H the trivial group, the Klein group, or the quaternion group, as long as q is suﬃciently large, coprime to 3 , and congruen t to 3 mo d 4 , there exist inﬁnitely many cyclic cubic extensions K/ F q ( t ) with Gal( K un , 2 /K ) ∼ = H . This result app ears to b e new. Many other existence results for function ﬁelds with Galois groups of maximal unramiﬁed extensions of a sp eciﬁc form follow from our main results. The extensive prior w ork on the distribution of class groups and their non-ab elian ana- logues, e.g. [CL84, CM90, Mal08, Mal10, Gar15, AM15, BW17, L T19, LST20, BB21, Liu22, L WZ24, L W25], has usually constructed probabilit y distributions either directly b y guessing a form ula for the probability of a giv en group or in terms of explicit constructions of random groups b y generators and random relations, such as the cok ernel of a random matrix. In the case of nonab elian groups, with the inﬂuence of ro ots of unity , it is not at all clear how to guess the probabilities or ﬁnd a suitable construction of random relations to accurately mo del the Galois group. Liu [Liu22] has given a Galois group mo del in one case, but it is not clear how to use this mo del to calculate the probability of obtaining a giv en group. Instead, w e use the moment method introduced by the authors in [SW22]. A classical problem in probability theory is to give criteria for when a distribution is uniquely detemined b y its moments. Subsequent w ork deﬁned a notion of moments for probability distributions on the set of isomorphism classes of ﬁnite ab elian groups, and ga v e criteria for when these probabilit y distributions are determined by their moments [EVW16, W o o17]. More recen tly , similar results were obtained for nonab elian groups [BBH17, Sa w20]. The pap er [SW22] generalizes those results and also strengthens them b y giving criteria for a distribution with a given moments to exist and, more crucially , a formula for the probabilit y that distribution assigns to a given group. Th us, to construct a probability distribution that conjecturally models the distribution of the Galois group, we need only conjecture a formula for the moments of the probability distribution, and apply the results of [SW22] to calculate the probabilities. The moments of the Galois group, in this sense, ma y b e calculated in the function ﬁeld setting in the q → ∞ limit, and the ﬁrst part of the pap er is dev oted to computing these momen ts, which ev entually gives a theorem instead of a conjecture in that case. Ho wev er, using the form ulas of [SW22] to compute the probabilities from the moments in this situation is highly nontrivial. It turns out to b e helpful in our arguments to consider the Galois group with an extra in v ariant, the Artin-V er dier fundamental class . The statement of Theorem 1.1 ab ov e do es not incorp orate this extra structure, but w e calculate the probabili- ties p H b y ﬁrst calculating probabilities of obtaining a group together with an Artin-V erdier fundamen tal class, and then summing o ver p ossible choices of Artin-V erdier fundamen tal class. A sum corresponding to p ossible v alues of this inv arian t naturally arises when one tries to computes the probabilities, and it do es not seem p ossible to give a formula for the probabilities in general without that sum, so it is natural to giv e a conceptual interpretation 3 of this sum b y incorp orating the Artin-V erdier fundamental class. The construction and consideration of this class along with the unramiﬁed Galois groups is one of the key new ideas of this paper. As w e will discuss later, this in v ariant reﬁnes several in v ariants that ha ve b een previously considered. Let K b e a n umber ﬁeld, let H be the Galois group of an unramiﬁed extension L/K and let n b e a positive integer. The extension L/K giv es a map H 3 ( H , Z /n ) → H 3 ( O K , Z /n ) of étale cohomology . If L/K is split at eac h inﬁnite place then it is easy to see that the comp osition H 3 ( H , Z /n ) → H 3 ( O K , Z /n ) → H 3 ( K v , Z /n ) v anishes for eac h inﬁnite place v of K . With sligh tly more w ork, one can see (in Lemma 2.4) that in the split at ∞ case, the map H 3 ( H , Z /n ) → H 3 ( O K , Z /n ) factors through a natural map H 3 ( H , Z /n ) → H 3 c ( O K , Z /n ) . If K contains the n th ro ots of unit y then, after ﬁxing a generator for the n th roots of unit y of K , w e ha ve a comp osed map H 3 ( H , Z /n ) → H 3 c ( O K , Z /n ) → H 3 c ( O K , µ n ) → Z /n where the ﬁrst arrow is from L/K as ab o ve, the second arro w arises from the map of sheav es Z /n → µ n sending 1 to our ﬁxed generator, and the last arro w comes from Artin-V erdier dualit y . W e call this map the Artin-V er dier tr ac e for L/K , and the corresp onding element in H 3 ( H , Z /n ) the A rtin-V er dier fundamental class for L/K . More details are given in Section 2. 1.1. The main theorems and conjectures. Our main conjecture, com bined with our main theorem, predict the distribution of Galois groups together with tw o t yp es of extra structure – ﬁrst, the natural action of Γ , and second, the Artin-V erdier fundamental class. T o express this extra structure, we work with the category of ﬁnite n -orien ted Γ -groups. W e ﬁx a p ositive integer n and a ﬁnite group Γ . W e deﬁne a Γ -gr oup to b e a group H together with an action of Γ on H . W e deﬁne a ﬁnite n -oriente d Γ -gr oup to b e a ﬁnite Γ -group H of order prime to | Γ | along with a Γ -inv arian t class in s H ∈ H 3 ( H , Z /n ) . A morphism of n -orien ted Γ -groups ( G, s G ) → ( H , s H ) is a Γ -equiv arian t group homomorphism f : G → H suc h that f ∗ s G = s H . W e deﬁne a proﬁnite n -orien ted Γ -group to b e a proﬁnite group X , the inv erse limit of ﬁnite groups of order prime to Γ , with an action of Γ by automorphisms, and a Γ -in v ariant class s X ∈ H 3 ( X , Z /n ) . W e deﬁne morphisms the same as in the ﬁnite case. W e write n -oriented Γ -groups with b oldface letters suc h as H , and write Sur( X , H ) for the num b er of surjections from X to H in the category of proﬁnite n -oriented Γ -groups, and Aut( H ) for the group of automorphisms in the same category . W e use the corresp onding non-b olded letter to denote the underlying group with a Γ -action, e.g. H for H . The conjecture is motiv ated by an analogous function ﬁeld theorem, whic h we now describ e the notation for and state. Let q b e a prime p o w er, which will alwa ys b e prime to | Γ | , and let F q b e the ﬁnite ﬁeld with q elemen ts. An extension of F q ( t ) , giv en as a Galois subﬁeld of the separable closure of F q ( t ) , together with a c hoice of isomorphism of the Galois group to Γ , is called a Γ -extension of F q ( t ) . F or K a Γ -extension of F q ( t ) , we let rDisc K b e the pro duct of the norms of the places of F q ( t ) that ramify in K (the norm of a place v of F q ( t ) b eing q deg v ). Let E Γ ( q m , F q ( t )) b e the set of Γ -extensions K of F q ( t ) , split at ∞ , suc h that rDisc K = q m . 4 F or K a Γ -extension of F q ( T ) , let K un , | Γ | ′ b e the comp osition of all ﬁnite Galois extensions of K with degree prime to Γ that are everywhere unramiﬁed and split ov er ∞ (the unique place of F q ( t ) that do es not arise from a prime ideal of F q [ T ] ). If F q ( t ) contains the n th ro ots of unity , i.e. n | q − 1 , then since K un , | Γ | ′ /K is an extension with Galois group Gal( K un , | Γ | ′ /K ) , it has an asso ciated Artin-V erdier fundamental class s ∈ H 3 (Gal( K un , | Γ | ′ /K ) Γ , Z /n ) Γ . This makes Gal ( K un , | Γ | ′ /K ) naturally an n -oriented Γ -group. (The action of Γ is by using a homomorphic section of Gal( K un , | Γ | ′ / F q ( t )) → Gal( K / F q ( t )) = Γ and conjugating. By the Sch ur–Zassenhaus theorem [RZ10, Theorem 2.3.15], this section is w ell-deﬁned up to conjugation by elements of Gal( K un , | Γ | ′ /K ) , and th us the resulting n -orien ted Γ -group is w ell-deﬁned up to isomorphism.) Theorem 1.2 (Theorem 4.1) . L et Γ b e a ﬁnite gr oup, n a p ositive inte ger c oprime to | Γ | , and H a ﬁnite n -oriente d Γ -gr oup with H Γ = 1 . L et q b e a prime p ower. As long as q satisﬁes the c ongruenc e c onditions ( q , | Γ || H | ) = 1 , q ≡ 1 mo d n , and ( q − 1 , | H | ) = ( n, | H | ) and q is suﬃciently lar ge dep ending on Γ , H we have lim b →∞ P m ≤ b P K ∈ E Γ ( q m , F q ( t )) Sur( Gal ( K un , | Γ | ′ /K ) , H ) P m ≤ b | E Γ ( q m , F q ( t )) | =   H Γ   | H 2 ( H ⋊ Γ , Z /n ) | | H || H 3 ( H ⋊ Γ , Z /n ) | . W e are now ready to state the main conjecture in the num b er ﬁeld case. Fix a num b er ﬁeld k and a ﬁnite group Γ . W e will assume that k lac ks a nontrivial everywhere unramiﬁed extension, whic h is satisﬁed for instance if k = Q . Let n b e the num b er of ro ots of unity of order prime to | Γ | contained in k . A Γ -extension of k is a Galois subﬁeld K of the algebraic closure of k together with an isomorphism Gal( K /k ) ∼ = Γ . F or eac h Γ -extension K of k , let K un , | Γ | ′ b e the maximal extension of K , unramiﬁed at all places including the inﬁnite ones, that is a limit of ﬁnite Galois extensions of degree prime to | Γ | . Then Gal ( K un , | Γ | ′ /K ) is a proﬁnite n -orien ted Γ -group. W e let rDisc K b e the norm of the radical of the ideal Disc( K/k ) , i.e. the pro duct of the norms of the ramiﬁed primes of K/k . F or each inﬁnite place v of k , ﬁx a conjugacy class γ v of Γ of order at most 2 if v is real or order 1 if v is complex. W e use γ to refer to this tuple of conjugacy classes γ v . Let E Γ ( D , k , γ ) b e the set of Γ -extensions K of k , such that the conjugacy class of complex conjugation at the place v is the elemen t γ v for all v , with rDisc K = D . W e write Sp ec( k ⊗ R ) for the set of inﬁnite places of k . F or H a Γ -group, we use H Γ to denote the Γ -coinv ariants of H . F or H a ﬁnite n -oriented Γ -group, let b H =   H Γ   | H 2 ( H ⋊ Γ , Z /n ) | | H 3 ( H ⋊ Γ , Z /n ) | Q v ∈ Spec( k ⊗ R ) | H γ v | if H Γ is trivial and b H = 0 if H Γ is nontrivial. Note that b H only dep ends on the underlying Γ -group H , not on the n -orientation. Conjecture 1.3. L et k b e a numb er ﬁeld that lacks any nontrivial everywher e unr amiﬁe d ﬁeld extension, Γ a ﬁnite gr oup, and ( γ v ) v ∈ Spec( k ⊗ R ) an assignment of a c onjugacy classes γ v as ab ove. L et n b e the numb er of r o ots of unity of or der prime to | Γ | c ontaine d in k . L et H b e a ﬁnite n -oriente d Γ -gr oup. 5 Then lim B →∞ P D ≤ B P K ∈ E Γ ( D,k ,γ ) Sur( Gal ( K un , | Γ | ′ /K ) , H ) P D ≤ B | E Γ ( D , k , γ ) | = b H . No w we explain how these moments determine a distribution. W e ﬁrst deﬁne the space our distributions will b e on. W e sa y a proﬁnite n -orien ted Γ -group is smal l if it is has ﬁnitely many op en subgroups of each index, i.e if the underlying proﬁnite group is small in the classical sense. A level L (of the category of ﬁnite Γ -groups) is a ﬁnitely generated formation of ﬁnite Γ - groups (a formation is a set of isomorphism classes of Γ -groups closed under taking quotients and ﬁnite sub direct pro ducts). F or example, the group Z /p Z (with trivial action) generates the lev el of elemen tary ab elian p -groups with trivial Γ -action. W e let P Γ ,n b e the set of isomorphism classes of small proﬁnite n -orien ted Γ -groups. A group X ∈ P Γ ,n has a pro- L completion X L whic h is the inv erse limit of the (contin uous) quotien ts of X in L , and this completion is a ﬁnite n -orien ted Γ -group since it is ﬁnite b y [SW22, Lemma 5.6] and inherits a homology class b y pushforward from X . W e consider a top ology on P Γ ,n generated b y { X | X L ≃ F } as L v aries through lev els and F v aries through ﬁnite n -orien ted Γ -groups whose Γ -group is in L . W e also consider the asso ciated Borel σ -algebra. In the function ﬁeld case, we ha v e the following probabilistic result. Theorem 1.4. Ther e exists a unique me asur e ν Γ ,n, { 1 } on P Γ ,n such that for every ﬁnite n -oriente d Γ -gr oup H Z X ∈P Γ ,n Sur( X , H ) dν Γ ,n = ( | H Γ || H 2 ( H ⋊ Γ , Z /n ) | | H || H 3 ( H ⋊ Γ , Z /n ) | if H Γ = 1 0 if H Γ  = 1 F or any level L of the c ate gory of ﬁnite Γ -gr oups, and ﬁnite n -oriente d Γ -gr oup H whose underlying Γ -gr oup is in L , the quantity ν Γ ,n, { 1 } ( { X | X L ≃ H } ) is describ e d by an explicit formula given later in §1.6. F urthermor e, in the same setting, letting M b e the pr o duct of al l primes dividing the or ders of elements of L , we have lim q →∞ ( q , | Γ | M )=1 ( q − 1 ,nM )= n lim sup b →∞ P m ≤ b    { K ∈ E Γ ( q m , F q ( t )) | Gal ( K un , | Γ | ′ /K ) L ≃ H }    P m ≤ b | E Γ ( q m , F q ( t )) | = ν Γ ,n, { 1 } ( { X | X L ≃ H } ) and lim q →∞ ( q , | Γ | M )=1 ( q − 1 ,nM )= n lim inf b →∞ P m ≤ b    { K ∈ E Γ ( q m , F q ( t )) | Gal ( K un , | Γ | ′ /K ) L ≃ H }    P m ≤ b | E Γ ( q m , F q ( t )) | = ν Γ ,n, { 1 } ( { X | X L ≃ H } ) The analogous result in the num b er ﬁeld case is as follo ws. Theorem 1.5. L et n b e a p ositive inte ger, Γ a ﬁnite gr oup of or der prime to n , and γ a tuple of c onjugacy classes of γ of or der at most 2 . Supp ose ther e exists a numb er ﬁeld k that has exactly n r o ots of unity of or der prime to | Γ | , and such that the elements of γ c an b e 6 plac e d in bije ction with the inﬁnite plac es of k such that al l the non-trivial c onjugacy classes c orr esp ond to r e al plac es of k . Then ther e exists a unique me asur e ν Γ ,n,γ on P Γ ,n such that for every ﬁnite n -oriente d Γ -gr oup H we have Z X ∈P Γ ,n Sur( X , H ) dν Γ ,n,γ = b H . F or any level L of the c ate gory of ﬁnite Γ -gr oups, and H a ﬁnite n -oriente d Γ -gr oups whose underlying Γ -gr oup is in L , the quantity ν Γ ,n,γ ( { X | X L ≃ H } ) is describ e d by an explicit formula given later in §1.6. F urthermor e, in the same setting, assuming Conje ctur e 1.3 we have lim B →∞ P D ≤ B    K ∈ E Γ ( D , k , γ ) | Gal ( K un , | Γ | ′ /K ) L ≃ H }    P D ≤ B | E Γ ( D , k , γ ) | = ν Γ ,n,γ ( { X | X L ≃ H } ) . The h yp othesis on k in Theorem 1.4 is only used to ensure that γ has at least one trivial elemen t. These theorems hav e three parts: The ﬁrst states that there exists a unique measure with the relev an t momen ts, the second gives a formula for that measure (the full details of which w e delay to §1.6), and the third sa ys that this measure gives the correct probabilities of obtaining a given group, in a double limit sense in the function ﬁeld case and conditionally on Conjecture 1.3 in the num b er ﬁeld case. W e p oint out that, even though the explicit formulas in §1.6 require splitting in to man y diﬀeren t cases, in each case, it is easy to determine when the probability is zero and when it is p ositive. Whether the probabilities are p ositiv e or zero is particularly imp ortan t b ecause, when the probability is p ositive, from Theorem 1.4 w e obtain an existence result for function ﬁelds with Gal ( K un , | Γ | ′ /K ) L ≃ H , and, when the probability is zero, Theorem 1.6 gives a corresp onding nonexistence result. 1.2. Relation with prior work. A n umber of prior works ha ve considered the distribution of the Galois group of the maximal unramiﬁed extension of a random num b er ﬁeld, and ev en more hav e considered the distribution of its abelianization, i.e., the class group. W e review here the relationship to that prior work, with more details giv en later in §3. Crucially , w e study the Galois group together with some extra data, the Artin-V erdier trace, whic h generalizes some extra data considered in prior w orks. In 1984, Cohen and Lenstra [CL84] gav e conjectures for the distribution of the o dd parts of class groups of imaginary and real quadratic ﬁelds, as well as for any ﬁnite ab elian group A , the prime-to- | A | part of class groups of totally real A -extensions of Q . Cohen and Martinet [CM90] generalized these conjectures to the situation of an arbitrary num b er ﬁeld K 0 as a base ﬁeld, and arbitrary group Γ , giving conjectures for distributions of part of class groups of order relatively prime to | Γ | among Γ -extensions of a ﬁxed K 0 with any ﬁxed b eha vior at the inﬁnite places of K 0 . Ho w ever, it w as noted that these conjectures app eared to b e wrong for the parts of class groups at primes dividing the n umber of ro ots of unit y in the base ﬁeld, in empirical num b er ﬁeld work of Malle [Mal08, Mal10] and theoretical fu nction ﬁeld w ork of A ch ter [Ac h06] and Garton [Gar15]. There hav e since b een man y pap ers aimed at correcting these class group distribution conjectures in the presence of ro ots of unit y , including work of Malle [Mal08, Mal10] in his original papers, Garton [Gar15], A dam and Malle [AM15], 7 Lipno wski, the ﬁrst author, and T simerman [LST20], and most recently of the tw o curren t authors [SW23]. See [SW23, Section 1.2] for more discussion of the relationship b etw een these pap ers. Our work [SW23] made conjectures for the | Γ | -prime part of the class group of Galois Γ -extensions of any ﬁxed n um b er ﬁeld, without any extra data b eyond the Γ action. The predictions of this pap er can b e compared with [SW23] by pro jecting the measure ν Γ ,n,γ on to ab elian groups and then forgetting the Artin-V erdier trace. This pro duces a distribu- tion whose moments agree with the moments conjectured in [SW23], and since it is sho wn in [SW23] that this distribution is determined b y its momen ts, it follows that the tw o dis- tributions agree. The pap er [SW23] itself built on the prior w ork describ ed abov e giving conjectures for distributions of class groups, agreeing with most of it but disagreeing with some, and it follows that the curren t pap er has the same relationships with prior conjectures, as describ ed in detail in [SW23, Section 12]. Lipno wski, Sawin, and T simerman [LST20] considered the class group with tw o inv arian ts, the ω and ψ inv arian ts, each describable in a diﬀerent wa y as bilinear forms on the dual of the class group, with a relationship b etw een them. W e show that b oth of these inv arian ts can b e calculated from the Artin-V erdier inv ariant – roughly , the Artin-V erdier inv ariant giv es a linear form on the third cohomology , and we can use this to obtain a bilinear form on the ﬁrst cohomology in tw o diﬀeren t w a ys, either cupping tw o classes and then taking a Bo c kstein homomorphism or cupping one class with a Bo ck stein homomorphism applied to the other. These t wo constructions turn out to giv e the tw o bilinear forms. This diﬀers from the approach in [LST20] to deﬁning these in v arian ts in that w ork also considered Bo ckstein homomorphisms in ﬂat cohomology arising from exact sequences of group schemes and here w e use only Bockstein homomorphisms in group cohomology arising from exact sequences of ﬁnite groups. Th us, the Artin-V erdier trace generalizes the ω and ψ inv arian ts and gives additional data even in the case of ab elian groups (since from the Artin-V erdier trace w e also obtain a trilinear form arising from cup pro duct of three classes in ﬁrst cohomology), whic h is not computable from only the ω and ψ in v ariants, though this does not occur in the Γ = Z / 2 setting considered in [LST20]. Morgan and Smith [MS24] obtained the ψ inv arian t as part of a general formalism of Cassels-T ate pairings. They also deﬁned [MS24, Notation 6.3 and Prop osition 6.4] an in v ari- an t that satisﬁes prop erties identical to the k ey prop erties of the ω in v arian t. They observed that it seems lik ely that this in v ariant is equal to the ω inv ariant, but did not pro ve this. Since w e prov e that a pairing deﬁned via Artin-V erdier dualit y agrees with the ω inv arian t, it also seems likely that this pairing agrees with the pairing CTP D ( m ) of [MS24, Prop osition 6.4], but we also do not pro v e this. There hav e also b een a num b er of pap ers on conjectures for the distribution of the non- ab elian generalization of the class group, i.e. the Galois group of the maximal unramiﬁed extension of a random num b er ﬁeld. Boston, Bush, and Ha jir [BW17, BB21] ga ve conjec- tures for the distribution of the p -class tow er group (i.e. the Galois group of the maximal unramiﬁed Galois extension of degree a p o wer of p ) of quadratic ﬁelds, for p o dd. Liu, the second author, and Zurieck-Bro wn [L WZ24] considered the Galois group of the maximal un- ramiﬁed extension of Γ -extensions of Q for an arbitrary ﬁnite group Γ , making conjectures similar to our o wn, except that they only considered extensions of degree relatively prime to the num b er of ro ots of unity in k (i.e., in the case k = Q , relatively prime to 2 ). In that 8 pap er, as in this one, the conjectures were justiﬁed by momen t theorems in a function ﬁeld mo del. Since the moments conjectured in this pap er agree with the moments conjectured in [L WZ24] in the relev ant sp ecial case, and w e c heck the distribution is uniquely determined b y its moments, it follows that the conjectural distributions from the tw o pap ers agree in this sp ecial case. A key diﬀerence is that [BW17, BB21, L WZ24] all construct their distributions from a mo del of random proﬁnite groups based on generators and explicit random relations, whereas our distribution is constructed using a general formalism to ﬁnd a distribution with giv en momen ts. In this sp ecial case of extensions of degree relatively prime to the num- b er of roots of unity in the base ﬁeld, the Artin-V erdier trace is trivial. Corresp ondingly , [BW17, BB21, L WZ24] did not consider an y extra data on their proﬁnite group b ey ond the Γ action. Boston and Bush [BBH21] considered the 2-class tow er group of cyclic cubic ﬁelds, ﬁnding empirical data and making conjectures based on that. They consider m ultiple v arian ts – the wide class group tow er where the extension is assumed split at inﬁnite places, the narrow class group tow er where the extension is not assumed split at inﬁnite places, and the joint distribution of the wide and narro w tow er as a pair of groups. Since w e consider extensions that are split at ∞ , our conjectures apply to the wide case. As mentioned ab o ve, w e c heck that the measure ν Z / 3 Z , Q , (1) arising from our Conjecture 1.3 agrees with Boston and Bush’s n umerical data for the smallest t wo rank tw o 2 -groups that can app ear. The Artin-V erdier trace in this setting is non trivial, so applying our conjecture requires summing o ver p ossible v alues of the Artin-V erdier trace. In the nonab elian setting, the study of extra inv ariants starts with unpublished w ork of Ellen b erg, V enk atesh, and W esterland. Building on work of Serre [Ser90] and F ried [F ri95], they deﬁned a lifting inv arian t which separates comp onents of a Hurwitz space parameter- izing cov ers of the pro jectiv e line. This lifting in v arian t was deﬁned in more detail and studied b y the second author in [W o o19, Section 3] and [W o o21]. In [W o o19] the second author gives conjectures on the momen ts of the Galois group of the maximal unramiﬁed extension of a random quadratic ﬁeld, including extensions of even degree, and th us in the case where ro ots of unity are relev ant. There are reﬁned conjectures in [W o o19, Section 5] that describ e the momen ts with a ﬁxed lifting inv arian t. These conjectures are based on function ﬁeld theorems. Liu [Liu22] deﬁned a ω -in v ariant of an unramiﬁed Galois extension L/K of a Γ -extension K / F q ( t ) , for a general ﬁnite group Γ , and sho w ed it agrees with the prime-to-the-order-of- Γ part of the lifting in v ariant of L/ F q ( t ) as deﬁned in [W o o21]. This allo wed Liu to prov e function ﬁeld theorems on counting unramiﬁed extensions with a ﬁxed ω -in v ariant and motiv ated reﬁned conjectures in the num b er ﬁeld case. W e sho w that the Artin-V erdier trace ma y b e used to calculate Liu’s ω -inv arian t. In the ab elian group case, all these inv ariants sp ecialize to the ω -in v ariant. As w e men tioned ab o v e, the Artin-V erdier trace gives additional data not computable from these in v ariants. Com bined with Liu’s re- sult, the Artin-V erdier trace ma y b e used to compute the prime-to-the-order-of- Γ part of the lifting in v ariant of [W o o21], and thus the Artin-V erdier trace reﬁnes and generalizes the previously studied inv ariants on Galois groups of maximal unramiﬁed extensions. 1.3. Evidence. In addition to the function ﬁeld evidence discussed ab o ve, we ﬁnd evidence for Conjecture 1.3 by calculating the probabilities ν Γ ,n,γ ( { X | X L ≃ H } , which that conjecture predicts is the probabilit y that H app ears as the maximal quotient in W of the Galois groups of maximal unramiﬁed extensions, and c hecking these predictions. When the predicted 9 probabilit y is zero, w e can c hec k the prediction b y pro ving that H do es not app ear as as the maximal quotien t in W of a Galois group of a maximal unramiﬁed extension. W e do this for ev ery group whose predicted probability is zero. When the predicted probability is nonzero, w e must instead compare against empirical data. W e do not generate new data in this pap er, but rather compare against some data produced in prior work. More precisely , in Prop osition 7.49 b elow, we ﬁnd necessary and suﬃcient conditions to ha ve ν Γ ,n,γ ( { X | X L ≃ H } = 0 . In Prop osition 8.15, we c hec k that groups violating these conditions in fact do not appear as Gal( K un , | Γ | ′ /K ) L (except if K has additional roots of unit y not con tained in k , which should happ en with density zero regardless). T o do this, we need the following, purely Galois-theoretic, statement. Theorem 1.6 (Theorems 8.6, 8.7, and 8.14) . L et k b e a numb er ﬁeld, Γ a ﬁnite gr oup, K an extension of k with Galois gr oup Γ . L et p b e a prime not dividing | Γ | and let V b e an irr e ducible r epr esentation of Gal( K un , | Γ | ′ /K ) ⋊ Γ over F p . Assume k admits no nontrivial unr amiﬁe d extensions. If k c ontains the p th r o ots of unity, then | H 1 (Gal( K un , | Γ | ′ /K ) ⋊ Γ , V ∨ ) || H 2 (Gal( K un , | Γ | ′ /K ) ⋊ Γ , V ) s K | ≤ | H 1 (Gal( K un , | Γ | ′ /K ) ⋊ Γ , V ) | Q v ar chime dean plac e of k | V Gal( k v ) | | V Gal( K un , | Γ | ′ /K ) ⋊ Γ | . If k do es not c ontain the p th r o ots of unity, and K do es not c ontain the p th r o ots of unity, we have | H 2 (Gal( K un , | Γ | ′ /K ) ⋊ Γ , V ) | ≤ | H 1 (Gal( K un , | Γ | ′ /K ) ⋊ Γ , V ) | Q v ar chime dean plac e of k | V Gal( k v ) | | V Gal( K un , | Γ | ′ /K ) ⋊ Γ | . This v eriﬁes a prediction of Conjecture 1.3 that is sp elled out in Theorem 1.5. The pro of of Theorem 1.6 is relativ ely short, and relies on comparing the Galois cohomol- ogy groups app earing in its statemen t to ﬂat cohomology groups of n um b er ﬁelds so that Artin-V erdier duality and Euler characteristic form ulas for ﬂat cohomology ma y b e applied. The signiﬁcance of Theorem 1.6 is that, according to Conjecture 1.3 and Theorem 1.5, it should b e the only restriction on Gal( K un , | Γ | ′ /K ) whic h is detectable b y examining ﬁnite quotien ts and holds for an arbitrary n umber ﬁeld. W e now turn to nonzero probabilities. First observe that Gal( K un , | Γ | ′ /K ) is the prime-to- | Γ | part of the class group of K , so Conjecture 1.3 implies a conjectural distribution for the prime-to- | Γ | part of the class group. As men tioned in §1.2, this agrees with the conjectural distribution of class groups in [SW23]. Malle [Mal08, Mal10] has previously pro duced a large amount of data on the distribution of the class groups of n umber ﬁelds containing ro ots of unit y , and [SW23] c hec ks that this data agrees with the conjectures of [SW23], and hence also with the conjectures of this pap er. F or the distribution with extra in v arian ts, note that Lipnowski, Sa win , and T simerman [LST20] pro duced data on the distribution of the ω and ψ in v ariants on class groups of quadratic extensions of Q ( µ 3 ) , chec king that this data agrees with the conjectures of [LST20], and hence also with the conjectures of this pap er. Gen uinely nonab elian data was pro duced b y Boston and Bush [BBH21], sp eciﬁcally on the Galois group of the maximal unramiﬁed Galois extension with degree a p o wer of 2 of cyclic cubic ﬁelds. As mentioned ab ov e, we chec k that our predicted probabilities agree with 10 Boston and Bush’s data for the tw o smallest 2 -groups considered in [BBH21]. Checking agreemen t for further groups w ould require more intricate calculations in group cohomology . 1.4. Ideas of the pro of. Our computation of the function ﬁeld moment builds on the strategy of prior works suc h as [L WZ24], [Liu22], and [LL25], suitably mo diﬁed to keep track of the Artin-V erdier trace. These works sho wed that the momen t may be interpreted as a ratio of counts of F q -p oin ts on certain Hurwitz spaces. In the large q limit, the n umber of p oints of the Hurwitz spaces may b e estimated using a Lang-W eil-lik e argumen t if one can coun t the geometrically irreducible connected components of Hurwitz spaces. This is done using top ological theorems to count the connected comp onen ts of Hurwitz spaces ov er C , deducing coun ts for the connected comp onen ts of Hurwitz spaces o ver F q , and then understanding the action of F rob enius on these comp onents to coun t the ﬁxed p oin ts, since F rob enius-ﬁxed p oints o ver F q corresp ond to geometrically irreducible comp onents o ver F q . F or ﬁxed q , Lang-W eil do es not suﬃce and control of the low-degree cohomology groups of individual comp onen ts is needed in addition to the count of the geometrically irreducible comp onen ts. F or Hurwitz spaces of arbitrary groups, this w as handled in the breakthrough w ork of [LL25], whic h enabled a more reﬁned counting theorem. W e c heck that the momen ts we are interested in can b e calculated as coun ts of F q -p oin ts on certain cov ers of Hurwitz spaces. In particular, w e require the use of new spaces in order to trac k the Artin-V erdier trace. T o coun t the geometrically irreducible connected comp onents of these co v ers of Hurwitz spaces, w e calculate the mono dromy group of this co vering. This ma y b e done b y reduction to C , whic h requires studying a purely top ological question. Fix a ﬁnite group H with an action of a ﬁnite group Γ and assume that H Γ = 1 . W e deﬁne an inv ariant, the braid fundamen tal class, asso ciated to a braid together with a homomor- phism from the fundamental group of the complemen t of that braid to H ⋊ Γ , that, for each strand of the braid, sends a small lo op around that strand to an elemen t of a conjugate of Γ . Using this data, we construct a 3 -manifold as co vering of S 3 , branc hed along the braid, with mono drom y group contained in Γ . This 3-manifold admits a natural homomorphism from its fundamental group to H , or in other w ords, a map to the classifying space B H . The braid fundamen tal class is the fundamen tal class of this 3-manifold inside the homology group H 3 ( H , Z ) . The braid fundamen tal class is Γ -in v ariant, and our ﬁrst key top ological theorem sho ws that every Γ -inv arian t class in H 3 ( H , Z ) is the braid fundamen tal class of some braid. T o pro ve this, w e ﬁrst show that every Γ -inv ariant class in H 3 ( H , Z ) is the fundamental class of an unbranc hed co vering with mono drom y group Γ of a 3-manifold. W e replace the base 3-manifold with S 3 b y Dehn surgery , which requires introducing curves where the cov ering branc hes. This pro duces a link in S 3 , whic h can b e expressed as the closure of the braid. Our second key top ological theorem computes the stable low-degree homology of certain co vers of Hurwitz space. These cov ers ma y also b e describ ed using the braid fundamen tal class: The fundamental group of a conﬁguration space of p oin ts in the plane is a braid group, Since the H × Γ -Hurwitz space is a co vering space of conﬁguration space, it fundamen tal group is a subgroup of the braid group. There is a natural homomorphism from the fundamental group of the complement of a braid in this subgroup to H ⋊ Γ . The braids in this subgroup with trivial braid fundamental class form a further subgroup, which corresp onds to a ﬁnite co vering of Hurwitz space. It is this cov ering w e compute the cohomology of. 11 W e use the results of [LL25] to compute the homology of this co ver of Hurwitz space. It is not immediately ob vious how to do this, since the rational homology of a space only gives a low er bound for the homology of its ﬁnite cov ering spaces. Ho wev er, w e sho w that this co vering of Hurwitz space is itself cov ered b y a diﬀeren t Hurwitz space of a larger group, generalizing an argument of [LST20]. This gives b oth upp er b ounds and low er b ounds for the homology , whic h match each other, giving a computation of the homology of the co ver. Since our co v ering has ab elian mono dromy group, its rational cohomology splits as a sum of the cohomology of certain rank one lo cal systems on Hurwitz space. Our homology com- putation is equiv alent to a v anishing result for the lo w-degree cohomology of the nontrivial rank one lo cal systems app earing in this decomp osition. V ery recen tly , similar results for lo w-degree cohomology of certain rank one local systems arising from 2-co cycles on rac ks w ere prov en by Ellen b erg and Shusterman [ES], as a sp ecial case of v anishing results for lo w- degree cohomology of braided vector spaces. How ever, these results do not seem to apply in our case, as the lo cal systems w e consider do not arise from 2-co cycles on rac ks. Thus w e require a diﬀeren t approac h, whic h the co vering by a Hurwitz space of a larger group pro vides. Our pro of of the probabilistic theorem requires ev aluating form ulas from [SW22] that express the probability of obtaining a group H as a w eigh ted sum of momen ts of v arious extensions G of H b y other groups F . Our form ula for the momen t of G inv olves the size of the group cohomology group H 2 ( G ⋊ Γ , Z /n ) . It is therefore natural to apply the Lyndon- Ho c hild-Serre sp ectral sequence, whic h computes the group cohomology of the extension G ⋊ Γ of H ⋊ Γ by F in terms of the cohomology of H and F : More precisely , it is a sp ectral sequence con verging to H p + q ( G ⋊ Γ , Z /n ) whose second page is H p ( H ⋊ Γ , H q ( F , Z /n )) . Applying the sp ectral sequence requires computing H p ( H ⋊ Γ , H q ( F , Z /n )) for p, q small and computing sev eral diﬀerentials. Computing these diﬀeren tials, and then using the dif- feren tials to calculate the moments, is the most technically diﬃcult part of the pap er. W e use a description of the diﬀeren tials due to Huebschmann [Hue81], but need to mak e this description more concrete and simplify it, taking adv antage of the fact that only certain groups ma y app ear as F in our setup. The reader interested in the pro of may b eneﬁt from ﬁrst reading our previous pap er [SW24] on the distribution of the fundamental groups of 3-manifolds. This pap er included a similar Lyndon-Ho c hschild-Serre sp ectral sequence argument, but it w as considerably simpler as sev eral diﬃculties that o ccur for Galois groups of num b er ﬁelds do not app ear for fundamental groups of 3-manifolds. First, we need to keep track of the action of Γ , which did not exist in the 3-manifold setting w e studied (though could, of course, if one studied 3 -manifolds with actions of Γ ). Second, there are multiple cases in this w ork dep ending on the num b er of roots of unity , but only one case needs to b e considered for 3-manifolds. (Roughly , an orien ted 3-manifold is analogous to a n um b er ﬁeld con taining all ro ots of unity that are p ossibly relev ant.) Third, in [SW24] w e used Poincaré duality to obtain strong restrictions on the fundamental group of a 3-manifold. W e only needed to consider groups H satisfying these restriction, whic h forced certain extension groups to v anish. In this pap er, several calculations in volv e these extension groups, which need not v anish. (Artin-V erdier duality can b e used to prov e restrictions on the Galois group of the maximal unramiﬁed extension of a n umber ﬁeld, as w e do in Lemma 8.7, but o wing to the inﬂuence of Arc himedean places, these restrictions are weak er than in the 3-manifold case, and do not show that these 12 extension groups v anish.) F or all these reasons, the pro of of [SW24] can serve as a simpler mo del for the pro ofs in this pap er. 1.5. Notation. Before giving the form ulas for the probabilities in our main theorems, w e m ust introduce some notation. In order to ha v e things in one place, we collect here notation that is used throughout the pap er. Throughout the pap er Γ is a ﬁnite group and n is a p ositiv e integer relatively prime to | Γ | . W e write C Γ ,n for the category of ﬁnite n -oriented Γ -groups. The letter q alwa ys denotes a prime p o wer and F q the ﬁnite ﬁeld of order q . F or an y ﬁeld K that contains the n th ro ots of unit y , w e ﬁx a generator ξ for the n th ro ots of unit y in K . Asymptotic notation: W e write f ( X ) = O ( g ( X )) to mean there exists a constant C suc h that | f ( X ) | ≤ C g ( X ) for all v alues of X , and f ( P , X ) = O P ( g ( P , X )) to mean that there is a function C ( P ) of parameters P suc h that | f ( P , X ) | ≤ C ( P ) g ( P , X ) for all v alues of X . Sometimes w e write O for O P when w e sp ecify in w ords what the constan t ma y dep end on. Group theory: Giv en a group Γ , a Γ -gr oup G is a group G with an action of Γ b y automorphisms. A morphism of Γ -gr oups is a Γ -equiv ariant morphism of the underlying groups, and w e write Aut Γ ( G ) for the automorphisms of a Γ -group G . When G is an n - orien ted Γ -group, we write Aut( G ) for the group of automorphisms in that category , without an y subscript since the b olding serv es as a reminder. A Γ -sub gr oup H of G is a subgroup of G closed under the Γ -action, and a Γ -gr oup quotient is the image of a surjectiv e Γ -group morphism. A Γ -group is a simple Γ -gr oup if it is nontrivial and con tains no prop er, nontrivial normal Γ -subgroup. A Γ -group is semisimple if it is a ﬁnite direct pro duct of simple Γ -groups. A [Γ] -gr oup is a group G together with a homomorphism ρ : Γ → Out( G ) , though we often lea v e ρ implicit. An isomorphism of [Γ] -gr oups from ( G, ρ ) to ( G, ρ ′ ) is given by an a group isomorphism f : G → G ′ suc h that the induced map f ∗ : Out( G ) ≃ Out( G ′ ) satisﬁes f ∗ ◦ ρ = ρ ′ . W e write Aut [Γ] ( G ) for the automorphisms of a [Γ] -group G . W e ha ve that Γ acts on the set of normal subgroups of an [Γ] -group G , and w e sa y a nontrivial [Γ] -group is simple if it has no non trivial prop er ﬁxed p oints for this action. A semisimple [Γ] -group is a ﬁnite direct product of simple [Γ] -groups. If w e ha ve an exact sequence of groups 1 → N → G → Γ → 1 , then N is naturally a [Γ] -group. A normal subgroup N ′ of N is ﬁxed b y the Γ -action if and only if it is normal in G . Hence N is a si mple (semisimple) G -group (via conjugation) if and only if N is a simple (semisimple) [Γ] -group. Simple ﬁnite Γ -groups or [Γ] -groups are characteristically simple as groups, and hence their underlying group is pro duct of isomorphic ﬁnite simple groups. W e write Out( G ) for the group of outer automorphisms of a group, Γ -group, or [Γ] -group G , where Out( G ) only dep ends on the underlying group structure of G . W e often work with proﬁnite groups. Whenever we refer to morphisms of proﬁnite groups, w e alwa ys mean contin uous homomorphisms. When considering the homology and coho- mology of proﬁnite groups, w e alw a ys mean the contin uous cohomology . When discussing quotien ts of proﬁnite groups, w e alwa ys mean quotients by closed normal subgroups, which are themselv es proﬁnite groups. F or an ab elian group G , we write G ∨ for G = Hom( G, Q / Z ) . In v ariants and coin v arian ts: F or a Γ -group G , we write G Γ for the inv arian ts, and for γ ∈ Γ , we write G γ for the elements ﬁxed by γ . W e write G Γ for the coinv ariants, i.e. 13 the maximal Γ -in v ariant quotien t of G , whic h is the quotient of G b y the normal subgroup generated b y the elemen ts g − 1 γ ( g ) for each g ∈ G and γ ∈ Γ . When G is ab elian and ( | G | , | Γ | ) = 1 , w e ha ve that the natural map G Γ → G Γ is an isomorphism, and ( G Γ ) ∨ ≃ ( G ∨ ) Γ . Pro ducts: By con ven tion, a product Q b k = a where a > b is 1 . q -series: W e deﬁne an adjusted q -Pochhammer sym b ol ( q ) n := Q n i =1 (1 − q − i ) for any p ositiv e integer n . By con ven tion ( q ) 0 = 1 . T ensor p o w ers, ∆ : If A is a ﬁnitely generated abelian group, there is an Aut( A ) - equiv ariant map ∆ : A ⊗ A → A ⊗ A given b y a ⊗ b 7→ a ⊗ b − b ⊗ a , which descends to an injection ∧ 2 A → A ⊗ 2 . W e deﬁne ∧ 2 A to be the the image of ∆ , and ∆ induces an isomorphism ∧ 2 A ≃ ∧ 2 A . Cohomology of sc hemes: When taking cohomology groups of a sc heme, we alwa ys mean étale cohomology groups, except in Sections 3 and 8 where we will consider ﬂat cohomology groups, denoted with the subscript fppf , as w ell. Aﬃne Symplectic Group: W e follow Gurevich and Hadani [GH12]. Giv en a ﬁnite dimensional F 2 -v ector space V , and a full rank form ω ∈ ∧ 2 V ∨ , w e let Sp( V ) b e the set of automorphisms of V preserving ω (and so this deﬁnition will alw a ys require an implicit ω , whic h should b e clear from context). If [ f ] ∈ H 2 ( V , Z / 4) is the unique class represented b y a bilinear form f such that ∆ f = ω (whic h exists by Lemma 6.1), then we deﬁne ASp( V ) to b e the automorphisms of the extension E of V b y Z / 4 corresp onding to f that ﬁx Z / 4 p oin twise and act on V through Sp( V ) . W e hav e an exact sequence 1 → Hom( V , F 2 ) → ASp( V ) → Sp( V ) → 1 . F or more ab out the aﬃne symplectic group, including the pro of that this exact sequence exists, see [SW24, §2.1]. This exact sequence represen ts a class Φ ∈ H 2 (Sp( V ) , Hom( V , F 2 )) whic h combined with ω − 1 : Hom( V , F 2 ) → V giv es a class ω − 1 ∗ (Φ) ∈ H 2 (Sp( V ) , V ) . Represen tations ov er ﬁnite ﬁelds: All representations of groups considered in this pap er are ﬁnite-dimensional. Let p b e a prime and V an irreducible represen tation of a ﬁnite group Π ov er F p , with κ = End Π ( V ) . (In most of this pap er, Π will b e H ⋊ Γ , where Γ is a ﬁnite group and H a ﬁnite Γ -group.) W e write V ∨ for the dual represen tation Hom( V , F p ) (whic h under the inclusion F p → Q / Z agrees with our notation ab o ve), and sa y V is self-dual if V ≃ V ∨ as Π -represen tations. Note that V is self-dual if and only if ( V ⊗ V ) Π  = 0 . The trace map κ → F p giv es an Π -equiv arian t isomorphism Hom κ ( W , κ ) ≃ W ∨ . W e call V unitary if ( V ⊗ V ) Π  = 0 but ( V ⊗ κ V ) Π = 0 (i.e. V is self-dual ov er F p but not o ver κ ). W e call V symmetric if ( V ⊗ κ V ) Π  = 0 , but ( ∧ 2 κ V ) Π = 0 . W e call V symple ctic if ( ∧ 2 κ V ) Π  = 0 . F or self-dual irreducible representations V , we deﬁne a mo diﬁed F rob enius-Sc hur indicator ϵ V . In c haracteristic 2 , the v alue of ϵ V will actually also dep end on whether 4 | n , and will not b e deﬁned for every representation. When p is o dd, we call all self-dual irreducible representations non-anomalous , and w e deﬁne ϵ V as follo ws: • ϵ V = 0 if V is unitary , • ϵ V = 1 if V is symmetric, and • ϵ V = − 1 if V is symplectic. 14 No w w e consider the case p = 2 . W e sa y V is A-symple ctic if V is symplectic and a map Π → Sp( V ) for an in v ariant form ϕ ∈ ∧ 2 κ V factors through Π → ASp( V ) . W e sa y V is interme diate if V is symplectic but not A -symplectic. W e sa y V is F 2 -ortho gonal if there is a non-zero inv ariant quadratic form in Sym 2 V . W e say V is anomalous • if 4 | n and V is intermediate or, • if 2 | n but 4 ∤ n , and V is intermediate and F 2 -orthogonal. An y other V is called non-anomalous . When p = 2 and 4 | n , we deﬁne ϵ V for non-anomalous self-dual V as follows: • ϵ V = 0 if V is unitary , • ϵ V = 1 if V is symmetric (which is the same as trivial), and • ϵ V = − 1 if V is A-symplectic. When p = 2 and 2 | n but 4 ∤ n , we deﬁne ϵ V for non-anomalous self-dual V as follows: • ϵ V = 0 if V is unitary and F 2 -orthogonal, • ϵ V = 1 if V is symmetric (which is the same as trivial) • ϵ V = 1 if V is non- F 2 -orthogonal, and • ϵ V = − 1 if V is A-symplectic and F 2 -orthogonal. F or anomalous V , ϵ V is not deﬁned. 1.6. Explicit form ulas for the probabilities. W e now describ e the terminology needed to give the form ulas for the measures ν Γ ,n, { 1 } and ν Γ ,n,γ . W e ﬁx a ﬁnite group Γ , a p ositive in teger n prime to | Γ | , and a m ultiset U of conjugacy classes of γ , whic h will sp ecialize to U = { 1 } in the function ﬁeld case or U = γ in the num b er ﬁeld case. Asso ciated to the m ultiset U w e ha ve a function on Γ -groups G · U := Q γ ∈ U | G γ | | G Γ | . Let L b e a level of the category of ﬁnite Γ -groups. Let H b e a ﬁnite n -orien ted Γ -group whose underlying Γ -group is in L . A ﬁnite simple ab elian H ⋊ Γ -group V is a group of the form F d ℓ , for some prime ℓ , with an irreducible action of H ⋊ Γ . W e sa y V is admissible if V ⋊ H , view ed as a Γ -group b y its natural em b edding in V ⋊ ( H ⋊ Γ) = ( V ⋊ H ) ⋊ Γ , is in L . Let V 1 , . . . V r b e represen tativ es of the isomorphism classes of admissible ﬁnite simple ab elian H ⋊ Γ -groups. W e say a ﬁnite simple nonab elian [ H ⋊ Γ] -group N , of order prime to | Γ | is admissible if for one, equiv alen tly , ev ery , lift of Γ to H ⋊ Γ × Out( N ) Aut( N ) , the complementary normal subgroup H × Out( N ) Aut( N ) is in L . (By Sch ur-Zassenhaus, diﬀerent lifts are conjugate and th us the complements are isomorphic as Γ -groups.) Let N 1 , . . . , N s b e represen tativ es of the isomorphism classes of admissible ﬁnite simple nonab elian [ H ⋊ Γ] -groups. Note our deﬁnitions of admissible implicitly dep end on L and H . Our form ula for ν ( { X | X L ∼ = H } ) will b e a pro duct of lo cal factors corresp onding to V 1 , . . . , V r and N 1 , . . . , N s . T o deﬁne these lo cal factors, we hav e the follo wing inv arian ts. F or V an admissible ﬁnite simple ab elian H ⋊ Γ -group, let κ V b e the ( H ⋊ Γ -equiv arian t) endomorphisms of V , nec- essarily a ﬁnite ﬁeld. Let q V = | κ V | . Let H 2 ( H ⋊ Γ , V ) L b e the set of cohomology classes 15 whose asso ciated extension of H b y V as a Γ -group lies in L (the asso ciation is describ ed explicitly in Lemma 5.8). Note that H 2 ( H ⋊ Γ , V ) L is a κ V -v ector space (Lemma 5.9). The orien tation s H ∈ H 3 ( H , Z /n ) induces a map H 3 ( H , Z /n ) → Z /n , whic h w e also denote b y s H (see Lemma 5.5). W e also write s H for the composite of this map with the pullbac k isomorphism H 3 ( H ⋊ Γ , Z /n ) → H 3 ( H , Z /n ) Γ (Lemma 5.4), giving a map s H : H 3 ( H ⋊ Γ , Z /n ) → Z /n . If κ V has characteristic dividing n and V is a nontrivial represen tation, let H 2 ( H ⋊ Γ , V ) L ,s H b e the set of cohomology classes α ∈ H 2 ( H ⋊ Γ , V ) L suc h that s H ( α ∪ β ) = 0 for all β in H 1 ( H ⋊ Γ , Hom( V , Z /n )) ≃ H 1 ( H ⋊ Γ , V ∨ ) . If V is a trivial representation F ℓ with characteristic dividing n , let H 2 ( H ⋊ Γ , V ) L ,s H b e the set of cohomology classes α ∈ H 2 ( H ⋊ Γ , V ) L suc h that s H ( α ∪ β ) = 0 for all β in H 1 ( H ⋊ Γ , Hom( V , Z /n )) ≃ H 1 ( H ⋊ Γ , V ∨ ) and s H ( B ( α )) = 0 for B : H 2 ( H ⋊ Γ , F ℓ ) → H 3 ( H ⋊ Γ , Z /n ) the connecting homomorphism asso ciated to the exact sequence 0 → Z /n → Z /nℓ → Z /ℓ → 0 . If κ V has c haracteristic not dividing n , let H 2 ( H ⋊ Γ , V ) L ,s H = H 2 ( H ⋊ Γ , V ) L . Let z V = dim κ V H 2 ( H ⋊ Γ , V ) L ,s H and h V = dim κ V H 1 ( H ⋊ Γ , V ∨ ) − dim κ H 1 ( H ⋊ Γ , V ) . Let u V b e suc h that q u V V =   V Γ   | V H ⋊ Γ | V · U . F or V non trivial, of characteristic prime to n , let w V = ∞ Y j =1  1 − q − j − u V V | H 2 ( H ⋊ Γ , V ) L | | H 1 ( H ⋊ Γ , V ) |  . F or V trivial, let w V = z V − 1 Y k =0 (1 − q k − u V v ) . If the characteristic of V divides n and V is admissible but V ∨ is not, let w V = ∞ Y j =1  1 − q h V + z V − j − u V V  . If the c haracteristic of V divides n and V ∨ is admissible but not isomorphic to V , then let w V = s ( q V ) ∞ ( q V ) 2 u ( q V ) u V − h V − z V ( q V ) u V + h V − z V ∨ if − u V + z V ∨ ≤ h V ≤ u V − z V and 0 otherwise. (Using that V is a non-trivial H ⋊ Γ - represen tation of c haracteristic not dividing | Γ | , it is straigh tforward to c hec k that h V ∨ = − h V and u V = u V ∨ , so that w V = w V ∨ .) 16 If the characteristic of V divides n and V is self-dual, non-trivial, and non-anomalous, let w V = ∞ Y k =0 (1 + q − k − ϵ V +1 2 − u V V ) − 1 z V − 1 Y k =0 (1 − q k − u V V ) . If the characteristic of V divides n and V is anomalous, there is a particular class ω − 1 ∗ (Φ) ∈ H 2 (Sp( V ) , V ) deﬁned in §1.5, which pulls back to a class in H 2 ( H ⋊ Γ , V ) . If ω − 1 ∗ (Φ) ∈ H 2 ( H ⋊ Γ , V ) L ,s H , w V = ∞ Y k =0 (1 + q − k − u V V ) − 1 z V − 1 Y k =0 (1 − q k − u V V ) If ω − 1 ∗ (Φ) ∈ H 2 ( H ⋊ Γ , V ) L ,s H , w V = ∞ Y k =0 (1 + q − k − 1 − u V V ) − 1 z V − 1 Y k =0 (1 − q k − u V V ) . One can observ e that we could deﬁne ϵ V to b e − 1 in case ω − 1 ∗ (Φ) ∈ H 2 ( H ⋊ Γ , V ) L ,s H , and 1 in case ω − 1 ∗ (Φ) ∈ H 2 ( H ⋊ Γ , V ) L ,s H , which would mak e the same form ula v alid in the anomalous and non-anomalous self-dual cases, but w e ha v e not made this deﬁnition as that w ould make ϵ V dep end on L and s H . F or N an admissible nonab elian group, let L N b e the n umber of lifts of H ⋊ Γ → Out( N ) to H ⋊ Γ → Aut( N ) . Giv en a c hoice of lift, N has the structure of a Γ -group, and so we can deﬁne N · U . Ho wev er, since | Γ | is relativ ely prime to | N | , any tw o lifts of Γ from Out( N ) to Aut( N ) are conjugates by an inner automorphism (whic h follows from the Sch ur-Zassenhaus Theorem applied to Aut( N ) × Out( N ) Γ ). Thus the Γ -groups pro duced by the t wo lifts are isomorphic, and in fact N · U dep ends only on the structure of N as an [ H ⋊ Γ] -group. Thus, w e will use the notation N · U without c ho osing a lift. If the natural map δ : H 2 ( N , Z /n ) H ⋊ Γ → H 3 ( H ⋊ Γ , Z /n ) from the sp ectral sequence computing H 3 (Aut( N ) × Out( N ) ( H ⋊ Γ) , Z /n ) , comp osed with the pullbac k H 3 ( H ⋊ Γ , Z /n ) → H 3 ( H , Z /n ) and then s H , is non trivial, then we deﬁne w N = 1 . If the map H ⋊ Γ → Out( N ) is trivial, then we deﬁne w N = 1 . In any other case, w e deﬁne w N = e − L N | H 2 ( N , Z /n ) H ⋊ Γ | | N || Z Out( N ) ( H ⋊ Γ) | N · U . Example 1.7. The map δ is trivial for most smal l ﬁnite simple gr oups, but one c an che ck it c an b e nontrivial for H = A 6 or H = P S U 3 ( F 8 ) . The ﬁrst example was p ointe d out to us by Pham Huu Tiep, and we found the se c ond using the A tlas of Finite Simple Gr oups. They c an b e r e c o gnize d in the Atlas fr om the fact that some c entr al extension of H do es not extend to a c entr al extension of some p articular S with H ⊂ S ⊆ Aut( H ) . This r eﬂe cts the fact that the map H 2 ( S, Z /n ) → H 2 ( H , Z /n ) is not surje ctive for some n , which implies the r elevant class in H 2 ( H , Z /n ) must not lie in the kernel of every diﬀer ential in the sp e ctr al se quenc e. With these deﬁnitions we hav e the follo wing. 17 Theorem 1.8. Fix a ﬁnite gr oup Γ , a p ositive inte ger n prime to | Γ | , and a multiset U of c onjugacy classes of Γ . Assume that e ach nonzer o r epr esentation of Γ of char acteristic dividing n c ontains a nonzer o ve ctor ﬁxe d by some element of U . F or a ﬁnite n -oriente d Γ -gr oup G , let M G = ( | H 2 ( G ⋊ Γ , Z /n ) | | H 3 ( G ⋊ Γ , Z /n ) G · U if G Γ = 1 0 if G Γ  = 1 . Ther e exists a unique me asur e ν Γ ,n,U on P Γ ,n such that for every ﬁnite n -oriente d Γ -gr oup G we have Z X ∈P Γ ,n Sur( X , G ) dν Γ ,n,U = M G . F or any level L of the c ate gory of ﬁnite Γ -gr oups, ﬁnite n -oriente d Γ -gr oup H whose underlying Γ -gr oup is in L , we have ν Γ ,n,U ( { X | X L ≃ H } ) = M H | Aut( H ) | r Y i =1 w V i s Y i =1 w N i wher e V 1 , . . . , V r ar e r epr esentatives of the isomorphism classes of admissible ﬁnite simple ab elian H ⋊ Γ -gr oups and N 1 , . . . , N s ar e r epr esentatives of the isomorphism classes of ad- missible ﬁnite simple non-ab elian [ H ⋊ Γ] -gr oups. 1.7. Outline of the pap er. In Section 2, w e deﬁne precisely the Artin-V erdier fundamental class for an extension L/K , and pro v e basic prop erties about this class. In Section 3, w e sho w how previously deﬁned additional structure on Galois groups of unramiﬁed extensions, including the bilinearly enhanced structure of Lipnowski-Sa win-T simerman [LST20] and the lifting inv ariant of [Liu22], is determined by the Artin-V erdier fundamen tal class. W e also w ork out explicit cases of our predictions for the 2 -class to wer group of cyclic degree 3 extensions, and compare to the data, results, and predictions of Boston and Bush [BBH21]. In Section 4, we prov e Theorem 1.2, our main result on function ﬁeld moments. In Section 5, w e collect some standard results in group theory that we use rep eatedly in our arguments. In Section 6, we analyze the E 3 0 , 2 term in the Lyndon-Hochsc hild-Serre sp ectral sequence, esp ecially for an extension of a group H by a semisimple abelian H -group. That is a k ey ingredien t in our argument for Theorems 1.1, 1.4, 1.5, and 1.8 in Section 7, where w e w ork out explicitly the distributions of oriented groups determined b y the moments we hav e found. In Section 8, w e prov e Theorem 1.6, giving num b er ﬁeld evidence for our conjecture in the form of non-existence pro ofs when our conejcture predicts probabilit y zero. A ckno wledgements. W e would lik e to thank Johan de Jong and Y uan Liu for helpful con versations and Aaron Landesman for helpful comments on an earlier v ersion of this man- uscript. The ﬁrst author was supp orted by a Cla y Researc h F ello wship, NSF DMS-2101491 and DMS-2502029 and a Sloan F ello wshp. The second author w as partially supp orted b y a P ack ard F ellowship for Science and Engineering, NSF DMS-2052036 and DMS-2140043, the Radcliﬀe Institute for A dv anced Study at Harv ard Universit y , and a MacArthur F ello wship. Contents 1. In tro duction 1 18 1.1. The main theorems and conjectures 4 1.2. Relation with prior work 7 1.3. Evidence 9 1.4. Ideas of the pro of 11 1.5. Notation 13 1.6. Explicit form ulas for the probabilities 15 1.7. Outline of the pap er 18 A ckno wledgements 18 2. The Artin-V erdier trace and fundamen tal class 20 3. Relations with prior inv arian ts 23 3.1. Bilinearly enhanced groups, after Lipnowski-Sa win-T simerman 24 3.2. The nonab elian ω -in v ariant, after Liu 33 3.3. 2-class ﬁeld tow ers of cyclic cubic ﬁelds, after Boston-Bush 37 4. Evidence in the function ﬁeld case 46 4.1. Algebraic geometry notation 48 4.2. Deﬁnition of Hurwitz schemes 48 4.3. Prop erties and applications of Hurwitz sc hemes 51 4.4. The braid fundamental class 55 4.5. Homology of cov ers of Hurwitz space 61 4.6. F rom c haracteristic 0 to characteristic p 64 4.7. Pro ofs of lemmas ab out in tegration along the ﬁb ers 69 5. Group theory preliminaries 80 5.1. Alternating tensor p ow ers 80 5.2. Represen tations ov er ﬁnite ﬁelds 80 5.3. Group homology and cohomology 82 5.4. Extensions b y center free groups 83 5.5. Extensions of Γ -groups 83 6. Analysis of E 3 0 , 2 in the Lyndon-Ho chsc hild-Serre sp ectral sequence 86 6.1. H 2 ( A, Z /n ) 86 6.2. F unctoriality 88 6.3. Analysis of the diﬀerentials 88 7. Measures from moments 103 7.1. T erminology of diamond categories 103 7.2. Distribution from the moments 105 7.3. Expressing the v W , H in terms of irreducibles 107 7.4. Non-ab elian groups 116 7.5. Preparation for extensions by ab elian groups 117 7.6. Represen tations of characteristic prime to n 118 7.7. Represen tations whose duals do not app ear of characteristic dividing n 120 7.8. Non-anomalous self-dual representations of characteristic dividing n 121 7.9. Non-self-dual represen tations whose duals app ear of characteristic dividing n 123 7.10. Anomalous self-dual representations 125 7.11. Pro of of Theorem 7.5 127 7.12. Criteria for nonzero probability 128 7.13. The main statements 129 19 8. Non-existence results 131 References 140 2. The Ar tin-Verdier tra ce and fund ament al class In this section we establish some of the basic prop erties of the comp onents of the Artin- V erdier trace map and fundamen tal class. Let K b e a global ﬁeld, of characteristic not dividing a p ositive integer n . If K is a n umber ﬁeld, let X K = Sp ec O K . If K is a function ﬁeld with ﬁeld of constan ts k , let X K b e the unique connected smo oth complete curv e o v er k ha ving K as its function ﬁeld. There is a natural isomorphism on the compactly supp orted étale cohomology [Mil06, Chapter I I, Section 3] (2.1) H 3 c ( X K , G m ) ≃ Q / Z , that is in particular in v ariant under automorphisms of K . The map of n th ro ots of unit y µ n → G m giv es a map H 3 c ( X K , µ n ) → H 3 c ( X , G m ) , and taking the comp osite with the map ab o ve we obtain a map D : H 3 c ( X K , µ n ) → 1 n Z / Z × n → Z /n . If K contains the n th ro ots of unit y then, we ﬁx throughout the pap er a generator ξ for the n th roots of unity of K , which in particular giv es a morphism H 3 c ( X K , Z /n ) → H 3 c ( X K , µ n ) that w e will use. Let L/K b e a Galois extension that is unramiﬁed everywhere, with H = Gal( L/K ) . W e will show in Remark 2.3 that there is a natural map H 3 ( H , Z /n ) → H 3 ( X K , Z /n ) . Moreov er, if L/K is split completely ov er all real places of K , then w e show in Lemma 2.4 that w e can reﬁne this to a map to cohomology with compact supp orts H 3 ( H , Z /n ) → H 3 c ( X K , Z /n ) . Then, asso ciated to this extension L/K , w e hav e a comp osed map H 3 ( H , Z /n ) → H 3 c ( X K , Z /n ) → H 3 c ( X K , µ n ) D → Z /n where eac h map is discussed ab ov e. W e call this map A V L/K : H 3 ( H , Z /n ) → Z /n the A rtin-V er dier tr ac e for L ov er K , and the corresp onding elemen t av L/K (via the dualit y in Lemma 5.5) in H 3 ( H , Z /n ) the A rtin-V er dier fundamental class for L ov er K . If K is a Galois extension of some other global ﬁeld F with Γ = Gal( K /F ) , such that L/F is Galois, we will show in Lemma 2.5 that the Artin-V erdier fundamental class for L/K is in H 3 ( H , Z /n ) Gal( K/F ) . Lemma 2.2. L et X b e a c onne cte d scheme, quasi-pr oje ctive over an aﬃne scheme. L et Y /X b e a ﬁnite étale morphism with a gr oup of automorphisms H of Y over X that acts simply tr ansitively on e ach ge ometric ﬁb er. L et p b e a non-ne gative inte ger. L et A b e an H -mo dule, viewe d as a r epr esentation of π 1 ( X ) and thus an étale she af on X via the c overing Y /X (and a choic e of a b asep oint in Y over a b asep oint in X , which gives a map π 1 ( X ) → H ). Then ther e is a homomorphism ϕ Y /X = ϕ Y /X,A : H p ( H , A ) → H p ( X , A ) . 20 If we have f : Z → X , such that Z is a c onne cte d scheme, quasi-pr oje ctive over an aﬃne scheme, then f ∗ ϕ Y /X = ϕ Z × X Y / Z (with c omp atible choic es of b asep oints). If A ′ is an H - mo dule and f : A → A ′ is an H -mo dule homomorphism, then we have f ∗ ϕ Y /X,A = ϕ Y /X,A ′ f ∗ . If A ′′ is another H -mo dule, and we let B denote the c onne cting homomorphism for an exact se quenc e of H -mo dules 0 → A → A ′ → A ′′ → 0 in either the c ohomolo gy of H or X , then we have ϕ Y /X,A B = B ϕ Y /X,A ′′ . If a ∈ H ∗ ( H , A ) and b ∈ H ∗ ( H , A ′ ) , then ϕ Y /X ( a ∪ b ) = ϕ Y /X ( a ) ∪ ϕ Y /X ( b ) ∈ H ∗ ( X , A ⊗ A ′ ) . If Y ′ /X is another ﬁnite étale morphism with a gr oup of automorphisms H ′ of Y ′ over X that acts simply tr ansitively on e ach ge ometric ﬁb er, with the map Y ′ → X factoring thr ough Y → X (and Y ′ → Y b asep oint pr eserving), and with a sso ciate d quotient q : H ′ → H inducing q ∗ : H p ( H , A ) → H p ( H ′ , A ) , then ϕ Y ′ /X q ∗ = ϕ Y /X . Pr o of. W e in terpret H ∗ ( H , A ) as the cohomology of the standard complex C ∗ ( H , A ) of ho- mogeneous co cycles, i.e. H -equiv arian t functions from H p +1 to A , where the action of H on H p is given by left multiplication on each co ordinate. W e will map the complex C ∗ ( H , A ) to the (étale) Čech complex C ∗ ( Y /X , A ) for the cov er Y /X . Cho ose a basepoint x ∈ X , and let F Y b e the ﬁber o v er x in Y . Pic k a basep oint y ∈ F Y . W e then ha v e a bijection F Y → H taking h − 1 y 7→ h for h ∈ H . The choice of basep oint y ∈ F Y also giv es an asso ciated group homomorphism π 1 ( X ) → H suc h that for g ∈ π 1 ( X ) , if g y = h − 1 g y for h g ∈ H , the g 7→ h g . By the fact that π 1 ( X ) is an automorphism of the ﬁb er functor, we hav e for g ∈ π 1 ( X ) , that g ( h − 1 y ) = ( h g h ) − 1 y for all h ∈ H . The sections of C ∗ ( Y /X , A ) are given by maps of π 1 ( X ) -sets from F p +1 Y → A , whic h under the identiﬁcation of F Y with H ab o ve are precisely maps of π 1 ( X ) -sets H p +1 → A , with π 1 ( X ) acting through the map π 1 ( X ) → H and then left m ultiplication on each comp onent. This gives a map C ∗ ( H , A ) → C ∗ ( Y /X , A ) taking a map H p +1 → A to the same map H p +1 → A . Moreov er thi s map manifestly respects the b oundary maps in these complexes. Thus we obtain a homomorphism from H p ( H , A ) to the cohomology of the complex C p ( Y /X , A ) , which maps to the Čech cohomology , and th us b y the assumption on X , the étale cohomology H p ( X , A ) . The compatibility on pul lbac ks can b e c heck ed directly from the construction. The com- patibilities with a c hange in co eﬃcients and the connecting homomorphisms are immediate from the construction. Also immediate from the construction is the compatibility of the cup pro duct on group cohomology and the cup pro duct on Čech co chains, which b y [Sw a99, Corollary 3.10] agrees with the cup pro duct in étale cohomology . The compatibility b etw een diﬀeren t cov ers is also immediate from the construction. (In the language of stac ks, the cov ering Y → X deﬁnes a map X → B H , and the group cohomology of H is the cohomology of B H , with the standard group cohomology co c hain complex corresp onding to the Cech complex for the cov ering pt → B H . In this language ϕ Y /X is just the pullback along X → B H .) □ R emark 2.3 . If L/K is a Galois extension that is unramiﬁed everywhere with H = Gal( L/K ) , then since H 3 ( H , Z /n ) is the direct limit of H 3 ( H i , Z /n ) ov er ﬁnite quotien ts H i of H , from Lemma 2.2 and the compatibility b etw een diﬀerent quotients, we get a map H 3 ( H , Z /n ) → H 3 ( X K , Z /n ) . 21 More generally , if V is a ﬁnite H -module, then H 3 ( H , V ) is the direct limit of H 3 ( H i , V ) o ver ﬁnite quotients H i of H suc h that the action of H on V factors through H i , and so we ha ve H 3 ( H , V ) → H 3 ( X K , V ) . When X K = Sp ec O K , for a num b er ﬁeld K , and the étale morphsim is from an extension of ﬁelds that is split completely at all real places, we can reﬁne the map ab ov e to a map to cohomology with compact supp orts. Lemma 2.4. L et L/K b e a Galois unr amiﬁe d extension of a numb er ﬁeld K , split c ompletely over al l r e al plac es of K . L et X K = Sp ec O K . L et A b e an Gal( L/K ) -mo dule. A choic e of algebr aic closur e ¯ K and a choic e of an emb e dding L → ¯ K , gives a homomorphism π 1 ( X ) → Gal( L/K ) , and thus A is also a a r epr esentation of π 1 ( X K ) and thus an étale she af on X K . F or al l p ≥ 2 , ther e is a homomorphism ϕ c L/K : H p (Gal( L/K ) , A ) → H p c ( X K , A ) . whose c omp osition with H p c ( X K , A ) → H p ( X K , A ) is the usual pul lb ack map ϕ Spec O L / Spec O K of L emma 2.2 (or R emark 2.3). This map has the fol lowing c omp atibilities: F or an inclusion of ﬁelds K → K ′ such that L ′ = L ⊗ K K ′ is a ﬁeld, this map forms a c ommutative squar e H p (Gal( L/K ) , A ) H p c ( X K , A ) H p (Gal( L ′ /K ′ ) , A ) H p c ( X K ′ , A ) ϕ c L/K ϕ c L ′ /K ′ with the natur al pul l-b acks. F or L ′ a subﬁeld of L that is Galois over K , this map forms a c ommutative triangle with the usual pul l-b ack map H p (Gal( L ′ /K ) , A ) → H p (Gal( L/K ) , A ) H p (Gal( L ′ /K ) , A ) H p c ( X K , A ) H p (Gal( L/K ) , A ) . ϕ c L ′ /K ϕ c L/K Pr o of. As in Remark 2.3, we can immediately reduce to s ho wing the case when L/K is ﬁnite, as the ﬁnal statemen t of the lemma giv es the compatibility in tow ers L/L ′ /K that we need to pass to the direct limit. F or p ≥ 1 , the compactly supp orted cohomology H p c ( X K , A ) is deﬁned [Mil06, Section I I:2] as the cohomology of the translated mapping cone of the Čech complex C ∗ ( X K , A ) represen ting the cohomology of X K to the sum o ver inﬁnite places v of K of the Čec h complex C ∗ (Sp ec K v , A ) . Let f b e the map ` v |∞ Sp ec K v → X K and let L v = K v ⊗ K L . A class in the translated mapping cone describ ed ab o ve is a pair ( a, b ) ∈ C ∗ ( X K , A ) × Q v |∞ C ∗− 1 (Sp ec K v , A ) , and it is a co cycle in particular if a is a co cycle in C ∗ ( X K , A ) suc h that f ∗ a = − d ( b ) , where d is the diﬀeren tial. 22 Let A ∗ := C ∗ ( X L /X K , A ) and B ∗ = ` v |∞ C ∗ (Sp ec L v / Sp ec K v , A ) . First, we will ﬁnd a homomorphism from the subgroup of co cycles in C ∗ (Gal( L/K ) , A ) to the subgroup of co cycles in the mapping cone of the Čec h complexes for co verings f ∗ : A ∗ → B ∗ . F rom the pro of of Lemma 2.2, we ha ve a homomorphism of complexes Φ : C ∗ (Gal( L/K ) , A ) → A ∗ . Since L/K is split completely at v , the complex B p is exact at p ≥ 1 . F rom this it follows, for p ≥ 2 , that given a co cycle f ∗ a ∈ A p , there is a b ∈ B p − 1 , unique up to cob oundaries, suc h that f ∗ a = − d ( b ) . So for p ≥ 2 , the map α 7→ (Φ α, b ) , such that f ∗ Φ α = − d ( b ) , gives us a homorphism from the subgroup of co cycles in C p (Gal( L/K ) , A ) to the quotien t of the group of elements ( a, b ) ∈ A p × B p − 1 suc h that f ∗ a = − d ( b ) by the group 0 × dB p − 2 . W e can c heck that a cob oundary dα is mapp ed to the cob oundary d ( − Φ α, 0) = ( d Φ α , − f ∗ Φ α ) , using the deﬁnition of the b oundary map of a mapping cone. The maps from Čec h complexes for cov erings to Čech complexes then give us a map of complexes from the mapping cone for f ∗ : A ∗ → B ∗ to the mapping cone for f ∗ : C ∗ ( X K , A ) → Q v |∞ C ∗ (Sp ec K v , A ) . Combining with the ab o ve, w e obtain a homomorphism H p (Gal( L/K ) , A ) → H p c ( X K , A ) . It is clear from the construction that the comp osition with H p c ( X K , A ) → H p ( X K , A ) giv es the map of Lemma 2.2. The compatibilities b oth follo w in a straigh tforw ard w ay from the compatibilit y of Φ with pullbac k. □ Lemma 2.5. L et n b e a p ositive inte ger. L et L/F b e a ﬁnite Galois extension of glob al ﬁelds, with K an interme diate ﬁeld such that K /F is Galois and K c ontains n th r o ots of unity. The A rtin-V er dier tr ac e and the A rtin-V er dier fundamental class for L/K ar e Gal( K/F ) - invariant. Pr o of. Let Γ = Gal( K/F ) . F or an y lift ˜ γ ∈ Gal( L/F ) of a γ ∈ Γ , we hav e ˜ γ : L → L , and ˜ γ : K → K , and thus b y Lemma 2.4 we hav e that the map H 3 ( H , Z /n ) → H 3 c ( X K , Z /n ) asso ciated to L/K is a Γ -equiv ariant map for the natural actions of Γ on the left and right (where Γ acts on H 3 ( H , Z /n ) b y lift to Gal( L/F ) and conjugation on H , not dep ending on the lift since H ’s self-conjugation acts trivially on H 3 ( H , Z /n ) ). The map H 3 c ( X K , Z /n ) → H 3 c ( X K , µ n ) is Γ -equiv arian t, and H 3 c ( X K , µ n ) D → Z /n is Γ -in v ariant. W e conclude that the Artin-V erdier trace H 3 (Gal( L/K ) , Z /n ) → Z /n and the Artin-V erdier fundamental class for L/K are Γ -inv ariant. □ 3. Rela tions with prior inv ariants This section will explain in detail how our w ork relates to prior pap ers. W e will explain in Section 3.1 ho w our Artin-V erdier fundamen tal class reﬁnes the bilinearly enhanced group structure that Lipno wski, T simerman, and the ﬁrst author [LST20] deﬁned in the case that | Γ | = 2 . In Section 3.2, we will explain ho w our Artin-V erdier fundamental class reﬁnes lifting in v ariants deﬁned b y Liu [Liu22] and the second author [W o o19, W o o21]. In Section 3.3, w e will explain how our conjectures relate to some conjectures made b y Boston and Bush [BBH21] in the case that Γ is the cyclic group of order 3 . The relationship with [Liu22] will b e used in subsequen t proofs, as we will cite some results from [Liu22], while the others are used only to provide context and numerical evidence. 23 3.1. Bilinearly enhanced groups, after Lipno wski-Sa win-T simerman. In the pap er [LST20], tw o in v ariants are deﬁned on the class group of a n umber ﬁeld, called the ω and ψ -in v ariants, whic h can b e calculated using the Artin-V erdier trace. In this subsection, w e ﬁrst review the deﬁnitions of the ω and ψ inv arian ts of [LST20], then explain how to deriv e these in v ariants from the Artin-V erdier trace, and ﬁnally pro ve some auxiliary results ab out the information captured by the Artin-V erdier trace but not the ω and ψ inv arian ts. Let K be a num b er ﬁeld and n b e the order of group group of ro ots of unit y in K . Let ℓ b e an o dd prime and let v ≥ 1 b e the ℓ -adic v aluation of n , so that K contains the ℓ v th ro ots of unit y . In particular, this implies that K has no real places. The pap er [LST20] deﬁnes t wo inv arian ts, the ω and ψ inv arian t, on the ℓ -part of the class group of K . W e will use in this subsection the same notation used in that pap er, except that their n is our v . In this section, we will consider ﬂat cohomology H i fppf (Sp ec O K , F ) along with étale coho- mology , as the deﬁnitions of [LST20] use ﬂat cohomology . Recall that when F is a smo oth, quasi-pro jectiv e, commutativ e group scheme, the ﬂat and étale cohomology groups coincide [Mil80, I II, Theorem 3.9]. W e will use this in particular when F = Z /k or G m . Since K has no real places, cohomology coincides with compactly supp ort cohomology in b oth the ﬂat and étale cases. W e write tr A V for the map (isomorphism) H 3 (Sp ec O K , G m ) → Q / Z [Mil06, Chapter I I, Section 3], and with a slight abuse of notation, also use tr A V to denote the maps H 3 (Sp ec O K , µ k ) → Q / Z and H 3 fppf (Sp ec O K , µ k ) → Q / Z obtained b y ﬁrst mapping to cohomology with co eﬃcients in G m and then applying the ab ov e map. The ω -inv arian t on the class group Cl( K ) of a num b er ﬁeld K containing the ℓ v ’th ro ots of unit y is an elemen t of ( ∧ 2 Cl( K ))[ ℓ v ] . T o deﬁne this inv ariant [LST20, Deﬁnition 4.5], one uses [LST20, Lemma 4.2 and Corollary 4.3], which show that such an elemen t is equiv alen t to a tuple of, for each m , a symplectic bilinear form ω m,K : Cl( K ) ∨ [ ℓ m ] × Cl( K ) ∨ [ ℓ m ] → Q / Z [ ℓ v ] , suc h that for all a ∈ Cl( K ) ∨ [ ℓ m ] , b ∈ Cl( K ) ∨ [ ℓ m +1 ] we ha ve ω m,K ( a, ℓb ) = ω m +1 ,K ( a, b ) . Let ˜ K b e the Hilb ert class ﬁeld of K , and w e use class ﬁeld theory to write Cl( K ) = Gal( ˜ K /K ) . F or any in teger k , w e write ϕ = ϕ ˜ K /K : H i (Cl( K ) , Z /k ) → H i (Sp ec O K , Z /k ) for the map from Lemma 2.2. Recall our ﬁxed generator ξ of the n th ro ots of unit y in K . Let ζ m ∈ H 1 fppf (Sp ec O K , µ ℓ m ) corresp ond to the µ ℓ m torsor consisting of the ℓ m th ro ots of the generator ξ n/ℓ v of µ ℓ v (as in [Mil80, I I I, Prop osition 4.6]). One constructs the pairing ω m,K b y the form ula ω m,K ( a, b ) = − 1 2 tr A V ( ζ m ∪ ϕ ( a ) ∪ ϕ ( b )) for a, b ∈ Cl( K ) ∨ [ ℓ m ] = H 1 (Cl( K ) , Z /ℓ m ) . Since ζ m ∈ H 1 fppf , the cup pro ducts are in ﬂat cohomology . In [LST20, Lemma 4.4], these are chec k ed to satisfy the conditions required to deﬁne an element of ( ∧ 2 Cl( K ))[ ℓ v ] . The ψ -in v ariant is a homomorphism ψ : Cl( K ) ∨ [ ℓ v ] → Cl( K )[ ℓ v ] . This data is equiv alen t b y dualit y to a pairing ⟨ , ⟩ : Cl( K ) ∨ [ ℓ v ] × Cl( K ) ∨ → Q / Z given b y the form ula ⟨ a, b ⟩ = b ( ψ ( a )) . Again, we use class ﬁeld theory to write Cl( K ) = Gal( ˜ K /K ) . W e deﬁne the ψ map [LST20, Deﬁnition 4.1] as the comp osition of ϕ : Cl( K ) ∨ [ ℓ v ] = H 1 (Cl( K ) , Z /ℓ v ) → H 1 (Sp ec O K , Z /ℓ v ) , the cup pro duct H 1 (Sp ec O K , Z /ℓ v ) → H 1 (Sp ec O K , µ ℓ v ) with the gen- erator ξ n/ℓ v of µ ℓ v , and the map H 1 (Sp ec O K , µ ℓ v ) → H 1 (Sp ec O K , G m ) = Cl( K ) from the map µ ℓ v → G m . Next, we will giv e form ulas for the bilinear forms ω m,K and ⟨ , ⟩ in terms of the Artin- V erdier trace and the Bo c kstein homomorphism B : H i (Cl( K ) , Z /ℓ m ) → H i +1 (Cl( K ) , Z /ℓ v ) 24 arising from the short exact sequence Z /ℓ v → Z /ℓ v + m → Z /ℓ m . Let s Cl( K ) ∈ H 3 (Cl( K ) , Z /n ) b e the Artin-V erdier fundamental class for ˜ K /K . Let s Cl( K ) : H 3 (Cl( K ) , Z /n ) → Z /n be the usual induced map, and let s ℓ Cl( K ) : H 3 (Cl( K ) , Z /ℓ v ) → Q / Z [ ℓ v ] b e the comp osition of the map H 3 (Cl( K ) , Z /ℓ v ) → H 3 (Cl( K ) , Z /n ) by m ultiplication by n/ℓ v , the map s Cl( K ) , and Z /n → Q / Z giv en by division b y n . Theorem 3.1. L et ℓ b e an o dd prime and K a numb er ﬁeld c ontaining the ℓ v th r o ots of unity, and not the ℓ v +1 the r o ots of unity, for some v ≥ 1 . F or al l p ositive i nte gers m and a, b ∈ H 1 (Cl( K ) , Z /ℓ m ) we have ω m,K ( a, b ) = − 1 2 s ℓ Cl( K ) ( B ( a ∪ b )) and for a ∈ H 1 (Cl( K ) , Z /ℓ v ) and b ∈ H 1 (Cl( K ) , Z /ℓ m ) we have ⟨ a, b ⟩ = s ℓ Cl( K ) ( a ∪ B ( b )) . Th us, the ω and ψ in v arian ts both can b e calculated by applying s Cl( K ) to classes in H 3 (Cl( K ) , Z /ℓ v ) arising from the cup pro duct and Bo c kstein homomorphism. Motiv ated b y Theorem 3.1, w e deﬁne ω and ψ inv arian ts for an oriented Γ -group H = ( H , s H ) b y the same form ulas: ω m ( a, b ) = − 1 2 s ℓ H ( B ( a ∪ b )) , for a, b ∈ H 1 ( H , Z /ℓ m ) ⟨ a, b ⟩ = b ( ψ ( a )) = s ℓ H ( a ∪ B ( b )) , for a ∈ H 1 ( H , Z /ℓ v ) and b ∈ H 1 ( H , Z /ℓ m ) in the ﬁrst case assuming that ℓ is an o dd prime. Note that the Γ action plays no role in the deﬁnitions, but the fact that s H is Γ -inv arian t do es imply that the pairings ω m and ⟨ , ⟩ are Γ -in v ariant. T o prov e Theorem 3.1, the k ey lemma is the follo wing, whic h generalizes [LST20, Lemma 6.18]. Lemma 3.2. L et X b e a scheme and let 0 → G 1 → G 2 → G 3 → 0 b e a short exact se quenc e of ﬁnite ﬂat c ommutative gr oup schemes over X . L et 0 → G ∨ 3 → G ∨ 2 → G ∨ 1 → 0 b e the Cartier dual se quenc e, which is exact (e.g. by [Oh17, Prop osition 1.2.2] ). L et B : H i fppf ( X , G 3 ) → H i +1 fppf ( X , G 1 ) b e the c onne cting homomorphism and similarly for B ∨ : H i fppf ( X , G ∨ 1 ) → H i +1 fppf ( X , G ∨ 3 ) . F or i, j nonne gative inte gers, α ∈ H i fppf ( X , G 3 ) ,and β ∈ H j fppf ( X , G ∨ 1 ) we have α ∪ B ∨ β = ( − 1) i +1 B α ∪ β . F or X the sp e ctrum of a ring of inte gers of the numb er ﬁeld, if i ≥ 2 , then the same claim is true for α ∈ H i fppf ,c ( X , G 3 ) and β ∈ H j fppf ( X , G ∨ 1 ) , wher e the cup pr o duct of a c omp actly- supp orte d c ohomolo gy class and an or dinary c ohomolo gy class is a c omp actly-supp orte d c o- homolo gy class. Pr o of. The crux of the argument is to consider Cech cohomology and then use the pro duct rule for cup pro ducts. Cho ose cocycles in the Cec h complex of a suitable h yp erco vering represen ting α and β [Sta18, 01GU]. Lift α to a co c hain ˜ α in C i ( X , G 2 ) and lift β to a co c hain ˜ β in C j ( X , G ∨ 2 ) . 25 Then ˜ α ∪ ˜ β is a co c hain in C i + j ( X , G m ) so d ( ˜ α ∪ ˜ β ) is a cob oundary and th us represen ts the zero class in cohomology . Hence d ( ˜ α ∪ ˜ β ) = d ˜ α ∪ ˜ β + ( − 1) i ˜ α ∪ d ˜ β represen ts the zero class in cohomology [Sta18, 01FP]. No w by the deﬁnition of the Bo ckstein homomorphism, d ˜ α is the pushforward of a co cycle represen ting B ˜ α along G 1 → G 2 . The multiplication map G 2 × G ∨ 2 → G m , restricted to G 1 × G ∨ 2 , factors through the multiplication map G 1 × G ∨ 1 → G m , so the multiplication map C i +1 ( X , G 2 ) × C j ( X , G ∨ 2 ) → C i + j +1 ( X , G m ) , restricted to C i +1 ( X , G 1 ) × C j ( X , G ∨ 2 ) , factors through the m ultiplication map C i +1 ( X , G 1 ) × C j ( X , G ∨ 1 ) → C i + j +1 ( X , G m ) and the pro jection C j ( X , G ∨ 2 ) → C j ( X , G ∨ 1 ) applied to ˜ β recov ers β . This implies d ˜ α ∪ ˜ β represents the cohomology class B α ∪ β . Symmetrical reasoning sho ws that ˜ α ∪ d ˜ β represents the cohomology class α ∪ B ∨ β . Com- bining these gives the statement. W e now consider the case where X is Sp ec O L , and α lies in the compactly supp orted cohomology , and i ≥ 2 . Compactly supp orted cohomology of the ring of in tegers of a num b er ﬁeld is deﬁned as the mapping cone of the natural map from the usual cohomology complex to a complex represen ting T ate cohomology of the Galois groups of the places at inﬁnity . Thus co c hains in degree i for the compactly supp orted cohomology consist of co c hains in degree i for the usual cohomology together with co chains in degree i − 1 for the T ate cohomology . Co c hains in degree i − 1 ≥ 1 for the T ate cohomology are the same as cochains in degree i − 1 for the ordinary group cohomology whic h may b e computed by Cech cohomology of the same hypercov ering. So we may c ho ose a co cycle representing the cohomology class α that is a pair consisting of a co chain in C i (Sp ec O L ; G 2 ) and a co chain in C i − 1 of the Cec h complex at the inﬁnite places. F or lifting to ˜ α , applying the cup pro duct, and applying the diﬀeren tial, we may pro ceed as ab o ve. □ Pr o of of The or em 3.1. W e apply Lemma 3.2 to X = Sp ec O K and G 1 = Z /ℓ v and G 2 = Z /ℓ m + v and G 3 = Z /ℓ m . Let B b e the connecting homomorphism of this short exact sequence and let B ∨ b e the connecting homomorphism of the dual short exact sequence 0 → µ ℓ m → µ ℓ v + m → µ ℓ v → 0 . F or i, j satisfying i + j = 2 and classes α ∈ H i fppf ( X , Z /ℓ m ) and β ∈ H j fppf ( X , µ ℓ v ) , this gives the identit y in H 3 fppf (Sp ec O K , G m ) (3.3) α ∪ B ∨ β = ( − 1) i +1 B α ∪ β . W e apply (3.3) to establish b oth parts of Theorem 3.1. The class called ζ m in the deﬁnition of ω m,K is the connecting homomorphism B ∨ applied to the generator ξ − n/ℓ v of H 0 fppf (Sp ec O K , µ ℓ v ) . Applying (3.3) with i = 2 w e see that ω m,K ( a, b ) = − 1 2 tr A V ( ζ m ∪ ϕ ( a ) ∪ ϕ ( b )) = 1 2 tr A V ( B ∨ ξ n/ℓ v ∪ ϕ ( a ) ∪ ϕ ( b )) = 1 2 tr A V ( ϕ ( a ) ∪ ϕ ( b ) ∪ B ∨ ξ n/ℓ v ) = − 1 2 tr A V ( B ( ϕ ( a ) ∪ ϕ ( b )) ∪ ξ n/ℓ v ) = − 1 2 tr A V ( ϕ ( B ( a ∪ b )) ∪ ξ n/ℓ v ) . 26 The last line follows from Lemma 2.2, whic h giv es the compatibility of ϕ with cup pro ducts and the Bo c kstein homomorphism. Finally , we ha ve a commutativ e diagram relating the deﬁnition of s ℓ Cl( K ) to the deﬁnition of s Cl( K ) (using the functoriality of ϕ in the co eﬃcien ts from Lemma 2.2) H 3 (Cl( K ) , Z /n ) H 3 (Sp ec O K , Z /n ) H 3 (Sp ec O K , µ n ) Q / Z Z /n H 3 (Cl( K ) , Z /ℓ v ) H 3 (Sp ec O K , Z /ℓ v ) H 3 (Sp ec O K , µ ℓ v ) Q / Z . ϕ 1 7→ ξ tr A V × n × 1 n ϕ × n ℓ v 1 7→ ξ n/ℓ v × n ℓ v tr A V The comp osite of the top line ab ov e is s Cl( K ) , and then going from the b ottom left to the b ottom righ t is s ℓ Cl( K ) , which is also the map x 7→ tr A V ( ϕ ( x ) ∪ ξ n/ℓ v ) . Thus we conclude that ω m,K ( a, b ) = − 1 2 s ℓ Cl( K ) ( B ( a ∪ b )) . No w w e consider the claim ab out the pairing ⟨ , ⟩ . First [LST20, Prop osition 6.3] im- plies that ⟨ a, b ⟩ for a ∈ Cl ∨ K [ ℓ v ] and b ∈ Cl ∨ K [ ℓ m ] can b e calculated b y ﬁrst cupping ϕ ( a ) with the ﬁxed generator ζ to obtain a class in H 1 fppf (Sp ec O K , µ ℓ v ) , then applying the map H 1 fppf (Sp ec O K , µ ℓ v ) → H 1 fppf (Sp ec O K , G m ) , then applying the Kummer map H 1 fppf (Sp ec O K , G m ) → H 2 fppf (Sp ec O K , µ ℓ m ) , and then taking the cup pro duct with ϕ ( b ) , and then taking Artin- V erdier trace tr A V . Note that the tw o maps H 1 fppf (Sp ec O K , µ ℓ v ) → H 1 fppf (Sp ec O K , G m ) and H 1 fppf (Sp ec O K , G m ) → H 2 fppf (Sp ec O K , µ ℓ m ) are b oth are part of the Kummer sequence but, ev en if v = m , these are not adjacen t arro ws in the Kummer sequence and so their comp osi- tion is not automatically zero. Instead, one arises from a map of group sc hemes µ ℓ v → G m and the next from a short exact sequence µ ℓ m → G m → G m . Their comp osition therefore arises from the pullbac k of the short exact sequence along the map of group sc hemes, whic h is µ ℓ m → µ ℓ v + m → µ ℓ v . In other words, the comp osition of the tw o maps is B ∨ . Using this, (3.3), and the compatibilit y of ϕ with cup pro ducts and the Bo c kstein homomorphism, we ha ve ⟨ a, b ⟩ = tr A V ( B ∨ ( ϕ ( a ) ∪ ξ n/ℓ v ) ∪ ϕ ( b )) = tr A V ( ϕ ( a ) ∪ ξ n/ℓ v ∪ B ( ϕ ( b ))) = tr A V ( ϕ ( a ∪ B ( b )) ∪ ξ n/ℓ v ) = s ℓ Cl( K ) ( a ∪ B ( b )) . □ The pap er [LST20, Lemma 6.20] prov ed a certain compatibilit y b et ween the ω and ψ in v ariants of class groups. W e will sho w this compatibilit y for general orien ted groups follo ws straigh tforwardly from our deﬁnition of the general v ersion of these in v ariants. The pro of follo ws the same idea as that in [LST20], but is a bit more transparen t as certain non-split group sc hemes app earing in [LST20] are replaced with cyclic groups. The upshot is that the data of a bilinearly-enhanced ab elian group from [LST20] all comes from the data of an orien ted group. Prop osition 3.4. L et H b e a ﬁnite gr oup, n a p ositive inte ger, s H ∈ H 3 ( H , Z /n ) a class. L et ℓ b e an o dd prime and v the ℓ -adic valuation of n . L et r b e a non-ne gative inte ger and let m = v + r . F or ω m and ⟨ , ⟩ the p airings deﬁne d in terms of s ℓ H , we have, for a, b ∈ H 1 ( H , Z /ℓ m ) and ¯ a, ¯ b ∈ H 1 ( H , Z /ℓ v ) their images under the map 1 7→ 1 on c o eﬃcients, ⟨ ¯ a, b ⟩ − ⟨ ¯ b, a ⟩ = 2 ω m ( a, b ) . 27 Pr o of. Cho ose cocycles represen ting a, b (and call these a, b b y a sligh t abuse of notation) and lift these to co chains ˜ a, ˜ b ∈ C 1 ( H , Z /ℓ v + m ) . Then ˜ a ∪ ˜ b ∈ C 2 ( H , Z /ℓ v + m ) lifts a ∪ b . Hence, b y the deﬁnition of Bo ckstein homomorphism, d (˜ a ∪ ˜ b ) is the pushforward from C 3 ( H , Z /ℓ v ) to C 3 ( H , Z /ℓ v + m ) of a cycle represen ting B ( a ∪ b ) . The Leibnitz formula for cup pro duct gives d (˜ a ∪ ˜ b ) = d ( ˜ a ) ∪ ˜ b − a ∪ d ( ˜ b ) . By the deﬁnition of Bo c kstein homomorphism, d ( ˜ α ) is the pushforward from C 2 ( H , Z /ℓ v ) of a co cycle represen ting B ( a ) . The multiplication map Z /ℓ v + m × Z /ℓ v + m → Z /ℓ v + m comp osed with the m ultiplication b y ℓ m map t : Z /ℓ v → Z /ℓ v + m giv es a bilinear form Z /ℓ v × Z /ℓ v + m → Z /ℓ v + m . This can be expressed as the comp osition of ﬁrst the mod ℓ v map r : Z /ℓ v + m → Z /ℓ v , then the multiplication map Z /ℓ v × Z /ℓ v → Z /ℓ v , follow ed b y the m ultiplication by ℓ m map t : Z /ℓ v → Z /ℓ v + m . Applying this identit y to co cycles, the cup pro duct map C 2 ( H , Z /ℓ v + m ) × C 1 ( H , Z /ℓ v + m ) → C 3 ( H , Z /ℓ v + m ) comp osed with t ∗ : C 2 ( H , Z /ℓ v ) → C 2 ( H , Z /ℓ v + m ) is the comp osition of r ∗ : C 1 ( H , Z /ℓ v + m ) → C 1 ( H , Z /ℓ v ) , the cup pro duct map C 2 ( H , Z /ℓ v ) × C 1 ( H , Z /ℓ v ) → C 3 ( H , Z /ℓ v ) , and t ∗ : C 3 ( H , Z /ℓ v ) → C 3 ( H , Z /ℓ v + m ) . In other words, for x ∈ C 2 ( H , Z /ℓ v ) and y ∈ C 1 ( H , Z /ℓ v + m ) , w e hav e t ∗ ( x ) ∪ y = t ∗ ( x ∪ r ∗ ( y )) . Applying this to ( t − 1 ) ∗ d (˜ a ) (whic h is a co chain representing B ( a ) ) and ˜ b , w e ha v e d (˜ a ) ∪ ˜ b = t ∗ (( t − 1 ) ∗ d (˜ a ) ∪ r ∗ ( ˜ b )) . W e observ e that since m = v + r ≥ v , the pro jection r ∗ : C 1 ( H , Z /ℓ v + m ) → C 1 ( H , Z /ℓ v ) factors through C 1 ( H , Z /ℓ m ) . Th us r ∗ ( ˜ b ) = ¯ b . So w e ha v e d (˜ a ) ∪ ˜ b = t ∗ (( t − 1 ) ∗ d (˜ a ) ∪ ¯ b ) and similarly ˜ a ∪ d ( ˜ b ) = t ∗ (¯ a ∪ ( t − 1 ) ∗ d ( ˜ b )) . Applying t − 1 ∗ to the Leibnitz form ula then giv es t − 1 ∗ ( d (˜ a ∪ ˜ b )) = ( t − 1 ) ∗ d (˜ a ) ∪ ¯ b − ¯ a ∪ ( t − 1 ) ∗ d ( ˜ b ) , whic h in cohomology giv es B ( a ∪ b ) = B ( a ) ∪ ¯ b − ¯ a ∪ B ( b ) . T aking s ℓ H of b oth sides we obtain s ℓ H ( B ( a ∪ b )) = s ℓ H ( ¯ b ∪ B ( a )) − s ℓ H (¯ a ∪ B ( b )) whic h by deﬁnition gives. − 2 ω m ( a, b ) = ⟨ ¯ b, a ⟩ − ⟨ ¯ a, b ⟩ . □ R emark 3.5 . W e remark on the case ℓ = 2 , not considered in [LST20]. The factor 1 2 in the form ula for ω m giving in Theorem 3.1 is unreasonable to include in this case, and can be remo ved (it exists only for consistency with [LST20], who included it for consistency with a certain random matrix mo del.) The pairing a, b 7→ s H ( B ( a ∪ b )) is clearly an tis ymmetric. F or pairings on 2 -p o wer torsion groups, b eing an tisymmetric is a strictly weak er condition than b eing alternating, i.e., the pairing of any elemen t with itself b eing 0 . It is straightforw ard to chec k the pairing a, b 7→ s H ( B ( a ∪ b )) is alternating. An y class a ∈ H 1 ( H , Z / 2 m ) is the pullbac k of a univ ersal class in H 1 ( Z / 2 m , Z / 2 m ) along a homomorphism H → Z / 2 m , and so B ( a ∪ a ) is the pullbac k of a class in the image of the Bo ckstein map H 2 ( Z / 2 m , Z / 2 m ) → H 3 ( Z / 2 m , Z / 2 v ) , but that map is zero, as ma y b e calculated using the standard p erio dic resolution for group cohomology of cyclic groups. 28 If this pairing w ere c heck ed to b e equiv alen t to the pairing constructed in [MS24, Prop o- sition 6.4], then the fact that this pairing is alternating w ould be equiv alent to [MS24, Prop osition 6.6], and the argument ab o v e would give a new pro of of that prop osition. It is straightforw ard to c heck that for an ab elian group H , the ω and ψ inv arian ts typically do not contain all the information included in s H . Example 3.6. L et ℓ b e an o dd prime, n = ℓ , and H = ( Z /ℓ ) d . Then the Künneth formula and the fact that H i ( Z /ℓ, Z /ℓ ) ∼ = Z /ℓ for al l i ≥ 0 gives H 3 ( H , Z /ℓ ) ∼ = ( Z /ℓ ) d ( d +1)( d +2) 6 . Thus ther e ar e ℓ d ( d +1)( d +2) 6 choic es of orientation on H . On the other hand, the ψ invariant is a line ar form ( Z /ℓ ) d → ( Z/ℓ ) d and the ω invariant is determine d by ω 1 and thus by ψ using Pr op osition 3.4. So the ω , ψ invariants ar e p ar ametrize d by Hom(( Z /ℓ ) d , ( Z /ℓ ) d ) ∼ = ( Z /ℓ ) d 2 . Thus the numb er of choic es of orientation s H for e ach ω , ψ is at le ast ℓ d ( d +1)( d +2) 6 − d 2 = ℓ d ( d − 1)( d − 2) 6 . In fact, for a ﬁnite ab elian group H , an orientation s H is determined b y ω and ψ together with certain trilinear forms on H 1 ( H , Z /ℓ m ) . Lemma 3.7. L et H b e a ﬁnite ab elian gr oup and n a nonne gative int. The gr oup H 3 ( H , Z /n ) is gener ate d by classes of the fol lowing thr e e typ es, over al l primes ℓ | n with v the ℓ -adic valuation v of n , under the × n/ℓ v map on c o eﬃcients H 3 ( H , Z /ℓ v ) → H 3 ( H , Z /n ) : (1) F or m ≤ v , the image under H 3 ( H , Z /ℓ m ) × ℓ v − m → H 3 ( H , Z /ℓ v ) of the cup pr o duct of thr e e classes in H 1 ( H , Z /ℓ m ) . (2) Classes of the form B ( a ∪ b ) for a, b ∈ H 1 ( H , Z /ℓ m ) . (3) Classes of the form a ∪ B ( b ) for a ∈ H 1 ( H , Z /ℓ v ) and b ∈ H 1 ( H , Z /ℓ m ) . In p articular, a class s H ∈ H 3 ( H , Z /n ) is uniquely determine d by its p airings with classes of these thr e e typ es. Pr o of. Since H 3 ( H , Z /n ) is generated by H 3 ( H , Z /ℓ v ) for ℓ dividing n , we may as w ell assume n = ℓ v . Since ℓ -p o wer cohomology classes of ﬁnite ab elian groups are alw ays pullbac ks from a ﬁnite ab elian ℓ -group, we may assume H is an ℓ -group. The short exact sequence 0 → Z → Z → Z /n → 0 induces a long exact sequence · · · → H 3 ( H , Z ) → H 3 ( H , Z /n ) → H 4 ( H , Z ) → H 4 ( H , Z ) → . . . and th us in particular an exact sequence H 3 ( H , Z ) /nH 3 ( H , Z ) t 0 → H 3 ( H , Z /n ) B v → H 4 ( H , Z )[ n ] → 0 . W e will chec k that classes of t yp e (2) span the image of t 0 , and that B v applied to classes of t yp e (1) and (3) generate H 4 ( H , Z )[ n ] . It immediately follo ws that classes of types (1), (2), and (3) together generate H 3 ( H , Z /n ) . T o do this, w e use a description of the in tegral cohomology ring of a ﬁnitely generated ab elian group due to Huebschmann [Hue91]. W e ﬁrst review this description, though we m ust adjust Huebschmann’s notation slightly to harmonize it with our o wn. W rite H = Q r i =1 Z /d i where d i | d i +1 . Let A ( H ) b e the graded algebra generated by v ariables x 1 , . . . , x r in degree 1 and ζ 1 , . . . , ζ r in degree 2 , sub ject to the relations that the ζ i comm ute with everything, the x i an ticommute with eac h other, x 2 i = d i ( d i − 1) 2 ζ i , and d i ζ i = 0 . Also for i 1 , . . . , i t a tuple of integers satisfying 1 ≤ i 1 < · · · < i t ≤ r , let Z i 1 ,i 2 ,...,i t = B d i 1 ( X i 1 ∪ · · · ∪ X i t ) ∈ H t +1 ( H , Z ) , 29 for certain the classes X i ∈ H 1 ( H , Z /d i ) . It is not necessary for our argument to know what classes these are, but presumably X i is dual to the standard generator of Z /d i , although it is not easy to chec k this from [Hue91]. Then [Hue91, Theorem B] states that the integral cohomology rin g H ∗ ( H , Z ) is isomorphic to the subring of A ( H ) generated by the expressions ˜ ζ x i 1 x i 2 ...x i t = t X s =1 ( − 1) s − 1 d s d 1 ζ i s x i 1 . . . x i s − 1 x i s +1 . . . x i t for i 1 , . . . , i t a tuple of integers satisfying 1 ≤ i 1 < · · · < i t ≤ r , and furthermore states that this isomorphism sends Z i 1 ,i 2 ,...,i t to ˜ ζ x i 1 x i 2 ...x i t . In particular, the cohomology ring is generated b y the classes Z i 1 ,i 2 ,...,i t in degree t + 1 . Since we ha ve generators in degree 2 , 3 , and 4 , but not 1 , the only wa y to obtain a class in degree 3 is from a generator Z i,j in degree 3 , and the only wa ys to obtain a class in degree 4 is from a generator Z i,j,k in degree 4 or the pro duct Z i Z j of tw o generators in degree 2 . In the last case we do not assume i < j but without loss of generalit y assume i ≤ j . T o calculate the n -torsion in H 4 ( H , Z ) , w e need to understand the relations b etw een these classes. Note that A ( H ) is generated as an ab elian group by monomials in the classes x 1 , . . . , x r , ζ 1 , . . . , ζ r where no x i app ears twice, sub ject to the relation that a monomial containing ζ i is d i -torsion. Th us a p olynomial is n -torsion if and only if each monomial app earing is n -torsion. Since no t wo of the classes ˜ ζ x i x j x k or ˜ ζ x i ˜ ζ x j share a monomial, a sum of them is n -torsion if and only if each summand is n -torsion. So the n -torsion in H 4 ( H , Z ) is generated by the classes d i gcd( d i ,n ) Z i,j,k and d i gcd( d i ,n ) Z i Z j . W e will now sho w that the images of classes Z i,j under t 0 are classes of t yp e (2) and the classes d i gcd( d i ,n ) Z i,j,k and d i gcd( d i ,n ) Z i Z j arise from applying B v to classes of t yp e (1) and (3). This requires using the compatibilit y of many diﬀerent Bo c kstein maps. Let B a,b denote the connecting Bo c kstein homomorphisms for the short exact sequence Z /ℓ b × ℓ a → Z /ℓ a + b → Z /ℓ a , and B a = B a, ∞ analogously for the sequence Z × ℓ a → Z → Z /ℓ a . Our previous map B is B m,v in this notation (though m can v ary). W e let t k : H ∗ ( H , Z /ℓ a ) → H ∗ ( H , Z /ℓ b ) denote the map induced b y the map of coeﬃcients Z /ℓ a → Z /ℓ b (or Z → Z /ℓ b ) sending 1 to ℓ k (for a + k ≤ b ), slightly o verloading the n otation. It is straigh tforw ard to c heck from the deﬁnitions that t k B a,b = B c,d t c + k − a for any a, b, c, d, k ≥ 0 with c + k − a ≥ 0 . (Roughly , B a,b comes ab out by lifting, taking d , and dividing b y ℓ a .) Note that the identit y map is a sp ecial case of t 0 . W e ﬁrst consider a class Z i,j . Let m b e such that ℓ m = d i . Let ¯ X j = t 0 ( X j ) ∈ H 1 ( H , Z /ℓ m ) . W e hav e Z i,j = B m ( X i ∪ X j ) = B m ( X i ∪ ¯ X j ) . Thus in H 3 ( H , Z /ℓ v ) we hav e t 0 ( Z i,j ) = t 0 B m ( X i ∪ ¯ X j ) = B m,v ( X i ∪ ¯ X j ) , a class of type (2) , as desired. W e now consider a generator d i gcd( d i ,n ) Z i,j,k of H 4 ( H , Z )[ n ] . Let m be such that ℓ m = d i . If m ≤ v then gcd( d i , n ) = ℓ m and d i gcd( d i ,n ) Z i,j,k = Z i,j,k . W e write ¯ X j and ¯ X k as ab o ve. W e hav e Z i,j,k = B m ( X i ∪ X j ∪ X k ) = B m ( X i ∪ ¯ X j ∪ ¯ X k ) = B v ( t v − m ( X i ∪ ¯ X j ∪ ¯ X k )) is the image of a class of type (1) under B v . If m > v then our generator of H 4 ( H , Z )[ n ] is d i gcd( d i ,n ) Z i,j,k = ℓ m − v Z i,j,k . W e write ¯ X i for t 0 ( X i ) ∈ H 3 ( H , Z /ℓ v ) , and similarly for ¯ X j , ¯ X k . W e hav e ℓ m − v Z i,j,k = t m − v B m ( X i ∪ X j ∪ X k ) = B v ( t 0 ( X i ∪ X j ∪ X k ) = B v ( ¯ X i ∪ ¯ X j ∪ ¯ X k ) . In particular ℓ m − v Z i,j,k is B v applied to a class of type (1) with m = v . 30 Finally , consider a class d i gcd( d i ,n ) Z i Z j . W e ﬁrst c ho ose a class a ∈ H 1 ( H , Z /ℓ v ) so that d i gcd( d i ,n ) Z i = B v ( a ) . If d i = ℓ c for c ≥ v then w e can take a = t 0 ( X i ) , and d i gcd( d i ,n ) Z i = ℓ c − v B c ( X i ) = B v ( a ) . If d i = ℓ c for c < v then w e can take a = t v − c ( X i ) , and d i gcd( d i ,n ) Z i = B c ( X i ) = B v ( a ) . W e let m b e such that ℓ m = d j and set b = X j ∈ H 1 ( H , Z /ℓ m ) , and in H 2 ( H , Z /ℓ v ) w e hav e that t 0 ( Z j ) = B m,v ( b ) = B ( b ) . Then to conﬁrm that d i gcd( d i ,n ) Z i Z j arises as th e image under B v of a class of type (3), it suﬃces to chec k that B v ( a ∪ B ( b )) = d i gcd( d i ,n ) Z i ∪ Z j . T o do this, c ho ose a co cycle in C 1 ( H , Z /ℓ v ) represen ting a and a lift to a co c hain ˜ a ∈ C 1 ( H , Z ) . Also choose a co cycle ∈ C 2 ( H , Z ) representing Z j , whic h we will refer to for simplicity as Z j . Then ˜ a ∪ Z j is a lift to C 3 ( H , Z ) of a co c hain represen ting a ∪ B ( b ) ∈ H 3 ( H , Z /ℓ v ) . Thus d ( ˜ a ∪ Z j ) is the image under the multiplication-b y- ℓ v map C 4 ( H , Z ) → C 4 ( H , Z ) of a cochain representing B v ( a ∪ B ( b )) ∈ H 4 ( H , Z ) . W e hav e d (˜ a ∪ Z j ) = d ( ˜ a ) ∪ Z j − ˜ a ∪ d ( Z j ) = d ( ˜ a ) ∪ Z j − ˜ a ∪ 0 = d ( ˜ a ) ∪ Z j since Z j is a cocycle. Now d (˜ a ) is the image under the m ultiplication b y ℓ v map of a co cycle representing d i gcd( d i ,n ) Z i , so that d ( ˜ a ) ∪ Z j is the image under multiplication b y ℓ v of a co c hain representing d i gcd( d i ,n ) Z i ∪ Z j . So d (˜ a ∪ Z j ) is the image under the m ultiplication- b y- ℓ v map C 4 ( H , Z ) → C 4 ( H , Z ) of b oth a co chain representing B v ( a ∪ B ( b )) and a co chain represen ting d i gcd( d i ,n ) Z i ∪ Z j . Since m ultiplication b y ℓ v is injective on cochains, this giv es that B v ( a ∪ B ( b )) = d i gcd( d i ,n ) Z i ∪ Z j . Th us, the classes d i gcd( d i ,n ) Z i Z j are in the image of classes of t yp e (3) under B v , completing the pro of. □ Ho wev er, when w e consider a group H ⋊ Γ with H ab elian, if Γ ∼ = Z / 2 then the trilinear forms on H 1 ( H , Z /ℓ m ) v anish so the ω and ψ inv arian ts do determine τ . Lemma 3.8. Assume Γ ∼ = Z / 2 and let H b e a ﬁnite ab elian Γ -gr oup of o dd or der such that H Γ is trivial. The gr oup H 3 ( H , Z /n ) Γ is gener ate d by classes of typ es (2) and (3) of L emma 3.7. In p articular, a cl ass s H ∈ H 3 ( H , Z /n ) Γ is uniquely determine d by its ω and ψ invariants. Pr o of. Since H has o dd order, H 3 ( H , Z /n ) has o dd order, and hence splits into summands where Γ acts with eigenv alue +1 and − 1 resp ectively . Since Γ acts on H with eigen v alue − 1 , it acts on H 1 ( H , Z /ℓ m ) with eigenv alue − 1 , and hence on cup pro ducts of three classes in H 1 ( H , Z /ℓ m ) with eigenv alue ( − 1) 3 = − 1 . Similarly , classes of types (2) and (3) ha ve eigen v alue ( − 1) 2 = 1 . So an y inv arian t class is a sum of classes of type (2) and (3). The last claim follows b ecause, again for odd order reasons H 3 ( H , Z /n ) Γ and H 3 ( H , Z /n ) Γ are dual. □ F urthermore, for L a lev el consisting only of ab elian Γ -groups, our form ula for the proba- bilit y depends only on ψ and the n umber of automorphisms of the orien ted group, as sho wn in the follo wing lemma. One result of this is that we see that diﬀeren t s H with the same ω , ψ do o ccur, and so w e ha ve found an inv ariant that carries more information than the billinearly enhanced group structure of [LST20], ev en on class groups. On the other hand, when for a giv en Γ -group H , the probabilities for all p ossibly orientations s H with a given ψ are summed, the following lemma gives that the resulting formula is still a nice pro duct. Th us for class groups the entire reﬁnement to s H , or even to the pair ω , ψ , is not necessary to obtain pro duct formula probabilities. 31 Lemma 3.9. L et L b e a level in the c ate gory of ﬁnite Γ -gr oups that c onsists only of ab elian Γ -gr oups (or, e quivalently, a level in the c ate gory of ﬁnite ab elian Γ -gr oups). L et W b e the set of ﬁnite n -oriente d Γ -gr oups whose underlying Γ -gr oup is in L . L et H ∈ L b e of or der r elatively prime to | Γ | . F or any n -orientation s H on H , let ν Γ ,n,U ( { X | X L ≃ ( H , s H ) } ) b e the pr ob ability fr om The or em 1.8, which is also c al le d v W , ( H,s H ) in The or em 7.6. Then | Aut( H , s H ) | v W , ( H,s H ) dep ends on s H only thr ough ψ : H ∨ [ ℓ v ] → H [ ℓ v ] for primes ℓ dividing n . Pr o of. Let H = ( H , s H ) . Theorem 1.8 expresses v W , H as M H | Aut( H ) | times a product of lo cal factors w V i and w N i . W e ha ve cancelled the | Aut( H ) | factor and the formula for M H mani- festly inv olves only H and not s H , so it suﬃces to chec k the w V i and w N i factors dep end on s H only through ψ . Also if H Γ is non trivial then M H = 0 by deﬁnition and thus v W , H = 0 indep enden tly of s H , so we may assume H Γ is trivial. Since L consists only of ab elian groups, there are certainly no admissible non-ab elian groups N i , and no N i terms. F urthermore, for V i a representation of H ⋊ Γ , if V i is admissible then V i ⋊ H ∈ L is ab elian so the action of H on V i is trivial, i.e., V i is a representation of Γ . In the expressions given for w V i in v arious cases b efore Theorem 1.8, the only dep endence on s H is through the subgroup H 2 ( H ⋊ Γ , V i ) L ,s H of H 2 ( H ⋊ Γ , V i ) , and only in cases when the c haracteristic of V i divides n . Thus it suﬃces to c heck H 2 ( H ⋊ Γ , V i ) L ,s H dep ends on s H only through ψ for eac h V i with ℓ := char V i dividing n . W e ﬁrst observ e that H 2 ( H ⋊ Γ , V i ) = H 2 ( H , V i ) Γ = ( H 2 ( H , F ℓ ) ⊗ V i ) Γ since the action of H on V i is trivial. Inside H 2 ( H , F ℓ ) we ha ve a subspace H 2 ( H , F ℓ ) B consisting of classes in the image of the Bo ckstein map B m, 1 : H 1 ( H , Z /ℓ m ) → H 2 ( H , F ℓ ) for at least one (equiv alently , all suﬃcien tly large) m . The subspace H 2 ( H , F ℓ ) B is clearly Γ -in v ariant so ( H 2 ( H , F ℓ ) B ⊗ V i ) Γ is a w ell-deﬁned subspace of ( H 2 ( H , F ℓ ) ⊗ V i ) Γ . Deﬁning H 2 ( H ⋊ Γ , V i ) L to consist of classes whose corresp onding Γ -extension (via Lemma 5.8) is in L , we next chec k that H 2 ( H ⋊ Γ , V i ) L ⊆ ( H 2 ( H , F ℓ ) B ⊗ V i ) Γ . T o do this, observe that a class α ∈ H 2 ( H ⋊ Γ , V i ) L can b e viewed as a ( Γ -inv arian t) dim V i - tuple of classes α j ∈ H 2 ( H , F ℓ ) , with the corresp onding extension of H by V i b eing the ﬁb er pro duct of the dim V i extensions of H by F ℓ , and with α ∈ ( H 2 ( H , F ℓ ) B ⊗ V i ) Γ if and only if eac h α j ∈ H 2 ( H , F ℓ ) B . Since the extension corresp onding to α lies in L , it is certainly ab elian, hence all the extensions in the ﬁb er pro duct are ab elian. So it suﬃces to chec k that if a class α i ∈ H 2 ( H , F ℓ ) corresp onds to an ab elian extension, then it lies in H 2 ( H , F ℓ ) B . F rom the deﬁnition of the Bo ckstein homomorphism H 1 ( H , Z /ℓ m ) → H 2 ( H , F ℓ ) , the ele- men ts in its image are represented by symmetric co chains, and thus corresp onding to ab elian extensions. Ab elian extensions of H by F ℓ are classiﬁed by Ext 1 ( H , F ℓ ) , which has size | H [ ℓ ] | . W e c ho ose an m large enough so that the image of H 1 ( H , Z /ℓ m ) → H 2 ( H , F ℓ ) is H 2 ( H , F ℓ ) B and so that ℓ m H has no ℓ -torsion. W e consider the exact sequence H 1 ( H , Z /ℓ m +1 ) → H 1 ( H , Z /ℓ m ) → H 2 ( H , F ℓ ) . One can c hec k that ℓ m H having no ℓ -torsion implies that the image of the ﬁrst map is ℓH 1 ( H , Z /ℓ m ) ⊂ H 1 ( H , Z /ℓ m ) , and th us H 2 ( H , F ℓ ) B also has size | H [ ℓ ] | . Th us all ab elian extension classes in H 2 ( H , F ℓ ) are in H 2 ( H , F ℓ ) B . W e no w handle the case that V i is trivial. In this case ( H 2 ( H , F ℓ ) B ⊗ V i ) Γ = ( H 2 ( H , F ℓ ) B ) Γ is the image of H 1 ( H , Z /ℓ m ) Γ (since Γ acts semisimply on ab elian groups of order a p ow er 32 of ℓ ). Ho wev er H 1 ( H , Z /ℓ m ) Γ = 0 since H Γ is trivial and any class in H 1 ( H , Z /ℓ m ) Γ w ould giv e a Γ -inv ariant quotient. Thus H 2 ( H ⋊ Γ , V i ) L ,s H = 0 regardless of s H in this case. In the case that V i is nontrivial, H 2 ( H ⋊ Γ , V i ) L ,s H consists of classes α ∈ H 2 ( H ⋊ Γ , V i ) L , suc h that s H ( α ∪ β ) = 0 for all β ∈ H 1 ( H ⋊ Γ , Hom( V i , Z /n )) . W e claim that for such α and β , the v alue of s H ( α ∪ β ) dep ends on s H only through ψ . One can easily reduce the pro of of this claim to the case that n is a p ow er of ℓ , so w e no w assume n = ℓ v . W e ma y compute s H ( α ∪ β ) b y pulling back α to H 2 ( H , V i ) ∼ = H 2 ( H , F ℓ ) dim V i and pulling β bac k to H 1 ( H , V ∨ i ) ∼ = H 1 ( H , F ℓ ) dim V i , where they b ecome tuples of classes α 1 , . . . , α dim V i ∈ H 2 ( H , F ℓ ) B and β 1 , . . . , β dim V i ∈ H 1 ( H , F ℓ ) resp ectiv ely (after choosing a basis of V i and the dual basis of V ∨ i ). Let t v − 1 : H ∗ ( H , F ℓ ) → H ∗ ( H , Z /ℓ v ) b e the map induced b y the inclusion F ℓ ⊂ Z /ℓ v whic h m ultiplies elements by ℓ v − 1 . Thus it suﬃces to sho w that s H ( t v − 1 ( α i ∪ β j )) dep ends on s H only through ψ . Let t 0 : H ∗ ( H , Z /ℓ v ) → H ∗ ( H , F ℓ ) b e the map induced by the quotient Z /ℓ v → F ℓ send- ing 1 to 1 . F or u ∈ H 2 ( H , Z /ℓ v ) and v ∈ H 1 ( H , F ℓ ) , it is straightforw ard to chec k that t v − 1 ( t 0 ( u ) ∪ v ) = u ∪ t v − 1 ( v ) . By deﬁnition of H 2 ( H , F ℓ ) B , w e hav e α i = B m, 1 ( a ) for some a ∈ H 1 ( H , Z /ℓ m ) . As in the pro of of Lemma 3.7, w e ha ve that B m, 1 = t 0 ◦ B m,v and so α i = t 0 ( B m,v ( a )) . Thus s H ( t v − 1 ( α i ∪ β j )) = s H ( t v − 1 ( t 0 ( B m,v ( a )) ∪ β j )) = s H ( B m,v ( a ) ∪ t v − 1 ( β j )) = a ( ψ ( t v − 1 ( β j ))) . Th us w e conclude that s H ( α ∪ β ) and hence H 2 ( H ⋊ Γ , V i ) L ,s H dep ends on s H only through ψ , and ha ve prov en the lemma. □ 3.2. The nonab elian ω -inv ariant, after Liu. Let H b e a ﬁnite group and s H ∈ H 3 ( H , Z /n ) an element. W e deﬁne the lifting invariant asso ciate d to s H to b e the image of s H in H 2 ( H , Z ) under the connecting homomorphism asso ciated to the short exact sequence 0 → Z → Z → Z /n → 0 . The lifting in v arian t deﬁnes a linear form on H 2 ( H , Q / Z ) . Dually this linear form ma y b e expressed as the comp osition H 2 ( H , Q / Z ) → H 3 ( H , Z /n ) → Z /n → Q / Z with the ﬁrst arrow the connecting homomorphism B ′ n asso ciated to the short exact sequence 0 → Z /n → Q / Z → Q / Z → 0 , the second arro w s H , and the third arrow division b y n . W e will c heck that this is compatible (up to a constant factor) with previously-deﬁned notions of the lifting inv arian t. In this section, we ﬁx compatibly a generator ξ m of µ m ( F q ) for eac h m coprime to q , thus giving an isomorphism Q / Z ℓ → µ ℓ ∞ . W e no w see how in the function ﬁeld setting, the lifting in v ariant comes from P oincaré duality on the geometric curv e. Lemma 3.10. L et K b e the function ﬁeld of a c onne cte d, smo oth, c omplete curve C over a ﬁnite ﬁeld F q of char acteristic p . L et L/K b e a ﬁnite Galois extension that is unr amiﬁe d everywher e, with H = Gal( L/K ) of or der prime to p . L et n b e a divisor of q − 1 and let s H ∈ H 3 ( H , Z /n ) b e the Artin-V er dier tr ac e for L/K , using ξ = ξ n . Then the lifting invariant line ar form is the c omp osition H 2 ( H , Q / Z ) → H 2 ( C, Q / Z ) → H 2 ( C F q , Q / Z ) → Q / Z wher e the ﬁrst map is ϕ L/K fr om L emma 2.2, the se c ond map is the usual pul lb ack, and the last map is the the sum over ℓ  = p of − q − 1 n times the c omp osite H 2 ( C F q , Q ℓ / Z ℓ ) → 33 H 2 ( C F q , µ ℓ ∞ ) → Q ℓ / Z ℓ of the map induc e d by the isomorphism Q ℓ / Z ℓ → µ ℓ ∞ sending ℓ − m 7→ ξ m and the tr ac e map asso ciate d to Poinc ar é duality in étale c ohomolo gy. W e could include a term for ℓ = p in the sum ab o v e but since H 2 ( C F q , Q p / Z p ) = 0 the zero map is the only p ossible c hoice of map H 2 ( C F q , Q p / Z p ) → Q p / Z p . Pr o of. Every class in H 2 ( H , Q / Z ) breaks into a sum of ℓ -p o wer torsion classes for primes ℓ  = p , and every ℓ -pow er torsion class in H 2 ( H , Q / Z ) arises from H 2 ( H , Z /ℓ m ) for some p ositiv e integer m , where the map H 2 ( H , Z /ℓ m ) → H 2 ( H , Q / Z ) is by division b y ℓ m . It suﬃces to prov e the stated iden tit y for eac h class arising from H 2 ( H , Z /ℓ m ) . Let v b e the ℓ -adic v aluation of n and let ˜ n b e the in verse of n/ℓ v mo dulo ℓ m + v . W e hav e a commutativ e diagram of short exact sequences Z /ℓ v Z /ℓ m + v Z /ℓ m Z /n Q / Z Q / Z × ℓ m × n ˜ n ℓ v × ˜ n ℓ m + v × 1 × 1 ℓ m × 1 n × n . The lifting in v ariant on classes that arise from H 2 ( H , Z /ℓ m ) is the comp osition of the follo wing maps H 2 ( H , Z /ℓ m ) H 2 ( H , Q / Z ) H 3 ( H , Z /n ) H 3 ( C, Z /n ) H 3 ( C, G m ) Q / Z , 1 7→ 1 ℓ m B ′ n ϕ L/K 1 7→ ξ A where A is the basic Artin-V erdier isomorphism of Equation (2.1). By the commutativ e diagram ab ov e, we can replace the ﬁrst t w o maps to obtain another description of the lifting in v ariant as H 2 ( H , Z /ℓ m ) H 3 ( H , Z /ℓ v ) H 3 ( H , Z /n ) H 3 ( C, Z /n ) H 3 ( C, G m ) Q / Z , B m,v 1 7→ n ˜ n ℓ v ϕ L/K 1 7→ ξ A where B m,v is the connecting homomorphism from the top ro w of the comm utative d iagram ab o ve. By Lemma 2.2, w e can replace the second, third, and fourth maps to get another description of the lifting inv arian t as H 2 ( H , Z /ℓ m ) H 3 ( H , Z /ℓ v ) H 3 ( C, Z /ℓ v ) H 3 ( C, G m ) Q / Z , B m,v ϕ L/K 1 7→ ξ n ˜ n ℓ v A and then replace the ﬁrst and second maps to get another description of the lifting inv ariant as H 2 ( H , Z /ℓ m ) H 2 ( C, Z /ℓ m ) H 3 ( C, Z /ℓ v ) H 3 ( C, G m ) Q / Z . ϕ L/K B m,v 1 7→ ξ n ˜ n ℓ v A Let ζ m b e the image of ξ − n ℓ v = ξ − 1 ℓ v in H 1 fppf ( C, µ ℓ m ) under the connecting homomorphism of the exact sequence µ ℓ m → µ ℓ m + v → µ ℓ v . By Lemma 3.2, we get another description of the lifting in v arian t as H 2 ( H , Z /ℓ m ) H 2 ( C, Z /ℓ m ) H 3 ( C, G m ) Q / Z . ϕ L/K ∪ ζ ˜ n m A 34 By [LST20, Proof of Lemma 7.5], using that ξ n = ξ and the compatibility of the ξ k , w e can replace the last tw o maps and the lifting inv arian t is H 2 ( H , Z /ℓ m ) H 2 ( C, Z /ℓ m ) H 2 ( C F q , Z /ℓ m ) H 2 ( C F q , µ ℓ ∞ ) Q ℓ / Z ℓ . ϕ L/K 1 7→ ξ ℓ m P.D . × ˜ n (1 − q ) ℓ v Since the map lands in ℓ m torsion, and n ˜ n is 1 mo d ℓ m + v , we can replace the last multipli- cation b y ˜ n (1 − q ) ℓ v = ˜ nn ℓ v (1 − q ) n with m ultiplication by (1 − q ) n . The lemma now follows from the fact that the maps induced by the map of co eﬃcients Z /ℓ m → Q / Z sending 1 7→ ℓ − m comm utes with ϕ L | K (Lemma 2.2) and with the pull back induced b y C F q → C . □ This lemma allows us to compare our lifting in v arian t with the notion of the lifting inv arian t deﬁned by Liu [Liu22], which contains essen tially the same information [Liu22, Prop osition 4.3] as the lifting in v ariant deﬁned by the second author [W o o21] based on work of Ellen b erg, V enk atesh, and W esterland, as well as the earlier lifting inv ariant for quadratic extensions [W o o19, Theorem 3.13]. W e next review Liu’s notion. Liu considers a Galois extension K/ F q ( t ) with Galois group Γ , corresp onding to a curve X/ P 1 F q of p ositiv e genus. She deﬁnes K ′ ∅ to b e the maximal extension of K F q that is ev ery- where unramiﬁed and a limit of ﬁnite Galois extensions of degree prime to | Γ | q . Liu ﬁrst deﬁnes an isomorphism ω K : ˆ Z (1) ( q | Γ | ) ′ → H 2 (Gal( K ′ ∅ / F q ( t )) , Z ) ( q | Γ | ) ′ whic h is characterized [Liu22, Corollary 2.12] by the following property: the inv erse map ω − 1 K : H 2 (Gal( K ′ ∅ / F q ( t ) , Z ) ( q | Γ | ) ′ → ˆ Z (1) ( q | Γ | ) ′ corresp onds to a class in H 2 (Gal( K ′ ∅ / F q ( t )) , ˆ Z (1)) ( q | Γ | ) ′ whose image in the étale cohomology H 2 ( X F q , ˆ Z (1) ( q | Γ | ) ′ ) has Poincaré dualit y trace −| Γ | ∈ ˆ Z ( q | Γ | ) ′ . (The subscripts denote a pro-prime-to- q | Γ | completion.) F or L a Galois extension of K , also Galois ov er F q ( t ) , that is ev erywhere unramiﬁed, split completely ov er ∞ , and has degree prime to q | Γ | , Liu [Liu22, Deﬁnition 2.13, Remark 2.16(2)] deﬁnes a homomorphism ω L/K : ˆ Z (1) ( q | Γ | ) ′ → ω K H 2 (Gal( K ′ ∅ / F q ( t )) , Z ) ( q | Γ | ) ′ → H 2 (Gal( L/ F q ( t )) , Z ) ( q | Γ | ) ′ where the second maps is induced b y the group homomorphism Gal( K ′ ∅ / F q ( t )) → Gal( L/ F q ( t )) arising from the inclusions F q ( t ) ⊂ F q ( t ) and L F q ( t ) ⊆ K ′ ∅ . Lemma 3.11. L et K / F q ( t ) b e a Galois extension with Galois gr oup Γ , c orr esp onding to a curve X/ P 1 F q of p ositive genus. L et L b e a Galois extension of K , also Galois over F q ( t ) , that is everywher e unr amiﬁe d, split c ompletely over ∞ , and has de gr e e prime to q | Γ | . L et n b e the maximal divisor of q − 1 prime to | Γ | . Our ﬁxe d system of r o ots of unity gives an isomorphism ˆ Z ( q | Γ | ) ′ → ˆ Z (1) ( q | Γ | ) ′ . L et ξ b e the image of 1 under this isomorphism, with image ξ m in µ m for al l m . L et n b e a divisor of q − 1 , and assume ξ n = ξ . Then q − 1 n | Γ | ω L/K ( ξ ) is e qual to the the image of the lifting invariant of the Artin-V er dier tr ac e in H 3 (Gal( L/K ) , Z /n ) under the natur al map H 2 (Gal( L/K ) , Z ) → H 2 (Gal( L/ F q ( t )) , Z ) . Pr o of. F or an abelian group A , we write A ( q | Γ | ) ′ to denote the subgroup of A of all el- emen ts of ﬁnite order that is relativ ely prime to q | Γ | . T o c heck that t wo elements of 35 H 2 (Gal( L/ F q ( t )) , Z ) ( q | Γ | ) ′ agree, it suﬃces to c heck that their pairings with an element α of H 2 (Gal( L/ F q ( t )) , Q / Z ) ( q | Γ | ) ′ agree. The pairing of ω L/K ( ξ ) and α ma y b e calculated b y pulling bac k α to ˆ α ∈ H 2 (Gal( K ′ ∅ / F q ( t )) , Q / Z ) ( q | Γ | ) ′ and then pairing with ω K ( ξ ) . Let ϵ ∈ H 2 (Gal( K ′ ∅ / F q ( t )) , ˆ Z (1) ( q | Γ | ) ′ ) b e a class corresp onding to ω − 1 K as in [Liu22, Remark 2.7]. In other w ords, this class has the prop erty that for a ∈ ˆ Z (1) ( q | Γ | ) ′ w e hav e ⟨ ω K ( a ) , ϵ ⟩ = a where ⟨ , ⟩ denotes the pairing H 2 ( Z ) × H 2 ( A ) → A for an y abelian group A . H ence if f : ˆ Z (1) ( q | Γ | ) ′ → Q / Z is a homomorphism, we hav e ⟨ ω K ( a ) , f ∗ ( ϵ ) ⟩ = f ( ⟨ ω K ( a ) , ϵ ⟩ ) = f ( a ) . Since H 2 (Gal( K ′ ∅ / F q ( t )) , Q / Z ) ( q | Γ | ) ′ is a direct limit of group cohomology of ﬁnite groups with Q / Z co eﬃcien ts, each of which is a direct limit of group cohomology with ﬁnite co eﬃcien ts, w e ha v e that ˆ α is in the image of H 2 (Gal( K ′ ∅ / F q ( t )) , µ m ) for some p ositiv e in teger m , relativ ely prime to q | Γ | , and injective map µ m → Q / Z . By [Liu22, Pro of of Lemma 2.2], w e hav e that the map induced by the natural quotient on coeﬃcients H 2 (Gal( K ′ ∅ / F q ( t )) , ˆ Z (1) ( q | Γ | ) ′ ) → H 2 (Gal( K ′ ∅ / F q ( t )) , µ m ) ≃ Z /m Z is surjectiv e. By [Liu22, Corollary 2.12], since ϵ has trace whic h is a unit in ˆ Z ( q | Γ | ) ′ , it is a top ological generator of H 2 (Gal( K ′ ∅ / F q ( t )) , ˆ Z (1) ( q | Γ | ) ′ ) . Thus, for some integer k , the map on f co eﬃcients that is the natural quotien t ˆ Z (1) ( q | Γ | ) ′ → µ m follo wed by multiplication by k and then our map µ m → Q / Z , induces a map on cohomology suc h that f ∗ ( ϵ ) = ˆ α . Th us the pairing of α with ω L/K ( ξ ) is equal to ⟨ ω K ( ξ ) , ˆ α ⟩ = ⟨ ω K ( ξ ) , f ∗ ( ϵ ) ⟩ and th us to f ( ξ ) for this v alue of f . T o calculate f ( ξ ) , w e use that ϵ has Poincaré duality trace −| Γ | so that the image of f ∗ ( ϵ ) ∈ H 2 (Gal( K ′ ∅ / F q ( t )) , Q / Z ) , under the map on co eﬃcients ϕ : Q / Z → Q ℓ ∤ q | Γ | µ ℓ ∞ sending 1 /m to ξ m , has P oincaré dualit y trace −| Γ | f ( ξ ) for the elemen t ξ ∈ ˆ Z (1) ( q | Γ | ) ′ with image ξ m in µ m for all m . In other words, ϕ ∗ ( ˆ α ) has Poincaré dualit y trace −| Γ | f ( ξ ) . Th us the pairing of α with q − 1 n | Γ | ω L/K ( ξ ) is q − 1 n | Γ | f ( ξ ) , whic h is − q − 1 n times the Poincaré duality trace of ϕ ∗ ( ˆ α ) . By Lemma 3.10, − q − 1 n times the Poincaré duality trace of ϕ ∗ ( ˆ α ) is precisely the ev aluation of the lifting in v ariant linear form on the image of α in H 2 (Gal( L/K ) , Q / Z ) . This ev aluation is also equal to the ev aluation on α itself of the linear form H 2 (Gal( L/ F q ( t )) , Q / Z ) → Q / Z corresp onding to the image of the lifting inv ariant under the natural map H 2 (Gal( L/K ) , Z ) → H 2 (Gal( L/ F q ( t )) , Z ) . W e conclude that q − 1 n | Γ | ω L/K ( ξ ) is the image of the lifting inv ariant under the natural map H 2 (Gal( L/K ) , Z ) → H 2 (Gal( L/ F q ( t )) , Z ) , since they hav e the same pairing with any α . □ Since q − 1 n | Γ | is divisible only b y primes dividing Γ , multiplying by it is inv ertible on an y ( q | Γ | ) ′ -mo dule, so Lemma 3.11 shows that the image under ω L/K of a generator, and hence ω L/K itself, is uniquely determined by the Artin-V erdier trace. In fact, even more precisely , w e hav e the following. Lemma 3.12. In the setup of L emma 3.11, the natur al map H 2 (Gal( L/K ) , Z ) Γ → Hom( ˆ Z (1) ( q | Γ | ) ′ , H 2 (Gal( L/ F q ( t )) , Z ) ( q | Γ | ) ′ ) that sends a class in H 2 (Gal( L/K ) , Z ) to the unique homomorphism sending a ﬁxe d top olo gi- c al gener ator of ˆ Z (1) ( q | Γ | ) ′ to the image of that class under the inﬂation map H 2 (Gal( L/K ) , Z ) → H 2 (Gal( L/ F q ( t )) , Z ) ( q | Γ | ) ′ is an isomorphism. 36 In particular, there is an isomorphism b et ween th e set in which our lifting in v arian t lies and the set in which ω L/K lies that sends one to the other. Pr o of. Since | Gal( L/K ) | is prime to q | Γ | , the order of H 2 (Gal( L/K ) , Z ) is prime to q | Γ | and so H 2 (Gal( L/K ) , Z ) ( q | Γ | ) ′ → H 2 (Gal( L/K ) , Z ) is an isomorphism. The inﬂation map H 2 (Gal( L/K ) , Z ) Γ ( q | Γ | ) ′ → H 2 (Gal( L/ F q ( t )) , Z ) ( q | Γ | ) ′ is an isomorphism b y taking the Lyndon- Ho c hschild-Serre sp ectral sequence computing H 2 (Gal( L/ F q ( t )) , Z )) from H a (Γ , H b (Gal( L/K ) , Z )) and observing that all terms with a > 0 are | Γ | -torsion so after taking q | Γ | ′ -parts the sp ectral sequence degenerates on the second page. Finally , for A a ﬁnite group of order prime to q | Γ | , the map A → Hom( ˆ Z (1) ( q | Γ | ) ′ , A ) sending an element to the unique homomorphism sending a generator to that element is an isomorphism since it is constructed as the in verse of the map Hom( ˆ Z (1) ( q | Γ | ) ′ , A ) → A giv en b y ev aluating at a ﬁxed generator whic h is an isomorphism since ev ery elemen t of A generates a ﬁnite cyclic group of order prime to q | Γ | and this ﬁnite cyclic group is a quotient of ˆ Z (1) ( q | Γ | ) ′ . □ 3.3. 2-class ﬁeld to w ers of cyclic cubic ﬁelds, after Boston-Bush. In this subsection, w e giv e an example of ho w to derive explicit probabilities from our conjectures. Our goal is to compute probabilities that the Galois group of the 2 -class ﬁeld tow er (i.e. the comp osition of all ev erywhere unramiﬁed Galois extensions of degree a pow er of 2 ) of a cyclic cubic ﬁeld is equal to a given 2 -group. In particular, w e will do this for t wo of the simplest p ossible 2 -groups, the Klein four group and the eigh t-elemen t quaternion group. Our c hoice of this example is motiv ated by w ork of Boston and Bush [BBH21], who collected numerical data and theoretical results on this problem. In particular, we observ e that our conjectural probabilities, based on function ﬁeld theorems and the moment metho d, agree with their conjectural probabilities, based on n umerical evidence in the num b er ﬁeld case. This pro vides some basic n umerical evidence for our conjectures. In addition, we explain the relationship b et ween our theorems and conjectures and some existen tial theorems and conjectures made b y Boston and Bush. W e begin with the follo wing result, which uses our calculation of the probabilit y of ob- taining a given orien ted Γ -group to giv e a formula for the probability of obtaining a giv en Γ -group without orientation. In the following statemen t we add the parenthetical notation ( s H ) to w V i and w N i to mak e clear that they dep end on s H , though in the rest of the pap er w e work with a ﬁxed ( H , s H ) and thus drop this notation for simplicit y . Lemma 3.13. L et U b e a multiset of elements of Γ . Assume al l nonzer o r epr esentations of Γ of char acteristic dividing n c ontain some nontrivial ve ctor ﬁxe d by at le ast one element of U . L et ν b e the me asur e of The or em 7.6. L et L b e a level in the c ate gory of ﬁnite Γ -gr oups. L et H b e a ﬁnite Γ -gr oup in L with H Γ = 1 . Then ν ( { X | X L ∼ = H } ) = | H 2 ( H ⋊ Γ , Z /n ) | | H 3 ( H ⋊ Γ , Z /n ) || Aut Γ ( H ) | H · U X s H ∈ H 3 ( H, Z /n ) Γ r Y i =1 w V i ( s H ) s Y i =1 w N i ( s H ) . If H Γ  = 1 , then ν ( { X | X L ∼ = H } ) = 0 . Pr o of. Let W b e the set of isomorphism classes of ﬁnite n -orien ted Γ -groups whose underlying Γ -group is in L . Then X L ∼ = H if and only if X W ∼ = ( H , s H ) for some s H ∈ H 3 ( H , Z /n ) Γ . 37 Since the condition X W ∼ = ( H , s H ) dep ends only on the isomorphism class of ( H , s H ) , w e ha ve by Theorem 7.6 and Corollary 7.25 ν ( { X | X L ∼ = H } ) = ν  [ s H ∈ H 3 ( H, Z /n ) Γ / Aut Γ ( H ) { X | X W ∼ = ( H , s H ) }  = X s H ∈ H 3 ( H, Z /n ) Γ / Aut Γ ( H ) ν ( { X | X W ∼ = ( H , s H ) } ) = X s H ∈ H 3 ( H, Z /n ) Γ / Aut Γ ( H ) v W , ( H,s H ) = X s H ∈ H 3 ( H, Z /n ) Γ / Aut Γ ( H ) M H | Aut( H , s H ) | r Y i =1 w V i ( s H ) s Y i =1 w N i ( s H ) . When H Γ  = 1 , then M H = 0 , and otherwise M H = | H 2 ( H ⋊ Γ , Z /n ) | | H 3 ( H ⋊ Γ , Z /n ) | H · U dep ends only on H and therefore can b e brough t outside the sum ov er s H . F urthermore w e ha ve X s H ∈ H 3 ( H, Z /n ) Γ / Aut Γ ( H ) 1 | Aut( H , s H ) | r Y i =1 w V i ( s H ) s Y i =1 w N i ( s H ) = 1 | Aut Γ ( H ) | X s H ∈ H 3 ( H, Z /n ) Γ r Y i =1 w V i ( s H ) s Y i =1 w N i ( s H ) since Aut( H , s H ) is the stabilizer of s H in Aut Γ ( H ) so b y the orbit-stabilizer theorem eac h Aut( H ) -orbit con tributes | Aut Γ ( H ) | | Aut( H , s H ) | r Y i =1 w V i ( s H ) s Y i =1 w N i ( s H ) to the sum P s H ∈ H 3 ( H, Z /n ) Γ Q r i =1 w V i ( s H ) Q s i =1 w N i ( s H ) . Com bining these, we obtain the statemen t. □ The follo wing lemma lets us see what terms app ear in the pro ducts. Lemma 3.14. L et p b e a prime not dividing | Γ | . L et L b e a level in the c ate gory of ﬁnite Γ -gr oups c onsisting only of p -gr oups with an action of Γ . L et H b e a ﬁnite Γ -gr oup in L . Then: (1) Ther e ar e no admissible nonab elian simple [ H ⋊ Γ] -gr oups N i and the only ab elian simple H ⋊ Γ -mo dules V i ar e the gr oups of the form F d p with a trivial H -action and an irr e ducible action of Γ . (2) A gr oup V i of the form F d p with a trivial H -action and an irr e ducible action of Γ is an admissible ﬁnite simple H ⋊ Γ -gr oup if and only if V i is in L . (3) A n admissible ﬁnite simple H ⋊ Γ -gr oup V i is never interme diate. Pr o of. F or a nonab elian simple [ H ⋊ Γ] -group N i to b e admissible, an extension of the form H × Out( N i ) Aut( N i ) of H b y N i m ust b e in L and in particular m ust b e a p -group, so N i m ust b e a p -group. But the commutator subgroup of a non-ab elian p -group is a nontrivial prop er normal subgroup which is characteristic and hence [ H ⋊ Γ] -inv arian t, so this contradicts simplicit y . 38 F or an ab elian simple H ⋊ Γ -group V i to b e admissible, V i ⋊ H m ust b e a p -group, so V i m ust b e a p -group. If H acts non trivially on V i then the maximal subgroup on whic h H acts trivially is a non trivial prop er subgroup of V i , ﬁxed as a set under the action of H ⋊ Γ , con tradicting simplicity , so H acts trivially and the action of H ⋊ Γ factors through Γ . This v eriﬁes (1). If V i is an admissible ﬁnite simp le H ⋊ Γ -group, then V i ⋊ H in L . If the action of H on V i is trivial, we hav e V i ⋊ H = V i × H . Since V i is a quotien t of V i × H , we hav e that V i × H ∈ L implies V i ∈ L . On the other hand, if V i ∈ L , then since V i × H is a ﬁb er pro duct of V i and H , w e hav e V i × H ∈ L as long as H ∈ L , whic h is true b y assumption. This veriﬁes (2). The in termediate case o ccurs only if p = 2 , and if the natural homomorphism H ⋊ Γ → Aut( V i ) factors through the symplectic group but not the aﬃne symplectic group. Since the homomophism H ⋊ Γ → Aut( V i ) factors through Γ , this can happ en only if Γ → Aut( V i ) factors through the symplectic group to the aﬃne symplectic group. The aﬃne symplectic group is an extension of the symplectic group by an ab elian 2 -group, and thus every map from a group with o dd order to the symplectic group lifts to the aﬃne symplectic group. Since p = 2 , we hav e that | Γ | is o dd, and thus every map from Γ to the symplectic group lifts to the aﬃne symplectic group. This v eriﬁes (3). □ The next lemma lets us apply our formulas to calculate the probabilit y that the maximum p -group quotien t of a random group is a given group. Lemma 3.15. L et p b e a prime not dividing | Γ | . L et H b e a ﬁnite p -gr oup with an action of Γ . L et L b e a level in the c ate gory of Γ -gr oups c onsisting only of p -gr oups with an action of Γ , including H and every extension of H by V i for V i any gr oup of the form F d p with a trivial action of H and an irr e ducible action of Γ . Then for X a pr oﬁnite Γ -gr oup, the maximal p -gr oup quotient of X is isomorphic to H if and only if X L ∼ = H . F urthermor e, in this c ase, for any gr oup V i of the form F d p with an irr e ducible action of Γ , we have H 2 ( H ⋊ Γ , V i ) L = H 2 ( H ⋊ Γ , V i ) . Pr o of. Certainly if the maximal p -group quotient of X is isomorphic to H , L consists only of p -groups, and H ∈ L , then the maximal quotient of X in L is isomorphic to H . Conv ersely , if X L ∼ = H but H is not the maximal p -group quotient of X , there exists another ﬁnite p -group with an action of Γ that admits a surjection from G and a surjection to H with non trivial kernel. Let G ′ b e the quotien t of G b y a maximal prop er normal subgroup of the k ernel. Then the kernel of G ′ → H is a simple G ′ ⋊ Γ -group, which since G and the k ernel are b oth p -groups implies that the kernel is a group of the form F d p with an irreducible action of Γ . Hence G ′ ∈ L whic h contradicts X L ∼ = H . A class is in H 2 ( H ⋊ Γ , V i ) L if and only if the asso ciated extension of H by V i is in L , whic h is true for all classes in H 2 ( H ⋊ Γ , V i ) b y assumption. □ Boston and Bush [BBH21] studied the statistics of the 2 -class ﬁeld tow ers of cyclic cubic ﬁelds. In other w ords, these are the comp osition of all everywhere unramiﬁed Galois ex- tensions of degree a p ow er of 2 . They consider tw o v arian ts, one in whic h C / R is treated as an unramiﬁed extension (the narro w class ﬁeld tow er) and one in which C / R is treated as ramiﬁed (the wide class ﬁeld tow er). In b oth v arian ts, they prov ed theorems restricting whic h groups can appear as the Galois group of the 2 -class ﬁeld to w er, made conjectures ab out the set of groups that app ear, and made conjectures, based on n umerical data, ab out 39 the frequency with whic h certain small ﬁnite groups app ear. These conjectures are fo cused on the sp ecial case where the Galois group of the 2 -class ﬁeld has exactly tw o generators (equiv alently , the class group of the cyclic cubic ﬁeld has 2 -rank tw o). Motiv ated by this w ork, let us sp ecialize further to Γ = Z / 3 , n = 2 , and U = { 1 } . These are the correct v alues to match k = Q , since Q has 2 ro ots of unity and the only p ossible c hoice of γ ∞ in Z / 3 of order 1 or 2 is 1 . F or k = Q and Γ = Z / 3 , the Γ -extensions K /k w e consider are cyclic cubic ﬁelds. The groups of the form F d 2 with an irreducible action of Γ are V 1 := F 2 with a trivial action of Γ and V 2 := F 2 2 with the unique-up-to-isomorphism nontriv ial action of Γ . Let H b e a ﬁnite Γ -group 2 -group such that H Γ = 1 . Let L b e a level con taining only of 2 -groups and con taining H and ev ery Γ -group extension of H b y V 1 or V 2 . The form ula of Lemma 3.13 sp ecializes to (3.16) ν ( { X | X L ∼ = H ) = | H 2 ( H ⋊ Γ , F 2 ) || H Γ | | H 3 ( H , F 2 ) Γ || Aut Γ ( H ) || H | X s H ∈ H 3 ( H ⋊ Γ , F 2 ) w V 1 ( s H ) w V 2 ( s H ) . W e now calculate w V 1 and w V 2 . W e hav e that V 1 = F 2 is trivial of c haracteristic dividing n . Th us we apply Lemma 7.42 to calculate w V 1 ( s H ) . W e ha ve κ 1 = F 2 and q 1 = 2 . W e hav e V · U ′ 1 = | F 2 | | F H ⋊ Γ 2 | = 2 2 = 1 . Th us for u 1 = 0 w e hav e q u 1 1 = V · U ′ 1 . Letting z 1 ( s H ) = dim κ 1 H 2 ( H ⋊ Γ , V 1 ) L ,s H = dim F 2 H 2 ( H ⋊ Γ , F 2 ) L ,s H w e hav e (3.17) w V 1 ( s H ) = z 1 ( s H ) − 1 Y k =0 (1 − q k − u 1 1 ) = z 1 ( s H ) − 1 Y k =0 (1 − 2 k ) = ( 1 if z 1 ( s H ) = 0 0 otherwise = ( 1 if H 2 ( H ⋊ Γ , F 2 ) L ,s H = 0 0 otherwise . W e now consider V 2 . By Lemma 3.14(3), V 2 is not in termediate and thus not anomalous. W e ha v e κ 2 = F 4 and q 2 = 4 . The represen tation V 2 is unitary and F 2 -orthogonal, so ϵ V 2 = 0 . W e ha v e V · U ′ 2 = | V 2 | | V H ⋊ Γ 2 | = 4 1 = 4 . Th us for u 2 = 1 w e hav e q u 2 2 = V · U ′ 2 . Letting z 2 ( s H ) = dim κ 2 H 2 ( H ⋊ Γ , V 2 ) L ,s H = dim F 4 H 2 ( H ⋊ Γ , F 2 ) L ,s H w e hav e b y Lemma 7.39 w V 2 ( s H ) = ∞ Y k =0 (1 + q − k − ϵ V 2 +1 2 − u 2 2 ) − 1 z 2 ( s H ) − 1 Y k =0 (1 − q k − u 2 2 ) = ∞ Y k =0 (1 + 4 − k − 1 2 − 1 ) − 1 z 2 ( s H ) − 1 Y k =0 (1 − 4 k − 1 ) 40 (3.18) = ∞ Y k =0 (1 + 4 − k − 3 2 ) − 1 ×      1 if z 2 ( s H ) = 0 3 4 if z 2 ( s H ) = 1 0 otherwise . Com bining (3.16), (3.17), and (3.18), w e see that the probability under our conjectured distribution that the 2 -class ﬁeld to wer of a cyclic cubic ﬁeld has Galois group H is p ositive if and only if there exists s H ∈ H 3 ( H , F 2 ) Γ suc h that tw o conditions are satisﬁed: First that the comp osition H 2 ( H ⋊ Γ , F 2 ) → H 3 ( H ⋊ Γ , F 2 ) → F 2 of the Bo c kstein homomorphism with the map induced s H is injectiv e, and second that the k ernel in H 2 ( H ⋊ Γ , V 2 ) of the pairing H 1 ( H ⋊ Γ , V 2 ) × H 2 ( H ⋊ Γ , V 2 ) → H 3 ( H ⋊ Γ , F 2 ) → F 2 arising from cup pro duct and s H has size at most 4 . The ﬁrst of these conditions, in diﬀerent language, w as highligh ted in the w ork of Boston and Bush. Lemma 3.19. L et Γ = Z / 3 and let H b e a 2 -gr oup with an action of Γ such that H Γ = 1 . Ther e exists s H ∈ H 3 ( H , F 2 ) Γ such that the c omp osition H 2 ( H ⋊ Γ , F 2 ) → H 3 ( H ⋊ Γ , F 2 ) → F 2 of the Bo ckstein homomorphism with the p airing with s H is inje ctive if and only if the Schur multiplier of H ⋊ Γ has or der 1 or 2 . Pr o of. Since H ∗ ( H ⋊ Γ , F 2 ) and H ∗ ( H ⋊ Γ , F 2 ) are dual vector spaces, there exists s H suc h that the comp osition H 2 ( H ⋊ Γ , F 2 ) → H 3 ( H ⋊ Γ , F 2 ) → F 2 is injectiv e if and only if H 2 ( H ⋊ Γ , F 2 ) has order at most 2 and the natural map H 2 ( H ⋊ Γ , F 2 ) → H 3 ( H ⋊ Γ , F 2 ) is injective. Since the Bo c kstein sequence is exact, this o ccurs if and only if H 2 ( H ⋊ Γ , F 2 ) has order at most 2 and the natural map H 2 ( H ⋊ Γ , Z / 4) → H 2 ( H ⋊ Γ , F 2 ) is zero. Since H Γ = 1 we hav e H 1 ( H ⋊ Γ , Z ) = Z / 3 and th us for every ﬁnite group A of order prime to 3 , H 2 ( H ⋊ Γ , A ) = Hom( H 2 ( H ⋊ Γ , Z ) , A ) . The map Hom( H 2 ( H ⋊ Γ , Z ) , Z / 4) → Hom( H 2 ( H ⋊ Γ , Z ) , F 2 ) is zero if and only if the 2 -Sylo w subgroup of H 2 ( H ⋊ Γ , Z ) is 2 -torsion, and additionally Hom( H 2 ( H ⋊ Γ , Z ) , F 2 ) has order at most 2 if and only if the 2 -Sylo w subgroup of H 2 ( H ⋊ Γ , Z ) has order at most 2 . But for p  = 2 , 3 the p -Sylow subgroup of H 2 ( H ⋊ Γ , Z ) is trivial since H ⋊ Γ has order prime to p , and the 3 -part of H 2 ( H ⋊ Γ , Z ) is the 3 -part of H 2 (Γ , Z ) which v anishes, so this o ccurs if and only if H 2 ( H ⋊ Γ , Z ) has order at most 2 . □ In [BBH21, Theorem 5.2], Boston and Bush prov e that ev ery 2-generator 2 -group that arises as the wide 2 -class tow er group of a cyclic cubic ﬁeld is 2 -select. Here 2 -select is a somewhat complicated group-theoretic notion. One ﬁrst deﬁnes sp ecial groups [BBH21, Deﬁnition 4.3] as 2-generator 2 -groups H with an action of Z / 3 suc h that the Sch ur multiplier of H ⋊ Γ is trivial. By [BBH21, Lemma 4.5], one can then deﬁne 2 -sp ecial groups as sp ecial groups with relation rank at most 4 . Finally one deﬁnes 2 -select groups [BBH21, Deﬁnition 5.1] as all groups arising as quotients of a 2 -sp ecial group b y the normal Γ -inv ariant subgroup generated b y an elemen t of order at most 2 . In [BBH21, Theorem 5.3], Boston and Bush prov e that for a 2-select group H , the group H ⋊ Γ has Sc h ur m ultiplier 1 or 2 . Combining this with [BBH21, Theorem 5.2], one obtains in particular that for H a wide 2 -class to wer group of a cyclic cubic ﬁeld, H ⋊ Γ has Sch ur 41 m ultiplier 1 or 2 . Combining Lemma 3.19 with Lemma 8.13 (with V = V 1 ), we obtain the same conclusion by diﬀerent means. This giv es the ov erlap b etw een our work and Boston and Bush’s. Boston and Bush further conjecture [BBH21, Conjecture 5.4] that every 2 -select group in fact app ears as a 2 -class ﬁeld to wer. This raises tw o questions, whic h we are not curren tly able to answ er. Do all 2-select groups ha ve p ositive probability under our conjectured distribution? Are all groups which hav e p ositiv e probabilit y 2-select? The ab o ve c haracterization of p ositiv e probabilit y makes these questions purely group-theoretic. Positiv e answers to these questions would demonstrate a compatibilit y b etw een [BBH21, Conjecture 5.4] and our conjectures Finally , we calculate our predicted probabilities for the ﬁrst t wo 2 -select 2 -groups with the n umerical evidence of Boston-Bush. Note that they compute frequencies only within the set of cyclic cubic ﬁelds whose 2-class ﬁeld tow er group has t wo generators, so w e ﬁrst calculate the probability that the 2 -class ﬁeld to wer group has tw o generators. T o do this, w e can take L = L elem to consist of elementary ab elian 2 -groups with an action of Γ and compute the probability in our distribution that the maximum quotien t in L elem of the Galois group is V 2 . Since ( V 2 1 ) Γ  = 1 , w e hav e ν ( { X | X L ∼ = ( V 1 ) 2 ) = 0 . Since H 2 ( V 2 ⋊ Γ , F 2 ) L = H 2 ( V 2 ⋊ Γ , V 2 ) L = 0 , w e hav e w V 1 ( s H ) = 1 and w V 2 ( s H ) = Q ∞ k =0 (1 + 4 − k − 3 2 ) − 1 indep enden tly of s H . Thus the sum ov er s H in (3.16) cancels with the H 3 term and we obtain ν ( { X | X L ∼ = V 2 ) = | H 2 ( V 2 ⋊ Γ , F 2 ) || V Γ 2 | | Aut Γ ( V 2 ) || V 2 | ∞ Y k =0 (1+4 − k − 3 2 ) − 1 = 2 · 1 3 · 4 ∞ Y k =0 (1+4 − k − 3 2 ) − 1 = 1 6 ∞ Y k =0 (1+4 − k − 3 2 ) − 1 . This agrees with the probability that a cyclic cubic ﬁeld has class group of 2 -rank tw o conjectured by Malle [Mal08, Equation (1) on p. 2827], and with the probabilit y conjectured in our prior w ork [SW23, Conjecture 1.1, Theorem 1.2]. (This was guaran teed since the conjecture in this pap er agrees with [SW23] for all ab elian groups, and [SW23] was already c heck ed to agree with Malle in many cases, including this one.) W e must divide the probability of obtaining a giv en 2 -generator 2 -group b y 1 6 Q ∞ k =0 (1 + 4 − k − 3 2 ) − 1 to obtain its exp ected frequency among 2 -generator 2 -groups. W e no w calculate the conjectural probability that the 2 -class ﬁeld to wer is the Klein four group, b efore doing the same for the eigh t-element quaternion group. In each case, w e b egin with a lemma ab out the cohomology of that group. Lemma 3.20. L et V 2 b e the Klein four gr oup, viewe d as a nontrivial r epr esentation of Γ = Z / 3 . • F or i ≤ 3 we have H i ( V 2 ⋊ Γ , F 2 ) =          F 2 if i = 0 0 if i = 1 F 2 if i = 2 F 2 2 if i = 3 . 42 • F or i ≤ 2 we have H i ( V 2 ⋊ Γ , V 2 ) =      0 if i = 0 F 2 2 if i = 1 F 2 2 if i = 2 • The Bo ckstein map H 2 ( V 2 ⋊ Γ , F 2 ) → H 3 ( V 2 ⋊ Γ , F 2 ) is nontrivial. • Ther e exist isomorphisms of of ab elian gr oups fr om H 3 ( V 2 ⋊ Γ , F 2 ) , H 2 ( V 2 ⋊ Γ , V 2 ) , and H 1 ( V 2 ⋊ Γ , V ∨ 2 ) to F 4 such that the cup pr o duct map H 2 ( V 2 ⋊ Γ , V 2 ) × H 1 ( V 2 ⋊ Γ , V ∨ 2 ) → H 3 ( V 2 ⋊ Γ , F 2 ) is given by a nontrivial sesquiline ar form F 4 × F 4 → F 4 . Pr o of. These follo w from the follo wing more basic facts. W e can express H ∗ ( V 2 , F 2 ) as the free ring o ver F 2 on generators x, y in degree 1 . (This follows from the Künneth form ula and the analogous description of the cohomology of F 2 .) A ﬁxed generator γ of Γ acts by sending x to y and y to x + y . So H 1 ( V 2 , F 2 ) ≃ V 2 as Γ -represen tations. The Bo ckstein map satisﬁes the Leibnitz rule and the identities B ( x ) = x 2 , B ( y ) = y 2 . Finally , for an y representation V of Γ w e hav e H i ( V 2 ⋊ Γ , V ) = H i ( V 2 , V ) Γ = ( H i ( V 2 , F 2 ) ⊗ V ) Γ ≃ (Sym i V 2 ⊗ V ) Γ . F rom this w e can easily compute the dimensions in the ﬁrst t wo claims, though we will ﬁnd elements more explicitly in order to prov e the last tw o claims. In particular, H i ( V 2 ⋊ Γ , F 2 ) is the space of Γ -inv arian t p olynomials of degree i o ver F 2 . The Bo ckstein map applied to the generator x 2 + xy + y 2 of the in v ariants in degree 2 gives 2 x 3 + x 2 y + xy 2 + 2 y 3 = xy ( x + y ) which is a non trivial inv ariant in degree 3 , proving the third claim. W e ﬁx a generator ω of F 4 satisfying ω 2 = ω + 1 . Let Γ act on a one-dimensional F 4 -v ector space V = F 4 , such that the generator γ acts by m ultiplication b y ω . As Γ -representations o ver F 2 , we hav e that V is isomorphic to V 2 . The Γ -in v ariants in H ∗ ( V 2 , V ) are a F 4 -v ector space. W e can chec k that ω x + y ∈ H 1 ( V 2 , V ) is Γ -inv ariant and th us generates the Γ - in v ariants as an F 4 -v ector space, and that ω x 2 + y 2 generates the Γ -in v ariants in H 2 ( V 2 , V ) as an F 4 -v ector space. The cup pro duct map H 2 ( V 2 ⋊ Γ , V 2 ) × H 1 ( V 2 ⋊ Γ , V ∨ 2 ) → H 3 ( V 2 ⋊ Γ , F 2 ) , expressed in terms of Γ -inv arian t p olynomials o ver F 4 , ma y b e obtained b y multiplying one p olynomial with the conjugate of the other and taking the trace from F 4 to F 2 , since the Γ -in v ariant F 2 -bilinear form F 4 × F 4 → F 2 is obtained by multiplying one element by the conjugate of the other and taking the trace. Applied to α ( ω x + y ) and β ( ω x 2 + y 2 ) w e obtain α ( ω x + y ) × β ( ωx 2 + y 2 ) = αβ ( x 3 + ω x 2 y + ω xy 2 + y 3 ) . If we choose an isomorphism b etw een F 4 and the ring of degree 3 inv ariant p olynomials o ver F 2 that sends 1 to tr( x 3 + ω x 2 y + ω xy 2 + y 3 ) = x 2 y + xy 2 = xy ( x + y ) and ω to tr( ω ( x 3 + ω x 2 y + ω xy 2 + y 3 )) = x 3 + xy 2 + y 3 then this map sends α ( ω x + y ) and β ( ω x 2 + y 2 ) to α β , v erifying the ﬁnal claim. □ Lemma 3.21. F or X distribute d ac c or ding to the me asur e ν , the pr ob ability that the maximal 2 -gr oup quotient of X is the Klein four gr oup is 1 12 Q ∞ k =0 (1 + 4 − k − 3 2 ) − 1 and the pr ob ability c onditional on the ab elianization of X having 2 -r ank 2 is 1 2 . 43 The probabilit y 1 2 agrees with [BBH21, T able 7, ro w 1]. Pr o of. Let L b e a lev el con taining every extension of V 2 b y V 1 or V 2 . By Lemma 3.15, the probabilit y that the maximal 2 -group quotien t of X is the Klein four group is ν ( { X | X L ∼ = H ) . W e calculate this using the formula of (3.16). Since H 3 ( H ⋊ Γ , F 2 ) ∼ = F 2 2 b y 3.20(1) there are four p ossible c hoices for s H , corresp onding to the four linear forms on F 2 2 . By Lemma 3.15 we hav e H 2 ( V 2 ⋊ Γ , V 1 ) L = H 2 ( V 2 ⋊ Γ , F 2 ) so H 2 ( V 2 ⋊ Γ , V 1 ) L ,s H is the k ernel of the Bo c kstein homomorphism H 2 ( V 2 ⋊ Γ , F 2 ) → H 3 ( V 2 ⋊ Γ , F 2 ) → s H F 2 . By Lemma 3.20(1,3), the source of this map is F 2 so the k ernel v anishes if and only if the comp osition is non trivial, i.e. if and only if the linear form asso ciated to s H v anishes on the image under the Bo c kstein homomorphism in H 3 ( V 2 ⋊ Γ , F 2 ) of the unique nontrivial element in H 2 ( V 2 ⋊ Γ , F 2 ) . Th us the kernel v anishes for t wo p ossible v alues of s H and do es not v anish for t wo other v alues of s H , including s H = 0 . By (3.17) we hav e w V 1 ( s H ) = 1 for the tw o v alues where the kernel v anishes and w V 1 ( s H ) = 0 for the tw o v alues where the kernel do es not v anish. By Lemma 3.15 we hav e H 2 ( V 2 ⋊ Γ , V 2 ) L = H 2 ( V 2 ⋊ Γ , V 2 ) so H 2 ( V 2 ⋊ Γ , V 2 ) L ,s H is the kernel of the cup pro duct bilinear form H 2 ( V 2 ⋊ Γ , V 2 ) × H 1 ( V 2 ⋊ Γ , V ∨ 2 ) → H 3 ( V 2 ⋊ Γ , F 2 ) → s H F 2 . As long as w V 1 ( s H )  = 0 , we hav e s H  = 0 , so by Lemma 3.20(4) no element of H 2 ( V 2 ⋊ Γ , V 2 ) maps H 1 ( V 2 ⋊ Γ , V ∨ 2 ) in to a 1 -dimensional subspace of H 3 ( V 2 ⋊ Γ , F 2 ) . Thus s H  = 0 imples the k ernel is trivial. By (3.18) w e ha v e w V 2 ( s H ) = Q ∞ k =0 (1 + 4 − k − 3 2 ) − 1 as long as w V 1 ( s H )  = 0 . Com bining these observ ations, w e see that X s H ∈ H 3 ( V 2 ⋊ Γ , F 2 ) w V 1 ( s H ) w V 2 ( s H ) = 2 · 1 · ∞ Y k =0 (1 + 4 − k − 3 2 ) − 1 . Hence b y (3.16) w e hav e ν ( { X | X L ∼ = V 2 ) = | H 2 ( V 2 ⋊ Γ , F 2 ) || H Γ | | H 3 ( V 2 ⋊ Γ , F 2 ) || Aut Γ ( V 2 ) || V 2 | 2 · 1 · ∞ Y k =0 (1 + 4 − k − 3 2 ) − 1 = 2 · 1 4 · 3 · 4 2 · 1 · ∞ Y k =0 (1 + 4 − k − 3 2 ) − 1 = 1 12 ∞ Y k =0 (1 + 4 − k − 3 2 ) − 1 . Dividing b y 1 6 Q ∞ k =0 (1 + 4 − k − 3 2 ) − 1 to compute the conditional probability , we obtain 1 2 . □ Lemma 3.22. L et Q 8 b e the eight-element quaternion gr oup, with the unique-up-to-isomorphism nontrivial action of Γ = Z / 3 . • F or i ≤ 3 we have H i ( Q 8 ⋊ Γ , F 2 ) =          F 2 if i = 0 0 if i = 1 0 if i = 2 F 2 if i = 3 . 44 • F or i ≤ 3 we have H i ( Q 8 ⋊ Γ , V 2 ) =          0 if i = 0 F 2 2 if i = 1 F 2 2 if i = 2 0 if i = 3 • The cup pr o duct map H 2 ( Q 8 ⋊ Γ , V 2 ) × H 1 ( Q 8 ⋊ Γ , V ∨ 2 ) → H 3 ( Q 8 ⋊ Γ , F 2 ) is a non-de gener ate F 2 -biline ar form. Pr o of. These follo w from the fact that Q 8 ⋊ Γ is a ﬁnite subgroup of S U (2) , being the in verse image of the group of rotational symmetries of the tetrahedron, and hence acts freely on S 3 . Thus for V a represen tation of Q 8 ⋊ Γ w e hav e a natural map H i ( Q 8 ⋊ Γ , V ) → H i ( S 3 /Q 8 ⋊ Γ , V ) where S 3 /Q 8 ⋊ Γ is a manifold. Let us c hec k that the natural map H i ( Q 8 ⋊ Γ , V ) → H i ( S 3 /Q 8 ⋊ Γ , V ) is an isomorphism for i ≤ 3 and an y irreducible represen tation V of characteristic 2 . Since S 3 has cohomology only in degrees 0 and 3 , the sp ectral sequence computing H p + q ( S 3 /Q 8 ⋊ Γ , V ) from H p ( Q 8 ⋊ Γ , H q ( S 3 , V )) degenerates to a long exact sequence H i − 4 ( Q 8 ⋊ Γ , V ) → H i ( Q 8 ⋊ Γ , V ) → H i ( S 3 /Q 8 ⋊ Γ , V ) → H i − 3 ( Q 8 ⋊ Γ , V ) . F or i ≤ 3 we ha v e H i − 4 ( Q 8 ⋊ Γ , V ) = 0 and w e hav e H i − 3 ( Q 8 ⋊ Γ , V ) = 0 unless i = 3 , in whic h case H i − 3 ( S 3 /Q 8 ⋊ Γ , V ) = V Q 8 ⋊ Γ v anishes unless V is the trivial rep esen tation. In the case that V is the trivial representation F 2 , the map H 3 ( S 3 /Q 8 ⋊ Γ , F 2 ) → H 0 ( Q 8 ⋊ Γ , F 2 ) → F 2 represen ts in tegration against the fundamental class of S 3 , whic h is trivial as the fundamental class of S 3 pushes forward to | Q 8 ⋊ Γ | = 24 ≡ 0 mo d 2 times the fundamen tal class of S 3 /Q 8 ⋊ Γ and thus has trivial pairing with the mo d 2 cohomology of S 3 /Q 8 ⋊ Γ . Because the map H i ( S 3 /Q 8 ⋊ Γ , V ) → H i − 3 ( Q 8 ⋊ Γ , V ) is zero regardless of whether V ∼ = F 2 , the natural map H i ( Q 8 ⋊ Γ , V ) → H i ( S 3 /Q 8 ⋊ Γ , V ) is an isomorphism, as desired. Ha ving c hec ked this, for any represen tation V , b y P oincaré dualit y we hav e H 2 ( Q 8 ⋊ Γ , V ) ∼ = H 1 ( Q 8 ⋊ Γ , V ∨ ) ∨ with the isomorphism arising from the cup pro duct bilinear form H 2 ( Q 8 ⋊ Γ , V ) × H 1 ( Q 8 ⋊ Γ , V ∨ ) → H 3 ( Q 8 ⋊ Γ , F 2 ) . This immediately prov es the third claim and reduces the calculation of H 2 to the calculation of H 1 , which is immediate from the fact that the ab elianization of Q 8 is F 2 2 , with Γ acting non trivially , and reduces the calculation of H 3 to the even easier calculation of H 0 . □ Lemma 3.23. F or X distribute d ac c or ding to the me asur e ν , the pr ob ability that the maximal 2 -gr oup quotient of X is the quaternion gr oup Q 8 is 7 2 5 · 3 Q ∞ k =0 (1 + 4 − k − 3 2 ) − 1 and the pr ob ability c onditional on the ab elianization of X having 2 -r ank 2 is 7 16 . The probabilit y 7 16 agrees with [BBH21, T able 7, ro w 2]. Pr o of. Let L b e a lev el con taining every extension of Q 8 b y V 1 or V 2 . By Lemma 3.15, the probabilit y that the maximal 2 -group quotient of X is Q 8 is ν ( { X | X L ∼ = H ) . W e calculate this using the formula of (3.16). Since H 3 ( Q 8 ⋊ Γ , F 2 ) ∼ = F 2 b y 3.22(1) there are tw o possible choices for s H , one corre- sp onding to a trivial linear form and one corresponding to a nontrivial form. By Lemma 3.22(1) w e hav e H 2 ( Q 8 ⋊ Γ , F 2 ) = 0 so certainly H 2 ( Q 8 ⋊ Γ , F 2 ) L ,s H = 0 . Hence b y (3.17) w e hav e w V 1 ( s H ) = 1 for either c hoice of s H . 45 By Lemma 3.15 w e ha v e H 2 ( Q 8 ⋊ Γ , V 2 ) L = H 2 ( Q 8 ⋊ Γ , V 2 ) so H 2 ( Q 8 ⋊ Γ , V 2 ) L ,s H is the kernel of the cup pro duct bilinear form H 2 ( Q 8 ⋊ Γ , V 2 ) × H 1 ( Q 8 ⋊ Γ , V ∨ 2 ) → H 3 ( Q 8 ⋊ Γ , F 2 ) → s H F 2 . By Lemma 3.22(3) this bilinear form is nondegenerate if s H is nonzero and zero if s H is zero. If s H is nonzero, this mak es the kernel trivial, so b y (3.18) we hav e w V 2 ( s H ) = Q ∞ k =0 (1 + 4 − k − 3 2 ) − 1 . If s H is zero, this makes the k ernel 2 -dimensional ov er F 4 , and hence 1 -dimenisonal ov er F 4 , so by (3.18) w e hav e w V 2 ( s H ) = 3 4 Q ∞ k =0 (1 + 4 − k − 3 2 ) − 1 . Com bining these observ ations, w e see that X s H ∈ H 3 ( Q 8 ⋊ Γ , F 2 ) w V 1 ( s H ) w V 2 ( s H ) = ∞ Y k =0 (1 + 4 − k − 3 2 ) + 3 4 ∞ Y k =0 (1 + 4 − k − 3 2 ) = 7 4 ∞ Y k =0 (1 + 4 − k − 3 2 ) . Hence b y (3.16) w e hav e ν ( { X | X L ∼ = Q 8 ) = | H 2 ( Q 8 ⋊ Γ , F 2 ) || H Γ | | H 3 ( Q 8 ⋊ Γ , F 2 ) || Aut Γ ( Q 8 ) || Q 8 | 7 4 ∞ Y k =0 (1 + 4 − k − 3 2 ) − 1 = 1 · 2 2 · 3 · 8 7 4 · ∞ Y k =0 (1 + 4 − k − 3 2 ) − 1 = 7 2 5 · 3 ∞ Y k =0 (1 + 4 − k − 3 2 ) − 1 . Dividing b y 1 6 Q ∞ k =0 (1 + 4 − k − 3 2 ) − 1 to compute the conditional probabilit y , we obtain 7 16 . □ 4. Evidence in the function field case The goal of this section is to prov e Theorem 4.1, a function ﬁeld analogue of Conjecture 1.3. W e ﬁx an n -orien ted Γ -group H throughout this section. Our function ﬁeld analogue esti- mates a certain sum o ver Γ -extensions K of F q ( t ) , split at ∞ , of the n um b er of surjections from the Galois group of the maximal unramiﬁed, split at ∞ , extension of K to H . Similar sums ha ve been considered in prior w ork [L WZ24, Liu22, LL25, L W25]. Such a sum was estimated in the large q limit by Liu, W o o d, and Z uriec k-Brown [L WZ24, Theorem 1.4], as- suming that | H | is prime to q − 1 , in whic h case the orien tation is alw ays trivial. Liu [Liu22] remo ved this assumption, introduced the ω -in v ariant, which we ha v e already seen includes some, but not all, of the information in the orientation, and prov ed a large q estimate k eeping trac k of the ω -in v arian t. A similar estimate in the more diﬃcult case of ﬁxed large q w as pro ven by Landesman and Levy [LL25, Theorem 1.3.2]. Finally , Liu and Willyard [L W25, Theorem 1.1] relaxed the assumption that the Γ -extension b e split at inﬁnity . These argumen ts all rely on certain top ological results. F or q → ∞ limits, it suﬃces to ha ve results describing the set of connected comp onen ts of Hurwitz space. T o handle the ﬁxed large q case, Landesman and Levy [LL25] pro ved breakthrough results con trolling the lo w-degree homology groups of Hurwitz space, generalizing and strengthening earlier work of Ellen b erg, V enk atesh, and W esterland [EVW16]. Our strategy will therefore b e similar to the strategy of these prior works, except accoun t- ing for the orien tation. This requires us to take a certain co v ering of Hurwitz space, and pro ve results on the set of comp onents of this cov ering and the low-degree homology groups of this cov ering. The description of the set of comp onen ts of this cov ering is essentially a large mono drom y result, and follows from classical results in lo w-dimensional top ology . The computation of the low-degree homology groups of the cov er of Hurwitz space pro ceeds 46 b y observing that this co ver is itself cov ered by another Hurwitz space, whose cohomology groups can b e controlled by applying the results of [LL25]. W e b egin with the statemen t of our main coun ting theorem. W e follo w with a brief general notation Subsection 4.1, and then Subsection 4.2 deﬁning the Hurwitz schemes w e will use in our pro of. In Subsection 4.3 we establish the b asic prop erties of Hurwitz schemes and prov e Theorem 4.1 conditional on a result comparing the num b er of F q -p oin ts on a comp onen t of Hurwitz space and a cov er of that comp onent. The remaining sections prov e that result, with Subsection 4.4 proving a top ological result ab out the braid fundamen tal class, Subsection 4.5 pro ving a top ological result ab out the homology of cov ers of Hurwitz space, and then Subsection 4.6 applying these results to Hurwitz spaces and their co vers o ver ﬁnite ﬁelds. F or K a Γ -extension of F q ( t ) , w e let rDisc K b e the pro duct of the norms of the places of F q ( t ) that ramify in K (the norm of a place v of F q ( t ) b eing q deg v ). Let K un , | Γ | ′ b e the comp osition of all ﬁnite Galois extensions of K with degree prime to Γ that are everywhere unramiﬁed and split ov er ∞ . If F q ( t ) con tains the n th ro ots of unity , i.e. n | q − 1 , then since K un , | Γ | ′ /K is an extension with Galois group Gal( K un , | Γ | ′ /K ) , it has an asso ciated Artin- V erdier fundamen tal class s ∈ H 3 (Gal( K un , | Γ | ′ /K ) Γ , Z /n ) Γ . This mak es Gal ( K un , | Γ | ′ /K ) naturally an n -oriented Γ -group. (The Γ action is by using a homomorphic section of Gal( K un , | Γ | ′ / F q ( t )) → Gal( K / F q ( t )) = Γ and conjugating. By the Sch ur–Zassenhaus the- orem [RZ10, Theorem 2.3.15], this section is w ell-deﬁned up to conjugation by elements of Gal( K un , | Γ | ′ /K ) , and thus the resulting n -oriented Γ -group is w ell-deﬁned up to isomor- phism.) Let E Γ ( q m , F q ( t )) b e the set of Γ -extensions K of F , split at ∞ , suc h that rDisc K = q m . The goal of this section will b e to pro ve the follo wing theorem. Theorem 4.1. L et Γ b e the ﬁnite gr oup and n the p ositive inte ger c oprime to | Γ | ﬁxe d thr oughout the p ap er, and H the ﬁnite n -oriente d Γ -gr oup ﬁxe d thr oughout this se ction. Sup- p ose that H Γ = 1 . L et q b e a prime p ower. As long as q satisﬁes the c ongruenc e c onditions that ( q , | Γ || H | ) = 1 and q ≡ 1 mo d n and ( q − 1 , | H | ) = ( n, | H | ) and q is suﬃciently lar ge dep ending on Γ , H we have lim b →∞ P m ≤ b P K ∈ E Γ ( q m , F q ( t )) Sur( Gal ( K un , | Γ | ′ /K ) , H ) P m ≤ b | E Γ ( q m , F q ( t )) | =   H Γ   | H 2 ( H ⋊ Γ , Z /n ) | | H || H 3 ( H ⋊ Γ , Z /n ) | . W e b egin with the follo wing analogue of [L WZ24, Lemma 9.3] used to translate surjection coun ting to extension coun ting. Lemma 4.2. L et q ≡ 1 mo d n b e a prime p ower. L et N ( H , Γ , q m , F q ( t )) b e the numb er of surje ctions Gal( F q ( t ) / F q ( t )) → H ⋊ Γ such that the c orr esp onding ( H ⋊ Γ) -extension K has rDisc K = q m , the asso ciate d H -extension is unr amiﬁe d everywher e, split c ompletely at al l plac es lying ab ove ∞ , the induc e d map Gal( K un , | Γ | ′ /K ) → H gives a homomorphism of n -oriente d gr oups Gal ( K un , | Γ | ′ /K ) → H , and the asso ciate d Γ -extension K H / F q ( t ) is split c ompletely ab ove ∞ . Then X K ∈ E Γ ( q m , F q ( t )) Sur( Gal ( K un , | Γ | ′ /K ) , H ) =   H Γ   | H | N ( H , Γ , q m , F q ( t )) . 47 Pr o of. F rom [L WZ24, Lemma 9.3], w e hav e X K ∈ E Γ ( q m , F q ( t )) Sur(Gal( K un , | Γ | ′ /K ) , H ) =   H Γ   | H | N ( H , Γ , q m , F q ( t )) where the surjections on the left-hand side need not resp ect the orien tation and N ( H , Γ , q m , F q ( t )) is deﬁned the same w ay as N ( H , Γ , q m , F q ( t )) but without the orien tation condition. The pro of of [L WZ24, Lemma 9.3] gives a | H | | H Γ | -to-one surjectiv e map from the set coun ted by N ( H , Γ , q m , F q ( t )) to the union ov er K ∈ E Γ ( q m , F q ( t )) of the set counted b y Sur(Gal( K un , | Γ | ′ /K ) , H ) . T o chec k our statemen t, it suﬃces to chec k that the constructed surjection sends orientation- compatible maps to surjections whose asso ciated H -extension has compatible orien tation. This is clear since the map sends a su rjection Gal( F q ( t ) / F q ( t )) → H ⋊ Γ associated to an extension K to a pair consisting of the Γ -extension K H and the surjection asso ciated to the Γ -extension K/K H , so the orientations considered on eac h side are the same. □ 4.1. Algebraic geometry notation. W e no w set some algebraic geometry notation to b e used in Section 4. F or a scheme X ov er Sp ec Z and a ring R , we write X R for the pullback to X from Sp ec Z to Sp ec R . W e write Z [ µ n ] for the ring Z [ x ] / Φ n ( x ) , where Φ n is the n th cyclotomic p olynomial. All cohomology of schemes is étale cohomology , and for a scheme X w e write π 1 ( X ) for the algebraic fundamental group (with a c hoice of geometric basep oin t suppressed). F or a sc heme X ov er Sp ec C , when w e write X ( C ) , we mean the complex p oints of X as a top ological space with their analytic top ology . F or a scheme X o ver F q , w e write F rob q for the automorphism of X ¯ F q giv en by the ring map a 7→ a q on aﬃne pieces. F or X a sc heme deﬁned o ver F q , and x ∈ X ( F q ) , w e write F rob q ,x for the image of F rob enius in the map π 1 (Sp ec F q ) → π 1 ( X ) asso ciated to x , whic h is only well-deﬁned up to conjugation since w e do not c ho ose base-p oints. F or a scheme X and a ﬁnite group G , the deﬁnition of the algebraic fundamen tal group asso ciates to eac h homomorphism π 1 ( X ) → G a degree | G | étale co ver Y → X with a given action of G on Y equiv arian t with the trivial action on X . In other w ords, Y /X is a G -torsor. 4.2. Deﬁnition of Hurwitz sc hemes. W e now deﬁne the Hurwitz schemes that are rele- v ant to the pro of, in particular b ecause they will b e mo duli spaces for the kinds of extensions w e wan t to count. F or G a ﬁnite group and m a nonnegative in teger, let H m G, ∗ b e the scheme o ver Z [ | G | − 1 ] whose S -p oin ts are given b y isomorphism classes of tame Galois co vers of P 1 with branc h lo cus of degree m , unramiﬁed ab ov e ∞ , with a mark ed p oin t on the ﬁb er o ver ∞ , and with an identiﬁcation of their automorphism group with G , deﬁned as “ Hur m G, ∗ ” after Remark 11.3 in [L WZ24]. W e call a Galois cov er of P 1 with an identiﬁcation of its automor- phism group with G a G -c overing . (See also [L WZ24, Section 11.1] for deﬁnitions of these terms.) The fact that H m G, ∗ exists (or, if the language is preferred, that the stack H m G, ∗ is a scheme) is [L WZ24, Lemma 11.4]. F or c a union of conjugacy classes of G , stable under the op eration of raising a conjugacy class to the d th p o wer for d prime to | G | , let H m G,c b e the union of connected comp onents of H m G, ∗ parameterizing cov ers all whose inertia groups are generated b y elements of c , as in [L WZ24, Section 11.4]. W e hav e ﬁxed n throughout the pap er. W e will w ork o ver Z [ µ n ] , and we deﬁne Hur m G,c to b e ( H m G,c ) Z [ µ n ] . Even though this diﬀers from the use of the notation in [L WZ24], there will b e little conﬂict, since we are mostly fo cused on the base c hange to a ﬁnite ﬁeld, and for an y scheme X and prime p ow er 48 q ≡ 1 (mo d n ) , we hav e ( X ) F q = ( X Z [ µ n ] ) F q . F or the rest of this section, we will only ever consider prime p ow ers q ≡ 1 (mo d n ) . Let c Γ = Γ \ { 1 } and let c H ⋊ Γ b e the set of all nontrivial elements of H ⋊ Γ conjugate to an elemen t of Γ . W e will mostly consider Hur m G,c in the case when ( G, c ) is ( H ⋊ Γ , c H ⋊ Γ ) . The denominator in Theorem 4.1 was already computed in [L WZ24] in terms of Hur m Γ ,c Γ . W e review this computation no w. F or q relativ ely prime to | Γ | and q ≡ 1 (mo d n ) we hav e (4.3) | E Γ ( q m , F q ( t )) | =   Hur m Γ ,c Γ ( F q )   b y [L WZ24, Lemma 10.2]. This enables the computation of | E Γ ( q m , F q ( t )) | via the étale cohomology of Hur m Γ ,c Γ . T o deriv e Theorem 4.1 we m ust obtain a similar result ab out N ( H , Γ , q m , F q ( t )) . Without the orien tation on H , [L WZ24, Lemma 10.2] shows that the analogous count is the num b er of F q -p oin ts of the Hurwitz space Hur H ⋊ Γ ,c H ⋊ Γ . W e will show that N ( H , Γ , q m , F q ( t )) is prop ortional to the n umber of F q -p oin ts on a cov ering of the Hurwitz space. W e will deﬁne this co vering comp onent b y comp onen t. Let F : Y → X b e a family of curv es (i.e. a smo oth prop er morphism of schemes of relativ e dimension 1 with connected geometric ﬁb ers) deﬁned o ver Z [ µ n , 1 /n ] . Then we hav e an “in tegration along the ﬁb ers” map R F : H i +2 ( Y , µ n ) → H i ( X , Z /n ) , deﬁned as follo ws. W e ha ve R j F ∗ µ n = 0 for j > 2 so there is a map of complexes RF ∗ µ n → R 2 F ∗ µ n [ − 2] inducing a map H i +2 ( X , RF ∗ µ n ) → H i ( X , R 2 F ∗ µ n ) . The isomorphism H i +2 ( Y , µ n ) → H i +2 ( X , RF ∗ µ n ) arises from the derived category version of the Lera y sp ectral sequence (i.e. the derived functor of a comp osition b eing the composition of derived functors). The trace morphism [A GV72, XVI I I, Theorem 2.9] deﬁnes a morphism R 2 F ∗ µ n → Z /n . (Much more generally , for F ﬂat prop er of relative dimension d and F a sheaf on X torsion of order inv ertible on x , [AGV72, XVII I, Theorem 2.9] deﬁnes a map R 2 d F ∗ R ∗ F ( d ) → F . The case d = 1 and F = Z /n is what w e use). By functorialit y of cohomology , this induces a map H i ( X , R 2 F ∗ µ n ) → H i ( X , Z /n ) . Comp osing these H i +2 ( Y , µ n ) → H i +2 ( X , RF ∗ µ n ) → H i ( X , R 2 F ∗ µ n ) → H i ( X , Z /n ) w e obtain the in tegration along the ﬁb ers map. Ov er Z [ µ n ] = Z [ x ] / Φ n ( x ) , there is a natural map of shea v es Z /n → µ n , which can pull bac k to an y scheme ov er Z [ µ n ] , including Y ab ov e, to obtain H i +2 ( Y , Z /n ) → H i +2 ( Y , µ n ) . W e comp ose this with the ab ov e to obtain an integration along the ﬁb ers map H i +2 ( Y , Z /n ) → H i ( X , Z /n ) . F or each q ≡ 1 (mo d n ) , we choose, for use throughout this pap er, a primitiv e n th ro ot of unit y in F q . When we base change a sc heme ov er Z [ µ n ] to F q , it is alw ays with the map Z [ µ n ] to F q sending x to that chosen ro ot of unit y . When we consider a global function ﬁeld K o ver F q and compute an Artin-V erdier trace, w e alwa ys use that same c hoice of ro ot of unity in F q . When applying this map, w e will alwa ys identify µ n with Z /n by the same choice of n th ro ot of unity we use when deﬁning the Artin-V erdier trace. W e now state some lemmas giving fundamental properties of the in tegration along the ﬁb ers map which we will use later. First, that pullbac k on cohomology is compatible with in tegration along the ﬁb ers. 49 Lemma 4.4. F or g : X 1 → X 2 a morphism with Y 2 → X 2 a family of curves deﬁne d over Z [ µ n , 1 /n ] and Y 1 = Y 2 × X 2 X 1 its pul lb ack, we have the c ommutative diagr am wher e the vertic al arr ows ar e inte gr ation along the ﬁb ers and the horizontal arr ows ar e pul lb ack H i +2 ( Y 2 , µ n ) H i +2 ( Y 1 , µ n ) H i ( X 2 , Z /n ) H i ( X 1 , Z /n ) . W e also ha ve a “pro jection form u la” sho wing that in tegration along the ﬁb ers is compatible with the cup pro duct. Lemma 4.5. L et F : Y → X b e a family of curves deﬁne d over Z [ µ n , 1 /n ] and α a class in H j ( X , Z /n ) , then for j ≤ 1 we have a c ommutative diagr am H i +2 ( Y , µ n ) H i + j +2 ( Y , µ n ) H i ( X , Z /n ) H i + j ( X , Z /n ) ∪ F ∗ α ∪ α Both Lemma 4.4 and Lemma 4.5 will b e prov ed later in §4.7. Let X b e a connected comp onen t of Hur m H ⋊ Γ ,c H ⋊ Γ . Let Z , o ver X , b e the universal family of ( H ⋊ Γ) -cov erings of P 1 , so that Z → X × P 1 is an H ⋊ Γ -co vering. Let Y b e the universal family of induced Γ -co verings of P 1 , i.e. the quotient of Z by H . Then Z → Y is a ﬁnite étale H -cov ering and thus, by Lemma 2.2, determines a map from group cohomology to étale cohomology H 3 ( H , Z /n ) → H 3 ( Y , Z /n ) . This, together with the in tegration map and the explicit ev aluation of étale H 1 , deﬁnes a comp osed homomorphism H 3 ( H , Z /n ) ϕ Z/ Y → H 3 ( Y , Z /n ) → H 1 ( X , Z /n ) ∼ = Hom( π 1 ( X ) , Z /n ) whic h w e denote by ˜ e . Since H 3 ( H , Z /n ) is dual to H 3 ( H , Z /n ) (see Lemma 5.5), w e let e b e the induced map π 1 ( X ) → Hom( H 3 ( H , Z /n ) , Z /n ) ∼ = H 3 ( H , Z /n ) . Since Z is an H ⋊ Γ -cov ering of X × P 1 , the morphism H 3 ( H , Z /n ) → H 3 ( Y , Z /n ) is Γ - equiv ariant for the Γ -group action on H and the action of Γ on Y by the base-c hange claim in Lemma 2.2. Since Y → X is Γ in v ariant, the in tegration map H 3 ( Y , Z /n ) → H 1 ( X , Z /n ) is Γ -inv ariant. Hence the comp osition ˜ e is Γ -in v ariant, and th us e is Γ -in v ariant as w ell. Th us the image of e lands in H 3 ( H , Z /n ) Γ . Hence e deﬁnes a ﬁnite étale H 3 ( H , Z /n ) Γ -torsor ov er X . W e let X 0 b e this étale cov ering and let Hur m, 0 H ⋊ Γ ,c H ⋊ Γ b e the union of X 0 o ver all connected comp onen ts X of Hur m H ⋊ Γ ,c H ⋊ Γ . F or q ≡ 1 (mo d n ) and s H ∈ H 3 ( H , Z /n ) Γ , w e now deﬁne a cov ering of a comp onent X q of (Hur m H ⋊ Γ ,c H ⋊ Γ ) F q using the homomorphism π 1 ( X q ) → H 3 ( H , Z /n ) Γ that sends an element σ ∈ π 1 ( X ) to e ( σ ) − s H deg σ , where deg σ ∈ ˆ Z is the image of σ under the natural homomorphism π 1 ( X q ) → Gal( ¯ F q / F q ) ∼ = ˆ Z that sends the F rob enius element to 1 . Let X s H , F q q b e the ﬁnite étale H 3 ( H , Z /n ) Γ -torsor o ver X q asso ciated to this homomorphism and let Hur m,s H , F q H ⋊ Γ ,c H ⋊ Γ b e the union of X s H , F q q o ver all connected components X q of (Hur m H ⋊ Γ ,c H ⋊ Γ ) F q . (W e put F q in the exp onent b ecause this construction is not compatible with base change: F or q ′ a p ow er 50 of q w e do not necessarily ha ve (Hur m,s H , F q H ⋊ Γ ,c H ⋊ Γ ) F q ′ ∼ = Hur m,s H , F q ′ H ⋊ Γ ,c H ⋊ Γ .) Note that when w e base c hange our scheme Hur m H ⋊ Γ ,c H ⋊ Γ to F q , w e hav e implicitly chosen a map Z [ µ n ] → F q . 4.3. Prop erties and applications of Hurwitz schemes. The next lemma, pro ved in §4.7, relates the Artin-V erdier fundamen tal class to the map e . Lemma 4.6. L et m b e a p ositive inte ger and X b e a c onne cte d c omp onent of Hur m H ⋊ Γ ,c H ⋊ Γ . L et Z /X b e the universal family of ( H ⋊ Γ) -c overings of P 1 and let Y /X b e the universal family of induc e d Γ -c overings of P 1 . L et q b e a prime p ower that is 1 mo d n and is r elatively prime to | H || Γ | . F or x ∈ X ( F q ) and Z x and Y x the ﬁb ers of Z and Y at x , r esp e cti vely, we have that e (F rob q ,x ) is the Artin-V er dier fundamental class for Z x / Y x . Recall N ( H , Γ , q m , F q ( t )) w as deﬁned in Lemma 4.2 to coun t certain H -extensions of F q ( t ) . Using Lemma 4.6, we relate N ( H , Γ , q m , F q ( t )) to the count of p oints on Hur m,s H , F q H ⋊ Γ ,c H ⋊ Γ . Lemma 4.7. F or q ≡ 1 mo d n r elatively prime to | H || Γ | and H = ( H , s H ) we have, for every p ositive inte ger m , | N ( H , Γ , q m , F q ( t )) | = 1 | H 3 ( H ⋊ Γ , Z /n ) |    Hur m,s H , F q H ⋊ Γ ,c H ⋊ Γ ( F q )    . Pr o of. The argumen t in [L WZ24, Pro of of Lemma 10.2 on p. 54] gives a bijection b etw een F q - p oin ts of Hur m H ⋊ Γ ,c H ⋊ Γ and surjections Gal( F q ( t ) / F q ( t )) → H ⋊ Γ suc h that the corresp onding H ⋊ Γ -extension K has rDisc K = q m , the asso ciated H -extension is unramiﬁed ev erywhere and split completely at all places lying ab ov e ∞ , and the asso ciated Γ -extension K H / F q ( t ) is split completely ab ov e ∞ . F urthermore N ( H , Γ , q m , F q ( t )) counts such surjections with the additional condition that the natural orientation on H arising from the Artin-V erdier fundamen tal class on the asso ciated H -extension is equal to s H . It suﬃces to prov e that the bijection of [L WZ24, Proof of Lemma 10.2 on p. 54] sends a surjection satisfying that additional condition to an F q -p oin t of Hur m H ⋊ Γ ,c H ⋊ Γ that admits exactly | H 3 ( H ⋊ Γ , Z /n ) | lifts to F q -p oin ts of Hur m,s H , F q H ⋊ Γ ,c H ⋊ Γ , and sends a surjection not satis- fying that additional condition to an F q -p oin t of Hur m H ⋊ Γ ,c H ⋊ Γ that do es not lift to a F q -p oin t of Hur m,s H , F q H ⋊ Γ ,c H ⋊ Γ . W e can c hec k whether a p oint lifts comp onen t-by-component and so ﬁx a comp onent X q of (Hur m H ⋊ Γ ,c H ⋊ Γ ) F q . By deﬁnition of Hur m,s H , F q H ⋊ Γ ,c H ⋊ Γ as the disjoint union of the X s H , F q q , the num b er of lifts of x ∈ X q ⊆ (Hur m H ⋊ Γ ,c H ⋊ Γ ) F q to Hur m,s H , F q H ⋊ Γ ,c H ⋊ Γ ( F q ) is equal to the n um b er of lifts of x to X s H , F q q . Since X s H , F q q is a ﬁnite étale cov er of X q of degree | H 3 ( H , Z /n ) Γ | , an F q -p oin t x of X q has exactly   H 3 ( H , Z /n ) Γ   = | H 3 ( H ⋊ Γ , Z /n ) | (Lemmas 5.4 and 5.5) lif ts to F q -p oin ts of X s H , F q q if the image of F rob q ,x in the asso ciated homomorphism π 1 ( X q ) → H 3 ( H , Z /n ) Γ is trivial and 0 otherwise. By deﬁnition, this homomorphism sends F rob q ,x to e (F rob q ,x ) − s H deg F rob q ,x = e (F rob q ,x ) − s H since deg F rob q ,x = 1 . This image is trivial if and only if e (F rob q ,x ) = s H whic h by Lemma 4.6 happ ens if and only if the Artin-V erdier fundamental class for Z x / Y x is s H . Under the bijection of [L WZ24, Pro of of Lemma 10.2 on p. 54], K H is the function ﬁeld of Y x and K 51 is the function ﬁeld of the asso ciated H -cov ering, so this o ccurs if and only if the surjection satisﬁes the additional condition, as desired. □ It remains to count p oints on Hur m,s H , F q H ⋊ Γ ,c H ⋊ Γ . Liu [Liu22] deﬁnes an ω -inv arian t asso ciated to an H ⋊ Γ extension. Liu’s ω -inv arian t is a certain homomorphism, but we chec k ed in Lemmas 3.11 and 3.12 that the set of homo- morphisms is isomorphic to H 2 ( H , Z ) Γ b y an isomorphism sending Liu’s ω -inv ariant to our lifting inv arian t. W e will app ly this isomorphism, treating Liu’s ω -inv ariant as an elemen t of H 2 ( H , Z ) Γ and th us as b eing equal to the lifting inv ariant. This will allo w us to mak e use of prior work [Liu22, LL25] which counts p oin ts on Hurwitz spaces with a given lifting in v ariant. Liu [Liu22, §4.2] shows that the ω -in v ariant is determined b y the lifting in v ariant deﬁned b y Ellenberg-V enk atesh-W esterland and the second author. Since this lifting in v ariant is constan t on comp onen ts of Hur m H ⋊ Γ ,c H ⋊ Γ b y [W o o21, Theorem 6.1], Liu’s ω and our lifting in- v ariant are determined b y the comp onen t. This, together with Lemma 4.6, has the follo wing consequence. Let ω H b e the image of s H in H 2 ( H , Z ) Γ . Lemma 4.8. L et m b e a p ositive inte ger, let q b e a prime p ower that is 1 mo d n , and is r elatively prime to | H || Γ | . L et X q b e a c onne cte d c omp onent of (Hur m H ⋊ Γ ,c H ⋊ Γ ) F q . If the c over X s H , F q q of X q has a F q -r ational p oint, then the lifting invariant of X q is ω H . Pr o of. An F q -p oin t of X s H , F q q lies o ver an F q -p oin t x ∈ X q . By deﬁnition, F rob enius acts on the ﬁb er ov er x by adding e (F rob q ,x ) − s H . F or the ﬁb er to contain a F q p oin t, w e m ust ha ve e (F rob q ,x ) − s H = 0 . Thus b y Lemma 4.6, x parameterizes a H ⋊ Γ -cov ering of P 1 with Artin-V erdier fundamental class s H . Thus its lifting inv arian t is ω H , and so the lifting in v ariant of the comp onen t X q is ω H . □ Motiv ated b y Lemma 4.8, let Hur m,ω H ⋊ Γ ,c H ⋊ Γ b e the union of the comp onen ts of Hur m H ⋊ Γ ,c H ⋊ Γ with lifting inv ariant ω . Using prior w ork, w e ma y no w con trol the num b er of p oin ts on “bad” comp onen ts of Hur m,ω H ⋊ Γ ,c H ⋊ Γ . Lemma 4.9. Supp ose that H Γ = 1 . L et q b e a prime p ower that is 1 mo d n , and is r elatively prime to | H || Γ | . Fix ω ∈ H 2 ( H , Z ) Γ [ q − 1] . L et a b e a p ositive inte ger. Then for b ≥ 0 an inte ger, if q and b ar e suﬃciently lar ge given H and Γ , the fr action of F q - p oints on S m ≤ b Hur m,ω H ⋊ Γ ,c H ⋊ Γ that lie on a c omp onent p ar ameterizing curves wher e at le ast one c onjugacy class in c has < a br anch p oints of that c onjugacy class is O ( a/b ) wher e the implicit c onstant may dep end on q , H, and Γ . Pr o of. In this pro of “suﬃcien tly large” alw ays meaning only dep ending on H , Γ , and the implicit constan ts in O notation dep end on q , H, Γ . Let d Γ ( q ) b e the n um b er of orbits of q th p o wering on non trivial conjugacy classes in Γ . Since eac h conjugacy class in c H ⋊ Γ con tains a unique non trivial conjugacy class of Γ , this is also the num b er of orbits of q th p o w ering on nontrivial conjugacy classes in c H ⋊ Γ . W e will show that the num b er of p oin ts of S m ≤ b Hur m,ω H ⋊ Γ ,c H ⋊ Γ is, for q and b suﬃciently large, greater than a p ositive constant dep ending on q , H , Γ times b d Γ ( q ) − 1 q b . W e will then sho w that the num b er of p oin ts that lie on a comp onent parameterizing curv es where at 52 least one conjugacy class in c has ≤ a branc h p oints of that conjugacy class is O ( ab d Γ ( q ) − 2 q b ) . Com bining these, we obtain the desired statemen t. By [LL25, Lemma 8.4.4], for q suﬃciently large, each geometrically connected comp onen t of Hur m H ⋊ Γ ,c H ⋊ Γ con tains at least q m / 2 p oin ts. The same is true for Hur m,ω H ⋊ Γ ,c H ⋊ Γ as it is a union of comp onents of Hur m H ⋊ Γ ,c H ⋊ Γ . F ollowing [Liu22, §4.3], let π ω H ⋊ Γ ,c H ⋊ Γ ( q , m ) be the n umber of geometrically connected comp onen ts of Hur m H ⋊ Γ ,c H ⋊ Γ with lifting inv ariant ω . By H Γ = 1 and [Liu22, Lemma 4.4(1)] (where w e hav e put lines ov er Liu’s v ariables a and b to av oid ambiguit y with our a and b ) there is a p ositive in teger M a , non-empt y set E a of residues mo d M a , and p ositive n um b er r ′ a,b for b ∈ E a , all dep ending only on H and Γ and the residue class a of q mo dulo | H ⋊ Γ | 2 , suc h that if m ≡ b ∈ E a mo dulo M a w e hav e π ω H ⋊ Γ ,c H ⋊ Γ ( q , m ) = r ′ a,b m d Γ ( q ) − 1 + O ( m d Γ ( q ) − 2 ) . Letting m b e the least integer ≤ b whose residue mo d M a lies in E a , which is alwa ys ≥ b − M a and thus each comp onen t con tributes at least q b − M a / 2 p oints, ensures that the n umber of comp onen ts is, for b suﬃciently large, greater than a p ositiv e constan t times m d Γ ( q ) − 1 and th us greater than a p ositive constan t times b d Γ ( q ) − 1 . Hence the total num b er of p oin ts of S m ≤ b Hur m,ω H ⋊ Γ ,c H ⋊ Γ , for q and b suﬃciently large, is greater than a p ositiv e constant times b d Γ ( q ) − 1 q b , as desired. Again b y [LL25, Lemma 8.4.4], for q suﬃciently large, eac h geometrically connected com- p onen t of Hur m H ⋊ Γ ,c H ⋊ Γ con tains at most 2 q m p oin ts. F ollo wing [L WZ24], let Z c H ⋊ Γ /H ⋊ Γ ≡ q b e the set of functions from conjugacy classes in c to nonnegativ e in tegers which tak e the same v alue on eac h comp onen t and its q th p o wer. Asso ciated to each comp onent is an element of Z c H ⋊ Γ /H ⋊ Γ ≡ q : The function whose v alue on each conjugacy class is the num b er of branch p oin ts with ramiﬁcation given b y that conjugacy class. F urthermore, this element of Z c H ⋊ Γ /H ⋊ Γ ≡ q is alw ays a function that sums to m . W e are interested in comp onen ts corresp onding to ele- men ts of Z c H ⋊ Γ /H ⋊ Γ ≡ q whic h take some v alue ≤ a . The n um b er of suc h elemen ts of Z c H ⋊ Γ /H ⋊ Γ ≡ q is O ( am d Γ ( q ) − 2 ) as they are determined b y d Γ ( q ) distinct v alues, one of which is b ounded b y a and the remainder of which are b ounded by m , but with a linear relation such that an y one is determined b y the remaining ones. By [L WZ24, Theorem 12.4] and [EV05, Lemma 3.3], the num b er of comp onents corresp onding to eac h element of Z c H ⋊ Γ /H ⋊ Γ ≡ q is O (1) . (This is prov ed ov er C but b y [L WZ24, Lemma 10.3] with i = 0 the same statement holds ov er ¯ F q .) Thus the total n umber of comp onen ts is O ( am d Γ ( q ) − 2 ) and hence the total n um b er of F q -p oin ts is O ( am d Γ ( q ) − 2 q m ) whic h summed o v er m ≤ b is O ( ab d Γ ( q ) − 2 q b ) , as desired. □ The k ey new result w e will need is the follo wing: Theorem 4.10. L et Γ b e the ﬁnite gr oup and n the p ositive inte ger c oprime to | Γ | ﬁxe d thr oughout the p ap er, and H the ﬁnite n -oriente d Γ -gr oup ﬁxe d thr oughout this se ction. Supp ose that H Γ = 1 . L et q b e prime p ower q ≡ 1 mo d n with gcd( q , | Γ || H | ) = 1 and gcd( q − 1 , | H | ) = gcd( n, H ) , L et m b e a p ositive inte ger and let X q b e a c omp onent of Hur m,ω H H ⋊ Γ ,c H ⋊ Γ , F q . L et a X q b e the minimum over c onjugacy classes in c H ⋊ Γ of the numb er of br anch p oints of that c onjugacy class in c overs p ar ameterize d by X q . 53 Ther e is δ > 0 , only dep ending on Γ , H , q , such that as long as q is suﬃciently lar ge in terms of H , Γ we have | X s H , F q q ( F q ) | = ( | H 2 ( H ⋊ Γ , Z /n ) | + O ( e − δ a X q )) | X q ( F q ) | , with the implicit c onstant in the O dep ending on Γ , H , q . W e now sho w ho w Theorem 4.10 implies Theorem 4.1. Pr o of of The or em 4.1. Through this pro of, the implicit constant in O notation dep ends on Γ , H , q . By Lemmas 4.2 and 4.7 we ha v e (4.11) X m ≤ b X K ∈ E Γ ( q m , F q ( t )) Sur( Gal ( K un , | Γ | ′ /K ) , H ) =   H Γ   | H || H 3 ( H ⋊ Γ , Z /n ) | X m ≤ b    Hur m,s H , F q H ⋊ Γ ,c H ⋊ Γ ( F q )    and b y (4.3) w e hav e (4.12) X m ≤ b | E Γ ( q m , F q ( t )) | = X m ≤ b   Hur m Γ ,c Γ ( F q )   . W e ha v e [LL25, (9.3)] (4.13) P m ≤ b    Hur m,ω H H ⋊ Γ ,c H ⋊ Γ ( F q )    P m ≤ b   Hur m Γ ,c Γ ( F q )   = 1 + O (1 /b ) . Com bining (4.11), (4.12) and (4.13), w e see that to establish Theorem 4.1 it suﬃces to sho w (4.14) P m ≤ b    Hur m,s H , F q H ⋊ Γ ,c H ⋊ Γ ( F q )    P m ≤ b    Hur m,ω H H ⋊ Γ ,c H ⋊ Γ ( F q )    =   H 2 ( H ⋊ Γ , Z /n )   + O (1 /b ) , or equiv alently (4.15) X m ≤ b    Hur m,s H , F q H ⋊ Γ ,c H ⋊ Γ ( F q )    −   H 2 ( H ⋊ Γ , Z /n )   X m ≤ b    Hur m,ω H H ⋊ Γ ,c H ⋊ Γ ( F q )    = O ( 1 b X m ≤ b    Hur m,ω H H ⋊ Γ ,c H ⋊ Γ ( F q )    ) . W e hav e (by Lemma 4.8) (4.16) X m ≤ b    Hur m,s H , F q H ⋊ Γ ,c H ⋊ Γ ( F q )    = X m ≤ b X X q ⊆ Hur m H ⋊ Γ ,c H ⋊ Γ component lifting inv ariant ω H   X F q ,s H q ( F q )   and (b y deﬁnition) (4.17) X m ≤ b    Hur m,ω H H ⋊ Γ ,c H ⋊ Γ ( F q )    = X m ≤ b X X q ⊆ Hur m H ⋊ Γ ,c H ⋊ Γ component lifting inv ariant ω H | X q ( F q ) | so the left hand side of (4.15) is 54 (4.18) X m ≤ b X X q ⊆ Hur m H ⋊ Γ ,c H ⋊ Γ component lifting inv ariant ω H    X F q ,s H q ( F q )   −   H 2 ( H ⋊ γ , Z /n )   | X q ( F q ) |  . F or eac h X q w e apply Theorem 4.10 and get (4.19)   X F q ,s H q ( F q )   −   H 2 ( H ⋊ Γ , Z /n )   | X q ( F q ) | = O ( e − δ a X q | X q ( F q ) | ) so (4.18) is O  X m ≤ b X X q ⊆ Hur m H ⋊ Γ ,c H ⋊ Γ component lifting inv ariant ω H e − δ a X q | X q ( F q ) |  . By Lemma 4.9, the terms with a X q = a on the righ t hand side of (4.19) contribute O  e − δ a · a b X m ≤ b    Hur m,ω H H ⋊ Γ ,c H ⋊ Γ ( F q )     so summing ov er a , (4.18) is O  ∞ X a =1 a b e − δ a X m ≤ b    Hur m,ω H H ⋊ Γ ,c H ⋊ Γ ( F q )     = O  1 b X m ≤ b    Hur m,ω H H ⋊ Γ ,c H ⋊ Γ ( F q )     since P ∞ a =0 ae − δ a = O (1) . Hence (4.18) is b ounded b y the right hand side of (4.15), so w e ha ve (4.15), which we already sa w suﬃces. □ Theorem 4.10 is trivial when X q is not geometrically irreducible as in that case neither X q nor X s H , F q q has any F q -p oin ts, an d also trivial if a ( X q ) is small. In the remaining case, w e will prov e Theorem 4.10 b y showing that X s H , F q q has | H 2 ( H ⋊ Γ , Z /n ) | comp onen ts and each comp onen t has (1 + O ( e − δ m )) | X q ( F q ) | F q -p oin ts. These will follo w from analogous statemen ts o ver the complex n umbers, one ab out the image of the map e from the fundamen tal group to H 3 ( H , Z /n ) , and one ab out the homology of connected comp onents of Hur m, 0 H ⋊ Γ ,c H ⋊ Γ . W e will pro ve these b y top ological metho ds. 4.4. The braid fundamen tal class. W e now explain the top ological analogue of our geo- metric setup. Consider a conﬁguration of m p oin ts in the (top ological) plane P . F or psyc hological con venience we may take the p oin ts to b e in a line. Let P ◦ b e the plane min us those p oints. The braid group Br m on m strands naturally maps to the mapping class group of P ◦ [FM11, Section 9.1.4], and th us also acts on the fundamental group π 1 ( P ◦ ) (taking a base p oint near inﬁnity). Given a ﬁnite group G , the braid group Br m then acts on the ﬁnite set of homomorphisms π 1 ( P ◦ ) → G. This action preserves the set of homomorphisms where the conjugacy class of the local mono drom y around eac h p oin t lies in a subset c of G that is closed under taking G -conjugates. Giv en a braid σ ∈ Br m , we ha ve a geometric realization of the braid as m sections s 1 , . . . , s m of the map P × [0 , 1] → [0 , 1] , suc h that the sections do not intersect, and the set s i (0) are our m chosen p oints in P , as is the set of s i (1) . W e can then complete P × [0 , 1] to S 2 × [0 , 1] , 55 and identify S 2 × 0 and S 2 × 1 . Then in the compact 3 -manifold S 2 × S 1 , our braids, along with ∞ × S 1 , describ e a link L σ . F or our giv en Γ and H , ﬁx a surjectiv e homomorphism f : π 1 ( P ◦ ) → H ⋊ Γ such that the lo cal mono drom y around each missing p oin t is in c H ⋊ Γ , and the lo cal mono dromy around ∞ is either in c H ⋊ Γ or trivial. Let σ ∈ Stab f ⊂ Br m . Then w e can chec k that there is a unique homomorphism ˜ f : π 1 ( S 2 × S 1 \ L σ ) → H ⋊ Γ that restricts to f on P ◦ × 0 and is trivial on the lo op ∞ × S 1 . W e let ˜ f Γ b e the comp osition of ˜ f with H ⋊ Γ → Γ . W e let M σ b e the compact oriented 3 -manifold such that ˜ f Γ corresp onds to a degree | Γ | branc hed cov ering u Γ : M σ → S 2 × S 1 , with an action of Γ . W e note by our mono dromy condition that the restriction of ˜ f : π 1 ( M σ \ u − 1 Γ L σ ) → H factors through a unique map π 1 ( M σ ) → H . W e can deﬁne a map from the stabilizer Stab f ⊂ Br m of f to H 3 ( H , Z ) Γ , by sending σ to the image of the fundamen tal class of M σ in H 3 ( H , Z ) Γ . In more detail, the fundamen tal class of M σ is an elemen t of H 3 ( M σ , Z ) , which maps to H 3 ( π 1 ( M σ ) , Z ) b y the usual map, and then we comp ose with H 3 ( π 1 ( M σ ) , Z ) → H 3 ( H , Z ) using the map π 1 ( M σ ) → H ab ov e, and ﬁnally we note that the image of the fundamen tal class of M σ in H 3 ( H , Z ) is Γ -in v arian t. W e call this map Stab f → H 3 ( H , Z ) Γ the br aid fundamental class map . W e now explain the relev ance of the braid fundamen tal class map to our problem. Let X b e a connected component of (Hur m H ⋊ Γ ,c H ⋊ Γ ) C . W e can express the top ological fundamen tal group X ( C ) in terms of the braid group on m strands. Let x b e a p oint of a comp onent V of Hur m H ⋊ Γ ,c H ⋊ Γ ( C ) , such that x corresp onds to a cov er unramiﬁed ov er P ◦ . Let f : π 1 ( P ◦ ) → H ⋊ Γ b e the map corresp onding to that cov er. Then the isomorphism Br m → π 1 (Conf m A 1 ( C )) takes Stab f to π 1 ( V ) . (See [FV, Section 1.4] and [L WZ24, Pro of of Theorem 12.4].) W e ha ve the following lemma, pro v en later in §4.7. Lemma 4.20. L et m b e a nonne gative inte ger and f and X as ab ove. Under the ab ove identiﬁc ation of Stab f and π 1 ( X ( C )) , for any br aid σ in the stabilizer of f , the r e duction mo d n of the br aid fundamental class of σ is the pr o duct of e ( σ ) by an element of ( Z /n ) × . It should b e p ossible to c hec k that the elemen t of ( Z /n ) × men tioned in Lemma 4.20 is 1 for any braid as long as the ﬁxed isomorphism µ n → Z /n sends e 2 π i/n to 1 , but this w ould b e unnecessary and add additional tec hnical details so we do not attempt it. The goal of this subsection is to prov e the follo wing result, which, b ecause of Lemma 4.20, will allo w us to con trol the image of e : Prop osition 4.21. L et m b e a nonne gative inte ger, and c onsider the (top olo gic al) plane P ◦ punctur e d at m p oints. L et f b e a surje ctive homomorphism f : π 1 ( P ◦ ) → H ⋊ Γ . Assume that the lo c al mono dr omy of f ar ound e ach punctur e is in c H ⋊ Γ , and the lo c al mono dr omy ar ound ∞ is either in c H ⋊ Γ or trivial. Assume that for e ach c onjugacy class of H ⋊ Γ , the numb er of p oints in the c onﬁgur ation sp ac e with lo c al mono dr omy in that c onjugacy class (i.e a smal l lo op ar ound that p oint go es to that c onjugacy class) is suﬃciently lar ge dep ending H and Γ . Then the br aid fundamental class map fr om the stabilizer Stab f in the br aid gr oup to H 3 ( H , Z ) Γ is surje ctive. T o pro ve Prop osition 4.21 w e deﬁne a more general fundamen tal class. Let M b e an orien ted compact 3-manifold, L an oriented link in M , and f : π 1 ( M \ L ) → H ⋊ Γ that, for eac h knot of L , sends a small lo op around that knot to an elemen t of a conjugate of Γ . Let 56 ˜ M b e the induced orien ted compact 3-manifold suc h that ˜ M → L is a branc hed Γ -co vering. W e hav e a morphism π 1 ( ˜ M ) → H and hence a map ˜ M → B H where B H is the classifying space of H . The fundamental class of ˜ M gives an element of H 3 ( H , Z ) which is manifestly Γ -in v ariant. Let [ M , L, f ] ∈ H 3 ( H , Z ) Γ b e the fundamental class of ˜ M . Lemma 4.22. L et m b e a nonne gative inte ger. L et P ◦ b e the top olo gic al plane punctur e d at m p oints, as ab ove. L et f b e a surje ctive homomorphism f : π 1 ( P ◦ ) → H ⋊ Γ such that the lo c al mono dr omy ar ound e ach punctur e is in c H ⋊ Γ and the lo c al mono dr omy at ∞ is either in c H ⋊ Γ or trivial. The br aid fundamental class is a homomorphism fr om the stabilizer of f in Br m to H 3 ( H , Z ) Γ . Pr o of. The pro of is essentially the same as [ SW24, Lemma 3.2]. W e use [SW24, Lemma 2.13] to in terpret the fundamental class of M σ in H 3 ( H , Z ) as the class of M σ in the b ordism group of B H . W e chec k that the braid fundamental class of a pro duct of t w o braids is the sum of the braid fundamen tal classes. The braid fundamen tal class is constructed by starting with a braid and producing a link in S 2 × S 1 . Giv en tw o braids whose pro duct is a third braid, we can construct a surface in the pro duct of S 2 with a pair of pants, whose restriction to eac h b oundary comp onent is one of the three links in S 2 × S 1 . T o construct the surface, w e view eac h braid as an elemen t of the fundamental group of the conﬁguration space of p oin ts in S 1 . W e then use the fact that when the pro duct of t w o fundamen tal group elements of some space is a third, we can ﬁnd a map from a pair of pants to the space whose b oundary consists of three lo ops representing the three elemen ts of the fundamen tal group. The complement of this surface in the pro duct of S 2 with a pair of pants is a ﬁbration o v er the pair of pants with ﬁb er a punctured sphere. Hence its fundamental group is an extension of the fundamental group of the pair of pain ts by the fundamen tal group of the punctured sphere. The fundamen tal group of the pair of pan ts is a free group on t wo generators, eac h corresp onding to one of the b oundary lo ops, and thus eac h corresp onding to a braid that lies in the stabilizer of f in Br m . W e can lift these t wo generators explicitly to the fundamen tal group of the surface complemen t by taking the pro duct of the corresp onding lo op in the pair of pants with the p oin t ∞ in P 1 . Since each generator of the free group stabilizes the homomorphism f from the fundamental group of the punctured sphere to H ⋊ Γ , we can extend the homomorphism f uniquely to a homomorphism from the fundamental group of the surface complement to H ⋊ Γ whic h is trivial on the tw o generators of the braid group. This giv es a Γ -cov ering of the pro duct of S 2 with a pair of pan ts branched at this surface, and a map from the branched cov ering to B H . This gives a four-manifold with b oundary witnessing the desired relation in the homology of B H b et ween the three braid fundamen tal classes. □ Lemma 4.23. Every class in H 3 ( H , Z ) Γ is of the form [ M , L, f ] for some 3 -manifold M , link L in M , and f : π 1 ( M \ L ) → H ⋊ Γ . In fact, we may take L empty. Pr o of. In the case L is empty , [ M ] represen ts a class in H 3 ( H ⋊ Γ , Z ) , from which [ M , L, f ] is obtained via the trace map H 3 ( H ⋊ Γ , Z ) → H 3 ( H , Z ) Γ . Because H and Γ ha v e coprime orders, the trace map H 3 ( H ⋊ Γ , Z ) → H 3 ( H , Z ) Γ is surjectiv e, so it suﬃces to sho w ev ery 57 class in H 3 ( H ⋊ Γ , Z ) is represen ted by a 3-manifold, which follo ws from [SW24, Lemma 2.13]. □ Lemma 4.24. The gr oup H 3 ( H , Z ) Γ is gener ate d by classes of the form [ S 3 , L, f ] wher e L is a link in S 3 . Pr o of. Fix an element of H 3 ( H ) Γ , which by Lemma 4.23 arises from a triple [ M , ∅ , f ] . By the Lick orish-W allace theorem, M arises from surgery on a link in S 3 . Th us, there exists a link L in S 3 suc h that when we remov e a tubular neighborho o d of L and glue in a new solid torus at each b oundary comp onent of the neigh b orho o d, w e obtain M . F urthermore, this is surgery with slop e ± 1 , so the meridian of the new solid torus is the sum of the meridian and standard longitude of the remov ed solid torus. How ever, it will b e more conv enien t for us to choose co ordinates in B 2 × S 1 for the remov ed solid torus where the new meridian is simply the longitude, which is p ossible as adding any in teger m ultiple of the meridian to a longitude giv es a new longitude. This isomorphism gives a homomorphism from the fundamen tal group of S 3 min us the tubular neigh b orho o d of L to H ⋊ Γ . W e can extend this to the complement of L , but there is no reason for the lo cal mono drom y elements to lie in Γ . How ever, since H Γ = 1 , the conjugates of Γ generate H ⋊ Γ , and thus we can write the lo cal mono drom y around eac h comp onen t K i of L as a pro duct of n i conjugates of elements of Γ . W e deﬁne a link L ′ whic h for eac h comp onent K i of L consists of n i lo ops in the tubular neighborho o d of K i , parallel to each other and to the meridian. (Inside B 2 × S 1 , this is the pro duct of n i p oin ts with S 1 ). W e can then extend f to f ′ : π 1 ( S 3 \ L ′ ) → H ⋊ Γ , i n suc h a wa y that the lo cal mono drom y around each new comp onent is the corresp onding conjugate of an element of Γ . Indeed, the fundamen tal group of B 2 × S 1 min us the pro duct of n i p oin ts with S 1 is the pro duct of the free group on n i letters with Z , where the lo op around each p oin t go es to the corresp onding letter, the meridian go es to the pro duct of the letters, and the longitude go es to a generator of Z . Restricted to the fundamental group of the torus b oundary , f sends the pro duct of the letters to the lo cal monodromy of f along K i and the generator of Z (the meridian curv e in M ) to the identit y , so b y writing the lo cal mono dromy as a pro duct, we can extend the homomorphism. Let us now calculate the diﬀerence [ M , ∅ , f ] − [ S 3 , L ′ , f ′ ] . The diﬀerence b etw een the fundamen tal class of ˜ M and the fundamen tal class of ˜ S 3 is the fundamental class of the Γ -co vering of the tubular neighborho o d of L we remov ed minus the Γ -cov ering of the solid tori w e added. In other words, it is the fundamen tal class of the Γ -cov ering of the manifold obtained b y gluing the tubuluar neighborho o d of L to the solid tori along their common b oundary . Since the union of these tw o tori is S 3 , this manifold is itself a disjoint union of Γ -co verings of copies of S 3 con taining links, so we can write [ M , ∅ , f ] as a sum and diﬀerence of classes of links, as desired. □ Lemma 4.25. L et p and m b e nonne gative inte gers. Fix a c onﬁgur ation of p p oints in the plane P and a homomorphism f fr om the fundamental gr oup of their c omplement in P to H ⋊ Γ , with lo c al mono dr omy ar ound e ach p oint in c H ⋊ Γ and lo c al mono dr omy ar ound ∞ either in c H ⋊ Γ or trivial. Fix a disc D in the plane, c ontaining m of the p oints. Consider a br aid σ in the br aid gr oup on m str ands, shifting only the p oints in D . L et σ ′ b e the br aid in the br aid gr oup on 58 p str ands obtaine d by extending σ so that the p oints outside D ar e unmove d. Assume that σ ′ ﬁxes f . L et L ( σ ) b e the link in D × S 1 ⊂ S 3 obtaine d by closing up the br aid σ . Ther e exists a unique homomorphism ˜ f fr om the fundamental gr oup of c omplement o f L ( σ ) in S 3 to H ⋊ Γ whose r estriction to D minus m p oints is e qual to the r estriction of f . Then [ S 3 , L ( σ ) , ˜ f ] is the br aid fundamental class of σ ′ (with r esp e ct to f ). Pr o of. T o show that ˜ f exists and is unique, we use the classical fact that the fundamen tal group of the complemen t of L ( σ ) is the quotient of the fundamen tal group of the m -times- punctured disc, i.e. the free group on generators x 1 , . . . , x m b y the relations x i = σ ( x i ) for i from 1 to m . This fact itself may b e c heck ed using the v an Kamp en theorem applied to the decomp osition of S 3 in to D × S 1 and a complementary solid torus, whose intersection is homotopic to a torus. The fundamental group of the complemen t of L ( σ ) in D × S 1 is the semidirect pro duct ⟨ x 1 , . . . , x m ⟩ ⋊ ⟨ σ ⟩ , and adding the solid torus pro duces the relation σ = 1 which gives the stated fundamental group. Since f is σ ′ -in v ariant, the restriction of f to the fundamental group of the m -times punctured disc is σ -in v ariant, and hence factors uniquely through the fundamental group of the complemen t of L ( σ ) in S 3 . W e must show t wo manifolds, one a branc hed Γ -co vering of S 3 and the other a branched Γ -co vering of S 2 × S 1 , with pro vided H -co verings, ha v e iden tical fundamen tal classes in H 3 ( H , Z ) . T o do this, we consider maps from eac h manifold to the classifying space B H of H . The fundamen tal classes in H 3 ( H , Z ) of these manifolds may b e represented by the images of these maps, viewed as c hains. Since the H ⋊ Γ -co verings of S 3 and S 2 × S 1 b ecome isomorphic when restricted to D × S 1 , we can c ho ose the maps from the Γ -cov erings to B H to agree on the Γ -cov erings of D × S 1 . Ha ving done this, the diﬀerence of the tw o fundamen tal classes is represented b y a map from a 3-manifold given b y gluing a Γ -cov ering of the complement of D × S 1 in S 3 to a Γ - co vering of the complement of D × S 1 in S 2 × S 1 . The relev ant cov ering of S 3 is unbranc hed in the complemen t of D × S 1 , whic h is a solid torus. The relev an t co vering of S 2 × S 1 is branc hed at the pro duct of p − m p oin ts outside D with S 1 , and p ossibly branched also at the pro duct of the p oin t at ∞ of S 1 . Thus the diﬀerence of the t wo fundamental classes is represen ted b y a map from a 3 -manifold giv en by gluing an un branc hed Γ -co vering of the complemen tary solid torus to D × S 1 in S 3 to a Γ -cov ering of ( S 2 \ D ) × S 1 branc hed at the pro duct of either p − m or p + 1 − m p oin ts with S 1 . W e claim that this Γ -co vering of S 3 ma y b e obtained by taking a n umber of copies of S 3 , remo ving balls, and gluing them together along their sphere b oundaries. It suﬃces to break S 3 in to pieces, with b oundary S 2 , such that the Γ -co vering of eac h piece is a disjoint union of copies of S 3 with balls remov ed, as w e can then glue the pieces together b y identifying some of their b oundary comp onen ts, all of which are spheres. T o do this, w e observe that the pro duct of either p − m or p + 1 − m points in S 2 \ D with S 1 inside ( S 2 \ D ) × S 1 ⊂ S 3 is a union of parallel circles inside the unknotted torus ( S 2 \ D ) × S 1 and th us a union of unknotted, unlink ed circles. W e may place each of these circles in to a disjoin t ball. Let C b e the complemen t of all these balls. Certainly C may be obtained from S 3 b y remo ving balls. Hence C is simply connected. The induced Γ -co v er of C is unbranc hed. Hence the induced Γ -cov er of C is a union of copies of C , and th us a union of complements of balls in S 3 . 59 Restricted to a ball con taining a single circle, the Γ -co v ering is b ranc hed only at that circle. The fundamental group of the complemen t of a single unknotted circle in a ball is Z , so the Γ -cov ering m ust b e a disjoin t union of cyclic cov erings, where the n -fold cyclic co vering is homeomorphic to the complemen t of n balls in S 3 . Th us the Γ -cov ering of S 3 ma y b e obtained by taking a n umber of copies of S 3 , removing balls, and gluing them together along their sphere boundaries. Any suc h manifold has fundamen tal group a free group by the group oid v an Kamp en theorem. Hence the natural map from the homology of this Γ -cov ering to H 3 ( H , Z ) factors through the H 3 of the free group and thus v anishes, so the fundamen tal class of this Γ -co v ering of S 3 v anishes, as desired. □ Lemma 4.26. L et p and m b e nonne gative inte gers. L et f b e a surje ctive homomorphism fr om the fundamental gr oup of an p -times punctur e d plane to H ⋊ Γ , with lo c al mono dr omies at e ach punctur e in c H ⋊ Γ and with suﬃciently many (dep ending on m ) punctur es with e ach p ossible c onjugacy class of lo c al mono dr omy. L et f ′ b e a homomorphism fr om the fundamental gr oup of an m -times punctur e d disc D to H ⋊ Γ , with lo c al mono dr omies in c H ⋊ Γ . Then ther e is a disc in the plane c ontaining m punctur es which is home omorphic to D by a home omorphism sending the r estriction of f to f ′ . Pr o of. It suﬃces to ﬁnd a a homomorphism from the fundamen tal group of the ( p − m ) - punctured disc D ∗ suc h that the homomorphism from the fundamen tal group of the p - punctured plane obtained b y gluing together D and D ∗ lies in the same braid group orbit as f , as then w e can choose a braid in the braid group sending this homeomorphism to f , tak e an asso ciated homeomorphism from the punctured plane to itself, and c ho ose the image of D under this homeomorphism to b e our disc in the plane. This follows from the statemen t [W o o21, Theorem 3.1] that braid group orbits of co vers of the p -punctured plane with suﬃciently many punctures of eac h lo cal monodromy class are classiﬁed by their lifting inv arian t, and that lifting in v ariants are additive when gluing discs: W e m ust subtract the lifting inv arian t of f ′ from the lifting inv arian t of f and conﬁrm a braid with this lifting in v ariant exists, whic h happ ens if the diﬀerence has co ordinates suﬃcien tly large, which o ccurs if the lifting inv arian t of f has co ordinates suﬃcien tly large, where these co ordinates are the num b er of punctures with lo cal mono dromy realizing each conjugacy class of c . □ Pr o of of Pr op osition 4.21. Fix ﬁnitely many links L i in S 3 and homomorphisms f i : π 1 ( S 3 \ L i ) → H ⋊ Γ , whose lo cal mono dromy around eac h comp onen t of L is conjugate to an elemen t of Γ , suc h that [ S 3 , L i , f i ] generate H 3 ( H ) Γ (using Lemma 4.24). F or eac h of these, b y Alexander’s theorem, L i can b e made b y closing up a braid B i in the disc, in which case f i restricts to a homomorphism from the fundamen tal group of the punctured disc. If w e no w tak e a conﬁguration + surjective homomorphism f with suﬃcien tly man y lo cal mono drom y elements in each conjugacy class, we can ﬁnd by Lemma 4.26 a disc D i in the plane, a voiding the k mark ed strands, where f restricts to the homomorphism f i from the fundamen tal group of a punctured disc to H ⋊ Γ . W e can then extend B i to a braid B ′ i in volving all the points in the plane b y lea ving all the p oin ts outside D i is unmo v ed. By Lemma 4.25, the braid fundamental class of B ′ i is [ S 3 , L i , f i ] . Comp osing the B ′ i s, w e thus can generate an arbitrary element in H 3 ( H ) Γ . □ 60 4.5. Homology of co v ers of Hurwitz space. Fix a connected comp onen t X of Hur m H ⋊ Γ ,c H ⋊ Γ , C . W e will compute the homology of the connected comp onents of the cov er X 0 ( C ) of X ( C ) aris- ing from the homomorphism π 1 ( X ( C )) → π 1 ( X ) → H 3 ( H , Z /n ) Γ , where the ﬁrst map is the map from the top ological fundamen tal group to the algebraic fundamen tal group, and the sec- ond map is the e deﬁned ab ov e. W e also call this comp osition e : π 1 ( X ( C )) → H 3 ( H , Z /n ) Γ . T o do this, we will sho w that this cov er is itself cov ered by a connected comp onen t of another Hurwitz space. W e ﬁrst in tro duce the notation that allo ws us to deﬁne this Hurwitz space. F or a group G and a union c of conjugacy classes of G , let Hur m, ∞ G,c ( C ) b e the top ological Hurwitz space allo wing arbitrarily ramiﬁcation at ∞ : That is, Hur m, ∞ G,c ( C ) is the co v ering of the conﬁguration space of m p oin ts in the plane corresponding to the action of the braid group on the set of m -tuples of elemen ts of conjugacy classes in c that generate G . (W e could deﬁne this space as a sc heme following [LL25], but don’t need to as we only need the manifold.) Let A = H 3 ( H ⋊ Γ , Z /n ) . W e hav e a short exact sequence of H ⋊ Γ -mo dules 0 → A → Ind H ⋊ Γ 1 A → Ind H ⋊ Γ 1 A/ A → 0 , where A is the trivial mo dule embedding diagonally in Ind H ⋊ Γ 1 . Since H 3 ( H ⋊ Γ , Ind H ⋊ Γ 1 A ) = 0 , w e thus ha v e a surjection H 2 ( H ⋊ Γ , Ind H ⋊ Γ 1 A/ A ) → H 3 ( H ⋊ Γ , A ) and the universal co eﬃcient theorem induces a surjection H 3 ( H ⋊ Γ , A ) → Hom( H 3 ( H ⋊ Γ , Z ) , A ) . Fix α ∈ H 2 ( H ⋊ Γ , Ind H ⋊ Γ 1 A/ A ) whose image in Hom( H 3 ( H ⋊ Γ , Z ) , A ) under the com- p osition of these t wo surjections is the natural map H 3 ( H ⋊ Γ , Z ) → H 3 ( H ⋊ Γ , Z /n ) = A (induced b y Z → Z /n on co eﬃcien ts). Lemma 4.27. L et m b e a nonne gative inte ger. Fix a c onne cte d c omp onent X of Hur m H ⋊ Γ ,c H ⋊ Γ , C . L et G b e the extension of H ⋊ Γ by Ind H ⋊ Γ 1 A/ A deﬁne d by the class α as deﬁne d ab ove. F or e ach sub gr oup G ′ of this G , let c G ′ b e the set of nontrivial elements of G ′ whose or der divides the or der of Γ . Ther e is a sub gr oup G ′ of G , whose image inside H ⋊ Γ is al l of H ⋊ Γ , such that the natur al map Hur m, ∞ G ′ ,c ′ G ( C ) → Hur m, ∞ H ⋊ Γ ,c H ⋊ Γ ( C ) has image c ontaining the c omp onent X ( C ) . F urthermor e, we c an cho ose a c omp onent Z of Hur m, ∞ G ′ ,c ′ G ( C ) whose image in Hur m H ⋊ Γ ,c H ⋊ Γ ( C ) is X ( C ) , such that for e ach c omp onent Y ( C ) of X 0 ( C ) , ther e exists a c overing sp ac e map Z → Y ( C ) such that the c omp osition Z → Y ( C ) → X ( C ) is the given map Z ( C ) → X ( C ) . Pr o of. The top ological space Hur m H ⋊ Γ ,c H ⋊ Γ ( C ) is the co vering of the conﬁguration space of m p oin ts in the plane corresp onding to the action of the braid group on the set of m -tuples of elemen ts in conjugacy classes in c H ⋊ Γ that m ultiply to 1 and generate H ⋊ Γ ([L WZ24, Section 11.3]). Eac h connected comp onen t corresp onds to a braid group orbit on the set of m -tuples. Let g 1 , . . . , g m b e a tuple in the orbit corresp onding to X ( C ) . T o pro ve the ﬁrst claim, we must c heck that the orbit of g 1 , . . . , g m is the image of some orbit of c G ′ -tuples for some G ′ . Eac h g i is conjugate to a nontrivial elemen t of Γ and so has order dividing | Γ | . Lift each elemen t g i from H ⋊ Γ to an element g ′ i of G . Since the kernel of G → H ⋊ Γ has order prime 61 to | Γ | , we can c ho ose the lift to ha v e the same order. Let G ′ b e the subgroup generated b y g ′ 1 , . . . , g ′ m . Then g ′ 1 , . . . , g ′ m lie in c G ′ and therefore the orbit of the tuple g ′ 1 , . . . , g ′ m deﬁnes a comp onen t of Hur m, ∞ G ′ ,c ′ G ( C ) whose image is X ( C ) . T ak e Z to b e this comp onent. W e now pro ve the second part of the claim. The top ological space X 0 ( C ) is the H 3 ( H , Z /n ) Γ - torsor o ver X 0 ( C ) corresp onding to the map e : π 1 ( X ( C )) → H 3 ( H , Z /n ) Γ . Hence all the connected comp onents of X 0 ( C ) are isomorphic to eac h other and the natural map Z ( C ) → X ( C ) lifts to one suc h comp onen t if the comp osition π 1 ( Z ) → π 1 ( X ( C )) → H 3 ( H , Z /n ) Γ v anishes. Th us it suﬃces to c heck this v anishing. Let σ b e an element of π 1 ( Z ) , whic h w e ma y view as an element of the braid group stabilizing a tuple g ′ 1 , . . . , g ′ m of elements of conjugacy classes in c G ′ . W e m ust chec k that e ( σ ) = 0 . By Lemma 4.20, it suﬃces to chec k that the image of the braid fundamental class of σ under H 3 ( H , Z ) → H 3 ( H , Z /n ) v anishes. By Lemma 4.25 with p = m , this braid fundamental class is given b y [ S 3 , L, f ] where L is the closure of the braid σ and f is a homomorphism π 1 ( S 3 \ L ) → H ⋊ Γ whose restriction to an m -punctured disc in S 3 is the homomorphism corresp onding to a p oin t of X ( C ) . The fundamental group of the complemen t of the braid closure in S 3 is the free group on generators x 1 , . . . , x m mo dulo the relation x i = σ ( x i ) for all i , and f is the homomorphism sending x i to g i . Th us the fact that σ ﬁxes the tuple g ′ 1 , . . . , g ′ m implies that f lifts to a homomorphism f ′ : π 1 ( S 3 \ L ) → G ′ sending each x i to g ′ i . Let H ′ b e the k ernel of the natural map G ′ → H ⋊ Γ → Γ . Let ˜ S 3 b e the Γ -co v ering of S 3 corresp onding to f . Then f ′ corresp onds to a H ′ -co vering of ˜ S 3 lifting the H -co v ering corresponding to f . Since the image of a lo op around each comp onen t of L under f ′ is an elemen t of g ′ i and thus has order dividing Γ and hence has order prime to H ′ , the H ′ -co vering m ust b e unbranc hed. Hence the class [ S 3 , L, f ] , which b y deﬁnition is the image of the fundamental class of ˜ S 3 in H 3 ( H , Z ) , lifts to H 3 ( H ′ , Z ) . Since this H ′ -co vering admits an action of Γ , this lift is Γ -in v arian t. Thus the braid fundamen tal class lies in the image of H 3 ( H ′ , Z ) Γ → H 3 ( H , Z ) Γ . The class α v anishes when pulled back to from H ⋊ Γ to G , and hence to H ′ , since the extension α splits o ver G . Since pullbac k is compatible with the connecting homomorphism, the pullback from H 3 ( H ⋊ Γ , A ) to H 3 ( H ′ , A ) of the class corresp onding to α v anishes. Since pullbac k is compatible with the universal co eﬃcient theorem, the comp osition of the natural homomorphism H 3 ( H ⋊ Γ , Z ) → H 3 ( H ⋊ Γ , Z /n ) = A with H 3 ( H ′ , Z ) → H 3 ( H ⋊ Γ , Z ) is zero. Thus the map along the b ottom ro w of the b elo w commutativ e diagram v anishes. H 3 ( H ′ , Z ) Γ H 3 ( H , Z ) Γ H 3 ( H , Z /n ) Γ H 3 ( H ′ , Z ) H 3 ( H ⋊ Γ , Z ) H 3 ( H ⋊ Γ , Z /n ) Since the rightmost arrow H 3 ( H , Z /n ) Γ → H 3 ( H ⋊ Γ , Z /n ) is an isomorphism, the comp o- sition of the top row H 3 ( H ′ , Z ) Γ → H 3 ( H , Z ) Γ → H 3 ( H , Z /n ) Γ v anishes. W e saw that the image of the braid fundamen tal class in H 3 ( H , Z /n ) Γ lies in the image of this comp osition, and hence v anishes, as desired. □ 62 Prop osition 4.28. Ther e exist p ositive inte gers I , J, K dep ending only on H and Γ such that for e ach p ositive inte ger m , e ach c onne cte d c omp onent X ( C ) of Hur m H ⋊ Γ ,c H ⋊ Γ ( C ) , e ach c omp onent Y ( C ) of X 0 ( C ) , and e ach nonne gative inte ger i we have: (1) dim H i ( Y ( C ) , Q ) ≤ K i +1 . (2) If, for e ach c onjugacy class in c , X ( C ) p ar ameterizes c overs with at le ast I i + J br anch p oints of that c onjugacy class, then the natur al map H i ( Y ( C ) , Q ) → H i ( X ( C ) , Q ) is an isomorphism. Pr o of. Cho ose a subgroup G ′ of G and a component Z of Hur m, ∞ G ′ ,c ′ G ( C ) satisfying the conclu- sion of Lemma 4.27. Let us chec k that there are ﬁnitely many p ossibilities for G ′ for a given H , Γ , ev en inde- p enden tly of n : A = H 3 ( H ⋊ Γ , Z /n ) has size b ounded by | H 3 ( H ⋊ Γ , Z ) || H 2 ( H ⋊ Γ , Z ) | and th us has ﬁnitely many p ossible v alues, then there are ﬁnitely many extensions of H ⋊ Γ b y A , and ﬁnitely many subgroups G ′ of eac h extension. Let g 1 , . . . , g v b e representativ es of the conjugacy classes of c H ⋊ Γ and let m 1 , . . . , m v b e the num b er of branc h p oin ts of each conjugacy class in co vers parameterized by X ( C ) . Then w e hav e a sequence of maps Z → Y ( C ) → X ( C ) → Conf m 1 ,...,m v ( C ) where Conf m 1 ,...,m v ( C ) parameterizes conﬁgurations of p oin ts in the plane with m 1 p oin ts lab eled by g 1 , m 2 p oin ts lab eled by g 2 , and so on, and the map X ( C ) → Conf m 1 ,...,m v ( C ) lab els eac h branc h p oin t by a represen tative of its conjugacy class. W e hav e the induced maps on homology H i ( Z, Q ) → H i ( Y ( C ) , Q ) → H i ( X ( C ) , Q ) → H i (Conf m 1 ,...,m v ( C ) , Q ) . All our maps b etw een spaces are co vering maps since the individual spaces are all connected co vering spaces of Conf m . Th us the induced maps on homology with rational co eﬃcients are injectiv e. P art (1) then follo ws from [LL25, Lemma 8.4.2] by dim H i ( Y ( C ) , Q ) ≤ dim H i ( Z, Q ) ≤ K i +1 where we choose K large enough to b e v alid for any of the ﬁnitely many p ossible v alues of G ′ . F or part (2), w e ﬁrst chec k that the natural map from the set c ′ G /G ′ of conjugacy classes in c G ′ to the set c H ⋊ Γ /H ⋊ Γ of conjugacy classes in c H ⋊ Γ is a bijection. This follows b ecause G ′ is the extension of Γ b y a group H ′ of order prime to Γ , hence is H ′ ⋊ Γ b y Sch ur-Zassenhaus, and thus every conjugacy class of G ′ of order dividing | Γ | is in fact the conjugacy class of an elemen t of Γ . Hence the natural map c ′ G /G ′ → c Γ / Γ is a bijection, and the same is true for c H ⋊ Γ /H ⋊ Γ → c Γ / Γ , so the natural map c G ′ /G ′ → c H ⋊ Γ /H ⋊ Γ is a bijection as well. It follows from [LL25, Theorem 1.4.2] that as long as all the m j are at least I i + J for some I , J dep ending only on G ′ that the natural map H i ( Z, Q ) → H i (Conf m 1 ,...,m v ( C ) , Q ) is an isomorphism. Thus, b ecause all the maps are injective, all the maps are isomorphisms, as desired. W e can c ho ose a single I , J that works for all the ﬁnitely many p ossible v alues of G ′ . □ 63 4.6. F rom characteristic 0 to characteristic p . W e will use Prop osition 4.21 and Prop o- sition 4.28 to prov e the lemmas whic h will b e the ﬁnal ingredients in the pro of of Theo- rem 4.10. T o do this, we relate the comp onen ts of Hurwitz space in characteristic 0 and c haracteristic p . While this was done in prior work, we do it in a sligh tly diﬀerent w a y that directly keeps trac k of the fu ndamen tal group of the comp onen ts. Our key lemma, Lemma 4.30, prov es several useful facts under the assumption that a map b etw een the spaces induces a surjection on their fundamental groups. Because of this, we will need Lemma 4.33, whic h sho ws that natural maps b etw een conﬁgurations spaces induce a surjection on fundamental groups, which we pro ve as a consequence of t w o general results on fundamen tal groups. It w ould lik ely b e p ossible to giv e a shorter pro of a voiding fundamen tal groups, but we do not do this as w e think the argument via fundamental groups is natural, and the results w e prov e ab out fundamen tal groups could b e useful later. Using Prop osition 4.21 and Lemma 4.20, w e obtain the follo wing. Lemma 4.29. L et X b e a c onne cte d c omp onent of (Hur m H ⋊ Γ ,c H ⋊ Γ ) C . The image of e : π 1 ( X ) → H 3 ( H , Z /n ) Γ is e qual to the image of H 3 ( H , Z ) Γ under the natur al map H 3 ( H , Z ) Γ → H 3 ( H , Z /n ) Γ , as long as the c omp onent p ar ametrizes c overs with suﬃciently many (given H , Γ ) br anch p oints of e ach c onjugacy class in c H ⋊ Γ . Pr o of. The fundamental group π 1 ( X ) is the completion of the top ological fundamental group of X ( C ) . Comp osition gives a homomorphism e : π 1 ( X ( C )) → H 3 ( H , Z /n ) and it suﬃces to sho w that this comp osition has image the image of H 3 ( H , Z ) Γ . Using Lemma 4.20, since the braid fundamen tal class map has image in H 3 ( H , Z ) Γ , w e can immediately conclude that the image of the geometric fundamental group is contained in the image of H 3 ( H , Z ) Γ . By Prop osition 4.21, the image of the braid fundamen tal class map is equal to the image of H 3 ( H , Z ) Γ as long as the comp onen t parameterizes cov ers with suﬃcien tly man y branch p oints in each conjugacy class. It then follo ws that the image of the geometric fundamental group is equal to the image of H 3 ( H , Z ) Γ : Indeed, for α in the image of H 3 ( H , Z ) Γ , α is the braid fundamental class of some σ , hence by Lemma 4.20 an in teger multiple of e ( σ ) , hence equal to e ( σ d ) for some d . □ Lemma 4.30. L et A → B b e a map of schemes such that π 1 ( A ) → π 1 ( B ) is surje ctive, and let C b e a ﬁnite étale c over of B . L et g b e the map fr om the set of c onne cte d c omp onents of A × B C to the set of c onne cte d c omp onents of C that sends a c omp onent X A of A × B C to the unique c onne cte d c omp onent of C c ontaining the image of X A in C . Then (1) g is a bije ction. (2) F or X A a c onne cte d c omp onent of A × B C and X B = g ( X A ) the c orr esp onding c omp o- nent of C , the map π 1 ( X A ) → π 1 ( X B ) induc e d by the map X A → X B is surje ctive. in p articular, any homomorphism fr om π 1 ( X B ) to a ﬁxe d gr oup G has the same image as its c omp osition with π 1 ( X A ) → π 1 ( X B ) . (3) F or X A a c onne cte d c omp onent of A × B C and X B = g ( X A ) the c orr esp onding c om- p onent of C , we have X A = A × B X B . Pr o of. The only fact ab out the étale fundamental group w e need for this is that the map of fundamen tal groups π 1 ( A ) → π 1 ( B ) is surjective if and only if the pullback of each connected ﬁnite étale cov er of B to A is connected [Gro03, V, Prop osition 6.9]. 64 Indeed, C is a ﬁnite union of connected ﬁnite étale co v ers C 1 , . . . , C k of B , those b eing the connected comp onen ts of C , and A × B C is the union of the pullbac ks A × B C i of these comp onen ts from B to A . Since the pullbac k A × B C i are connected, these are the connected comp onen ts of A × B C , and this gives a bijection b et ween the connected comp onen ts of A × B C and the connected comp onents of C , which b y deﬁnition has the prop erty (3). The image of a comp onen t of A × B C i in C is certainly contained in the comp onen t C i , so this bijection is the same as g . F or C i a comp onent of C , a connected ﬁnite étale cov er of C i is itself a connected ﬁnite étale cov er of B , and its pullback from C i to A × B C i agrees with its pullback from B to A and hence is connected, showing that π 1 ( A × B C i ) → π 1 ( C i ) is surjective. Finally , precomp osition of π 1 ( C i ) → G with the surjectiv e morphism π 1 ( A × B C i ) → π 1 ( C i ) certainly preserv es the image in G . □ Lemma 4.31. L et R b e a strict Henselian lo c al ring of generic char acteristic 0 . L et s b e a ge ometric sp e cial p oint of R and η a ge ometric generic p oint. L et X b e a scheme over R which is the c omplement in a smo oth pr op er scheme X over R of a normal cr ossings divisors D . Then the natur al maps π 1 ( X η ) → π 1 ( X ) and π 1 ( X s ) → π 1 ( X ) ar e surje ctive. Pr o of. Indeed, note ﬁrst that since R has generic characteristic 0 , the generic p oin t of each comp onen t of D has characteristic 0 , so the lo cal rings at these p oints hav e residue charac- teristic 0 , and thus all ﬁnite étale extensions o ver the ﬁeld of fractions of these lo cal rings are tamely ramiﬁed. Hence b y [Gro03, XII I, Deﬁnition 2.1.1], all ﬁnite étale cov ers of X are tamely ramiﬁed and th us by [Gro03, XI II, Deﬁnition 2.1.3] the natural map from the usual fundamen tal group of X to the tame fundamental group π t 1 ( X ) is an isomorphism. It is sho wn in [Gro03, XI I I, p. 48, l. 3-4] that π t 1 ( X s ) → π t 1 ( X ) is an isomorphism which combined with the surjectivity of π 1 ( X s ) → π t 1 ( X s ) [Gro03, XI I I, 2.1.3] gives that π 1 ( X s ) → π t 1 ( X ) = π 1 ( X ) is surjectiv e. F or the morphism π 1 ( X η ) → π 1 ( X ) , it suﬃces to sho w that for eac h ﬁnite group G , the natural map from isomorphism classes of G -torsors o ver X to isomorphism classes of G -torsors o ver X η is injectiv e. Since all G -torsors o v er X are tame, it suﬃces to sho w the injectivit y of the analogous map on tame torsors. This is the conten t of [Gro03, XII I, Corollary 2.8] for F the constant sheaf G , taking s 1 = η and s 2 the geometric sp ecial p oint asso ciated to the algebraic closure of the residue ﬁeld, once w e make several observ ations: the condition that X → Sp ec R is lo cally acyclic is satisﬁed b y [Gro03, XI I I, Remark 1.17]. The fact that ( R 1 t f ∗ F ) η agrees with the set of tame G -torsors o ver X η follo ws from the stated compatibilit y of that pushforward with arbitrary c hange of base applied to the change of base η → Sp ec R . Since R is strict Henselian so Sp ec R is the strict lo calization of Sp ec R at s 2 , b y [Gro03, XI I I, (2.1.2.1)] ( R 1 t f ∗ F ) s 2 agrees with the set of tame G -torsors ov er Sp ec R . □ Lemma 4.32. L et R b e a ring and z a p oint of Sp ec R . L et X b e a scheme over R , let G b e a ﬁnite gr oup acting fr e ely on X , and let Y = X/G b e the quotient. If the natur al map π 1 ( X z ) → π 1 ( X ) is surje ctive, then the natur al map π 1 ( Y z ) → π 1 ( Y ) is surje ctive. Pr o of. Note that X → Y is a ﬁnite étale cov er with Galois group G . Th us for C a connected ﬁnite étale co ver of Y , C × Y X is a ﬁnite étale cov er of Y with an action of G , where G acts transitiv ely on the connected comp onents. Since the inv erse image of each connected comp onen t of C × Y X in the pullback C × Y X × X X z = C × Y X z = C × Y Y z × Y z X z 65 is connected by [Gro03, V, Prop osition 6.9], G also acts transitiv ely on the connected com- p onen ts of C × Y Y z × Y z X z , and thus C × Y Y z is connected, showing that π 1 ( Y z ) → π 1 ( Y ) is surjectiv e by [Gro03, V, Prop osition 6.9], as desired. □ Lemma 4.33. L et R b e the ring of Witt ve ctors of ¯ F q , and ﬁx an emb e dding R → C . The maps π 1 (Conf m ¯ F q ) → π 1 (Conf m R ) and π 1 (Conf m C ) → π 1 (Conf m R ) ar e e ach surje ctive. Pr o of. R is a strict Henselian lo cal ring. The ﬁxed embedding R → C deﬁnes a geometric generic p oin t of R , and R → ¯ F q giv es a sp ecial p oin t. Since PConf m R is the complemen t of a normal crossings divisor in the smo oth pro jectiv e scheme M 0 ,m ( P 1 , 1) R , we ma y apply Lemma 4.31 to get that π 1 (PConf m ¯ F q ) → π 1 (PConf m R ) and π 1 (PConf m C ) → π 1 (PConf m R ) are eac h surjective. Because Conf m is the quotien t of PConf m b y a free action of S m , w e ma y apply Lemma 4.32 to b oth p oin ts to obtain the desired conclusion. □ Lemma 4.34. L et U b e a ﬁnite étale c over of Conf m R . Then dim H 2 m − i c ( U ¯ F q , Q ℓ ) = dim H i ( U C ( C ) , Q ) . Pr o of. This follo ws from [L WZ24, pro of of Lemma 10.3 on p. 55]: This pro of is stated for a sp eciﬁc ﬁnite étale cov ering of Conf m , but works in general. □ Lemma 4.35. L et q ≡ 1 (mo d n ) b e a prime p ower r elatively prime to | H || Γ | . L et X ¯ F q b e a c onne cte d c omp onent of (Hur m H ⋊ Γ ,c H ⋊ Γ ) ¯ F q . The image of e : π 1 ( X ¯ F q ) → H 3 ( H , Z /n ) Γ is e qual to the image of H 3 ( H , Z ) Γ under the natur al map H 3 ( H , Z ) Γ → H 3 ( H , Z /n ) Γ , as long as X ¯ F q p ar ametrizes c overs with suﬃciently many (given H , Γ ) br anch p oints of e ach c onjugacy class in c H ⋊ Γ . Pr o of. In view of Lemma 4.29 it suﬃces to show for eac h connected comp onen t X ¯ F q of (Hur m H ⋊ Γ ,c H ⋊ Γ ) ¯ F q that there exists a connected comp onen t X C of (Hur m H ⋊ Γ ,c H ⋊ Γ ) C , with the same n umber of branc h p oin ts of eac h conjugacy class, suc h that the image of π 1 ( X C ) in H 3 ( H , Z ) Γ is equal to the image of π 1 ( X ¯ F q ) , as we may then apply Lemma 4.29 to that comp onen t. The space (Hur m H ⋊ Γ ,c H ⋊ Γ ) ¯ F q is the ﬁb er pro duct of (Hur m H ⋊ Γ ,c H ⋊ Γ ) R o ver Conf m R with Conf m ¯ F q . Similarly , (Hur m H ⋊ Γ ,c H ⋊ Γ ) C is the ﬁb er pro duct of (Hur m H ⋊ Γ ,c H ⋊ Γ ) R o ver Conf m R with Conf m C By Lemma 4.33 the maps π 1 (Conf m ¯ F q ) → π 1 (Conf m R ) and π 1 (Conf m C ) → π 1 (Conf m R ) are eac h surjectiv e, as then applying Lemma 4.30 t wice giv es us t wo bijections, ﬁrst b et ween comp o- nen ts of (Hur m H ⋊ Γ ,c H ⋊ Γ ) ¯ F q and comp onen ts of (Hur m H ⋊ Γ ,c H ⋊ Γ ) R , and then b et ween comp onents of (Hur m H ⋊ Γ ,c H ⋊ Γ ) R and comp onents of (Hur m H ⋊ Γ ,c H ⋊ Γ ) C , which eac h preserv e the image of the homomorphism e to H 3 ( H , Z ) Γ . Since the n um b er of branch p oin ts of each conjugacy class is constan t on comp onen ts of (Hur m H ⋊ Γ ,c H ⋊ Γ ) R , b oth bijections preserve this quantit y . □ Lemma 4.36. Ther e exist p ositive inte gers I , J, K dep ending only on H and Γ such that for e ach m , e ach ge ometric al ly c onne cte d c omp onent X q of Hur m H ⋊ Γ ,c H ⋊ Γ , F q , e ach c onne cte d c omp onent Y ¯ F q of ( X s H , F q q ) ¯ F q , and e ach nonne gative inte ger i we have: (1) dim H 2 m − i c ( Y ¯ F q , Q ℓ ) ≤ K i +1 . (2) If, for e ach c onjugacy class in c , X q p ar ameterizes c overs with at le ast I i + J br anch p oints of that c onjugacy class, then t he natur al map H 2 m − i c (( X q ) ¯ F q , Q ℓ ) → H 2 m − i c ( Y ¯ F q , Q ℓ ) is an isomorphism. 66 Pr o of. The pro of is similar to the pro of of Lemma 4.35. Giv en a comp onen t X q , the homomorphisms deﬁning the cov erings X s H , F q q , for v arying s H , all agree on restriction to the geometric fundamen tal group of X q , and th us the co verings agree after base change from F q to ¯ F q . In other words, we hav e ( X s H , F q q ) ¯ F q ∼ = ( X 0 , F q q ) ¯ F q = ( X 0 q ) ¯ F q . So it suﬃces to prov e the same result for comp onents Y of ( X 0 q ) ¯ F q , which are all connected comp onen ts of Hur m, 0 H ⋊ Γ ,c H ⋊ Γ , ¯ F q . Lemmas 4.30 and 4.33 give a bijection b et ween connected comp onen ts Y ¯ F q of Hur m, 0 H ⋊ Γ ,c H ⋊ Γ , ¯ F q and connected comp onen ts Y C of Hur m, 0 H ⋊ Γ ,c H ⋊ Γ , C . By Lemma 4.30(3), comp onents Y ¯ F q and Y C related by this bijection are resp ectively the base changes to ¯ F q and C of a single comp onent Y R of Hur m, 0 H ⋊ Γ ,c H ⋊ Γ ,R , whic h is a ﬁnite étale co ver of Conf m R . By Lemma 4.34 it follows that dim H 2 m − i c ( Y ¯ F q , Q ℓ ) = dim H i ( Y C ( C ) , Q ) . W e apply Prop osition 4.28 to control the homology groups of Y C ( C ) . P art (1), b ounding the dimension, immediately translates to a b ound on dim H 2 m − i c ( Y ¯ F q , Q ℓ ) . F or part (2), w e ha ve to chec k that the n umber of branc h p oin ts of eac h conjugacy class is the same, but this follows b ecause the num b er of branch p oin ts of each conjugacy class is constant on comp onen ts of (Hur m H ⋊ Γ ,c H ⋊ Γ ) R , and then we only get from Prop osition 4.28 and Lemma 4.34 that dim H 2 m − i c ( Y ¯ F q , Q ℓ ) = dim H 2 m − i c (( X q ) ¯ F q , Q ℓ ) . Ho wev er, b ecause π : Y → X q is a ﬁnite étale co v er b etw een geometrically connected v ari- eties, the induced map on compactly supp orted cohomology π ◦ : H 2 m − i c ( X q , Q ℓ ) → H 2 m − i c ( Y ¯ F q , , Q ℓ ) is an injection. (Since π is ﬁnite, w e ha ve π ! = π ∗ and R j π ! Q ℓ = 0 for j > 0 so the Lera y sp ectral sequence with compact supp orts gives H 2 m − i c ( Y ¯ F q , Q ℓ ) = H 2 m − i c (( X q ) ¯ F q , π ∗ Q ℓ ) . In this p ersp ectiv e, the natural map π ◦ arises from the adjunction map Q ℓ → π ∗ Q ℓ whic h has a left inv erse given by the trace map π ∗ Q ℓ → Q ℓ (after division by the degree of π ) by [AGV72, XVI I I, Theorem 2.9 (V ar 4)(I)], inducing a left inv erse to π ◦ , showing that π ◦ is injective.) Th us b ecause the source and target of π ◦ ha ve the same dimension, π ◦ is an isomorphism, completing the pro of of part (2). □ W e can no w prov e Theorem 4.10. Pr o of of The or em 4.10. Fix a comp onen t X q of Hur H ⋊ Γ ,c H ⋊ Γ , F q . If X q has no F q -p oin ts then the same is true for X s H , F q q , and the theorem is automatically true. Thus w e assume that X q has a F q -p oin t. Since X q is smo oth, it further follows that X q is geometrically irreducible. If a ( X q ) is low er than any ﬁxed function of Γ , H , w e can mak e the statement true by taking the implicit constan t suﬃciently large, and using the trivial b ound | X s H , F q q ( F q ) | = O ( | X q ( F q ) | ) , so we may assume a ( X q ) is larger than any desired function of Γ , H . By deﬁnition X s H , F q q is the ﬁnite étale Galois cov er of X q asso ciated to the homomorphism that sends an elemen t σ ∈ π 1 ( X q ) to e ( σ ) − s H deg σ . Restricted to the geometric fundamental group, this homomorphism is e and hence b y Lemma 4.35, since w e can assume a ( X q ) is suﬃcien t large in terms of H and Γ , has image the image of H 3 ( H , Z ) Γ inside H 3 ( H , Z /n ) Γ . The comp onen ts of ( X s H , F q q ) F q are naturally a principal homogenous space for the quotient of H 3 ( H , Z /n ) Γ b y this image, which b y the long exact sequence is H 2 ( H , Z ) Γ [ n ] . The long 67 exact sequence asso ciated to 0 → Z → Z → Z /n expresses H 2 ( H , Z /n ) as the extension of H 1 ( H , Z )[ n ] by H 2 ( H , Z ) /nH 2 ( H , Z ) , and ( | H | , | Γ | ) = 1 implies that the same is true if we tak e Γ -inv arian ts of everything. F urther, H Γ = 1 implies that H 1 ( H , Z ) Γ = 0 . Th us   H 2 ( H , Z ) Γ [ n ]   =   H 2 ( H , Z ) Γ /nH 2 ( H , Z ) Γ   =   H 2 ( H , Z /n ) Γ   =   H 2 ( H ⋊ Γ , Z /n )   , using Lemmas 5.4 and 5.5 in the last step. Thus the num b er of comp onen ts o ver F q of X s H , F q q is | H 2 ( H ⋊ Γ , Z /n ) | . Next w e chec k that the action of F rob q on the set of components is trivial. The action of F rob q on the set of comp onen ts is the same as the action of the F rob enius element σ at an arbitrary F q -p oin t of X q . It follo ws from Lemmas 3.11, 3.12, and 4.6 that the pro jection of e ( σ ) to H 2 ( H , Z ) Γ [ n ] is the ω -in v ariant of the comp onen t, which w e ha ve assumed is ω H . The degree of a F rob e- nius elemen t is 1 so the pro jection of s H deg σ to H 2 ( H , Z ) Γ [ n ] is the pro jection of s H to H 2 ( H , Z ) Γ [ n ] , whic h by deﬁnition is ω H . These t wo terms cancel and thus the action of F rob q on the set of comp onents is trivial, so eac h of these comp onen ts is deﬁned o ver F q . Fix a component Y of X s H , F q q , necessarily geometrically connected. The Lefschetz ﬁxed p oin t formula gives (4.37) | X q ( F q ) | = 2 m X i =0 ( − 1) i tr(F rob q , H 2 m − i c (( X q ) ¯ F q , Q ℓ )) and (4.38) | Y ( F q ) | = 2 m X i =0 ( − 1) i tr(F rob q , H 2 m − i c ( Y ¯ F q , Q ℓ )) By Lemma 4.36(2), for i ≤ a ( X q ) − J I the F rob enius-equiv ariant natural map H 2 m − i c (( X q ) ¯ F q , Q ℓ ) → H 2 m − i c ( Y ¯ F q , Q ℓ ) is an isomorphism and thus the terms with i ≤ a ( X q ) − J I in the ab o ve tw o for- m ulas agree. Hence || Y ( F q ) | − | X q ( F q ) || ≤ X i> a ( X q ) − J I ( | tr(F rob q , H 2 m − i c (( X q ) ¯ F q , Q ℓ )) | + | tr(F rob q , H 2 m − i c ( Z ¯ F q , Q ℓ )) | ) ≤ X i> a ( X q ) − J I q m − i 2 ( | dim H 2 m − i c (( X q ) ¯ F q , Q ℓ ) | + | dim( H 2 m − i c ( Z ¯ F q , Q ℓ ) | ) ≤ X i> a ( X q ) − J I q m − i 2 2 K i +1 ≤ 2 K a ( X q ) − J I q m − a ( X q ) − J 2 I ∞ X i =0 K i +1 q − i 2 = 2 K a ( X q ) − J I q m − a ( X q ) − J 2 I K 1 − K √ q = O ( e − δ a ( X q ) ) q m . b y Deligne’s Riemann h yp othesis and Lemma 4.36(1) (whose n = 1 case handles X q ), taking δ ≤ 1 I  log q 2 − log K  , whic h w e can alw ays do as long as q is suﬃciently large in terms of H , Γ . 68 Since this is true for ev ery component and there are | H 2 ( H ⋊ Γ , Z /n ) | components, w e get | X s H , F q q ( F q ) |−   H 2 ( H ⋊ Γ , Z /n )   | X q ( F q ) | = O (   H 2 ( H ⋊ Γ , Z /n )   e − δ a ( X q ) q m ) = O ( e − δ a ( X q ) | X q ( F q ) | ) , since | X q ( F q ) | ≥ q m / 2 for q suﬃcien tly large in terms of Γ , H . b y [LL25, Lemma 8.4.4] and w e can absorb the factors of 2 and | H 2 ( H ⋊ Γ , Z /n ) | in to the big O . This is the desired statemen t. □ 4.7. Pro ofs of lemmas ab out integration along the ﬁb ers. W e ﬁrst pro ve the com- patibilit y of integration along ﬁb ers with pullbback, and then the pro jection formula. Pr o of of L emma 4.4. In this setting [AGV72, XVI I I, Theorem 2.9 (V ar 2)] giv es the com- m utative diagram, where F ′ : Y 1 = Y 2 × X 2 X 1 → X 1 is the second pro jection and g ′ : Y 1 = Y 2 × X 2 X 1 → Y 2 is the ﬁrst pro jection, R 2 F ′ ∗ µ n R 2 F ′ ∗ g ′ ∗ µ n g ∗ R 2 F ∗ µ n Z /n g ∗ Z /n T r F ′ g ∗ T r F where the top-righ t arrow is a base c hange map, the b ottom arro w is the isomorphism g ∗ Z /n ∼ = Z /n , and the top-right arrow arises from the isomorphism g ′ ∗ µ n ∼ = µ n . Applying the functor H i ( X 1 , · ) to this diagram, w e obtain the comm utative diagram H i ( X 1 , R 2 F ′ ∗ µ n ) H i ( X 1 , R 2 F ′ ∗ g ′ ∗ µ n ) H i ( X 1 , g ∗ R 2 F ∗ µ n ) H i ( X 2 , Z /n ) H i ( X 1 , g ∗ Z /n ) H i (T r F ′ ) H i ( g ∗ T r F ) W e can expand this diagram as follows, where “pullbac k” alw a ys denotes the natural map from the cohomology of a sheaf to the cohomology of the pullback of the sheaf. H i +2 ( Y 1 , µ n ) H i +2 ( Y 1 , g ′ ∗ µ n ) H i +2 ( Y 2 , µ n ) H i +2 ( X 1 , R F ′ ∗ µ n ) H i +2 ( X 1 , R F ′ ∗ g ′ ∗ µ n ) H i +2 ( X 1 , g ∗ RF ∗ µ n ) H i +2 ( X 2 , R F ∗ µ n ) H i ( X 1 , R 2 F ′ ∗ µ n ) H i ( X 1 , R 2 F ′ ∗ g ′ ∗ µ n ) H i ( X 1 , g ∗ R 2 F ∗ µ n ) H i ( X 2 , R 2 F ∗ µ n ) H i ( X 1 , Z /n ) H i ( X 1 , g ∗ Z /n ) H i ( X 2 , Z /n ) Leray Leray Leray pullback pullback H i (T r F ′ ) H i ( g ∗ T r F ) H i (T r F ) pullback pullback It suﬃces to c hec k that this diagram is commutativ e, since the left and right sides are by deﬁ- nition the in tegration along the ﬁb ers map and the top and b ottom sides are the maps giving functorialit y of cohomology in the space. Since we hav e c heck ed the b ottom-left rectangle is comm utative, it remains to c heck the other squares and rectangles are commutativ e. 69 In the b ottom-righ t square, the horizontal arrows are the map on derived functors induced b y a natural transformation on the original fun ctors, that b eing the natural map from the global sections of a sheaf to the global sections of the pullbac k. The vertical arrows are the maps on derived functors in tro duced b y a map of ob jects. Hence, this diagram expresses that a natural transformation on functors induces a natural transformation on the deriv ed functors, whic h is [Ho v99, Lemma 1.3.9]. In the top-left square and center-left square, the horizontal arrows are maps induced by an isomorphism of sheav es, with the vertical arrows induced b y applying the same construction to eac h of the tw o isomorphic sheav es. The cohomology , the deriv ed pushforw ard, and the maps relating them may all b e computed in terms of an injectiv e resolution of the sheaf. Since the t w o shea v es are isomorphic, w e ma y simply c ho ose the same complex as the injectiv e resolution of b oth, making the vertical maps identical and the horizontal maps iden tities so that the diagram commutes. In the top-right rectangle, all the arrows are natural transformations of derived functors, or comp ositions of derived functors, induced from natural transformations of the original functors, or comp ositions of functors. T o chec k that the deriv ed diagram comm utes, it suﬃces to c hec k that the original diagram comm utes (since a commutativ e diagram of functors b et ween ab elian categories induces a comm utative diagram of functors on the category of c hain complexes and then we can apply [Hov99, Theorem 1.4.3 and Deﬁnition 1.4.2.6]). The original diagram states that if we take a global section of a sheaf F on Y 2 , pull bac k to a global section of g ′ ∗ F , then view as a global section of F ′ ∗ g ′ ∗ F , we obtain the same global section as if w e view the original section as a section of F ∗ F , pull back to a section of g ∗ F ∗ F , and apply a base change map g ∗ F ∗ F → F ′ ∗ g ′ ∗ F . This can be c heck ed directly using the deﬁnition of the base change map. In the cen ter square, all the arro ws are maps on cohomology of X 1 with co eﬃcien ts in a complex of sheav es induced by maps b et w een complexes of sheav es, so it suﬃces to sho w the follo wing diagram of maps b et ween complexes of shea v es commutes. RF ′ ∗ g ′ ∗ µ n g ∗ RF ∗ µ n R 2 F ′ ∗ g ′ ∗ µ n [ − 2] X 1 , g ∗ R 2 F ∗ µ n [ − 2] Since the complexes ha ve cohomology in degree ≤ 2 , w e can equiv alently express the v ertical maps as the canonical map from a complex to its canonical truncation in degree ≥ 2 , so it suﬃces to chec k comm utativit y of the diagram RF ′ ∗ g ′ ∗ µ n g ∗ RF ∗ µ n τ ≥ 2 RF ′ ∗ g ′ ∗ µ n τ ≥ 2 g ∗ R 2 F ∗ µ n Since the b ottom horizon tal arro w is the truncation of the top horizon tal arro w, this can b e c heck ed directly from the deﬁnition of the canonical truncation of a complex. In the cen ter-right square, the v ertical arrows b oth arise from the map of complexes g ∗ RF ∗ µ n → g ∗ R 2 F ∗ µ n [ − 2] while the horizontal arrows arise from the map from the cohomol- ogy of a complex on X 2 to the cohomology of its pullbac k to X 1 . Thus the square expresses 70 that the map from a cohomology of a complex on X 2 to the cohomology of its pullback to X 1 is a natural transformation. This again follo ws from [Hov99, Lemma 1.3.9]. □ Pr o of of L emma 4.5. The assumption that j ≤ 1 is only used in one step of the pro of, and can probably b e remo ved. F or this reason we hav e written the rest of the pro of for arbitrary j , though w e only need the j = 1 case. The relev an t commutativ e diagram, where w e ha ve broken the cup pro duct map in to t w o steps for clarity , is as follows: H i +2 ( Y , µ n ) ⊗ H j ( X , Z /n ) H i + j +2 ( Y , µ n ⊗ F ∗ Z /n ) H i + j +2 ( Y , µ n ) H i + j +2 ( X , RF ∗ ( µ n ⊗ F ∗ Z /n )) H i + j +2 ( X , RF ∗ µ n ) H i +2 ( X , RF ∗ µ n ) ⊗ H j ( X , Z /n ) H i + j +2 ( X , RF ∗ µ n ⊗ Z /n ) H i ( X , R 2 F ∗ µ n ) ⊗ H j ( X , Z /n ) H i + j ( X , R 2 F ∗ µ n ⊗ Z /n ) H i + j ( X , R 2 F ∗ µ n ) H i ( X , Z /n ) ⊗ H j ( X , Z /n ) H i + j ( X , Z /n ⊗ Z /n ) H i + j ( X , Z /n ) ∪ Leray Leray Leray ∪ H i (T r F ) ∪ H i + j (T r F ⊗ id ) H i + j (T r F ) ∪ All tensor pro ducts are o v er Z /n . Here all the horizontal and diagonal arrows on the right side of the diagram are induced b y isomorphisms of the corresp onding complexes arising from the fact that the tensor pro duct (o v er Z /n ) of a complex of n -torsion groups with Z /n reco vers the original complex. Let us see wh y the individual p olygons comm ute. W e b egin with the b ottom-righ t square and the quadrilateral ab o ve it. Both of these diagrams arise from commutativ e diagrams of complexes of shea ves on X , where the vertical arrows on the right arise from certain maps of complexes and the v ertical arro ws on the left arise from the same maps tensored with Z /n . So these diagrams express the statemen t that isomorphism b et w een the tensor pro duct of a complex with Z /n and the complex itself is a natural transformation, which follows by [Ho v99, Lemma 1.3.9] from deriving the natural transformation of functors from the category of n -torsion sheav es to the category of sheav es giv en b y A ⊗ Z /n → A . All the arro ws in the top-righ t square are natural transformations b et w een comp ositions of deriv ed functors induced b y natural transformations of the underlying functors (each functor is applied to the sheaf µ n ). By [Ho v99, Theorem 1.4.3 and Deﬁnition 1.4.2.6], to chec k the comm utativity of the diagrams it suﬃces to chec k the comm utativity of the corresp onding diagrams for sheav es. This can b e d one directly from the deﬁnitions. F or the left-hand side, it is conv enient to think of tensor pro duct as a derived functor in the mo del category sense of the tensor product functor from the pro duct of t w o copies of the category of complexes, with the pro duct of the usual mo del structure, to the category of complexes. The cup pro duct is then the natural transformations on derived functors induced b y the natural transformation Γ( F ) ⊗ Γ( G ) → Γ( F ⊗ G ) . Since replacing both complexes 71 of shea ves with injective resolutions is a righ t Quillen replacement functor, this agrees with the deﬁnition of [Swa99]. That b eing done, all the arro ws in the noncon vex hexagon are natural transformations of deriv ed functors of maps on categories of complexes induced by natural transformations of functors on the original ab elian categories. How ev er, since the tensor pro duct is a left deriv ed functor and not a righ t derived functor, w e cannot directly apply [Ho v99, Theorem 1.4.3]. It ma y be p ossible to do this with the theory of [Shu11], but we prov e commutativit y b y hand. The k ey thing that makes this calculation easier is that given a complex of shea v es, for example any resolution of a sheaf, not necessarily injective, a co cycle in the global sections of the complex represen ts a class in the hypercohomology of the complex, or, in the example, in the cohomology of the sheaf, whic h may b e expressed in terms of an injective resolution b y mapping this resolution to the injective one. W e start with a cohomology class α in H i +2 ( Y , µ n ) and a cohomology class β in H j ( X , µ n ) . T o represen t them, we take an injective resolution A · of µ n on Y and a resolution B · of Z /n on X . Rather than choosing B · to b e an injectiv e resolution, we choose it to b e a ﬁnite resolution of ﬂat Z /n -mo dules such that H 0 ( X , B j ) admits a co cycle representing β . The fact that it is a ﬁnite complex of ﬂat Z /n - mo dules means that tensor pro duct with this complex computes the derived tensor pro duct. W e ma y do this for j ≤ 1 since if j = 0 we can take the complex to simply b e Z /n and if j = 1 then β represen ts an extension E of Z /n by Z /n as sheav es on X and then E → Z /n giv es the desired complex. Then F ∗ B · giv es a resolution of Z /n on Y where the same cycle represen ts the pullbac k of β . The top horizon tal arrow is obtained by taking the co cycle α ⊗ β in the tensor pro duct of resolutions A · ⊗ F ∗ B · , whic h is a resolution of µ n ⊗ F ∗ Z /n . W e can map α ⊗ β to a class in an injectiv e resolution C · of µ n ⊗ F ∗ Z /n . The top-right down w ards Lera y arrow then corresp onds to taking the induced section in H 0 ( X , F ∗ C i + j +2 ) . Since F ∗ C · is a complex represen ting RF ∗ ( µ n ⊗ F ∗ Z /n ) , this giv es a cohomology class in H i + j +2 ( X , RF ∗ ( µ n ⊗ F ∗ Z /n ) . Equiv alently , this is induced by taking the section α ⊗ β of F ∗ ( A ⊗ F ∗ B ) and mapping along the map F ∗ ( A ⊗ F ∗ B ) → F ∗ C induced by A ⊗ F ∗ B → C . The right w ard arro w on the righ t-hand side can b e obtained b y using the isomorphism µ n ⊗ F ∗ Z /n → µ n to view C as a resolution of µ n , and interpreting the same co cycle as a class in the cohomology of F ∗ µ n . F or the left vertical arrow, w e map α to an elemen t F ∗ α of H 0 ( X , F ∗ A i +2 ) , where F ∗ A i +2 is a complex represen ting H i +2 ( X , RF ∗ µ n ) . F or the b ottom horizon tal arro w, we observ e that F ∗ α ⊗ β gives an elemen t in H 0 ( X , ( F ∗ A ⊗ B ) i + j +2 ) which induces a class in the cohomology of R F ∗ µ n ⊗ Z /n . F or the b ottom-right diagonal arro w, w e observe that F ∗ A → ( F ∗ A ) ⊗ B is a quasi-isomorphism so F ∗ A ⊗ B is also a resolution of F ∗ A , and hence α ⊗ β induces a class in the hypercohomology of F ∗ A , i.e. in H ∗ ( X , RF ∗ µ n ) . In summary , the upw ard transit inv olves taking the classes α and β , obtaining a co cycle F ∗ ( α ⊗ F ∗ β ) in F ∗ ( A ⊗ F ∗ B ) and obtaining a co cycle in F ∗ C , which since the comp osition of the tw o vertical arrows on the righ t side of the b elo w diagram is a quasi-isomorphism (b eing the functor F ∗ applied to a quasi-isomorphism b et w een complexes of injectiv es) gives a class in the cohomology of F ∗ A . The down w ard transit in v olves obtaining a co cycle F ∗ α ⊗ β in F ∗ A ⊗ B , which since the vertical arro w on the left side of the b elow diagram is a quasi- isomorphism giv es a class in the cohomology of F A . 72 F ∗ A F ∗ A F ∗ A ⊗ B F ∗ ( A ⊗ F ∗ B ) F ∗ C id The horizontal arrow ( F ∗ A ⊗ B ) → F ∗ ( A ⊗ F ∗ B ) may b e constructed for arbitrary complexes of shea ves A, B as part of the pro jection formula. It is deﬁned to send the tensor pro duct of a section s 1 of F ∗ A and a section s 2 of B on the op en set U , with s 1 arising from a section s ′ 1 of A on F − 1 ( U ) , to the section s ′ 1 ⊗ s 2 of A ⊗ F ∗ B on F − 1 ( U ) . T o chec k the tw o co cycles constructed from the t w o transits induce the same class in cohomology , w e must c heck that they pro duce the same co cycle in F ∗ C , which reduces us to c hecking that the map on shea v es F ∗ A ⊗ B → F ∗ ( A ⊗ F ∗ B ) sends F ∗ α ⊗ β to F ∗ ( α ⊗ F ∗ β ) . This is true by deﬁnition. F or the cen ter-left square, the v ertical arro ws arise from the same map of complexes RF ∗ µ n → R 2 F ∗ µ n [ − 2] (and the iden tity map Z /n → Z /n ) while the horizon tal arro ws are the derived functors of the natural transformation Γ( F ) ⊗ Γ( G ) → Γ( F ⊗ G ) . The comm utativity of this quadilateral follows from the fact that the derived functor of a natural transformation is a natural transformation [Hov99, Lemma 1.3.9]. A similar argument work for the b ottom-left square, as b oth vertical arro ws arise from the map (T r F , id ) : ( R 2 f ∗ µ n , Z /n ) → ( Z /n, Z /n ) in the pro duct of tw o deriv ed categories, and the horizon tal arrow is the deriv ed functor of the natural transformation Γ( F ) ⊗ Γ( G ) → Γ( F ⊗G ) . Naturalit y of the deriv ed functor gives commutativit y of the diagram. □ Before proving Lemma 4.6 relatin g the Artin-V erdier trace and the map e , w e need the follo wing calculational to ol: Lemma 4.39. L et F b e a she af in the étale top olo gy on a smo oth pr oje ctive curve Y x over a ﬁeld. L et y b e a p oint of Y x and let j : Y x \ y → Y x b e the op en immersion. L et i ≥ 0 b e an inte ger. The c omp osition (4.40) H i ( k y , F ) → H i +1 c ( Y x \ y , j ∗ F ) → H i +1 c ( Y x , F ) of maps deﬁne d in [Mil06, I I, Section 2] may b e c ompute d on Ce ch c o cycles in the fol lowing way: L et α b e a c o cycle in C i +1 ( Y x , j ! j ∗ F ) . L et β b e a c o chain in C i ( Y x \ y , j ∗ F ) with dβ = j ∗ α . L et γ b e a c o chain in C i ( O k y , j ! j ∗ F ) with dγ e qual to the pul lb ack of α to C i +1 ( O k y , j ! j ∗ F ) . Then we c an pul l b oth β and γ b ack to C i ( k y , j ∗ F ) wher e dβ = dγ so β − γ is a c o cycle. The c omp osition (4.40) sends the class of β − γ to the image of the class of α under the c ounit map C i +1 ( Y x , j ! j ∗ F ) → C i +1 ( Y x , F ) . Pr o of. This can b e pro v ed by examining how the maps making up (4.40) are deﬁned in [Mil06, I I, Section 2]. The map from compactly supp orted cohomology to ordinary cohomology is obtained b y comp osing the isomorphism [Mil06, Prop osition 2.3(d)] betw een the compactly supp orted cohomology group H i +1 c ( Y x \ y , j ∗ F ) and the cohomology of the extension by zero H i +1 ( Y x , j ! j ∗ F ) (n the reference b oth groups are compactly supp orted, but since Y x is prop er, we ma y drop the c there) and the map H i +1 ( Y x , j ! j ∗ F ) → H i +1 ( Y x , F ) app earing in 73 the cohomology long exact sequence, whic h is the map induced from the counit j ! j ∗ F → F . So it suﬃces to chec k that the comp osition H i ( k y , F ) → H i +1 c ( Y x \ y , j ∗ F ) → H i +1 ( Y x , j ! j ∗ F ) sends the class of β − γ to the class of α . The compactly supp orted cohomology group H i +1 c ( Y x \ y , j ∗ F ) is deﬁned as the i + 1 st cohomology of the mapping cone of the map from C ∗ ( Y x , j ∗ F ) to C ∗ ( k y , j ∗ F ) . T o map this to the cohomology of the extension by zero, Milne uses [Mil06, Lemma 2.4(d)], which in our case giv es a long exact sequence · · · → H i ( k y , j ∗ F ) → H i +1 ( Y x , j ! j ∗ F ) → H i +1 ( Y x \ y , j ∗ F ) → H i +1 ( k y , j ∗ F ) → . . . Since H i +1 c ( Y x , j ∗ F ) is deﬁned to b e the i +1 st cohomology of the mapping cone of C ∗ ( Y x , j ! j ∗ F ) → C ∗ ( Y x \ y , j ∗ F ) , which is quasi-isomorphic to C ∗ ( Y x , j ! j ∗ F ) by the ab ov e exact sequence, this giv es an isomorphism b etw een H i +1 c ( Y x \ y , j ∗ F ) and H i +1 ( Y x , j ! j ∗ F ) . The isomorphism con- structed this w a y sends the natural map from H i ( k y , j ∗ F ) to H i +1 ( Y x , j ! j ∗ F ) to the map H i ( k y , j ∗ F ) → H i +1 ( Y x , j ! j ∗ F ) from the exact sequence ab ov e, since the natural map to the cohomology of the mapping cone of tw o complexes from the cohomology of one complex is one of the maps in the long exact sequence asso ciated to the mapping cone. So it suﬃces to c heck that the map H i ( k y , j ∗ F ) → H i +1 ( Y x , j ! j ∗ F ) of [Mil06, Lemma 2.4(d)] sends the class of β − γ to the class of α . The cohomology of the pair H ∗ y ( Y x , · ) is deﬁned as the derived functor of the functor that tak es a sheaf to its global sections on Y x that v anish on the complement of y . One can chec k that this derived functor is represented b y the functor that takes a sheaf G to the cohomology of the mapping cone of C ∗ ( Y x , G ) → C ∗ ( Y x \ y , j ∗ G ) since this mapping cone has the correct v alue in degree 0 , its higher cohomology v anishes for injectiv e shea ves, and it is compatible with short exact sequences of sheav es since C ∗ ( Y x , G ) and C ∗ ( Y x \ y , j ∗ G ) b oth are. The long exact sequence of the pair [Mil80, I II, Prop osition 1.25] H i ( Y x \ y , j ∗ G ) → H i +1 y ( Y x , G ) → H i +1 ( Y x , G ) → H i +1 ( Y x \ y , G ) → includes maps H i ( Y x \ y , j ∗ G ) → H i +1 y ( Y x , G ) and H i +1 y ( Y x , G ) → H i +1 ( Y x , G ) → H i +1 ( Y x \ y , G ) , deﬁned as maps of Ext groups induced by a short exact sequence of shea ves. W e can c heck that these agree with the natural maps b et ween the cohomology of the mapping cone of t w o complexes and the cohomology of the original complexes by chec king that the natural maps of the mapping cone are the correct maps on H 0 and compatible with connecting homomorphisms of short exact sequences of shea v es, b oth of whic h are straightforw ard. In particular, the map H i +1 y ( Y x , j ! j ∗ F ) → H i +1 ( Y x , j ! j ∗ F ) from the long exact sequence of the pair, expressed in terms of the mapping cone, is equiv alent to forgetting C ∗ ( Y x \ y , j ∗ F ) , so the pair ( α , β ) represen ts a class in H i +1 y ( Y x , j ! j ∗ F ) , whose image in H i +1 ( Y x , j ! j ∗ F ) is α . So it suﬃces to chec k that the class represen ted by ( α, β ) in H i +1 y ( Y x , j ! j ∗ F ) is sen t by the isomorphism H i +1 y ( Y x , j ! j ∗ F ) ∼ = H i ( k y , j ∗ F ) to β − γ . The isomorphism H i +1 y ( Y x , j ! j ∗ F ) ∼ = H i ( k y , j ∗ F ) is obtained b y comp osing an isomorphism H i +1 y ( Y x , j ! j ∗ F ) → H i +1 y ( O k y , j ! j ∗ F ) arising from excision [Mil80, I I I.1.28] with an isomor- phism H i +1 y ( O k y , j ! j ∗ F ) → H i y ( j y , j ∗ F ) of [Mil06, II, Prop osition 1.1(a)]. The map pro ved to b e an isomorphism in [Mil80, I I I, Corollary 1.28] is simply the pullback map, as one can 74 tell from [Mil80, I I I, pro of of Corollary 1.28 and statement of Proposition 1.27]. W e can c heck that pullbac k commutes with the identiﬁcation of relativ e cohomology with a mapping cone since pullback is a natural transformation of derived functors induced by the natural transformation of original functors, natural transformations b et ween deriv ed functors whic h are compatible with the connecting homomorphism of a short exact sequence are uniquely determined b y their v alue on R 0 , and the pullback map on the cohomology of the mapping cone is compatible with the connecting homomorphism of a short exact sequence and takes the correct v alue on R 0 . Th us the pullback map H i +1 y ( Y x , j ! j ∗ F ) → H i +1 y ( O k y , j ! j ∗ F ) sends the class represen ted by α and β to the class represented b y the pullback of α and the pull- bac k of β . The isomorphism H i +1 y ( O k y , j ! j ∗ F ) → H i y ( k y , j ∗ F ) is deﬁned to b e the inv erse of the connecting morphism H i y ( k y , j ∗ F ) → H i +1 y ( O k y , j ! j ∗ F ) of the long exact sequence of the pair H i ( O k y , j ! j ∗ F ) → H i y ( k y , j ∗ F ) → H i +1 y ( O k y , j ! j ∗ F ) → H i +1 ( O k y , j ! j ∗ F ) → . . . whic h is an isomorphism since H i ( O k y , j ! j ∗ F ) = 0 for all i by [Mil06, I I, Prop osition 1.1(b)]. The connecting homomorphism sends the class represented b y β − γ to the class represented b y (0 , β − γ ) (as we c heck ed ab o ve) which is equiv alent to the class represented by ( α, β ) since ( α, β ) − (0 , β − γ ) = ( α, γ ) = ( dγ , γ ) = d ( γ , 0) is a cob oundary in the mapping cone. □ Pr o of of L emma 4.6. It suﬃces to chec k that for each α ∈ H 3 ( H , Z /n ) , its image ˜ e ( α ) ∈ Hom( π 1 ( X ) , Z /n ) tak es v alue on F rob q ,x equal to A V Z x / Y x ( α ) , that is, the Artin-V erdier trace for Z x / Y x applied to α . Let f be the map Y x → Sp ec F q . Because ϕ Z/ Y , the in tegration map and the explicit calculation of étale H 1 are compatible with pullbac k, we can compute ˜ e ( α )(F rob q ,x ) up on pullback to Y x , i.e. taking the image of α under the composite map H 3 ( H , Z /n ) ϕ Z x / Y x → H 3 ( Y x , Z /n ) R f → H 1 ( F q , Z /n ) ∼ = Hom(Gal( F q ) , Z /n ) ev al F rob q − → Z /n. W e ha ve tw o maps H 3 ( H , Z /n ) → Z /n (the abov e composite and A V Z x / Y x ( α ) ), b oth factoring through H 3 ( H , Z /n ) ϕ Z x / Y x → H 3 ( Y x , Z /n ) , so we can c heck they are the same just by chec king that the corresp onding maps H 3 ( Y x , Z /n ) → Z /n are the same. Since H 3 ( Y x , Z /n ) ∼ = Z /n (by [Mil06, Prop osition 2.6] and the fact that on Y x w e hav e Z /n = µ n ), it suﬃces to chec k this on a generator. The Kummer map H 1 ( Y x , G m ) → H 2 ( Y x , µ n ) applied to the class of a degree 1 line bundle L in H 2 ( Y x , µ n ) , gives, using our identiﬁcation µ n ≃ Z /n Z , a class [ L ] ∈ H 2 ( Y x , Z /n ) . W e consider the cup pro duct of [ L ] with the pullback from F q of the class β of a Z /n -torsor on F q on whic h F rob q acts b y +1 . W e can chec k R f [ L ] = deg L = 1 as follows. By compatibility of integration with pullbac k, b ecause H 0 ( F q , Z /n ) → H 0 ( ¯ F q , Z /n ) is an isomorphism, we can reduce to the case where the base is Sp ec ¯ F q . When the base is the sp ectrum of a algebraically closed ﬁeld, the in tegration-along the ﬁb ers map reduces to the trace map from the second cohomology of the ﬁb er to Z /n . When the base is the sp ectrum of an algebraically closed ﬁeld and the ﬁb er is a connected curve, this map is given explicitly in [AGV72, XVI I I, 1.1.3.4]. This explicit map is given b y the in v erse of the Kummer map H 1 ( Y x , G m ) → H 1 ( Y x , µ n ) comp osed with the iden tiﬁcation betw een the Picard group of a curve mo dulo n and Z /n arising in [A GV72, IX, Corollary 4.7] from tensoring with Z /n the exact sequence [AGV72, IX, (4.8)] 0 → Pic 0 ( Y x ) → Pic( Y x ) → Z → 0 . Since the map Pic( Y x ) → Z in that exact sequence is 75 the degree map (this is not quite sp eciﬁed in [A GV72, IX, (4.8)], which only says to recall this exact sequence, but seems to b e what is meant), it follo ws that the trace map sends [ L ] to the degree of L , mo dulo n . The fact that this trace agrees with the more general notion of trace is [A GV72, XVI I I, Prop osition 2.10], which states that in the case the that ﬁb er is a curv e, the trace map of [A GV72, XVI I I, Theorem 2.9] agrees with the trace map of [AGV72, XVI I I, Prop osition 1.1.6], which is itself deﬁned to agree when the base is the sp ectrum of an algebraically closed ﬁeld with the trace map of [AGV72, XVI I I 1.1.3.4]. By the pro jection form ula (Lemma 4.5) Z f ([ L ] ∪ f ∗ β ) = ( Z f [ L ]) ∪ β = 1 ∪ β = β . In particular, ev al F rob q ( R f ([ L ] ∪ f ∗ β )) = 1 . Also, from this it follo ws that [ L ] ∪ f ∗ β is a generator of H 3 ( Y x , Z /n ) . Th us, w e will ha ve prov en the lemma if we can show that the image of [ L ] ∪ f ∗ β under the follo wing map used in deﬁning the Artin-V erdier trace (4.41) H 3 ( Y x , Z /n ) → H 3 ( Y x , µ n ) D → Z /n is 1. It suﬃces to chec k, for each p oint y , that the image of [ O Y x ( y )] ∪ f ∗ β under (4.41) is deg y , as we can take a p oin t y 1 of degree d and y 2 of degree d + 1 for d suﬃciently large, and then choose L = O Y x ( y 2 − y 1 ) . The deﬁnition of the map D comes from the isomorphism H 3 c ( Y x , G m ) ≃ Q / Z constructed in [Mil06, Chapter I I, Section 3]. In particular it follows from this construction that, for a p oin t y ∈ Y x with lo cal ﬁeld k y , the comp osition of D with H 2 ( k y , µ n ) → H 3 c ( Y x \ y , µ n ) → H 3 ( Y x , µ n ) agrees with the inv arian t map on the Brauer group of k y . W e ha ve a diagram H 2 ( k y , µ n ) H 3 c ( Y x \ y , µ n ) H 3 ( Y x , µ n ) H 1 ( k y , µ n ) H 2 c ( Y x \ y , µ n ) H 2 ( Y x , µ n ) H 0 ( k y , G m ) H 1 ( Y x \ y , G m ) H 1 ( Y x , G m ) where the v ertical arrows in the top ro w are cup pro ducts with f ∗ β and the v ertical arrows in the b ottom row are the Kummer map. W e ﬁrst chec k that the horizon tal arro ws in the b ottom row send the uniforrmizer in H 0 ( k y , G m ) = k × y to O Y x ( y ) ∈ H 1 ( Y x , G m ) . Ho w ever, the diagram do es not quite comm ute. Instead, w e c heck that the class obtained b y sending the uniformizer along the b ottom ro w and then the right column diﬀers from the class obtained by sending the uniformizer along the left column and top ro w by a sign of − 1 . F rom this it follows that the image of [ O Y x ( y )] ∪ f ∗ β under the map in (4.41) is the Brauer in v arian t of the cup pro duct of the Kummer class of the inverse of the uniformizer with f ∗ β , which is deg y by [LST20, Lemma 6.1] since the F rob enius at a degree deg y p oin t is the deg y th p ow er of F rob q . This w as the desired statemen t, so when w e chec k this, the pro of is complete. T o calculate the horizontal arro ws, we use Lemma 4.39. Let π ∈ k ( Y x ) v anish to order 1 at y and therefore function as a uniformizer. Let U 1 = Y x \ y and let U 2 b e obtained from Y x 76 b y removing all zero es and p oles of π other than y , so that U 1 ∪ U 2 is a Zariski op en cov er of Y x . W e start with the Cec h co cycle in a 1 ∈ C 1 ( Y x , j ! G m ) with underlying co vering U 1 ∪ U 2 deﬁned by the section that sends U 1 × U 1 and U 2 × U 2 to 1 , U 1 × U 2 to π , and U 2 → U 1 to π − 1 . Here we express sections of j ! j ∗ G m as sections of G m , i.e. in v ertible functions, that happ en to b e trivial, i.e. equal to 1 , on U 1 × U 1 , the only op en set under consideration containing y . The image of a 1 under C 1 ( Y x , j ! G m ) → C 1 ( Y x , G m ) is expressed using the same co cycle. One immediately c hec ks this is the co cycle arising from the line bundle O ( y x ) , with the trivialization on U 1 giv en b y the natural map O → O ( y ) and the trivialization on U 2 giv en b y the natural map O → O ( y ) divided b y π . W e can ﬁnd b 0 ∈ C 0 ( Y x \ y , G m ) with db 0 = j ∗ a 1 with underlying cov ering the pullbac k of U 1 ∪ U 2 , that sends the pullback of U 1 to 1 and the pullback of U 2 to π . W e can ﬁnd c 0 in C 0 ( O k y , j ! G m ) with dc 0 the pullback of a 1 with underlying cov ering the pullbac k of U 1 ∪ U 2 , that sends the pullback of U 1 to π − 1 and the pullback of U 2 to 1 . Then b 0 − c 0 is the co cycle in C 0 ( k y , G m ) with underlying cov ering the pullbac k of U 1 ∪ U 2 , that sends b oth U 1 and U 2 to π . This represents the class π , so b y Lemma 4.39 the b ottom ro w sends π to O Y x ( y ) . W e no w apply the Kummer map to all this data. T o do this, w e obtain by [Mil80, II I, Lemma 2.19] étale cov erings V 1 and V 2 of U 1 and U 2 suc h that π admits an n ’th ro ot π 1 /n on V 1 × V 2 . W e ha ve a co chain in a 1 /n ∈ C 1 ( Y x , j ! G m ) with underlying cov ering V 1 ∪ V 2 , deﬁned by the function that sends V 1 × V 1 and V 2 × V 2 to 1 , V 1 × V 2 to π 1 /n and V 2 × V 1 to π − 1 /n . The co c hain a 1 /n is no longer a co cycle, but its cob oundary d ( a 1 /n ) tak es v alues in µ n , and therefore deﬁnes a co cycle a 2 ∈ C 2 ( Y x , j ! µ n ) . The pro jection of a 2 to C 2 ( Y x , µ n ) is obtained from a 1 b y exactly the snak e lemma construction that is used to deﬁne the connecting homomorphism of the Kummer exact sequence and therefore is the Kummer class [ O Y x ( y )] corresp onding to O Y x ( y ) . W e tak e b 0 /n ∈ C 0 ( Y x \ y , G m ) , with underlying cov ering the reﬁnemen t of the pullback of V 1 ∪ V 2 giv en b y the pullback of V 1 ∪ V 1 × V 2 , deﬁned by the function that sends V 1 to 1 and V 1 × V 2 to π 1 /n . Then ( a 1 /n ) − d ( b 0 /n ) takes v alues in µ n , and therefore deﬁnes a co cycle b 1 ∈ C 1 ( Y x \ y , µ n ) . W e ha ve db 1 = d (( a 1 /n ) − d ( b 0 /n )) = d ( a 1 /n ) − d 2 ( b 0 /n ) = a 2 − 0 = a 2 . W e take c 0 /n in C 0 ( O k y , j ! G m ) with underlying co vering the reﬁnement of the pullback of V 1 ∪ V 2 giv en by the pullbac k of V 1 × V 2 ∪ V 2 , deﬁned by the function that sends V 1 × V 2 to π 1 /n and V 2 to 1 . Then ( a 1 /n ) − d ( c 0 /n ) tak es v alues in µ n , and therefore deﬁnes a co cycle c 1 ∈ C 1 ( O k y , j ! G m ) . Identical reasoning gives dc 1 = a 2 . Then b 1 − c 1 = d ( c 0 /n ) − d ( b 0 /n ) is obtained from c 0 − b 0 b y exactly the snak e lemma construction that is used to deﬁne the connecting homomorphism of the Kummer exact sequence and is therefore the Kummer class corresp onding to π − 1 . W e no w take the cup pro duct of ev erything with f ∗ β . T o do this, we can choose a Cech co cycle represen ting f ∗ β , arising from a ﬁnite étale cov ering of Sp ec F q , and use the formula [Sta18, 07MB] for the cup pro duct of Cech co chains to cup a 2 with f ∗ β , obtaining a 3 , and similarly cup b 1 and c 1 with f ∗ β , obtaining b 2 and c 2 . W e hav e db 2 = a 3 and dc 2 = a 3 since the diﬀeren tial is a deriv ation with resp ect to cup pro duct [Sta18, 01FP]. Then a 3 represen ts the cup pro duct of a 2 with f ∗ β (by [Sw a99, Corollary 3.10]) and b 2 − c 2 represen ts the cup pro duct of b 1 − c 1 with f ∗ β , and th us Lemma 4.39 sho ws that the cup pro duct of the Kummer 77 class of π − 1 with f ∗ β is sent to the cup pro duct of the Kummer class of O Y x ( y ) with f ∗ β , completing the pro of. □ Pr o of of L emma 4.20. Let Z , ov er X , b e the univ ersal family of ( H ⋊ Γ) -co verings of P 1 C , so that Z → X × P 1 is an H ⋊ Γ -cov ering. Let Y b e the univ ersal family of induced Γ -co v erings of P 1 , i.e. the quotien t of Z by H . Then Z → Y is a ﬁnite étale H -cov ering and th us, b y Lemma 2.2, determines a map from group cohomology to étale cohomology H 3 ( H , Z /n ) → H 3 ( Y , Z /n ) , and from étale cohomology to singular cohomology H 3 ( Y ( C ) , Z /n ) . Since the in tegration along the ﬁb ers map is compatible wi th pullbac ks, the map e : π 1 ( X ) → H 3 ( H , Z /n ) is induced b y the comp osition H 3 ( H , Z /n ) → H 3 ( Y , Z /n ) → H 1 ( X , Z /n ) → Hom( π 1 ( X ( C )) , Z /n ) . Let us discuss how to deﬁne an “integration along the ﬁb ers” map in the singular coho- mology setting. This can b e deﬁned as a comp osition H i +2 ( Y ( C ) , µ n ) → H i +2 ( X ( C ) , R F ′ ∗ µ n ) → H i ( X ( C ) , R 2 F ′ ∗ µ n ) → H i ( X ( C ) , Z /n ) where F ′ is the map Y ( C ) → X ( C ) induced b y the map Y → X once w e ha v e a trace map R 2 F ′ ∗ µ n → Z /n of sheav es on the top ological space X ( C ) . W e choose this trace map to b e compatible with our original trace map, in the sense of the commutativ e diagram (4.42) R 2 F ′ ∗ µ n R 2 F ′ ∗ g ′ ∗ µ n g ∗ R 2 F ∗ µ n Z /n g ∗ Z /n T r F ′ g ∗ T r F where g is the map from the site of op en sets on the top ological space X ( C ) to the étale site of X and g ′ is the analogous map for Y . W e can alwa ys choose a trace map to mak e this diagram commute, and furthermore, this trace map is an isomorphism. T o chec k this, it suﬃces to c hec k that every other arro w in the diagram is an isomorphism. F or the top-left arrow and b ottom arrow, this is immediate from the deﬁnition. F or the top-righ t arro w this is a case of the comparison theorem [A GV72, XVI, Theorem 4.1] b et ween étale cohomology and singular cohomology . F or the right vertical arro w, w e must c heck that the trace map in étale cohomology is an isomorphism. It suﬃces to c heck it is an isomorphism on stalks. T o do this, we use [AGV72, XVI I I, Proposition 2.10], whic h states that in the case the that ﬁb er is a curve, the trace map of [AGV72, XVI I I, Theorem 2.9] agrees with the trace map of [A GV72, XVI I I, Prop osition 1.1.6], which is itself deﬁned to agree when the base is the sp ectrum of an algebraically closed ﬁeld (in our case, C ) with the trace map of [A GV72, 1.1.3.4]. So it suﬃces to consider the trace map of [AGV72, XVI I I, 1.1.3.4]. This is deﬁned, in the case of a pro jectiv e curv e Y x , b y taking the Kummer isomorphism b etw een H 2 ( Y x , µ n ) and Pic( Z x ) /n , observing that Pic( Y x ) /n is isomorphic to ( Z /n ) c , c the n um b er of comp onen ts, and taking a sum map to Z /n , weigh ted b y the m ultiplicity of the comp onents. When the curv e is smo oth pro jectiv e irreducible, as it is in our case, there is only one comp onen t with m ultiplicity one, so the weigh ted sum is simply the identit y map Z /n → Z /n , and thus the trace map is an isomorphism. Using the “integration along the ﬁb ers” map, w e can deﬁne an analogue of the map e in singular cohomology the same w a y w e deﬁned the original map, as the map π 1 ( X ( C )) → 78 H 3 ( H , Z /n ) induced b y the comp osition H 3 ( H , Z /n ) → H 3 ( Y ( C ) , Z /n ) → H 1 ( X ( C ) , Z /n ) → Hom( π 1 ( X ( C )) , Z /n ) . inducing a homomorphism π 1 ( X ( C ) → H 3 ( H , Z /n ) . Let us chec k that the analogue agrees with e after comp osition with the map π 1 ( X ( C )) → π 1 ( X ) from the top ological to the étale fundamen tal group. T o pro ve this, w e need to c heck the integration-along-the-ﬁbers map is compatible with the comparison b et ween étale cohomology in c haracteristic zero and singular cohomology . This follo ws by an argument iden tical to Lemma 4.4, except that w e use pullback from the étale site to the analytic site instead of pullback b etw een diﬀeren t schemes, and we use (4.42) as our starting p oint. F or σ in the fundamen tal group of a comp onent of Hur m H ⋊ Γ ,c H ⋊ Γ ( C ) corresp onding to a braid, a lo op in Hur m H ⋊ Γ ,c H ⋊ Γ ( C ) represen ting σ is a lift from Conf m ( C ) to Hur m H ⋊ Γ ,c H ⋊ Γ ( C ) of a geometric realization of that braid. The pullback of the univ ersal family Y to that lo op is exactly the cov ering ˜ M of S 2 × S 1 constructed from that braid. Let S 1 b e the lo op, ˜ M b e the pullback of the universal family to that lo op, and ˜ F : ˜ M → S 1 the map betw een th em. Let ˜ g : S 1 → X ( C ) and ˜ g ′ : ˜ M → Y ( C ) b e the maps induced by the em b edding of the lo op S 1 in to X . Again, w e can deﬁne a trace map R 2 ˜ F ∗ µ n → Z /n to make the diagram (4.43) R 2 ˜ F ∗ µ n R 2 ˜ F ∗ ˜ g ′ ∗ µ n ˜ g ∗ R 2 F ′ ∗ µ n Z /n ˜ g ∗ Z /n T r ˜ F ˜ g ∗ T r ′ F comm ute, and again this map is an isomorphism: The right v ertical arrow is an isomorphism b y what we chec k ed b efore, that the the top-left horizon tal arro w and b ottom horizontal ar- ro w are isomorphisms follows from the deﬁnitions, and the top-righ t vertical arro w is an isomorphism b ecause of the prop er base c hange theorem in the setting of singular cohomol- ogy . Again this deﬁnition gives an in tegration-along-the-ﬁb ers map and a compatibility of in tegration-along-the-ﬁb ers with pullback. Hence e ( σ ) is giv en by the comp osition H 3 ( H , Z /n ) → H 3 ( ˜ M , Z /n ) → H 1 ( S 1 , Z /n ) ∼ = Z /n where we now consider the ﬁbration ˜ M → S 1 and its associated in tegration-along-the-ﬁb ers map. The braid fundamental class of σ , on the other hand, is giv en by comp osing H 3 ( H , Z /n ) → H 3 ( ˜ M , Z /n ) with the fundamen tal class isomorphism H 3 ( ˜ M , Z /n ) → Z /n . Hence to chec k our desired statemen t that the braid fundamen tal class is equal to e ( σ ) up to multiplica- tion by an elemen t of ( Z /n ) × , it suﬃces to c hec k that the in tegration-along-the-ﬁb ers map H 3 ( ˜ M , Z /n ) → H 1 ( S 1 , Z /n ) is an isomorphism. This map is the comp osition H 3 ( ˜ M , Z /n ) → H 3 ( S 1 , R ˜ F ∗ Z /n ) → H 1 ( S 1 , R 2 ˜ F ∗ Z /n ) → H 1 ( S 1 , Z /n ) . The ﬁrst map is an isomorphism b y the deriv ed category version of the Lera y sp ectral sequence. The second map is an isomorphism since, in the sp ectral sequence computing H p + q ( S 1 , R ˜ F ∗ Z /n ) from H p ( S 1 , R q ˜ F ∗ Z /n ) , every diﬀerential to or from H 1 ( S 1 , R 2 ˜ F ∗ Z /n ) after the second page must v anish since H p ( S 1 , R q ˜ F ∗ Z /n ) = 0 for p > 0 or q > 0 . The third 79 map is an isomorphism since we already sa w that the trace map is an isomorphism, and we are done. □ 5. Gr oup theor y preliminaries In this section, we give many results in group theory that we will use rep eatedly to deter- mine our distribution from its momen ts. These results may b e considered well-kno wn, but w e give them here for completeness and to ﬁx notation. 5.1. Alternating tensor p o wers. Lemma 5.1. F or a ﬁnite ab elian gr oup A and p ositive inte ger n , ther e is a homomorphism ∆ : Hom( A ⊗ A, Z /n ) → Hom( A ⊗ A, Z /n ) f 7→ ( a ⊗ b 7→ f ( b, a ) − f ( a, b )) , whose image ∆ Hom( A ⊗ A, Z /n ) is the set of g ∈ Hom( A ⊗ A, Z /n ) such that g ( a ⊗ a ) = 0 for al l a ∈ A ( alternating maps). A lso, ker ∆ is the set of g ∈ Hom( A ⊗ A, Z /n ) such that g ( a ⊗ b ) = g ( b ⊗ a ) for al l a, b ∈ A ( symmetric maps). Pr o of. Clearly all maps in ∆ Hom( A ⊗ A, Z /n ) are alternating. W e can chec k, using bases, that ev ery alternating g ∈ Hom( A ⊗ A, Z /n ) is in ∆ Hom( A ⊗ A, Z /n ) . □ W e write ∧ Hom( A ⊗ A, Z /n ) for the quotient Hom( A ⊗ A, Z /n ) / ker ∆ of Hom( A ⊗ A, Z /n ) b y the symmetric maps k er ∆ . Note that Lemma 5.1 implies we hav e an isomorphism ∆ : ∧ Hom( A ⊗ A, Z /n ) ≃ ∆ Hom( A ⊗ A, Z /n ) . 5.2. Represen tations o ver ﬁnite ﬁelds. W e explain in detail the relationships b et w een the diﬀeren t types of self-dual representations V and the prop erties of an inv ariant element of V ∨ ⊗ V ∨ . Lemma 5.2. L et V b e an irr e ducible self-dual r epr esentation of a ﬁnite gr oup Π over a ﬁeld F p for some prime p . L et κ = End Π ( V ) . L et ω ∈ ( V ∨ ⊗ V ∨ ) Π \ 0 . Then ther e is a unique automorphism σ of κ , such that for al l k ∈ κ , we have ( k ⊗ 1) ω = (1 ⊗ σ ( k )) ω . F urther, σ 2 = 1 . L et ω t b e the image of ω under switching factors. Ther e is a λ ∈ κ such that λσ ( λ ) = 1 and ω t = ( λ ⊗ 1) ω . Then • V is unitary if and only if σ is non-trivial, • for o dd p : V is symmetric if and only if σ is trivial and λ = 1 , • for p = 2 : V is symmetric if and only if σ is trivial and V is trivial, • for o dd p : V is symple ctic if and only if σ is trivial and λ = − 1 , and • for p = 2 : V is symple ctic if and only if σ is trivial and V is non-trivial. A lso, • if V is unitary, then dim F p ( ∧ 2 V ∨ ) Π = 1 2 dim F p κ , • if V is symmetric, then ∧ 2 V ∨ = 0 , and • if V is symple ctic, then dim F p ( ∧ 2 V ∨ ) Π = dim F p κ . 80 Pr o of. Since V is irreducible and self-dual, w e hav e 0  = ω ∈ ( V ∨ ⊗ V ∨ ) Π and ( κ ⊗ 1) ω = (1 ⊗ κ ) ω = ( V ∨ ⊗ V ∨ ) Π . Thus or a k ∈ κ , w e ha ve ( k ⊗ 1) ω = (1 ⊗ σ ( k )) ω , for some p erm utation σ of κ , and we hav e ω t = ( λ ⊗ 1) ω for some λ ∈ κ . It is immediate from the bilinearit y of ω that σ is an automorphism. By comparing the transp oses of ( k ⊗ 1) ω and (1 ⊗ σ ( k )) ω , w e ﬁnd that σ 2 = 1 , and by considering the fact that ( ω t ) t = ω , w e hav e that λσ ( λ ) = 1 . W e hav e isomorphisms of Π -representations V ∨ ⊗ V ∨ ≃ Hom( V , V ∨ ) ≃ Hom( V , V ∨ ) u ⊗ v 7→ ( a 7→ u ( a ) v ) ( a 7→ v ( a ) u ) f 7→ ( a 7→ ( b 7→ f ( b )( a ))) where for M ∈ V ∨ ⊗ V ∨ , w e write M L for the image in the ﬁrst Hom( V , V ∨ ) and M R for the image in the second Hom( V , V ∨ ) , and for M ∈ Hom( V , V ∨ ) , w e write M t for the image in the other Hom( V , V ∨ ) . F or λ ∈ {± 1 } , w e ha v e that M ∈ V ∨ ⊗ V ∨ has M t = λM if and only if M L = λM R , or equiv alently M t L = λM L . There is an entirely analogous situation when V ∨ is replaced b y Hom κ ( V , κ ) , and the tensor pro ducts and Hom ’s are ov er κ . The trace isomorphism of H -represen tations T r : Hom κ ( V , Hom κ ( V , κ )) → Hom κ ( V , V ∨ ) has the prop ert y that for λ ∈ {± 1 } and M ∈ Hom κ ( V , Hom κ ( V , κ )) , we ha v e M t = λM if and only if (T r M ) t = λ T r M . When σ is not the identit y , then ω L giv es an isomorphism from V to V ∨ as representations of Π o v er F p . Moreov er, all such isomorphisms are k ω L for some k ∈ κ , and hence none are κ -equiv ariant, and V and V ∨ are not isomorphic as Π -represen tations o ver κ . Thus V and Hom κ ( V , κ ) are not isomorphic represen tations o ver κ , and hence ( V ⊗ κ V ) H = 0 , and V is unitary . Next w e consider the case when σ is the iden tity . Then ω L giv es a κ -equiv ariant and Π -equiv ariant isomorphism V → V ∨ , and since V ∨ ≃ Hom κ ( V , κ ) , this implies V is self- dual o ver κ. By the remarks abov e, there is a non-zero Π -in v ariant elemen t T r − 1 ω L ∈ Hom κ ( V , Hom κ ( V , κ )) suc h that (T r − 1 ω L ) t = λ T r − 1 ω L , and hence, a non-zero element in (Hom κ ( V , κ ) ⊗ κ Hom κ ( V , κ )) Π with the same prop erty . Using that V is self-dual o v er κ , we ha ve a non-zero elemen t M ∈ ( V ⊗ κ V ) Π suc h that M t = λM . When p is o dd, we ha ve an isomorphism of Π -representations V ⊗ κ V ≃ Sym 2 κ V × ∧ 2 κ V , where Sym 2 κ V are the ﬁxed p oin ts of the factor switc hing and ∧ 2 κ V is the − 1 eigenspace of the factor switc hing. Since V is irreducible, if V ⊗ κ V has an y Π -in v ariants, it has a one dimensional κ vector space of H -inv arian ts, whic h is either in Sym 2 κ V or ∧ 2 κ V . Thus, for o dd p , when λ = 1 we ha v e that V is symmetric and when λ = − 1 w e hav e that V is symplectic. When p = 2 , we hav e ha ve an exact sequence of represen tations 0 → V → Sym 2 κ V → ∧ 2 κ V → 0 sending v ∈ V to v ⊗ v ∈ Sym 2 κ V , and [ v ⊗ w ] 7→ [ v ⊗ w ] for the second map. Th us if V is a non-trivial representation, self-dual o v er κ , then it is symplectic, and if V is trivial then it is symmetric. F or the claims ab out dim F p ( ∧ 2 V ∨ ) , w e ha v e ∧ 2 V ∨ ≃ ∧ 2 V as H -representations. W e hav e dim F p ( ∧ 2 V ) Π = dim κ ( ∧ 2 κ ( V ⊗ F p κ )) Π . W e ha ve V ⊗ F p κ is the sum of dim F p κ absolutely 81 irreducible represen tations o ver κ , whic h are dual in pairs if V is unitary , and self-dual and symplectic if V is symplectic, and the claims ab out dim F p ( ∧ 2 V ∨ ) follo w. □ Lemma 5.3. L et Π b e a ﬁnite gr oup. L et V b e an irr e ducible F 2 -ortho gonal r epr esentation of Π over F 2 . Then the map induc e d by ∆ (Sym 2 V ) Π → ( ∧ 2 V ) Π is surje ctive. Pr o of. W e ha ve an exact sequence 0 → V → Sym 2 V ∆ → ∧ 2 V → 0 , where the ﬁrst map sends v 7→ [ v ⊗ v ] . If V is a trivial representation, then the lemma follows. If V is non-trivial, then V Π = 0 implies that (Sym 2 V ) Π → ( ∧ 2 V ) Π is an injection. So let Q be a non-zero elemen t of (Sym 2 V ) Π , and then ω = ∆ Q is a non-zero elemen t of ( ∧ 2 V ) H . F or k ∈ κ , we hav e that ( k ⊗ k ) Q ∈ (Sym 2 V ) Π and maps under ∆ to ( k ⊗ k ) ω . W e will show the elements ( k ⊗ k ) ω are all the elements of ( ∧ 2 V ) Π . In the language of Lemma 5.2, if σ is non-trivial, then we are considering the elemen ts ( k σ ( k ) ⊗ 1) ω , which giv es | κ | 1 / 2 elemen ts of ( ∧ 2 V ) Π , which by Lemma 5.2 m ust b e all of them. If σ = 1 , then we are considering the elemen ts ( k 2 ⊗ 1) ω , which giv es | κ | elements of ( ∧ 2 V ) Π , which b y Lemma 5.2 must b e all of them. □ 5.3. Group homology and cohomology. When we write group homology or cohomology with Z /n co eﬃcients, we alwa ys mean for the trivial action of the group on Z /n . Lemma 5.4. F or a ﬁnite gr oup Γ and a ﬁnite Γ -gr oup G , and ﬁnite ab elian gr oup A of or der r elatively prime to | Γ | , the pul l-b ack map H k ( G ⋊ Γ , A ) → H k ( G, A ) Γ is an isomorphism for al l k ≥ 0 . Pr o of. In Lyndon-Ho c hschild-Serre sp ectral sequence computing H p + q ( G ⋊ Γ , A ) from H p (Γ , H q ( G, A )) the only non-zero terms are when p = 0 , and so the edge maps to those terms are isomor- phisms. □ Lemma 5.5 ([Sta18, 03VI]) . L et n b e a p ositive inte ger. Ther e is an natur al isomorphism of functors on the c ate gory of pr oﬁnite gr oups fr om H n ( − , Z /n ) to Hom( H n ( − , Z /n ) , Z /n ) . Via Lemma 5.5, we often view elemen ts of H n ( G, Z /n ) as homomorphisms H n ( − , Z /n ) → Z /n. Lemma 5.6. If A is an ab elian gr oup and n a p ositive inte ger, then ther e is an Aut( A ) - e quivariant homomorphism ∆ : H 2 ( A, Z /n ) → Hom( A ⊗ A, Z /n ) [ f ] 7→ ( a ⊗ b 7→ f ( b, a ) − f ( a, b )) , wher e f is any normalize d 2 -c o cycle. The image of the map ∆ ab ove is the same as ∆ Hom( A ⊗ A, Z /n ) fr om L emma 5.1. Pr o of. W e claim for a normalized co cycle f , the map a ⊗ b 7→ f ( b, a ) − f ( a, b ) is a homomor- phism from A ⊗ A to Z /n . This can b e c heck ed by noting it is the map that sends a ⊗ b to the comm utator of lifts of a, b in the extension corresp onding to f . This will also b e conﬁrmed in the pro of of Lemma 6.4 b elo w. 82 The map in the lemma is w ell-deﬁned on H 2 b ecause an y cob oundary is a symmetric function of the t wo inputs, and thus is mapp ed to 0 . Given that it is well-deﬁned, it is clearly a homomorphism. Since an elemen t of Hom( A ⊗ A, Z /n ) is a cocycle, the image from H 2 ( A, Z /n ) is at least as large as ∆ Hom( A ⊗ A, Z /n ) . The image from H 2 ( A, Z /n ) is a subset of the set alternating g , and w e conclude the ﬁnal statemen t of the lemma, using Lemma 5.1. □ 5.4. Extensions b y cen ter free groups. W e see there are unique extensions by cen ter free [ H ] -groups. Lemma 5.7. L et H b e a gr oup and let N b e a [ H ] -gr oup with trivial c enter. The c onjugation map N → Aut( N ) gives an exact se quenc e 1 → N → Aut( N ) × Out( N ) H → H → 1 , and the induc e d [ H ] -structur e on N agr e es with the given one. L et 1 → N → G → H → 1 b e an exact se quenc e of gr oups, such that the induc e d [ H ] -structur e on N agr e es with the given one. Then ther e is a unique isomorphism G → Aut( N ) × Out( N ) H c omp atible with the identity maps on N and H . Pr o of. Since N has trivial center N → Aut( N ) × Out( N ) H is an injection, and the quotient b y N is H , whic h prov es the ﬁrst claim. F or the second claim, we send g ∈ G to α ( G ) := ( c g , ¯ g ) ∈ Aut( N ) × Out( N ) H , where c g is the automorphism of N given by conjugation b y g and ¯ g is the image of g in H . They m ust hav e the same image in Out( G ) by the condition on the exact sequence. This map is compatible with the iden tity maps on N and H and it follo ws (via the short ﬁve lemma in the category of groups) that α : G → Aut( N ) × Out( N ) H is an isomorphism. An y other compatible automorphism G → Aut( N ) × Out( N ) H must hav e the form α ◦ β for β : G → G is an isomorphism compatible with the iden tit y maps on N and H . Then for g ∈ G and n ∈ N , w e ha ve g ng − 1 = β ( gng − 1 ) = β ( g ) nβ ( g ) − 1 . Also ¯ g = ¯ β g . Hence α ( g ) = α ( β ( g ) so α = α ◦ β . □ 5.5. Extensions of Γ -groups. Let Γ b e a group, H a Γ -group, and F a [ H ⋊ Γ] -group. W e consider an exact sequence 1 → F → G π → H → 1 of Γ -groups, in which the [ H ⋊ Γ] structure on F from the exact sequence agrees with the giv en [ H ⋊ Γ] structure. W e call such a thing an extension of Γ -groups of H by F . T wo such extensions 1 → F → G π → H → 1 and 1 → F → G ′ π ′ → H → 1 are isomorphic if there is an isomorphism of Γ -groups from G to G ′ that restricts to the iden tit y of F and H . W e write Ext Γ ( H , F ) for the set of isomorphism classes of suc h extensions. W e often write ( G, π ) for suc h an extension (even though the map F → G is also part of the data, we leav e it implicit). W e write Aut F,H ( G ) for the isomorphisms of such an extension, i.e. Γ -automorphisms of G that restrict to the identit y on F and H . 5.5.1. A b elian extensions. If Γ w ere trivial, then isomorphism classes of extensions of a group H by an ab elian H -group A exactly corresp ond to classes of H 2 ( G, A ) . W e seek a similar c haracterization of Γ -extensions b y cohomology , which turns out to b e simpliﬁed when | Γ | is relativ ely prime to | H | and | F | . 83 Lemma 5.8. L et Γ b e a gr oup, H a Γ -gr oup, and A an ab elian ( H ⋊ Γ) -gr oup. Supp ose | Γ | is r elatively prime to | H | and | A | . Ther e ther e is a bije ction b etwe en Ext Γ ( H , A ) and H 2 ( H ⋊ Γ , A ) taking A → G → H to A → G ⋊ Γ → H ⋊ Γ , with the natur al maps. Pr o of. W e consider an in verse to the giv en map. Let A → E π → H ⋊ Γ b e an extension of groups giving the correct ( H ⋊ Γ) -action on A . Let G = k er( E → Γ) . Then there is a splitting s : Γ → E by the Sc hur-Zassenhaus theorem. Moreo ver, by that theorem the image of the splitting in H ⋊ Γ must b e a conjugate of the trivial splitting, and so b y conjugating our original splitting, we can assume s lifts the trivial splitting of H ⋊ Γ → Γ . W e then ha ve group homomorphisms A → G → H . The Γ action on G given by conjuga- tion by s ( γ ) in E makes the map G → H Γ -equiv ariant. Moreo v er, A has the intended H ⋊ Γ action. Th us we ha ve given a map from H 2 ( H ⋊ Γ , A ) to Ext Γ ( H , A ) , though we should c hec k it is w ell-deﬁned. W riting E = G ⋊ Γ using our original section, if an alternate section also agrees with the trivial section to H ⋊ Γ , it must land in A ⋊ Γ . Hence b y Sc hur-Zassenhaus w e ha ve an a ∈ A suc h that our alternate section sends γ 7→ ( aγ ( a ) − 1 , γ ) . Th en, w e can c heck that conjugation b y a giv es a map G → G , that is the iden tity on A and H , taking the original Γ -action to the Γ -action that is conjugation b y this alternate section. So w e hav e constructed the same isomorphism class in Ext Γ ( H , A ) . It is straightforw ard to chec k that these tw o maps are inv erses, and hence we’v e given a bijection from Ext Γ ( H , A ) to H 2 ( H ⋊ Γ , A ) . □ In the setting of Lemma 5.8, we then often consider H 2 ( H ⋊ Γ , A ) as parametrizing the elemen ts of Ext Γ ( H , A ) . F or a set L of isomorphism classes of Γ -groups, w e write H 2 ( H ⋊ Γ , A ) L for the elements of H 2 ( H ⋊ Γ , A ) that corresp ond to Γ -group extensions of H b y A whose underlying Γ -group is in L . The following lemma is straigh tforward to chec k. Lemma 5.9. L et Γ b e a gr oup and H a Γ -gr oup. L et L b e a set of isomorphism classes of Γ -gr oups, close d under taking ﬁb er pr o ducts and quotients, L et A b e a simple ab elian ( H ⋊ Γ) -gr oup. Then κ := End H ⋊ Γ ( A ) (i.e. ( H ⋊ Γ) -gr oup morphisms fr om A to A ) is a ﬁeld. Mor e over, H 2 ( H ⋊ Γ , A ) is natur al ly a κ -ve ctor sp ac e thr ough the κ action on A and H 2 ( H ⋊ Γ , A ) L is a κ -subsp ac e of H 2 ( H ⋊ Γ , A ) as long as it is nonempty. L et B and C b e ab elian ( H ⋊ Γ) -gr oups. The isomorphism H 2 ( H ⋊ Γ , B × C ) → H 2 ( H ⋊ Γ , B ) × H 2 ( H ⋊ Γ , C ) takes H 2 ( H ⋊ Γ , B × C ) L to H 2 ( H ⋊ Γ , B ) L × H 2 ( H ⋊ Γ , C ) L . Pr o of. κ is a ﬁeld since any non-zero non-inv ertible element of κ would ha ve image a nontrivial prop er subspace, whic h is imp ossible, so κ is a division algebra, and every ﬁnite division algebra is a ﬁeld. The bijection b etw een H 2 ( H ⋊ Γ , A ) and Γ -group extensions of H b y A sends addition on H 2 ( H ⋊ Γ , A ) to the Baer sum, deﬁned by taking ﬁber pro ducts of H and quotienting by the an tidiagonally embedded A . Since L is closed under taking ﬁb er pro ducts and quotien ts, thet Baer sum is in L if the individual extensions are. Th us H 2 ( H ⋊ Γ , A ) L is a subspace of a ﬁnite group closed under addition, hence it is a subgroup unless it is empt y . The action of inv ertible elemen ts of κ on H 2 ( H ⋊ Γ , A ) sends an extension to an extension whic h is isomorphic as a Γ -group but with the map from A , part of the data of the extension, 84 comp osed with an inv ertible endomorphism of A . Since the extension is isomorphic as a Γ - group to the original, it is in L if the original extension is. Th us H 2 ( H ⋊ Γ , A ) L is κ -stable, and hence a κ -subspace, as long as H 2 ( H ⋊ Γ , A ) L is nonempt y . The isomorphism H 2 ( H ⋊ Γ , B × C ) → H 2 ( H ⋊ Γ , B ) × H 2 ( H ⋊ Γ , C ) sends an extension of H b y B × C to a pair consisting of its quotient by C and its quotient by H . The in verse map sends a pair of extension b y B and C resp ectively to their ﬁb er pro ducts. Since L is stable under ﬁb er pro ducts and quotients, the extension by B × C is in L if and only if b oth the extension by B and the extension b y C are. □ Lemma 5.10. If Γ is a gr oup, and H is a Γ -gr oup with H Γ = 1 , and A is an ab elian ( H ⋊ Γ) -gr oup, and A → G → H is an extension of Γ -gr oups such that the induc e d H ⋊ Γ action on A is as given, then G Γ is a quotient of A H ⋊ Γ . Pr o of. Since H Γ = 1 , it follows that the normal subgroup A of G surjects on to G Γ , and in particular that G Γ is ab elian. Hence any a ∈ A , and lift ˜ h to G of an element h ∈ H , h a ve images that comm ute in G Γ . So the image of A in G Γ factors through A H ⋊ Γ , and is all of G Γ . □ 5.5.2. Non-ab elian extensions of Γ -gr oups. Lemma 5.11. L et Γ b e a gr oup, H b e a Γ -gr oup, and N a [ H ⋊ Γ] -gr oup with trivial c enter. L et 1 → N → G → H → 1 b e an exact se quenc e of gr oups with induc e d [ H ] -gr oup structur e on N agr e eing with the given [ H ] -gr oup structur e. Ther e is a bije ction fr om (1) Γ actions on G such that the given map G → H is Γ - e quivariant and the induc e d Γ -gr oup structur e on N r e duc es to the given [Γ] -structur e on N , and (2) lifts of the given map Γ → Out( N ) to Γ → Aut( N ) (r esp e cting the map Aut( N ) → Out( N ) ). This bije ction is given by taking the induc e d Γ -gr oup structur e on N . Pr o of. By Lemma 5.7, w e may assume G = Aut( N ) × Out( N ) H . W e giv e an in v erse to the giv en map. Given a map Γ → Aut( N ) , lifting Γ → Out( N ) , w e get a homomorphism Γ → Aut( N ) × Out( N ) ( H ⋊ Γ) , using the identit y map of Γ for the second co ordinate. The group G is a normal subgroup of ˜ G := Aut( N ) × Out( N ) ( H ⋊ Γ) , and thus w e get an action of Γ on G by conjugation in ˜ G . W e can c heck that this action mak es G → H a Γ -equiv arian t map, and the induced Γ -group structure on N is compatible with the giv en [Γ] -group structure. W e now chec k these constructions are in verses. Let α : Γ → Aut( G ) b e an action as in (1). If α ( γ )( c n , 1) = ( c β ( γ )( n ) , 1) for γ ∈ Γ and n ∈ N , then β : Γ → Aut( N ) is the given map in (2). Then w e use β to obtain α ′ : Γ → Aut( G ) such that for γ ∈ Γ and σ ∈ Aut( N ) and h ∈ H , w e hav e α ′ ( γ )( σ , h ) = ( β ( γ ) , γ )( σ , h )( β ( γ ) − 1 , γ − 1 ) = ( β ( γ ) σ β ( γ ) − 1 , γ ( h )) . If σ = c n for some n ∈ N , note that β ( γ ) c n β ( γ ) − 1 = c β ( γ )( n ) W e claim α ( γ ) and α ′ ( γ ) are the same element in Aut( G ) . This is b ecause they give the same action on H , and the same action on N , and so by Lemma 5.7 they m ust b e the same. F or the other direction, we start with β : Γ → Aut( N ) , and obtain α ′ as ab o ve. W e’v e already seen abov e that the action coming from α ′ on N agrees with β . This shows that the given construction is a bijection. □ 85 6. Anal ysis of E 3 0 , 2 in the L yndon-Hochschild-Serre spectral sequence In this section, we giv e a detailed analysis of the E 3 0 , 2 term in the Lyndon-Ho chsc hild- Serre sp ectral sequence, esp ecially for an extension of H by a semisimple ab elian H -group. This will b e crucial for our determination of a distribution from its moments, but strictly sp eaking, the analysis we do is not restricted to the context of this problem. 6.1. H 2 ( A, Z /n ) . First, we must carefully understand H 2 ( A, Z /n ) for an ab elian group A of exp onen t dividing n . Recall the deﬁnition of ∆ and ∧ Hom( A ⊗ A, Z /n ) from Lemma 5.1. Lemma 6.1. L et n b e a p ositive inte ger and let A b e a ﬁnite ab elian gr oup of exp onent dividing n , and such that 4 do es not divide the exp onent of A . Then ther e is an Aut( A ) - e quivariant inje ctive Bo ckstein homomorphism B : Hom( A, Z /n ) → H 2 ( A, Z /n ) , and an Aut( A ) -e quivariant morphism using biline ar forms as c o chains C : Hom( A ⊗ A, Z /n ) → H 2 ( A, Z /n ) . When n is o dd or 4 | n , these maps induc e an isomorphism B × ¯ C : Hom( A, Z /n ) × ∧ Hom( A ⊗ A, Z /n ) ≃ H 2 ( A, Z /n ) . If n = 2 , the map C induc es an isomorphism Sym 2 Hom( A, Z / 2) ≃ H 2 ( A, Z / 2) . F or any n , the map C induc es an isomorphism ¯ ¯ C : ∧ Hom( A ⊗ A, Z /n ) → H 2 ( A, Z /n ) / im B and the map ∆ induc es an isomorphism ∆ : H 2 ( A, Z /n ) / im B → ∆ Hom( A ⊗ A, Z /n ) . Pr o of. If for eac h elemen t of s ∈ Z /n w e c ho ose a lift ˜ s ∈ Z /n 2 , then for a elemen t ϕ ∈ Hom( A, Z /n ) , w e can send v to a 2 -co cycle f ( a, b ) := g ϕ ( a )+ g ϕ ( b ) − ^ ϕ ( a + b ) n . This induces the map H 1 ( A, Z /n ) → H 2 ( A, Z /n ) from the Bo ckstein homomorphism for the short exact sequence 1 → Z /n → Z /n 2 → Z /n → 1 , and th us B is injectiv e since the exp onen t of A divides n . An y element of Hom( A ⊗ A, Z /n ) naturally gives a 2 -co chain A 2 → Z /n , whic h is au- tomatically a 2 -co cycle, giving the map Hom( A ⊗ A, Z /n ) → H 2 ( A, Z /n ) . The 2 -co c hains A 2 → Z /n that are cob oundaries are symmetric in the t wo A co ordinates. Th us an y bilinear form in the kernel of C is symmetric since it is a coboundary of a 1- co cycle. W e claim that if n is o dd or 4 | n then any symmetric elemen t of Hom( A ⊗ A, Z /n ) is in the kernel of Hom( A ⊗ A, Z /n ) → H 2 ( A, Z /n ) , and that if n ≡ 2 (mo d 4) then any symmetric elemen t of Hom( A ⊗ A, Z /n ) is in the k ernel of Hom( A ⊗ A, Z /n ) → H 2 ( A, Z /n ) / im B . By factoring Z /n Z we may assume n is either o dd or a p o w er of 2 . Giv en M 0 ∈ Hom( A ⊗ A, Z /n ) , then if we let M ( a ⊗ b ) := M 0 ( a ⊗ b ) + M 0 ( b ⊗ a ) for a, b ∈ A , then M ( a ⊗ b ) = − M 0 ( a ⊗ a ) − M 0 ( b ⊗ b ) + M 0 (( a + b ) ⊗ ( a + b )) and so M is a cob oundary . If n is o dd, for any symmetric elemen t M ∈ Hom( A ⊗ A, Z /n ) , w e can take M 0 = (1 / 2) M , and so M is a cob oundary . If n is even, let A 2 b e the maximal ab elian 2 -group quotient of A , which b y assumption is elemen tary ab elian. W e c ho ose a basis of A 2 , and if the matrix corresp onding to M is 0 along the diagonal then w e can choose 86 M 0 to b e the upp er triangular part of M and see that M is a cob oundary . So the claim will follo w if w e can sho w that the M 1 ∈ Hom( A 2 ⊗ A 2 , Z /n ) whose matrix has a single n/ 2 in the upp er left corner (and all other entries 0 ) maps to 0 in H 2 ( A, Z /n ) or H 2 ( A, Z /n ) / im B (as required). If 4 | n , w e let g : A → Z /n Z b e such that g ( a ) = n/ 4 for a ∈ A whose image in A 2 , written in our chosen basis, contains a copy of the ﬁrst basis v ector e 1 , and g ( a ) = 0 otherwise. W e can c heck, for all a, b ∈ A , that g ( a ) + g ( b ) − g ( a + b ) = M 1 ( a, b ) b y considering separately the cases that the images of a, b in A 2 b oth con tain e 1 , neither con tain e 1 , or exactly one of them con tains e 1 . Thus if 4 | n , w e ha ve that M 1 maps to 0 in H 2 ( A, Z /n ) . If n = 2 , then we can let ϕ ∈ Hom( A, Z /n ) b e the homomorphism suc h that ϕ ( a ) is 1 if a con tains e 1 and 0 otherwise. W e pic k a lift Z /n → Z /n 2 Z such that e 0 = 0 and e 1 = 1 . W e can chec k, for all a, b ∈ A , that g ϕ ( a ) + g ϕ ( b ) − ^ ϕ ( a + b ) n = M 1 ( a, b ) b y considering separately the cases that the images of a, b in A 2 b oth con tain e 1 , neither con tain e 1 , or exactly one of them contains e 1 . Th us if 2 | n , but 4 ∤ n ,we hav e that M 1 maps to im B in H 2 ( A, Z /n ) . If n is o dd or 4 | n , we ha ve seen ab ov e that the kernel of C is precisely the symmetric forms and hence ¯ C is injective. Also im ¯ C do es not con tain the class of any symmetric non trivial 2 -co cycle and th us im ¯ C ∩ im B = 0 . When 2 | n, but 4 ∤ n , we consider C ′ : Hom( A ⊗ A, Z /n ) → H 2 ( A, Z /n ) / im B . An y 2 -co cycle represent ing an element trivial on the righ t-hand side must b e symmetric, since it is a cob oundary plus an element of im B . W e ha ve also seen that all the symmetric forms in Hom( A ⊗ A, Z /n ) are in the kernel of C ′ , so the k ernel must b e precisely the symmetric forms in Hom( A ⊗ A, Z /n ) . This pro ves that ¯ ¯ C is injective. F or any n , w e can use the univ ersal co eﬃcient theorem to compute | H 2 ( A, Z /n ) | = | Hom( A, Z /n ) || ∧ Hom( A ⊗ A, Z /n ) | . When n is o dd or 4 | n , we then see b y counting that B × ¯ C is an isomorphism. F or any n , w e hav e b y counting that ¯ ¯ C is an isomorphism. W e hav e from Lemma 5.1 that the comp osite ∧ Hom( A ⊗ A, Z /n ) ¯ ¯ C → H 2 ( A, Z /n ) / im B → H 2 ( A, Z /n ) / im B ∆ → ∆ Hom( A ⊗ A, Z /n ) . is an isomorphism, which implies that ∆ : H 2 ( A, Z /n ) / im B → ∆ Hom( A ⊗ A, Z /n ) is an isomorphism. When n = 2 , w e ha v e that Sym 2 Hom( A, Z / 2 Z ) is the quotien t of Hom( A ⊗ A, Z / 2) = Hom( A, Z / 2 Z ) ⊗ Hom( A, Z / 2 Z ) by ∧ 2 Hom( A, Z / 2 Z ) , the elements corresponding to ma- trices o ver F 2 that are symmetric and 0 along the diagonal. W e ha ve seen ab o ve that elemen ts corresp onding to matrices o ver F 2 that are symmetric and 0 along the diagonal are in ker C . Moreov er, all elemen ts in ker C corresp ond to symmetric matrices, and it follo ws from that ab o ve that an y non-zero ρ ∈ Hom( A, Z / 2 Z ) ⊗ Hom( A, Z / 2 Z ) corresp ond- ing to a non-zero diagonal matrix has C ( ρ ) equal to a non-zero element of im B . Thus k er C = ∧ 2 Hom( A, Z / 2 Z ) , whic h implies that C induces an injection Sym 2 Hom( A, Z / 2 Z ) ⊗ 87 Hom( A, Z / 2 Z ) → H 2 ( A, Z / 2) , and the injection m ust be an isomorphism since the t w o groups are the same size. □ 6.2. F unctorialit y. W e will need to use the functoriality of the Lyndon-Hochsc hild-Serre sp ectral sequence. Lemma 6.2. L et n, m b e inte gers with a given map π : Z /m → Z /n. L et 1 F G H 1 1 F ′ G ′ H ′ 1 ρ ψ b e a c ommutative diagr am of gr oups with b oth r ows exact. Then the diﬀer entials in the Lyndon-Ho chschild-Serr e sp e ctr al se quenc es ( E , d ) , ( E ′ , d ′ ) c omputing H p + q ( G, Z /n ) and H p + q ( G ′ , Z /m ) , r esp e ctively, fr om H p ( H , H q ( F , Z /n )) and H p ( H , H q ( F ′ , Z /m )) , r esp e ctively, ar e c omp atible with the pul lb ack map τ induc e d by ρ , ψ , and π , τ : H p ( H ′ , H q ( F ′ , Z /m )) → H p ( H , H q ( F , Z /n )) , i.e. for al l r ≥ 2 , τ ( d ′ ) p,q r = d p,q r τ , wher e pul lb ack maps τ on p ages p ast the se c ond p age ar e wel l-deﬁne d by the c ommutativity of these diagr ams on pr evious p ages. Pr o of. The lemma follo ws in a straigh tforward wa y from the deﬁnition of the sp ectral se- quence, e.g. see [SW24, Lemma 7.11]. □ 6.3. Analysis of the diﬀeren tials. W e will work to relate the diﬀeren tials to concrete constructions. The ﬁrst tw o lemmas will describ e d 0 , 2 2 . Recall a 2 -co chain f is normalize d if f ( a, a ′ ) = 0 whenever a = 0 or a ′ = 0 . Lemma 6.3. L et H b e a ﬁnite gr oup, n a p ositive inte ger, and A a ﬁnite H -mo dule, of exp onent dividing n . L et f : A 2 → Z /n Z b e a normalize d 2-c o cycle such that its class [ f ] ∈ H 2 ( A, Z /n Z ) is H -invariant. F or e ach h ∈ H , let c h : A → Z /n b e a normalize d c o chain such that f ( h − 1 ( a ) , h − 1 ( b )) − f ( a, b ) = c h ( a ) + c h ( b ) − c h ( a + b ) for al l a, b ∈ A , and c 1 ( a ) = 0 for al l a ∈ A , and for h, i ∈ H with the same action on A , we have c h = c i . L et E b e the extension of A by Z /n c orr esp onding to the class [ f ] . L et A b e the gr oup of au- tomorphisms of E that ﬁx Z /n p ointwise and act on A thr ough the image ¯ H of H in Aut( A ) . Then A is an extension of ¯ H by Hom( A, Z /n ) with extension class in H 2 ( ¯ H , Hom( A, Z /n )) r epr esente d by the c o cycle ( h, i ) 7→ c h + c i ◦ h − 1 − c hi , with the action on ¯ H on Hom( A, Z /n )) induc e d by the action of H on A . W e will pro ve Lemma 6.3 in the course of the pro of of Lemma 6.4. W e let d 0 , 2 2 , 2 : H 2 ( A, Z /n Z ) H → H 2 ( H , Hom( A, Z /n )) denote the homomorphism sending [ f ] to the co cycle ( h, i ) 7→ c h + c i ◦ h − 1 − c hi from Lemma 6.3. Note that d 0 , 2 2 , 2 v anishes on classes represen ted by H -in v ariant co cycles. Also, note that d 0 , 2 2 , 2 only dep ends on H , A , n , and the H -action on A . 88 Lemma 6.4. L et n b e a p ositive inte ger. L et 1 → A → G → H → 1 b e a short exact se quenc e of gr oups with A ab elian of exp onent dividing n , and let G have extension class in H 2 ( H , A ) r epr esente d by a c o cycle α : H 2 → A. L et f : A 2 → Z /n Z b e a normalize d 2-c o cycle such that its class [ f ] ∈ H 2 ( A, Z /n Z ) is H -invariant. W e deﬁne a c o chain d 0 , 2 2 , 1 ( f ) : H 2 → Hom( A, Z /n ) : d 0 , 2 2 , 1 ( f )( h, i )( a ) := f ( a, α ( h, i )) − f ( α ( h, i ) , a ) = ∆ f ( α ( h, i ) , a ) , which induc es a homomorphism d 0 , 2 2 , 1 : H 2 ( A, Z /n Z ) H → H 2 ( H , Hom( A, Z /n )) . In the Lyndon-Ho chschild-Serr e sp e ctr al se quenc e c omputing H ∨ ( G, Z /n ) fr om H p ( H , H q ( A, Z /n )) , the diﬀer ential d 0 , 2 2 : H 2 ( A, Z /n ) H → H 2 ( H , Hom( A, Z /n )) satisﬁes d 0 , 2 2 = d 0 , 2 2 , 1 + d 0 , 2 2 , 2 . If α = 0 , then d 0 , 2 3 = 0 . L et B : H 2 ( H , Z /n ) → H 3 ( H , Z /n ) , b e the Bo ckstein map, which is deﬁne d as the b oundary map c oming fr om the exact se quenc e 0 → Z /n → Z /n 2 → Z /n → 0 of c o eﬃcients. If we use the inje ction B : Hom( A, Z /n ) H → H 2 ( A, Z /n ) H fr om L emma 6.1, for ϕ ∈ Hom( A, Z /n ) H , we have d 0 , 2 2 ( B ( ϕ )) = d 0 , 2 2 , 1 ( B ( ϕ )) = d 0 , 2 2 , 2 ( B ( ϕ )) = 0 and d 0 , 2 3 ([ B ( ϕ )]) is the image of [ α ] ∈ H 2 ( H , A ) under ϕ ∗ : H 2 ( H , A ) → H 2 ( H , Z /n ) and then B . R emark 6.5 . W e will iden tify Hom( A ⊗ A, Z /n Z ) with Hom( A, Hom( A, Z /n )) such that f ∈ Hom( A ⊗ A, Z /n Z ) sends a to the map that sends b to f ( a ⊗ b ) . Recall from Lemma 5.6, for f ∈ H 2 ( A, Z /n ) , w e ha v e ∆ f ∈ Hom( A ⊗ A, Z /n ) , whic h w e now also view as an element of Hom( A, Hom( A, Z /n )) . Then [ d 0 , 2 2 , 1 ( f )] ∈ H 2 ( H , Hom( A, Z /n )) is the image of α under the induced map (∆ f ) ∗ : H 2 ( H , A ) → H 2 ( H , Hom( A, Z /n )) . Pr o of. Huebschmann [Hue81] describ es some of the diﬀeren tials in the sp ectral sequence in a more general situation than we ha v e here. Let (6.6) 0 → Z /n → E → A → 1 b e the cen tral extension corresp onding to [ f ] ∈ H 2 ( A, Z /n ) H . Since f is normalized, and we can write E , as a set, as Z /n × A , where for j, k ∈ Z /n and a, b ∈ A , w e hav e ( j, a ) · E ( k , b ) = ( j + k + f ( a, b ) , a + b ) . Let ¯ H b e the subgroup of Aut( A ) that is the image of the action of H on A from the giv en exact sequence. Using Huebsc hmann’s deﬁnition [Hue81, Section 2.1] of Aut( A, Z /n ) , when A acts trivially on Z /n , w e can see that Aut( A, Z /n ) = Aut( A ) × Aut( Z /n ) . Also, Huebsc hmann’s Aut G ( A, Z /n ) is ¯ H ⊂ Aut( A ) × 1 ⊂ Aut( A, Z /n ) . F urther, Huebschmann’s Aut Z /n H ( E ) is the automorphisms of E that are the iden tity on Z /n and act on A via ¯ H . The kernel of the map Aut Z /n H ( E ) → ¯ H (pushing forw ard the automorphism to A ) is Hom( A, Z /n ) (as in Lemma 7.11), so w e hav e an exact sequence 1 → Hom( A, Z /n ) → Aut Z /n H ( E ) → ¯ H → 1 . 89 Since c h only dep ends on the action of h on A , w e can extend the notation so that for σ ∈ ¯ H we write c σ := c h for any h ∈ H with the same action on A as σ . F urther, for g ∈ G , we let c g := c ¯ g , where ¯ g is the image of g in H (or ¯ H ). W e ha ve a set-theoretic section ¯ H → Aut Z /n H ( E ) , taking σ ∈ ¯ H to the map taking ( j, a ) 7→ ( j + c σ ( σ ( a )) , σ ( a )) for all ( j, a ) ∈ E . W e can c heck this map is an automorphism of E using the deﬁnition of c h . This allow us to write Aut Z /n H ( E ) as a set theoretic pro duct Hom( A, Z /n ) × ¯ H , and for ϕ ∈ Hom( A, Z /n ) and σ ∈ Aut H , w e write [ ϕ, σ ] for the element of Aut Z /n H ( E ) suc h that [ ϕ, σ ]( j, a ) = ( j + ϕ ( σ ( a )) + c σ ( σ ( a )) , σ ( a )) for all ( j, a ) ∈ E . F or ϕ, ϕ ′ ∈ Hom( A, Z /n ) and σ, σ ′ ∈ Aut H , we can compute [ ϕ, σ ][ ϕ ′ , σ ′ ] b y seeing where it sends (0 , a ) ∈ E . W e hav e [ ϕ, σ ][ ϕ ′ , σ ′ ](0 , a ) = [ ϕ, σ ]( ϕ ′ ( σ ′ ( a )) + c σ ′ ( σ ′ ( a )) , σ ′ ( a )) = ( ϕ ′ ( σ ′ ( a )) + c σ ′ ( σ ′ ( a )) + ϕ ( σ ( σ ′ ( a ))) + c σ ( σ ( σ ′ ( a ))) , σ ( σ ′ ( a ))) = [ ϕ ′ σ − 1 + ϕ + c σ ′ σ − 1 + c σ − c σ σ ′ , σ σ ′ ](0 , a ) , and th us [ ϕ, σ ][ ϕ ′ , σ ′ ] = [ ϕ ′ σ − 1 + ϕ + c σ ′ σ − 1 + c σ − c σ σ ′ , σ σ ′ ] . (This pro ves Lemma 6.3.) W e next chec k how elemen ts of E act on E by conjugation. F or a ∈ A and (0 , a ) ∈ E , w e ha ve (0 , a ) − 1 = ( − f ( a, − a ) , − a ) and so for a, b ∈ A , (0 , a ) · E (0 , b ) · E (0 , a ) − 1 = ( f ( a, b ) , a + b ) · E ( − f ( a, − a ) , − a ) = ( f ( a, b ) − f ( a, − a )+ f ( a + b, − a ) , b ) . By the co cycle condition and f b eing normalized, f ( a, − a ) − f ( b + a, − a ) − f ( b, a ) = 0 . Let F a ( b ) := f ( a, b ) − f ( b, a ) for all a, b ∈ A . Then conjugation b y (0 , a ) , written as an element of Aut Z /n H ( E ) , is [ F a , 1] . (This prov es the claim in Lemma 5.6 that a ⊗ b 7→ f ( b, a ) − f ( a, b ) is a homomorphism.) Since Z /n is central in E , conjugation by ( j, a ) is the same as conjugation b y (0 , a ) . Huebsc hmann’s Aut G ( e ) := Aut Z /n H ( E ) × ¯ H G , is a set theoretic pro duct Hom( A, Z /n ) × G and has elements { ϕ, g } for ϕ ∈ Hom( A, Z /n ) and g ∈ G , with m ultiplication { ϕ, g }{ ϕ ′ , g ′ } = { ϕ ′ g − 1 + ϕ + c g ′ g − 1 + c g − c g g ′ , g g ′ } . This uses our earlier expression of Aut Z /n H ( E ) as a set-theoretic pro duct of Hom( A, Z /n ) and H . The action of an element of Aut G ( e ) on E is given by the form ula { ϕ, g } ◦ ( j, a ) = ( j + ϕ ( g ( a )) + c g ( g ( a )) , g ( a )) . Using that h 1 = 1 and c 1 = 0 , w e can chec k that { 0 , g } − 1 = {− c g g − c g − 1 , g − 1 } . Huebsc hmann’s β : E → Aut G ( e ) sends ( j, a ) ∈ E to the element of Aut Z /n H ( E ) × ¯ H G whose ﬁrst co ordinate is conjugation b y ( j, a ) and whose second co ordinate is a . In our notation ab o ve, this means β ( j, a ) = { F a , a } . 90 Huebsc hmann’s Out G ( e ) := Aut G ( e ) /β ( E ) is then the extension 1 → Hom( A, Z /n ) → Out G ( e ) → H → 1 that giv es the image of d 0 , 2 2 according to [Hue81, Theorem 1]. Let s : H → G b e a set theoretic section of the quotient G → H suc h that for h, i ∈ G , we ha ve s ( h ) s ( i ) = α ( s, i ) s ( hi ) . Then for h ∈ H , we can lift it to { 0 , s ( h ) } β ( E ) in Out G ( e ) . Then w e can compute a co cycle representing d 0 , 2 2 ([ f ]) by computing, for h, i ∈ H , β ( E ) { 0 , s ( h ) } β ( E ) { 0 , s ( i ) } β ( E ) { 0 , s ( hi ) } − 1 = β ( E ) { c i h − 1 + c h − c hi , s ( h ) s ( i ) }{ 0 , s ( hi ) } − 1 = { c i h − 1 + c h − c hi , s ( h ) s ( i ) }{− c hi hi − c ( hi ) − 1 , s ( hi ) − 1 } = β ( E ) {− c hi − c ( hi ) − 1 ( hi ) − 1 + c i h − 1 + c h − c hi + c ( hi ) − 1 ( hi ) − 1 + c hi − c α ( h,i ) , α ( h, i ) } = β ( E ) {− c hi + c i h − 1 + c h , α ( h, i ) } = β ( E ) { F α ( h,i ) , α ( h, i ) }{− F α ( h,i ) − c hi + c i h − 1 + c h , 1 } = β ( E ) {− F α ( h,i ) − c hi + c i h − 1 + c h , 1 } . W e used ab o ve that α ( h, i ) has image 1 in H , and so it acts trivially on A and c α ( h,i ) = 0 . W e th us conclude d 0 , 2 2 = d 0 , 2 2 , 1 + d 0 , 2 2 , 2 . The fact that d 0 , 2 2 and d 0 , 2 2 , 2 are well-deﬁned on classes in H 2 ( A, Z /n ) H (and not just co cycles) implies the same for d 0 , 2 2 , 1 . The fact that d 0 , 2 2 and d 0 , 2 2 , 2 ha ve image in H 2 ( H , Hom( A, Z /n )) implies the same for d 0 , 2 2 , 1 . No w supp ose that d 0 , 2 2 ([ f ]) = 0 . Then we can c ho ose a normalized co chain H → Hom( A, Z /n ) taking h 7→ v h suc h that for all h, i ∈ H , we hav e − v i h − 1 − v h + v hi = − F α ( h,i ) − c hi + c i h − 1 + c h . Then w e hav e a section Ψ : H → Aut G ( e ) taking h 7→ { v h , s ( h ) } . W e will chec k that the induced map ¯ Ψ : H → Out G ( e ) := Aut G ( e ) /β ( E ) is a homomorphism. F or h, i ∈ H , w e ha ve Ψ( h )Ψ( i ) = { v h , s ( h ) }{ v i , s ( i ) } = { v h + v i h − 1 + c i h − 1 + c h − c hi , s ( h ) s ( i ) } whereas β (0 , α ( h, i ))Ψ( hi ) = β (0 , α ( h, i )) { v hi , s ( hi ) } = { F α ( h,i ) , α ( h, i ) }{ v hi , s ( hi ) } = { F α ( h,i ) + v hi , α ( h, i ) s ( hi ) } . Th us Ψ( h )Ψ( i ) = β (0 , α ( h, i ))Ψ( hi ) b y our choice of v , and ¯ Ψ is a homomorphism. Using ¯ Ψ , we can form a ﬁb er pro duct Aut G ( e ) × Out G ( e ) H , which ﬁts in an exact sequence 0 → Z /n → E β → Aut G ( e ) × Out G ( e ) H → H → 1 . The result [Hue81, Theorem 3] sho ws that the abov e is a crossed 2 -fold extension, where Aut G ( e ) × Out G ( e ) H acts on E via the Aut G ( e ) co ordinate and its pro jection onto Aut Z /n H ( E ) . F urther, by the corresp ondence b etw een crossed 2 -fold extensions and H 3 ( H , Z /n Z ) , [Hue81, Theorem 3] says the ab ov e corresp onds to d 0 , 2 3 ([ f ]) . There is a co cycle description ♣♣♣ Melanie: [to check Huebschmann agrees] of the class in H 3 ( H , Z /n ) asso ciated to any crossed 2 -fold extension given in [Bro82, Chapter 4, Section 91 5]. W e use the section of Aut G ( e ) × Out G ( e ) H → H sending h 7→ (Ψ( h ) , h ) as Bro wn’s s . Note that, for h, i ∈ H , in Aut G ( e ) × Out G ( e ) H , w e hav e (Ψ( h ) , h )(Ψ( i ) , i ) = (Ψ( h )Ψ( i ) , hi ) = ( β (0 , α ( h, i ))Ψ( hi ) , hi ) = β (0 , α ( h, i ))(Ψ( hi ) , hi ) and so Bro wn’s F ( h, i ) can b e taken to b e (0 , α ( h, i )) . Then the co cycle H 3 → Z /n asso ciated to our extension ab ov e, and hence d 0 , 2 3 ([ f ]) , for g , h, i ∈ H sends ( g , h, i ) 7→ ( { v g , s ( g ) } ◦ (0 , α ( h, i )))(0 , α ( g , hi ))(0 , α ( g h, i )) − 1 (0 , α ( g , h )) − 1 (6.7) =( v g ( g ( α ( h, i ))) + c g ( g ( α ( h, i ))) , g ( α ( h, i )))(0 , α ( g , hi ))(0 , α ( g h, i )) − 1 (0 , α ( g , h )) − 1 . The individual terms of the ab ov e pro duct are in E , but α b eing a 2 -co cycle from H 2 → A implies that the pro duct has trivial image in A and hence is in Z /n Z . In particular, we see that if α = 0 , then d 0 , 2 3 ([ f ]) = 0 . F or f in the image of Hom( A, Z /n ) H : If f is the image of ϕ ∈ Hom( A, Z /n ) H under the map Hom( A, Z /n ) H → H 2 ( A, Z /n ) H from Lemma 6.1, then note that w e can take c h = 0 for all h ∈ H . Thus d 0 , 2 2 , 2 ( f ) = 0 . F urther, we ha ve F a = 0 for all a ∈ A . Th us from the description of d 0 , 2 2 ab o ve w e ha v e d 0 , 2 2 ([ f ]) = 0 , and w e can take v h = 0 for all h ∈ H . Note an in v erse form ula for E is giv en by ( j, a ) − 1 = ( − j − f ( a, − a ) , − a ) . Then the co cycle (6.7) sends ( g , h, i ) to (0 , g ( α ( h, i )))(0 , α ( g , hi ))(0 , α ( g h, i )) − 1 (0 , α ( g , h )) − 1 =(0 , g ( α ( h, i )))(0 , α ( g , hi )) · ( − f ( α ( g h, i ) , − α ( g h, i )) , − α ( g h, i ))( − f ( α ( g , h ) , − α ( g , h )) , − α ( g , h )) . Let a = g ( α ( h, i )) and b = α ( g , hi ) and c = − α ( g h, i ) and d = − α ( g , h ) . Since the image of the ab o ve co cycle (6.7) is in Z /n , we hav e a + b + c + d = 0 . The image of the ab o ve co cycle (6.7) has Z /n Z co ordinate f ( a, b ) + f ( a + b, c ) + f ( a + b + c, d ) − f ( − c, c ) − f ( − d, d ) . Since f ( x, y ) = d ϕ ( x )+ d ϕ ( y ) − \ ϕ ( x + y ) n for some lift k 7→ b k from Z /n Z to Z /n 2 Z with b 0 = 0 , we ha v e f ( a, b ) + f ( a + b, c ) + f ( a + b + c, d ) − f ( − c, c ) − f ( − d, d ) = d ϕ ( a ) + d ϕ ( b ) − \ ϕ ( − c ) − \ ϕ ( − d ) n . Since ϕ is H -inv ariant, we ha ve ϕ ( a ) = ϕ ( α ( h, i )) . Thus the co cycle asso ciated to the crossed 2 -fold extension is (6.8) ( g , h, k ) 7→ \ ϕ ( α ( h, i )) + \ ϕ ( α ( g , hi )) − \ ϕ ( α ( g h, i )) − \ ϕ ( − α ( g , h )) n On the other hand, if w e start with α : H 2 → A , and then apply ϕ ∈ Hom( A, Z /n Z ) to obtain ϕ ∗ ( α ) ∈ H 2 ( H , Z /n Z ) , and then apply the Bo c kstein connecting homomorphism to ϕ ∗ ( α ) , w e obtain the co cycle ( g , h, k ) 7→ \ ϕ ( α ( h, i )) − \ ϕ ( α ( g h, i )) + \ ϕ ( α ( g , hi )) − \ ϕ ( α ( g , h )) n , agreeing with Equation (6.8) and proving the claim ab out d 0 , 2 3 ([ f ]) for f in the image of Hom( A, Z /n ) H . □ 92 W e now relate the v anishing of d 0 , 2 2 , 2 to a group theoretic condition. Recall the map from Lemma 5.6, ∆ : H 2 ( V , Z /n Z ) → ∆ Hom( V ⊗ V , Z /n Z ) . If V is an F 2 v ector space, then any homomorphism V ⊗ V → Z /n Z lands in ( n/ 2) Z /n Z ≃ F 2 , and th us we ha v e an isomorphism ∆ Hom( V ⊗ V , Z /n Z ) ≃ ∧ 2 V ∨ . W e use ∆ to also denote the comp osite H 2 ( V , Z /n Z ) → ∧ 2 V ∨ . Lemma 6.9. L et H b e a ﬁnite gr oup, V a ﬁnite-dimensional r epr esentation of H over F 2 , and let ω ∈ ( ∧ 2 V ∨ ) H b e non-zer o. Henc e H ’s action on V is thr ough Sp( V ) . L et n b e a p ower of 2 . If n = 2 , we also assume V is F 2 -ortho gonal. L et ∆ − 1 ( ω ) b e a pr eimage of ω in H 2 ( V , Z /n ) H . Then for the map d 0 , 2 2 , 2 : H 2 ( V , Z /n Z ) H → H 2 ( H , Hom( V , Z /n )) deﬁne d ab ove, d 0 , 2 2 , 2 (∆ − 1 ( ω )) = 0 if and only if the map H → Sp( V ) factors thr ough the aﬃne symple ctic gr oup ASp( V ) → Sp( V ) . R emark 6.10 . If V is an irreducible representation of H o v er F 2 , then for any tw o non- zero ω, ω ′ ∈ ( ∧ 2 V ∨ ) H , there is an H -automorphism of V taking ω to ω ′ (see the pro of of Prop osition 6.15). Thus the question of whether H → Sp( V ) factors through ASp( V ) → Sp( V ) do es not dep end on the choice of ω for an irreducible represen tation V . Pr o of. Since im B is symmetric, d 0 , 2 2 , 2 is trivial on k er ∆ = im B . Thus d 0 , 2 2 , 2 (∆ − 1 ( ω )) do es not dep end on which preimage w e tak e. If 4 | n, b y Lemma 6.1, w e can take a c hoice [ g ] ∈ H 2 ( V , Z /n ) H represen ted by a co cycle g that is a bilinear form and ∆[ g ] = ω . If n = 2 , then by Lemmas 6.1 and 5.3 we can tak e a choice [ g ] ∈ H 2 ( V , Z / 2) H represen ted b y a co cycle g that is a bilinear form and ∆[ g ] = ω . F or an y n , w e can use the bilinear form g to obtain a class [ g ] ∈ H 2 ( V , Z / 4) , which is necessarily H -inv ariant by Lemma 6.1, since ∆[ g ] = ω . W e consider the Lyndon-Hochsc hild-Serre sp ectral sequence for V → V ⋊ H → H . By Lemma 6.4, in this case d 0 , 2 2 = d 0 , 2 2 , 2 . By the functoriality of the sp ectral sequence (Lemma 6.2) in the Z /n term, w e hav e that d 0 , 2 2 , 2 ([ g ]) ∈ H 2 ( H , V ∨ ) do es not dep end on which n w e take, and so we may assume n = 4 . By Lemma 6.4, d 0 , 2 3 = 0 . Th us [ g ] ∈ E 0 , 2 2 = H 2 ( V , Z / 4) H surviv es to E 0 , 2 ∞ if and only if d 0 , 2 2 , 2 ([ g ]) = 0 . By the edge maps of the sp ectral sequence, this happ ens if and only if [ g ] pulls bac k from H 2 ( V ⋊ H , Z / 4) , which is equiv alent to the extension Z / 4 → E → V asso ciated to [ g ] pulling bac k from an extension Z / 4 → E → V ⋊ H . If the extension E pulls back from Z / 4 → E → V ⋊ H , then E is the preimage of V in E . Then H acts on E b y lifting to E and conjugating, which preserves Z / 4 p oin twise and acts on V through Sp( V ) . This means the map H → Sp( V ) factors through ASp( V ) . Con versely , if H → Sp( V ) factors through A Sp( V ) , then the H action on V lifts to an action on E that ﬁxes Z / 4 p oin twise. Hence Z /n → E ⋊ H → V ⋊ H is an extension from whic h Z / 4 → E → V pulls bac k. In conclusion, Z / 4 → E → V pulls back from an extension Z /n → E → V ⋊ H if and only if the map H → Sp( V ) factors through ASp( V ) → Sp( V ) . Th us d 0 , 2 2 , 2 (∆ − 1 ( ω )) = 0 if and only if the map H → Sp( V ) factors through ASp( V ) → Sp( V ) . □ Next w e show the v anishing of d 0 , 2 3 on classes from bilinear forms. Lemma 6.11. L et V b e a ﬁnite dimensional ve ctor sp ac e over F 2 and and let ω b e a ful l r ank element of ∧ 2 V ∨ . L et n b e a p ositive inte ger with 4 | n . L et f ∈ Hom( V ⊗ V , Z /n ) b e a biline ar form such that ∆ f = ω and [ f ] ∈ H 2 ( V , Z /n ) Sp( V ) . W e view ASp( V ) as 93 an extension V → ASp( V ) → Sp( V ) using the isomorphism V → V ∨ induc e d by ω . Then in the Lyndon-Ho chschild-Serr e sp e ctr al se quenc e c omputing H ∗ (ASp( V ) , Z /n ) fr om H p (Sp( V ) , H q ( V , Z /n )) we have d 0 , 2 3 ([ f ]) = 0 . Pr o of. W e ﬁrst red uce to the case n = 4 . The inclusion Z / 4 → Z /n b y m ultiplication by n 4 induces a map H 2 ( A, Z / 4) → H 2 ( A, Z /n ) , which is compatible with ∆ and by Lemma 6.1 is an isomorphism on Sp( V ) inv arian t classes in im C . In particular any c hoice of f ′ ∈ Hom( V ⊗ V , Z /n ) suc h that ∆ f ′ = ω and [ f ′ ] ∈ H 2 ( V , Z /n ) Sp( V ) is n 4 f , where f ∈ Hom( V ⊗ V , Z / 4) suc h that ∆ f = ω and [ f ] ∈ H 2 ( V , Z / 4) Sp( V ) . So by Lemma 6.2, d 0 , 2 3 ([ f ′ ]) in the general n case is the image of d 0 , 2 3 ([ f ]) in the n = 4 case under the homorphism H 3 (Sp( V ) , Z / 4) → H 3 (Sp( V ) , Z /n ) . So it suﬃces, for pro ving the lemma, to show that d 0 , 2 3 ([ f ]) = 0 in the n = 4 case. W e will sho w that the class [ f ] pulls bac k from a class in H 2 (ASp( V ) , Z / 4) , whic h will imply d 0 , 2 3 ([ f ]) = 0 by consideration of the edge maps of the sp ectral sequence. In [GH12, Theorem 1.3], Gurevich and Hadani construct a group AMp( V ) as an extension of ASp( V ) by µ 4 , giving an extension class in H 2 (ASp( V ) , µ 4 ) . W e will sho w that this class is the desired class, when comp osed with either of the t wo isomorphisms µ 4 ∼ = Z / 4 . T o b egin the pro of, we must explain the deﬁning prop ert y of the group AMp( V ) of V . Let E be the extension of V b y Z / 4 Z given by [ f ] . Let ψ : Z / 4 → C × b e a faithful char- acter. The result [GH12, Theorem 1.1] states that there exists a unique faithful irreducible represen tation of E with central character ψ , and denotes the underlying v ector space of this represen tation b y H and the homomorphism E → GL ( H ) by π . Next [GH12, §1.3.2] de- ﬁnes a homomorphism ˜ ρ : ASp( V ) → P GL ( H ) by deﬁning ˜ ρ ( g ) as the unique-up-to-scalars elemen t of GL ( H ) solving ˜ ρ ( g ) π ( h ) ˜ ρ ( g ) − 1 = π ( g ( h )) for all h ∈ E . The statement of [GH12, Theorem 1.3] is that there exists a homomorphism ρ : AMp( V ) → GL ( H ) lying o ver ˜ ρ . Also b efore giving the pro of, w e note that the pullback of the extension class α of AMp( V ) to an extension class in H 2 ( V , µ 4 ) is in v ariant under the conjugation action of ASp( V ) /V ∨ , whic h is the usual action of Sp( V ) . Also [ f ] ∈ H 2 ( V , Z / 4) Sp( V ) . Ho wev er, the group of Sp( V ) -inv ariant classes in H 2 ( V , Z / 4) is isomorphic to F 2 since, b y Lemma 6.1, H 2 ( A, Z / 4) Sp( A ) ∼ = ( ∧ 2 V ∨ × V ∨ ) Sp( V ) = ( ∧ 2 V ∨ ) Sp( V ) × ( V ∨ ) Sp( V ) ∼ = F 2 × 1 = F 2 . Th us it suﬃces to sho w that the pullbac k of the extension class α to H 2 ( V , µ 4 ) ∼ = H 2 ( V , Z / 4) is not trivial. T o show an extension class of ab elian groups is not trivial, it is enough to sho w that t w o elemen ts hav e a non-identit y commutator, b ecause a split extension would b e ab elian and so all commutators would b e the identit y . Since comm utators are preserved by restrictions of extensions, it suﬃces to ﬁnd tw o elemen ts in the image of V inside ASp( V ) whose lifts to AMp( V ) ha v e nontrivial commutator. Let π : V → P GL ( H ) b e obtained from composing π with the pro jection GL ( H ) → P GL ( H ) and using that elemen ts in Z / 4 are sen t to scalars by π . The restriction of ˜ ρ to V is simply π , b ecause for g ∈ V and g ′ a lift of g to E , π ( g ′ ) solves the equation π ( g ′ ) π ( h ) π ( g ′ ) − 1 = π ( g ( h )) , since the action of g on E is by conjugation, and th us mo dulo scalars w e hav e ˜ ρ ( g ) = π ( g ′ ) = π ( g ) . The commutator of tw o elements in GL ( H ) is inv arian t under multiplying each element by scalars and thus may b e computed from their image in P GL ( H ) . Let g and h b e tw o elements 94 of V on which the bilinear form ω is nontrivial, let g ′ and h ′ b e their lifts to AMp( V ) , and let g ∨ and h ∨ b e their lifts to E . Then ρ ([ g ′ , h ′ ]) = [ ρ ( g ′ ) , ρ ( h ′ )] = [ ˜ ρ ( g ) , ˜ ρ ( h )] = [ π ( g ) , π ( h )] = [ π ( g ∨ ) , π ( h ∨ )] = π ([ g ∨ , h ∨ ]) . Since ω ([ g , h ])  = 0 , w e ha ve [ g ∨ , h ∨ ] a nonzero element of Z / 4 ⊂ E . Since π restricted to Z / 4 is the faithful character ψ , this implies π ([ g ∨ , h ∨ ])  = 1 so ρ ([ g ′ , h ′ ])  = 1 and hence [ g ′ , h ′ ]  = 1 , as desired. □ Lemma 6.12. L et V b e a ﬁnite dimensional ve ctor sp ac e over F 2 and let q b e an element of Sym 2 V ∨ whose asso ciate d (symple ctic) biline ar form deﬁne d by ω ( a, b ) = q ( a + b ) − q ( a ) − q ( b ) is nonde gener ate. L et O( V ) b e the gr oup of automorphisms of V pr eserving q , which maps to Sp( V ) sinc e automorphisms pr eserving the quadr atic form q pr eserve the symple ctic form ω . L et Ps( V ) b e the ﬁb er pr o duct of ASp( V ) with O( V ) over Sp( V ) . L et f ∈ Hom( V ⊗ V , Z / 2) b e a biline ar form whose pr oje ction to Sym 2 V ∨ (as in L emma 6.1) is q . L et [ f ] ∈ H 2 ( V , Z / 2) O ( V ) b e the asso ciate d class. W e view Ps( V ) as an extension V → Ps( V ) → O( V ) using the isomorphism V → V ∨ in- duc e d by ω . Then in the Lyndon-Ho chschild-Serr e sp e ctr al se quenc e c omputing H ∨ (Ps( V ) , Z / 2) fr om H p (O( V ) , H q ( V , Z /n )) we have d 0 , 2 3 ([ f ]) = 0 . Pr o of. W e will sho w that the class [ f ] pulls back from a class in H ∨ (Ps( V ) , Z / 2) , which will imply d 0 , 2 3 ([ f ]) = 0 by consideration of the edge maps of the sp ectral sequence. The group Ps( V ) is giv en an explicit description in [W ei64, §31] as the group of pairs σ, f with σ ∈ O( V ) and Q a quadratic form on V satisfying Q ( a + b ) − Q ( a ) − Q ( b ) = f ( σ ( a ) , σ ( b )) − f ( a, b ) . Here the m ultiplication ( σ 1 , Q 1 )( σ 2 , Q 2 ) = ( σ 1 σ 2 , Q 2 + Q 1 ◦ σ 2 ) . (A ctually , this is the rev erse order of W eil’s m ultiplication con v en tion – we hav e switc hed this since W eil uses right multiplication for group actions but the switch is irrelev an t since a group is isomorphic to its opp osite group.) W e chec k that this deﬁnition agrees with the deﬁnition as a ﬁb er pro duct. T o do this, realize the extension E of V by Z / 4 asso ciated to the bilinear form f as the group of pairs ( v , t ) with v ∈ V and t ∈ Z 4 under comp osition ( v 1 , t 2 )( v 2 , t 2 ) = ( v 1 + v 2 , t 1 + t 2 + 2 f ( v 1 , v 2 )) . Then let ( σ, Q ) act on E by ( σ, Q )( v , t ) = ( σ ( v ) , t + 2 Q ( v )) . This is indeed an automorphism since ( σ, Q )( v 1 , t 2 ) · ( σ, Q )( v 2 , t 2 ) = ( σ ( v 1 ) , t 1 + 2 Q ( v 1 )) · ( σ ( v 2 ) , t 2 + 2 Q ( v 2 )) = ( σ ( v 1 )+ σ ( v 2 ) , t 1 +2 Q ( v 1 )+ t 2 +2 Q ( v 2 )+2 f ( σ ( v 1 ) , σ ( v 2 ))) = ( σ ( v 1 + v 2 ) , t 1 + t 2 +2 Q ( v 1 +1 2 )+2 f ( v 1 , v 2 )) = ( σ, Q )( v 1 + v 2 , t 1 + t 2 + 2 f ( v 1 , v 2 )) = ( σ, Q )(( v 1 , t 1 ) · ( v 2 , t 2 )) and the map Ps( V ) → Aut( E ) is a homomorphism since ( σ 1 , Q 1 )(( σ 2 , Q 2 )( v , t )) = ( σ 1 , Q 1 )( σ 2 ( v ) , t + 2 Q 2 ( v )) = ( σ 1 ( σ 2 ( v )) , t + 2 Q 2 ( v ) + 2 Q 1 ( σ 2 ( v ))) = ( σ 1 σ 2 , Q 2 + Q 1 ◦ σ 2 )( t, v ) = (( Q 1 , σ 1 )( Q 2 , σ 2 ))( t, v ) . This homomorphism is an isomorphism since it is a homomorphism b etw een extensions of O( V ) b y V ∨ that induces isomorphisms on b oth O( V ) and V ∨ . In [Bla93, Prop osition b.i. on p. 18], Blásco demonstrates the existence of a certain group extension c Ps of Ps( V ) by ± 1 , giving an extension class in H 2 (Ps( V ) , Z / 2) (using the unique isomorphism Z / 2 ∼ = ± 1 ). W e will show that this class is the desired class. W e b egin b y describing the deﬁning prop erty of the group c Ps . 95 Blásco [Bla93, §1.2] ﬁrst deﬁnes the Heisen b erg group H ( f ) asso ciated to the bilinear form f , an extension of V b y F 2 . T ak e ψ the unique non trivial character of F 2 . Then Blásco deﬁnes a v ector space V and homomorphism ρ : H ( f ) → GL ( V ) as the unique irreducible represen tation with central character ψ . (Blásco also considers the case of a more general ﬁeld, where there can b e m ultiple choices of ψ . Since F 2 suﬃces, w e will drop the dep en- dence on ψ in all subsequent notation.) Next Blasco [Bla93, §2.1] deﬁnes a homomorphism ω : Ps( V ) → P GL ( V ) by setting ω ( s ) to b e the unique-up-to-scalars isomorphism b etw een ρ and the comp osition of ρ with the automorphism s of H ( f ) , i.e. as the unique solution to the equation ω ( s ) ρ ( h ) ω ( s ) − 1 = ρ ( s ( h )) . He next deﬁnes the extension f Ps as the ﬁb er pro duct of Ps( V ) with GL ( V ) o ver P GL ( V ) , th us an extension of Ps( V ) by G m ( C ) that admits a homomorphism ω to GL ( V ) lifting ω . Finally , [Bla93, Prop osition b.i. on p. 18] states the existence of a subgroup c Ps of f Ps whose pro jection to Ps is surjectiv e with kernel ± 1 . F urthermore, b efore giving the pro of, w e observ e that the pullbac k of the extension class of c Ps to V is in v arian t under the conjugation action of Ps( V ) , and th us inv ariant under the usual action of O ( V ) . By Lemma 6.1 w e ha ve H 2 ( V , Z / 2) O ( V ) = (Sym 2 V ∨ ) O ( V ) ∼ = F 2 , and [ f ] is a nontrivial class in H 2 ( V , Z / 2) O ( V ) , hence to show that the pullbac k of the extension class is [ f ] it suﬃces to show it is nontrivial. It suﬃces to show that tw o elements of c Ps hav e nontrivial commutator. Since c Ps is a subgroup of f Ps with surjectiv e pro jection to Ps , every element of f Ps is the pro duct of an elemen t of c Ps with a (central) element of G m and therefore it suﬃces to ﬁnd t wo elements of f Ps with non trivial commutator. F urthermore, it suﬃces to ﬁnd tw o elemen ts in the image of V inside Ps( V ) whose in verse images in f Ps( V ) ha ve nontrivial commutator. Let ρ : V → P GL ( V ) b e obtained by comp osing ρ with the pro jection GL ( V ) → P GL ( V ) and using that elements of Z / 2 are sen t to scalars by ρ . The restriction of ω to V is simply ρ , b ecause for s ∈ V and s ′ a lift of s to H ( f ) , ρ ( s ′ ) solv es the equation ω ( s ) ρ ( h ) ω ( s ) − 1 = ρ ( s ( h )) since the action of s on H ( f ) is by conjugation, and th us mo dulo scalars w e hav e ω ( s ) = ρ ( s ′ ) = ρ ( s ) . The commutator of tw o elements in GL ( H ) is inv ariant under multiplying eac h element b y scalars and th us may b e computed from their image in P GL ( H ) . Let g and h be t wo elemen ts of V on which the bilinear form ω is nontrivial, let g ′ and h ′ b e their lifts f Ps , and let g ∨ and h ∨ b e their lifts to H ( f ) . Then ω ([ g ′ , h ′ ]) = [ ω ( g ′ ) , ω ( h ′ )] = [ ω ( g ) , ω ( h )] = [ ρ ( g ) , ρ ( h )] = [ ρ ( g ∨ ) , ρ ( h ∨ )] = ρ ([ g ∨ , h ∨ ]) . Since ω ( g , h )  = 0 , we hav e [ g ∨ , h ∨ ] a nonzero element of Z / 2 ⊂ H ( f ) . Since ρ restricted to Z / 2 is the faithful character ψ , this implies ρ ([ g ∨ , h ∨ ])  = 1 so ω ([ g ′ , h ′ ])  = 1 and hence [ g ′ , h ′ ]  = 1 , as desired □ Finally , we will pro ve the key information w e need ab out E 0 , 2 3 . Deﬁnition 6.13 . Let V b e an irreducible represen tation of a ﬁnite group H o ver F p for some prime p . Let κ = End H ( V ) . F or α = ( α 1 , . . . , α e ) ∈ H 2 ( H , V e ) = H 2 ( H , V ) ⊗ κ e , w e let im α ⊂ κ e b e the image of the corresp onding map Hom κ ( H 2 ( H , V ) , κ ) → κ e , or 96 equiv alently the ( a 1 , . . . , a e ) ∈ κ e suc h that for every ( k 1 , . . . , k e ) ∈ κ e with P i k i α i = 0 , w e ha ve P i k i a i = 0 , or equiv alently , the minimal κ -subspace W of κ e suc h that the image of α is 0 in H 2 ( H , V e ) ⊗ ( κ e /W ) . W e let span( α ) b e the κ -linear span of the α i in H 2 ( H , V ) , or equiv alently the image of the map κ e → H 2 ( H , V ) corresp onding to α . R emark 6.14 . Let τ : V → V ′ b e an isomorphism of represen tations of G ov er F p . Then V ′ inherits a natural κ -action suc h that τ is an isomorphism o v er κ . Let M ∈ Hom( κ e , κ e ′ ) . Then M with co ordinates m ij giv es a map of representations M τ : V e → ( V ′ ) e ′ sending the j th cop y of V to the i th cop y of V ′ via m ij τ for all i, j . This induces a map ( M τ ) ∗ : H 2 ( H , V e ) → H 2 ( H , ( V ′ ) e ′ ) . W e can c hec k that for α ∈ H 2 ( H , V e ) , w e hav e that ( M τ ) ∗ ( α ) = 0 if and only if M (im α ) = 0 . Prop osition 6.15. L et p b e a prime and n b e a p ositive inte ger divisible by p . L et e b e a p ositive inte ger. L et V b e an irr e ducible self-dual r epr esentation of a ﬁnite gr oup H over F p . L et κ = End H ( V ) . L et W = V e . L et 1 → W → G → H → 1 b e a short exact se quenc e of gr oups c omp atible with the given H action on W . L et G have extension class α ∈ H 2 ( H , W ) and let r = dim κ im α . Consider the Lyndon-Ho chschild-Serr e sp e ctr al se quenc e c omputing H ∨ ( G, Z /n ) fr om H p ( H , H q ( W , Z /n )) . If V is non-anomalous, then | ker( d 0 , 2 2 ) | = | V H | e | κ | ( e − r )( e − r − ϵ V ) 2 . If V is interme diate (and if 4 ∤ n , also F 2 -ortho gonal), let f ∈ H 2 ( V , Z /n ) H with ω = ∆ f ∈ ∧ 2 V ∨ ⊂ Hom( V , V ∨ ) , wher e ω  = 0 , and let Φ = d 0 , 2 2 , 2 ( f ) ∈ H 2 ( H , V ∨ ) , which gives ω − 1 ∗ (Φ) ∈ H 2 ( H , V ) . Then we have | ker( d 0 , 2 2 ) | = ( 2 | κ | ( e − r )( e − r +1) 2 if ω − 1 ∗ (Φ) ∈ span( α ) | κ | ( e − r )( e − r − 1) 2 if ω − 1 ∗ (Φ) ∈ span( α ) . Using the map B fr om L emma 6.1, d 0 , 2 3 ( E 0 , 2 3 ) = d 0 , 2 3 ( B (Hom( W, Z /n ) H )) . Pr o of. Since the sp ectral sequence en tirely factors according to a factorization of Z /n Z in to cyclic groups of prime p ow er order, we can reduce to the case that n is a p o w er of p . W e change basis, relab elling V e so that im α is the κ r ⊂ κ e corresp onding to the last r co or- dinates. Let d = dim F 2 κ . W e will iden tify Hom( W, Z /n Z ) with W ∨ , and Hom( W ⊗ W, Z /n Z ) with ( W ⊗ W ) ∨ , which we identify with W ∨ ⊗ W ∨ . Then we iden tify the corresp onding sub- groups ∆ Hom( W ⊗ W, Z /n Z ) and ∧ 2 W ∨ . W e also iden tify W ∨ ⊗ W ∨ with Hom( W, W ∨ ) , where u ⊗ v is iden tiﬁed with the map that tak es w to u ( w ) v . Lemma 6.4 tells us that d 0 , 2 2 , and indeed d 0 , 2 2 , 1 and d 0 , 2 2 , 2 , factor through ∆ : H 2 ( W , Z /n ) H → ( ∧ 2 W ∨ ) H . W e will next identify ( ∧ 2 W ∨ ) H with a certain set of matrices ov er κ , and then compute d 0 , 2 2 in terms of these matrices. I. Iden tiﬁcation of ( ∧ 2 W ∨ ) H with matrices o ver κ : W e take a ω ∈ ( V ∨ ⊗ V ∨ ) H as in Lemma 5.2, and take ω in the subspace ∧ 2 V ∨ if p ossible (i.e. unless V is symmetric). The elemen ts of ( W ∨ ⊗ W ∨ ) H corresp ond to e × e matrices o ver κ as follows. A matrix M with entries m ij ∈ κ corresp onds to m := P ij ( m ij ⊗ 1) ω ij ∈ ( W ∨ ⊗ W ∨ ) H , where ω ij is the copy of ω from the j th copy of V ∨ on the left tensored with the i th copy of V ∨ on the right. As a map W → W ∨ , the element ω ij maps the j th copy of 97 V to the i th copy of V ∨ via ω : V → V ∨ . The switching factors action on W ∨ ⊗ W ∨ sends M ∈ M e × e ( κ ) to λσ ( M t ) . W e can c ho ose an F p basis of V ∨ and a corresp onding basis w i of W ∨ , and then ∧ 2 W ∨ is the set of elemen ts of the form P i 0 . So in particular, w e hav e an H ⋊ Γ - equiv ariant isomorphism H q ( F , Z ) ≃ H q ( F a , Z ) × H q ( F b , Z ) × ker j q . An analogous statemen t is true with Z /n Z co eﬃcients, with maps i n q , j n q . F or an y ﬁnite group A , we ha v e functorial isomorphisms H 2 ( A, Z ) ≃ H 1 ( A, Q / Z ) ≃ Hom( A, Q / Z ) . It follows that the map j 2 : H 2 ( F , Z ) → H 2 ( F a , Z ) × H 2 ( F b , Z ) is an isomorphism. F or an y ﬁnite group A , w e ha v e H 1 ( A, Z ) = 0 . Th us b y the Künneth form ula with principal ideal domain co eﬃcients, we hav e a functorial exact sequence, 0 → H 3 ( F a , Z ) × H 3 ( F b , Z ) i 3 → H 3 ( F , Z ) → T or 1 ( H 2 ( F a , Z ) , H 2 ( F b , Z )) → 0 and th us an H ⋊ Γ -equiv arant isomorphism k er j 3 ≃ T or 1 ( H 2 ( F a , Z ) , H 2 ( F b , Z )) . Applying the exact sequence 0 → Z → Z → Z /n → 0 , we hav e exact sequences 0 → H 2 ( F , Z ) /n → H 2 ( F , Z /n ) → H 3 ( F , Z )[ n ] → 0 0 → H 2 ( F a , Z ) /n × H 2 ( F b , Z ) /n → H 2 ( F a , Z /n ) × H 2 ( F b , Z /n ) → H 3 ( F a , Z )[ n ] × H 3 ( F b , Z )[ n ] → 0 . Using the map b et ween these exact sequences giv en by F a → F and F b → F , from the snak e lemma and observ ations abov e it follo ws that we ha v e an H ⋊ Γ -equiv ariant isomorphism k er j n 2 ≃ T or 1 ( H 2 ( F a , Z ) , H 2 ( F b , Z ))[ n ] . Thus, from the ab ov e observ ations, w e hav e an H ⋊ Γ - equiv ariant isomorphism H 2 ( F , Z /n ) ≃ H 2 ( F a , Z /n ) × H 2 ( F b , Z /n ) × T or 1 ( H 2 ( F a , Z ) , H 2 ( F b , Z ))[ n ] . Since H 2 ( F a , Z ) ≃ Hom( F a , Q / Z ) and H 2 ( F b , Z ) are pro ducts of vector spaces o ver ﬁnite ﬁelds, there is a functorial isomorphism from their T or 1 to their tensor pro duct. W e claim H 2 ( F a , Z ) ⊗ H 2 ( F b , Z ) contains no nontrivial element that is b oth n -torsion and H ⋊ Γ - in v ariant. Suc h an element would give a nontrivial H ⋊ Γ -inv arian t Z /n -v alued bilinear form on F a × F b , which cannot exist b ecause of our assumption on the irreducible factors of F a and F b . Thus we conclude that the map in Equation (7.16) is an isomorphism. 112 More straigh tforwardly , the pro duct of natural pullbac k maps H 1 ( F a , Z /n ) × H 1 ( F b , Z /n ) → H 1 ( F a × F b , Z /n ) is an isomorphism b ecause these cohomology groups are the same as sets of homomorphisms to Z /n , hence (7.17) H p ( H ⋊ Γ , H 1 ( F a , Z /n )) × H p ( H ⋊ Γ , H 1 ( F b , Z /n )) → H p ( H ⋊ Γ , H 1 ( F ab , Z /n )) is an isomorphism for all p . W e adopt an analogous notation for the diﬀeren tials as for the terms of the spectral sequences. Using (7.16), the p = 2 case of (7.17), and Lemma 6.2, it follows that d 0 , 2 2 ,G : H 0 ( H ⋊ Γ , H 2 ( F , Z /n )) → H 2 ( H ⋊ Γ , H 1 ( F , Z /n )) is the pro duct of d 0 , 2 2 ,G a and d 0 , 2 2 ,G b . Hence E 0 , 2 3 ,G = k er d 0 , 2 2 ,G is the pro duct of the k ernel E 0 , 2 3 ,G a of d 0 , 2 2 ,G a and the kernel E 0 , 2 3 ,G b of d 0 , 2 2 ,G b , v erifying (7.15). Recall b y Lemma 7.13, w e hav e s H ∈ im π ∗ if and only if s H ◦ d 1 , 1 2 ,G = 0 and s H ◦ d 0 , 2 3 ,G = 0 , and the the analogous statements are true for G a and G b . Using Lemma 6.2 and the p = 1 case of (7.17), the map d 1 , 1 2 ,G is the sum of d 1 , 1 2 ,G a and d 1 , 1 2 ,G b , hence s H ◦ d 1 , 1 2 ,G = 0 if and only if s H ◦ d 1 , 1 2 ,G a = 0 and s H ◦ d 1 , 1 2 ,G b = 0 . Similarly , using Lemma 6.2 and (7.15), d 0 , 2 3 ,G is the sum of d 0 , 2 3 ,G a and d 0 , 2 3 ,G b , hence τ ◦ d 0 , 2 3 ,G = 0 if and only if τ ◦ d 0 , 2 3 ,G a = 0 and τ ◦ d 0 , 2 3 ,G b = 0 . Thus w e conclude that s H ∈ im π ∗ if and only if s H ∈ im( π a ) ∗ and s H ∈ im( π b ) ∗ . Finally , if G Γ = 1 then ( G a ) Γ = 1 and ( G b ) Γ = 1 since G a and G b are quotien ts of G . Supp ose ( G a ) Γ = 1 and ( G b ) Γ = 1 , whic h in particular implies H Γ = 1 . The image of F a and F b in G Γ are b oth normal subgroups. If w e quotien t G Γ b y the image of F a , we obtain ( G b ) Γ = 1 (and similarly for F b ), and so image of F a in G Γ is all of G Γ and similarly for F b . So giv en t wo elements of G Γ , we can write one as an image of an elemen t of F a and the other as an image of an element of F b to see they comm ute, and hence G Γ is ab elian. So the image of F a in G Γ factors through F ab a (where ab denotes the abelianization), and the images of an y element of F and any elemen t of G commute in G Γ , so the image of F a in G Γ factors through ( F ab a ) H ⋊ Γ . Each simple factor of F a that is not ab elian and a trivial represen tation of H ⋊ Γ is sent to 1 in the map to ( F ab a ) H ⋊ Γ . All of this is also true for F b . So, since for each prime p , one of F a or F b do es not contain a characteristic p trivial representation of H ⋊ Γ , w e conclude G Γ = 1 . W e hav e sho wn that G Γ = 1 if and only if ( G a ) Γ = 1 and ( G b ) Γ = 1 , whic h was the last statemen t needed to prov e the lemma. □ Lemma 7.14 will allow us to factor M ( e, f ) , and we now will give notation for the factors. W e let V · U ′ i := | V Γ i | | V H ⋊ Γ i | V · U i = 1 | V H ⋊ Γ i | Y γ ∈ U | V γ i | . F or V i of c haracteristic not dividing n , let M i ( e i ) = 1 | GL e i ( q i ) || H 1 ( H ⋊ Γ , V i ) | e i ( V · U ′ i ) e i X ( G,π ) ∈ E L ( H ,V e i i ) | E 0 , 2 3 | . 113 F or V i self-dual of characteristic dividing n , let M i ( e i ) = 1 | GL e i ( q i ) | ( V · U ′ i ) e i | V H ⋊ Γ i | e i X ( G,π ) ∈ E L ( H ,V e i i ) | E 0 , 2 3 | . F or V i of c haracteristic dividing n suc h that V ∨ i ≃ V j for an y j , let M i ( e i ) = | H 1 ( H ⋊ Γ , V ∨ i ) | e i | GL e i ( q i ) || H 1 ( H ⋊ Γ , V i ) | e i ( V · U ′ i ) e i X ( G,π ) ∈ E L ( H ,V e i i ) | E 0 , 2 3 | . If V i falls in to one of the ab o ve three categories, i.e. char( V i ) ∤ n , or V i is self-dual, or V ∨ i is not among the V j , w e say that V i is solo . F or V i and V i ′ of c haracteristic dividing n , dual to each other, and non-isomorphic, deﬁne M i,i ′ ( e i , e i ′ ) = | H 1 ( H ⋊ Γ , V i ) | e i ′ − e i | H 1 ( H ⋊ Γ , V i ′ ) | e i − e i ′ | GL e i ( q i ) |   GL e i ′ ( q i ′ )   ( V · U ′ i ) e i ( V · U ′ i ′ ) e i ′ X ( G,π ) ∈ E L ( H ,V e i i × V e i ′ i ′ ) | E 0 , 2 3 | . F or any N i , let η i ( f i ) = 1 | N i | f i   Z Out( N i ) ( H ⋊ Γ)   f i f i !( N · U i ) f i X ( G,π ) ∈ E L ( H ,N f i i ) | E 0 , 2 3 | . Lemma 7.18. Given L , W , and H as at the start of Se ction 7.3, and non-ne gative inte gers e 1 , . . . , e r , f 1 , . . . , f s , M ( e, f ) = M H Y i ∈{ 1 ,...,r } V i solo M i ( e i ) Y { i,i ′ }⊆{ 1 ,...,r } i  = i ′ V i ∼ = V ∨ i ′ char( V i ) | n M i,i ′ ( e i , e i ′ ) s Y i =1 η i ( f i ) . Pr o of. Recall that M H := | H 2 ( H ⋊ Γ , Z /n ) | | H 3 ( H ⋊ Γ , Z /n ) | H · U . By Lemma 7.9, it suﬃces to c heck that X ( G,π ) ∈ E L ( H ,F ) 1 | Aut [ H × Γ] ( F ) || Aut F,H ( G ) | | H 1 ( H ⋊ Γ , H 1 ( F , Z /n )) || E 0 , 2 3 | | H 1 ( F , Z /n ) H ⋊ Γ | F · U = Y i ∈{ 1 ,...,r } V i solo M i ( e i ) Y { i,i ′ }⊆{ 1 ,...,r } i  = i ′ V i ∼ = V ∨ i ′ char( V i ) | n M i,i ′ ( e i , e i ′ ) s Y i =1 η i ( f i ) . Lemmas 7.11 and 7.12 give factorizations of the | Aut [ H × Γ] ( F ) | and | Aut F,H ( G ) | terms: 1 | Aut [ H × Γ] ( F ) || Aut F,H ( G ) | = r Y i =1 | V H ⋊ Γ i | e i | GL e i ( q i ) || H 1 ( H ⋊ Γ , V i ) e i || V Γ i | e i s Y i =1 1 | N i | f i   Z Out( N i ) ( G )   f i f i ! . 114 Lemma 7.14 gives us a factorization of the sum ov er | E 0 , 2 3 | . W e no w split the remaining terms as pro ducts: H 1 ( F , Z /n ) = Y i ∈{ 1 ,...,r } char( V i ) | n ( V ∨ i ) e i so (7.19) | H 1 ( H ⋊ Γ , H 1 ( F , Z /n )) | = Y i ∈{ 1 ,...,r } char( V i ) | n H 1 ( H ⋊ Γ , V ∨ i ) e i and (7.20) | H 1 ( F , Z /n ) H ⋊ Γ | = Y i ∈{ 1 ,...,r } char( V i ) | n V i trivial H ⋊ Γ rep | V i | e i . Finally (7.21) F · U = r Y i =1 ( V · U i ) e i s Y i =1 ( G · U i ) f i . W e can chec k in each case that all the factors asso ciated to i in these pro ducts hav e b een absorb ed in to the deﬁnition of M i ( e i ) , M i,i ′ ( e i , e i ′ ) , or N i ( f i ) , as appropriate. □ No w we will giv e notation for the factors of the v W , H and ˜ v W , H . W e deﬁne, for a solo V i , (7.22) w V i := X e i ≥ 0 ( − 1) e i q ( e i 2 ) i M i ( e i ) and ˜ w V i := X e i ≥ 0 q ( e i 2 ) i M i ( e i ) , for a V i , V i ′ for non-isomorphic dual representations of characteristic dividing n , we deﬁne w V i w V i ′ := X e i ,e i ′ ≥ 0 ( − 1) e i + e i ′ q ( e i 2 ) + ( e i ′ 2 ) i M i,i ′ ( e i , e i ′ ) and (7.23) ˜ w V i ˜ w V i ′ := X e i ,e i ′ ≥ 0 q ( e i 2 ) + ( e i ′ 2 ) i M i,i ′ ( e i , e i ′ ) . (This is a slight abuse of notation, as we nev er deﬁne w V i and w V i ′ separately , b ecause they alwa ys appear together in all formulas, and similarly for ˜ w V i and ˜ w V i ′ . W e could deﬁne each of them to b e the square-root of the relev ant expression abov e.) F or eac h N i , w e deﬁne (7.24) w N i := X f i ≥ 0 ( − 1) f i η i ( f i ) and ˜ w N i := X f i ≥ 0 8 f i η i ( f i ) . Corollary 7.25. Given L , W , and H as at the start of Se ction 7.3, v W , H = M H | Aut( H ) | r Y i =1 w V i s Y i =1 w N i and ˜ v W , H ≤ 8 r M H | Aut( H ) | r Y i =1 ˜ w V i s Y i =1 ˜ w N i . 115 Pr o of. By Lemma 7.8. v W , H = 1 | Aut( H ) | X e 1 ,...,e r ,f 1 ,...,f s ≥ 0 ( − 1) P i f i r Y i =1 ( − 1) e i q ( e i 2 ) i M ( e, f ) . Lemma 7.18 gives a factorization of the M ( e, f ) term, which gives the corollary . The argu- men t is the same for ˜ v W , H . □ In the next several subsections we explicitly compute the terms M i ( e i ) , M i,i ′ ( e i , e i ′ ) , and then w V i , w N i . F rom these calculations we will b e able to use Corollary 7.25 to prov e Theo- rem 7.5 on the ﬁniteness of ˜ v W , H and the non-negativit y of v W , H . F rom the deﬁnition of the v W , H , ˜ v W , H and the M H , we can see that v W , H = ˜ v W , H = 0 when H Γ  = 1 . So w e will restrict our w ork in the follo wing subsections to the case that H Γ = 1 . 7.4. Non-ab elian groups. Recall from Lemma 5.7 that Aut( N i ) × Out( N i ) ( H ⋊ Γ) is the unique extension of H by the [ H ⋊ Γ] -group N i . Consider the Lyndon-Ho c hschild-Serre sp ectral sequence computing H ∗ (Aut( N i ) × Out( N i ) ( H ⋊ Γ) , Z /n )) with E p,q 2 = H p ( H ⋊ Γ , H q ( N i , Z /n )) . Since N i , as a group, is a pro duct of non-ab elian simple groups, we ha ve H 1 ( N i , Z /n ) = 0 and hence d 0 , 2 2 = d 1 , 1 2 = 0 . Let δ N i = d 0 , 2 3 : H 2 ( N i , Z /n ) H ⋊ Γ → H 3 ( H ⋊ Γ , Z /n ) . Giv en a [ H ⋊ Γ] -group N , let L N b e the n umber of lifts of H ⋊ Γ → Out( N ) to H ⋊ Γ → Aut( N ) . Lemma 7.26. L et L , W , and H b e as at the start of Se ction 7.3, and further assume H Γ = 1 . If H ⋊ Γ → Out( N i ) is trivial or s H ◦ δ N i is nontrivial, η i ( f i ) = ( 1 f i = 0 0 f i > 0 . F or any other i fr om 1 to s , η i ( f i ) = L N i | H 2 ( N i , Z /n ) H ⋊ Γ | | N i |   Z Out( N i ) ( H ⋊ Γ)   N · U i ! f i 1 f i ! . Pr o of. Recall η i ( f i ) = 1 | N i | f i   Z Out( N i ) ( H ⋊ Γ)   f i f i !( N · U i ) f i X ( G,π ) ∈ E L ( H ,N f i i ) | E 0 , 2 3 | and E L ( H , F ) is the set of ( G, π ) ∈ Ext Γ ( H , F ) such that G ∈ L , and G Γ = 1 , and s H ∈ im π ∗ . W e ha ve H 1 ( N f i i , Z /n ) = 0 so H p ( H , H 1 ( N f i i , Z /n )) = 0 for all p . Thus the diﬀeren tials d 1 , 1 2 and d 0 , 2 2 v anish. By Lemma 7.13, w e hav e s H ∈ im π ∗ if and only if s H ◦ d 0 , 2 3 = 0 . By Equation (7.16), d 0 , 2 3 is the sum of f i copies of δ N i . So s H ◦ δ N i  = 0 then w e ha ve s H ∈ im π ∗ if and only if f i = 0 , in whic h case η i ( f i ) = ( 1 f i = 0 0 f i > 0 , while if s H ◦ δ N i = 0 then w e alwa ys ha ve s H ∈ im π ∗ . 116 When f i > 0 , w e claim G Γ = 1 if and only if H ⋊ Γ → Out( N i ) is nontrivial. If H ⋊ Γ → Out( N i ) is trivial, then G = N i × ( H ⋊ Γ) . Since ( | N | , | Γ | ) = 1 , w e m ust also ha ve Γ → Aut( N i ) is trivial, and so G Γ = N i × H Γ , whic h is non trivial. No w w e assume G Γ is nontrivial. Since H Γ = 1 , we ha ve that the image of N = N f i i m ust generate G Γ , and so some copy of N i m ust hav e nontrivial image in G Γ . The kernel of the map from N i to its image in G Γ is an intersection of tw o normal subgroups of G , so it is a normal subgroup of G , and hence ﬁxed b y the H -action on normal subgroups of N i . This k ernel is also ﬁxed by the Γ -action on normal subgroups of N i . Since N i is a simple [ H ⋊ Γ] -group, it follows this kernel is either trivial or N i , and since we assumed the image w as non trivial, the k ernel m ust b e trivial. It follows that Γ acts trivially on N i , which impli es that H ⋊ Γ → Out( N i ) factors through H Γ = 1 , and hence H ⋊ Γ → Out( N i ) is trivial. Hence, w e conclude the statemen t of the lemma when H ⋊ Γ → Out( N i ) is trivial. By deﬁnition of the N i , the extension of H b y N i is in L and thus the extension of H by N f i i is the f i -fold ﬁb er pro duct of the extension of H by N i , and hence also in L since L is join-closed. Next w e coun t | Ext Γ ( H , N f i i ) | . By Lemma 5.7, there is a unique group extension G of H b y N f i i , and b y Lemma 5.11, the n umber of Γ -group structures on G compatible with the structures on H and N f i i is L f i N i . Since d 0 , 2 2 = 0 , E 0 , 2 3 = H 0 ( H ⋊ Γ , H 2 ( N f i i , Z /n )) = H 0 ( H ⋊ Γ , H 2 ( N i , Z /n )) f i . Putting this all into the deﬁnition of η i ( f i ) , w e conclude the lemma. □ Lemma 7.27. L et L , W , and H b e as at the start of Se ction 7.3, and further assume H Γ = 1 . F or any N i , we have w N i = e − L N i | H 2 ( N i , Z /n ) H ⋊ Γ | | N i || Z Out( N i ) ( H ⋊ Γ) | N · U i . if H ⋊ Γ → Out( N i ) is nontrivial and s H ◦ δ N i = 0 , and 1 if either c ondition fails. F urther, ˜ w N i is ﬁnite. Pr o of. This follows immediately from the deﬁnitions of w N i , ˜ w N i and Lemma 7.26. □ 7.5. Preparation for extensions b y ab elian groups. In the sums for the M i ( e i ) , we will need to sum o v er E L ( H , V e i i ) , the set of ( G, π ) ∈ Ext Γ ( H , V e i i ) such that G ∈ L , and G Γ = 1 , and s H ∈ im π ∗ . Ev en though V i w as c hosen b ecause there is some extension 1 → V i → G → H → 1 suc h that G ∈ L , it do es not follow that every suc h extension of H b y V i is in L . W e will mak e a deﬁnition b elo w that captures G ∈ L and s H ∈ im π ∗ . Deﬁnition 7.28 . F or an ab elian H ⋊ Γ -group A , w e write H 2 ( H ⋊ Γ , A ) L for the subset of classes in H 2 ( H ⋊ Γ , A ) whose corresp onding Γ -group extension 1 → A → G → H → 1 via Lemma 5.8 has G ∈ L . W e write H 2 ( H ⋊ Γ , A ) L ,s H for the subset of classes α ∈ H 2 ( H ⋊ Γ , A ) L whose corresp onding Γ -extension G satisﬁes • s H ( α ∪ β ) = 0 for all β ∈ H 1 ( H ⋊ Γ , Hom( A, Z /n )) , and • for any H ⋊ Γ -equiv ariant homomorphism A → Z /n , we ha ve that the image of α in H 2 ( H ⋊ Γ , Z /n ) is in the kernel of the comp osite of Bo ckstein map B : H 2 ( H ⋊ Γ , Z /n ) → H 3 ( H ⋊ Γ , Z /n ) given by the connecting map in the long exact sequence 117 for 0 → Z /n → Z /n 2 → Z /n → 0 , and s H : H 3 ( H ⋊ Γ , Z /n ) → Z /n (view ed as a homomorphism using Lemmas 5.4 and 5.5). It is straightforw ard to chec k that this deﬁnition agrees with the deﬁnition in the in tro- duction. After taking the comp osition with the map H 2 ( H ⋊ Γ , F ℓ ) × n/ℓ → H 2 ( H ⋊ Γ , Z /n ) , the Bo ckstein msp from 0 → Z /n → Z /n 2 → Z /n → 0 agrees with the Bo c kstein map from 0 → Z /ℓ → Z /nℓ → Z /ℓ → 0 . Lemma 7.29. F or ( G, π ) ∈ Ext Γ ( H , V e i i ) , we have G ∈ L and s H ∈ im π if and only if ( G, π ) c orr esp onds via L emma 5.8 to a class in H 2 ( H ⋊ Γ , V e i i ) L ,s H . Pr o of. By Lemma 7.13, w e ha ve s H ∈ im π if and only if s H ◦ d 1 , 1 2 = 0 and s H ◦ d 0 , 2 3 = 0 . The map d 1 , 1 2 : H 1 ( H ⋊ Γ , Hom( F , Z /n )) → H 3 ( H ⋊ Γ , Z /n Z ) is given b y the cup pro duct with the class of the extension in H 2 ( H ⋊ Γ , F ab ) [NSW00, Theorem 2.4.4]. So s H ◦ d 1 , 1 2 = 0 if and only if s H ( α ∪ β ) = 0 for all β ∈ H 1 ( H ⋊ Γ , Hom( F , Z /n )) . Lemma 6.1 deﬁnes a homomorphism B : Hom( V e i i , Z /n ) H ⋊ Γ → H 2 ( V e i i , Z /n ) H ⋊ Γ . Lemma 6.4 says that d 0 , 2 3 ([ B ( ϕ )]) is the image of [ α ] ∈ H 2 ( H ⋊ Γ , V e i i ) under ϕ ∗ : H 2 ( H ⋊ Γ , V e i i ) → H 2 ( H ⋊ Γ , Z /n ) and then B . By Lemma 6.15, im d 0 , 2 3 = d 0 , 2 3 (im B ) . Th us s H ◦ d 0 , 2 3 = 0 if and only if the second bullet p oin t ab ov e holds for the extension class. □ Lemma 5.9 tells us that H 2 ( H ⋊ Γ , V e i i ) L is a κ i -subspace of H 2 ( H ⋊ Γ , V e i i ) and that H 2 ( H ⋊ Γ , V e i i ) L = ( H 2 ( H ⋊ Γ , V i ) L ) e i . F urther, we can chec k that H 2 ( H ⋊ Γ , V e i i ) L ,s H is a κ i -subspace of H 2 ( H ⋊ Γ , V e i i ) and that H 2 ( H ⋊ Γ , V e i i ) L ,s H = ( H 2 ( H ⋊ Γ , V i ) L ,s H ) e i . Also, w e ha v e that, the composite of H 2 ( A, F p ) → H 2 ( A, Z /n ) with B is the Bo c kstein map H 2 ( A, F p ) → H 3 ( A, Z /n ) for 0 → Z /n → Z /np → F p → 0 . W e deﬁne an adjusted q -Pochhammer symbol ( q ) n := Q n i =1 (1 − q − i ) for use in the next sev eral sections. By con v ention ( q ) 0 = 1 . 7.6. Represen tations of c haracteristic prime to n . Note if F = V e i i , where V i is of c haracteristic prime to n , then H p ( H ⋊ Γ , H q ( F , Z /n )) = 0 for q > 0 . In particular, this implies that | E 0 , 2 3 | = 1 and π ∗ : H 3 ( H ⋊ Γ , Z /n ) → H 3 ( G ⋊ Γ , Z /n ) is an isomorphism and hence b y Lemma 5.4, π ∗ : H 3 ( G, Z /n ) Γ → H 3 ( H , Z /n ) Γ is an isomorphism. Lemma 7.30. L et L , W , and H b e as at the start of Se ction 7.3, and further assume H Γ = 1 . If char( V i ) ∤ n and V i is nontrivial, M i ( e i ) = 1 | GL e i ( q i ) |  | H 2 ( H ⋊ Γ , V i ) L | | H 1 ( H ⋊ Γ , V i ) | V · U ′ i  e i . Pr o of. By deﬁnition, M i ( e i ) = 1 | GL e i ( q i ) || H 1 ( H ⋊ Γ , V i ) | e i ( V · U ′ i ) e i X ( G,π ) ∈ E L ( H ,V e i i ) | E 0 , 2 3 | , where E L ( H , V e i i ) is the set of ( G, π ) ∈ Ext Γ ( H , V e i i ) suc h that G ∈ L , and G Γ = 1 , and s H ∈ im π ∗ . By Lemma 5.10, w e hav e G Γ = 1 . Since π ∗ is an isomorphism, we alw ays ha ve s H ∈ im π ∗ . 118 W e then ha ve M i ( e i ) = 1 | GL e i ( q i ) || H 1 ( H ⋊ Γ , V i ) | e i ( V · U ′ i ) e i X ( G,π ) ∈ E L ( H ,V e i i ) | E 0 , 2 3 | = 1 | GL e i ( q i ) | | H 1 ( H ⋊ Γ , V i ) | e i ( V · U ′ i ) e i X ( G,π ) ∈ E L ( H ,V e i i ) 1 = 1 | GL e i ( q i ) |  | H 2 ( H ⋊ Γ , V i ) L | | H 1 ( H ⋊ Γ , V i ) | V · U ′ i  e i . □ Lemma 7.31. L et L , W , and H b e as at the start of Se ction 7.3, and further assume H Γ = 1 . If char( V i ) ∤ n and V i is nontrivial, w V i = ∞ Y j =1  1 − q − j i | H 2 ( H ⋊ Γ , V i ) L | | H 1 ( H ⋊ Γ , V i ) | V · U ′ i  . F urther, ˜ w V i is ﬁnite. Pr o of. W e ha ve by deﬁnition and Lemma 7.30, w V i = X e i ≥ 0 ( − 1) e i q ( e i 2 ) i M i ( e i ) = X e i ≥ 0 ( − 1) e i q ( e i 2 ) i 1 | GL e i ( q i ) |  | H 2 ( H ⋊ Γ , V i ) L | | H 1 ( H ⋊ Γ , V i ) | V · U ′ i  e i . Applying the inﬁnite q -binomial theorem, (7.32) ∞ Y j =1 (1 + q − j v ) = ∞ X k =0 q ( k 2 ) − k 2 v k ( q ) k , giv es the ﬁrst statement of the lemma. An analogous calculation without the sign gives that ˜ w V i is ﬁnite. □ Lemma 7.33. L et L , W , and H b e as at the start of Se ction 7.3, and further assume H Γ = 1 . If char( V i ) ∤ n and V i is trivial, M i ( e i ) = | Surj( H 2 ( H ⋊ Γ , V i ) L , V i e i ) | | GL e i ( q i ) | ( V · U ′ i ) e i . Pr o of. The pro of is similar to that of Lemma 7.30. One can chec k that G Γ  = 1 if and only if there is no prop er sub-representation W ⊂ V e i i , such that G/W is a trivial extension of H . Suc h extensions G are those that corresp ond to elemen ts of H 2 ( H ⋊ Γ , V e i i ) = H 2 ( H ⋊ Γ , V i ) e i whose e i co ordinates in H 2 ( H ⋊ Γ , V i ) are linearly indep endent. The num b er of suc h extensions in L is hence the num b er of surjections | Surj( H 2 ( H ⋊ Γ , V i ) L , V i e i ) | (as ab elian groups). Since H Γ = 1 , we hav e H 1 ( H ⋊ Γ , V i ) = Hom( H ⋊ Γ , V i ) = 0 . Putting this together, we conclude the lemma. □ 119 Lemma 7.34. L et L , W , and H b e as at the start of Se ction 7.3, and further assume H Γ = 1 . L et z = dim κ i H 2 ( H ⋊ Γ , V i ) L and u b e such that q u i = V · U ′ i . If V i is trivial, and c har( V i ) ∤ n , then w V i = z − 1 Y k =0 (1 − q k − u i ) . F urther, ˜ w V i is ﬁnite. If U is nonempty, this expr ession is 0 if z ≥ u + 1 and p ositive otherwise. Pr o of. Let q = | V i | . By deﬁnition and Lemma 7.33, w e hav e w V i = X e i ≥ 0 ( − 1) e i q ( e i 2 ) i | Surj( H 2 ( H ⋊ Γ , V i ) L ,s H , V i e i ) | | GL e i ( q i ) | ( V · U i ) e i = z X e =0 ( − 1) e q ( e 2 ) q z e ( q ) z / ( q ) z − e q e 2 + ue ( q ) e = z X e =0 q ( e 2 ) + z e − e 2 ( − q − u ) e ( q ) z ( q ) z − e ( q ) e = z − 1 Y k =0 (1 − q k − u ) , where the last equality is by the q -binomial theorem (7.35) n − 1 Y k =0 (1 + q k v ) = n X k =0 q ( k 2 ) + nk − k 2 v k ( q ) n ( q ) n − k ( q ) k . An analogous calculation without the sign giv es that ˜ w V i is ﬁnite. If U is nonempt y then V · U ′ i ≥ 1 so u ≥ 0 . F urthermore u is an in teger. The expression (1 − q k − u i ) = 0 if k = u , which happ ens for some k if z ≥ u + 1 , and p ositive if k < u , which happ ens for all k if z < u + 1 . □ 7.7. Represen tations whose duals do not app ear of c haracteristic dividing n . Lemma 7.36. L et L , W , and H b e as at the start of Se ction 7.3, and further assume H Γ = 1 . If V i ≃ V ∨ j for any j , and char( V i ) | n , then M i ( e i ) = 1 | GL e i ( q i ) |  | H 1 ( H ⋊ Γ , V ∨ i ) || H 2 ( H ⋊ Γ , V i ) L ,s H | | H 1 ( H ⋊ Γ , V i ) | V · U ′ i  e i . Pr o of. By deﬁnition, M i ( e i ) = | H 1 ( H ⋊ Γ , V ∨ i ) | e i | GL e i ( q i ) || H 1 ( H ⋊ Γ , V i ) | e i ( V · U ′ i ) e i X ( G,π ) ∈ E L ( H ,V e i i ) | E 0 , 2 3 | , where E L ( H , V e i i ) is the set of ( G, π ) ∈ Ext Γ ( H , V e i i ) suc h that G ∈ L , and G Γ = 1 , and s H ∈ im π ∗ . Lemma 6.1 giv es an exact sequence 0 → im B → H 2 ( V e i i , Z /n ) → ∧ 2 ( V ∗ i ) e i → 0 , where B is a homomorphism from Hom( V e i i , Z /n ) and ∧ 2 A is deﬁned as the subgroup of A ⊗ A generated b y elements of the form a ⊗ b − b ⊗ a for a, b ∈ A . 120 Since V i is not-self dual, we ha v e ( ∧ 2 ( V ∗ i ) e i ) H ⋊ Γ ⊆ ((( V ∗ i ) e i ) ⊗ 2 ) H ⋊ Γ = 0 . This, with V i non trivial and so (im B ) H ⋊ Γ = 0 , implies that H 2 ( V e i i , Z /n ) H ⋊ Γ = 0 . Th us | E 0 , 2 3 | = 1 . F or ( G, π ) ∈ Ext Γ ( H , V e i i ) , b y Lemma 5.10 and the fact that V i is not self-dual and thus not trivial, we hav e G Γ = 1 . The lemma then follows from Lemma 7.29. □ Lemma 7.37. L et L , W , and H b e as at the start of Se ction 7.3, and further assume H Γ = 1 . If V i ≃ V ∨ j for any j , and char( V i ) | n , then w V i = ∞ Y j =1  1 − q − j i  | H 1 ( H ⋊ Γ , V ∨ i ) || H 2 ( H ⋊ Γ , V i ) L ,s H | | H 1 ( H ⋊ Γ , V i ) | V · U ′ i  . F urther, ˜ w V i is ﬁnite. Pr o of. The lemma follo ws by deﬁnition, Lemma 7.36, and the q -binomial theorem, as in the pro of of Lemma 7.31. □ 7.8. Non-anomalous self-dual represen tations of c haracteristic dividing n . Lemma 7.38. L et L , W , and H b e as at the start of Se ction 7.3, and further assume H Γ = 1 . If V i is self-dual, nontrivial, non-anomalous, and char( V i ) | n , then M i ( e i ) = 1 | GL e i ( q i ) | ( V · U ′ i ) e i X α ∈ ( H 2 ( H ⋊ Γ ,V i ) L ,s H ) e i q ( e i − r α )( e i − r α − ϵ V i ) 2 i , wher e r α := dim κ i im α in the sense of Deﬁnition 6.13. Pr o of. By deﬁnition, M i ( e i ) = 1 | GL e i ( q i ) | ( V · U ′ i ) e i   V H ⋊ Γ i   e i X ( G,π ) ∈ E L ( H ,V e i i ) | E 0 , 2 3 | , where E L ( H , V e i i ) is the set of ( G, π ) ∈ Ext Γ ( H , V e i i ) suc h that G ∈ L , and G Γ = 1 , and s H ∈ im π ∗ . By Lemma 5.10, w e ha ve G Γ = 1 since V i is non trivial. Also since V i is non trivial w e ha ve   V H ⋊ Γ i   e i = 1 . By Prop osition 6.15, we hav e | E 0 , 2 3 | = | κ i | ( e i − r α )( e i − r α − ϵ V i ) 2 . The lemma then follo ws from Lemma 7.29. □ Lemma 7.39. L et L , W , and H b e as at the start of Se ction 7.3, and further assume H Γ = 1 . L et Z = H 2 ( H ⋊ Γ , V i ) L ,s H , and z = dim κ i Z , and let u b e such that q u i = V · U ′ i . Assume that V i c ontains a nontrivial ve ctor ﬁxe d by at le ast one element of U . If char( V i ) | n , and V i is self-dual, nontrivial, and non-anomalous, then w V i = ∞ Y k =0 (1 + q − k − ϵ V i +1 2 − u i ) − 1 z − 1 Y k =0 (1 − q k − u i ) . F urther, ˜ w V i is ﬁnite. If z ≥ u + 1 , then w V i = 0 , and otherwise w V i is p ositive. Pr o of. Let κ = κ V i , and q = | κ | , and ϵ = ϵ V i . W e are going to in terc hange an order of summation, which will b e justiﬁed later b y considering the same manipulations if no signs 121 app eared to see that the sum is absolutely con vergen t. Letting e = e i − r α , by deﬁnition and Lemma 7.38 w V i = ∞ X e i =0 ( − 1) e i q ( e i 2 ) | GL e i ( q ) | ( V · U ′ i ) e i X α ∈ ( H 2 ( H ⋊ Γ ,V i ) L ,s H ) e i q ( e i − r α )( e i − r α − ϵ ) 2 = ∞ X e =0 ∞ X r =0 X α ∈ ( H 2 ( H ⋊ Γ ,V i ) L ,s H ) e + r r α = r ( − 1) e + r q − ( e + r +1 2 ) + e ( e − ϵ ) 2 − u ( e + r ) ( q ) e + r . The n um b er of α ∈ Z e + r with r α = r , is the num b er of r -dimensional subspaces of κ e + r , whic h is q er ( q ) e + r ( q ) e ( q ) r , times the n umber of surjections from the κ -dual of Z to an r -dimensional κ v ector space, whic h is q zr ( q ) z ( q ) z − r when r ≤ z and 0 otherwise. W e then hav e w V i = ∞ X e =0 z X r =0 q er ( q ) e + r ( q ) e ( q ) r q z r ( q ) z ( q ) z − r ( − 1) e + r q − ( e + r +1 2 ) + e ( e − ϵ ) 2 − u ( e + r ) ( q ) e + r . = ∞ X e =0 ( − 1) e q − ( ϵ +1) e 2 − ue ( q ) e z X r =0 ( − 1) r q ( r 2 ) − ur + z r − r 2 ( q ) z ( q ) z − r ( q ) r . Note that our assumption on U implies that V · U ′ i > 1 so that u > 0 . By the same calculation without the signs, since u > 0 , w e see that ˜ w V i is ﬁnite and the sum ab ov e is absolutely con vergen t. W e apply the inﬁnite q -binomial theorem for negativ e p o wers (7.40) ∞ Y k =0 1 1 − q − k v = ∞ X k =0 v k ( q ) k to the sum ov er e and the q -binomial theorem (7.35) to the sum ov er r , and obtain w V i = ∞ Y k =0 (1 + q − k − ϵ +1 2 − u ) − 1 z − 1 Y k =0 (1 − q k − u ) . Using that u > 0 , the pro duct on the right is 0 if z − 1 − u ≥ 0 , and is otherwise p ositiv e. □ Lemma 7.41. L et L , W , and H b e as at the start of Se ction 7.3, and further assume H Γ = 1 . If V i is trivial, and char( V i ) | n , then M i ( e i ) = | Surj( H 2 ( H ⋊ Γ , V i ) L ,s H , V i e i ) | | GL e i ( q i ) | ( V · U ′ i ) e i . Pr o of. By deﬁnition, M i ( e i ) = 1 | GL e i ( q i ) | ( V · U ′ i ) e i   V H ⋊ Γ i   e i X ( G,π ) ∈ E L ( H ,V e i i ) | E 0 , 2 3 | , where E L ( H , V e i i ) is the set of ( G, π ) ∈ Ext Γ ( H , V e i i ) suc h that G ∈ L , and G Γ = 1 , and s H ∈ im π ∗ . As in the pro of of Lemma 7.33, there are | Surj( H 2 ( H ⋊ Γ , V i ) L ,s H , V i e i ) | choices of such ( G, π ) that hav e G Γ = 1 , and all of these ha v e r α = e i . By Prop osition 6.15, we hav e | E 0 , 2 3 | = | V i | e i | κ i | ( e i − r α )( e i − r α − ϵ V i ) 2 = | V i | e i =   V H ⋊ Γ i   e i . □ 122 Lemma 7.42. L et L , W , and H b e as at the start of Se ction 7.3, and further assume H Γ = 1 . L et z = dim κ i H 2 ( H ⋊ Γ , V i ) L ,s H and u b e such that q u i = V · U ′ i . If V i is trivial, and c har( V i ) | n , then w V i = z − 1 Y k =0 (1 − q k − u i ) , which is 0 if z ≥ u + 1 and p ositive otherwise. F urther, ˜ w V i is ﬁnite. If U is nonempty, this expr ession is 0 if z ≥ u + 1 and p ositive otherwise. Pr o of. The pro of is the same as the pro of of Lemma 7.34. □ 7.9. Non-self-dual represen tations whose duals app ear of characteristic dividing n . Lemma 7.43. L et L , W , and H b e as at the start of Se ction 7.3, and further assume H Γ = 1 . If V i and V i ′ ar e dual, with i  = j , and char( V i ) | n , then M i,i ′ ( e i , e i ′ ) = | H 1 ( H ⋊ Γ , V i ) | e i ′ − e i | H 1 ( H ⋊ Γ , V i ′ ) | e i − e i ′ | GL e i ( q i ) |   GL e i ′ ( q i ′ )   ( V · U ′ i ) e i ( V · U ′ i ′ ) e i ′ X ( α i ,α i ′ ) ∈ Z q ( e i − r α i )( e i ′ − r α i ′ ) i , wher e Z = ( H 2 ( H ⋊ Γ , V i ) L ,s H ) e i × ( H 2 ( H ⋊ Γ , V i ′ ) L ,s H ) e i ′ and r α := dim κ im α in the sense of Deﬁnition 6.13. Pr o of. Let F = V e i i × V e i ′ i ′ . Recall the deﬁnition, M i,i ′ ( e i , e i ′ ) = | H 1 ( H ⋊ Γ , V i ) | e i ′ − e i | H 1 ( H ⋊ Γ , V i ′ ) | e i − e i ′ | GL e i ( q i ) |   GL e i ′ ( q i ′ )   ( V · U ′ i ) e i ( V · U ′ i ′ ) e i ′ X ( G,π ) ∈ E L ( H ,F ) | E 0 , 2 3 | , where E L ( H , F ) is the set of ( G, π ) ∈ Ext Γ ( H , F ) such that G ∈ L , and G Γ = 1 , and s H ∈ im π ∗ . By Lemma 5.10, we hav e G Γ = 1 since V i and V j are nontrivial. By Lemma 7.13, s H ∈ im π ∗ if and only if s H ◦ d 1 , 1 2 = 0 and s H ◦ d 0 , 2 3 = 0 . By Prop osition 6.17, d 0 , 2 3 = 0 and | E 0 , 2 3 | = | κ i | ( e i − r α i )( e i ′ − r α i ′ ) . The classes ( G, π ) ∈ Ext Γ ( H , F ) suc h that G ∈ L and s H ◦ d 1 , 1 2 = 0 corresp ond via Lemma 5.8 to classes in H 2 ( H ⋊ Γ , V e i i ) L ,s H . □ Lemma 7.44. L et L , W , and H b e as at the start of Se ction 7.3, and further assume H Γ = 1 . L et char( V i ) | n and V i ∼ = V ∨ i ′ but i  = i ′ . L et u b e such that q u = V · U ′ i and z = dim κ V i H 2 ( H ⋊ Γ , V i ) L ,s H , and z ′ = dim κ V i H 2 ( H ⋊ Γ , V i ′ ) L ,s H . L et h = dim κ H 1 ( H ⋊ Γ , V i ′ ) − dim κ H 1 ( H ⋊ Γ , V i ) . Then w V i w V i ′ = ( q i ) ∞ ( q i ) 2 u ( q i ) u − h − z ( q i ) u + h − z ′ , if − u + z ′ ≤ h ≤ u − z and 0 otherwise. F urther, ˜ w V i ˜ w V i ′ is ﬁnite. Pr o of. Let κ = κ V i , and q = | κ | . Let Z = ( H 2 ( H ⋊ Γ , V i ) L ,s H ) e i × ( H 2 ( H ⋊ Γ , V i ′ ) L ,s H ) e i ′ . Since the characteristic of V i do es not divide | Γ | , for any subgroups Γ ′ of Γ , we ha ve V i and V i ′ are pro ducts of irreducible Γ ′ represen tations, and V Γ ′ i and V Γ ′ i ′ are comprised of the trivial factors and hence the same size. Th us V · U ′ i = ( V ∨ i ) · U ′ . 123 W e hav e by deﬁnition and Lemma 7.43, w V i w V i ′ = X e i ,e i ′ ≥ 0 ( − 1) e i + e i ′ q ( e i 2 ) + ( e i ′ 2 ) + h ( e i − e i ′ ) − u ( e i + e i ′ ) | GL e i ( q i ) |   GL e i ′ ( q i ′ )   X ( α i ,α i ′ ) ∈ Z q ( e i − r α i )( e i ′ − r α i ′ ) i . As in Lemma 7.39, the num b er of α i ∈ ( H 2 ( H ⋊ Γ , V i ) L ,s H ) e i with r α i = r is q ( e i − r ) r + z r ( q ) e i ( q ) z ( q ) e i − r ( q ) r ( q ) z − r if r ≤ z and r ≤ e i , and similarly for i ′ . So w V i w V i ′ = X e i ,e i ′ ≥ 0 ( − 1) e i + e i ′ q ( e i 2 ) − e 2 i + ( e i ′ 2 ) − e 2 i ′ + h ( e i − e i ′ ) − u ( e i + e i ′ ) ( q ) e i ( q ) e i ′ × min( e i ,z ) X r =0 min( e i ′ ,z ′ ) X r ′ =0 q ( e i − r ) r + z r ( q ) e i ( q ) z ( q ) e i − r ( q ) r ( q ) z − r q ( e i ′ − r ′ ) r ′ + z ′ r ′ ( q ) e i ′ ( q ) z ′ ( q ) e i ′ − r ′ ( q ) r ′ ( q ) z ′ − r ′ q ( e i − r i )( e i ′ − r i ′ ) . W e are going to interc hange an order of summation, whic h will b e justiﬁed later by consider- ing the same manipulations if no signs app eared to see that the sum is absolutely con v ergent. Letting e = e i − r and e ′ = e i ′ − r ′ , w V i w V i ′ = X e,e ′ ≥ 0 z X r =0 z ′ X r ′ =0 ( − 1) e + e ′ + r + r ′ q ( e 2 ) − e 2 + ( e ′ 2 ) − ( e ′ ) 2 + h ( e − e ′ ) − u ( e + e ′ )+ ee ′ − u ( r + r ′ )+ h ( r − r ′ )+ ( r 2 ) − r 2 + ( r ′ 2 ) − ( r ′ ) 2 + z r + z ′ r ′ ( q ) z ( q ) z ′ ( q ) e ( q ) e ′ ( q ) r ( q ) z − r ( q ) r ′ ( q ) z ′ − r ′ . W e can factor out z X r =0 ( − 1) r q hr − ur + ( r 2 ) − r 2 + z r ( q ) z ( q ) r ( q ) z − r = z − 1 Y k =0 (1 − q k + h − u ) , with the equalit y b y the q -binomial theorem (7.35), as well as the analogous sum o ver r ′ (whic h has a − in fron t of the h ). What remains is (7.45) X e,e ′ ≥ 0 ( − 1) e + e ′ q ( e 2 ) − e 2 + ( e ′ 2 ) − ( e ′ ) 2 + h ( e − e ′ ) − u ( e + e ′ )+ ee ′ ( q ) e ( q ) e ′ . W e now argue that the sums w e ha v e b een considering conv erge absolutely , equiv alen tly that ˜ w V i ˜ w V i ′ is ﬁnite. If we had been considering the same sums without signs, the ab ov e argumen t would reduce the ﬁniteness of ˜ w V i ˜ w V i ′ to the absolute conv ergence of (7.45). If f is a quadratic p olynomial with negativ e leading co eﬃcient and f ( Z ) ⊂ Z , then P e q f ( e ) ≤ 4 q m , where m is the maxim um v alue taken by f . This is b ecause the sum on either side of the maxim um is b ounded b y a geometric series summing to q m (1 − q − 1 ) , and q ≥ 2 . Th us X e,e ′ ≥ 0 q ( e 2 ) − e 2 + ( e ′ 2 ) − ( e ′ ) 2 + h ( e − e ′ ) − u ( e + e ′ )+ ee ′ ( q ) e ( q ) e ′ ≤ 4 (2) 2 ∞ X e,e ′ ≥ 0 q − e 2 2 +( − 1 2 + h − u + e ′ ) e +( − h − u − 1 2 ) e ′ − ( e ′ ) 2 2 ≤ 4 (2) 2 ∞ X e ′ ≥ 0 q ( − 1 2 + h − u + e ′ ) 2 2 +( − h − u − 1 2 ) e ′ − ( e ′ ) 2 2 . 124 The last sum ab o ve is a geometric series with ratio q − 2 u − 1 and hence con verges, showing that the sums we hav e b een considering conv erge absolutely , and ˜ w V i ˜ w V i ′ is ﬁnite. Returning to (7.45), we pull out the sum (7.46) X e ≥ 0 ( − 1) e q ( e 2 ) − e 2 + he − ue + ee ′ ( q ) e = ∞ Y j =1 (1 − q − j + h − u + e ′ ) = ( 0 if h − u + e ′ > 0 ( q ) ∞ ( q ) u − h − e ′ if h − u + e ′ ≤ 0 , using the inﬁnite q -binomial theorem (7.32) for the ﬁrst equalit y . Th us, by the q -binomial theorem (7.35), the sum in (7.45) is equal to u − h X e ′ =0 ( − 1) e ′ q ( e ′ 2 ) − ( e ′ ) 2 − he ′ − ue ′ ( q ) ∞ ( q ) e ′ ( q ) u − h − e ′ = ( q ) ∞ ( q ) u − h u − h − 1 Y k =0 (1 − q k − 2 u ) if h ≤ u and 0 if h > u . Note the pro duct ab ov e is 0 when − h > u . Thus, w e conclude w V i w V i ′ = ( q ) ∞ ( q ) 2 u ( q ) u − h ( q ) u + h z − 1 Y k =0 (1 − q k − u + h ) z ′ − 1 Y k =0 (1 − q k − u − h ) . if − u ≤ h ≤ u and 0 otherwise. W e note the ﬁnal pro ducts are non-zero if and only if h ≤ u − z and − h ≤ u − z ′ , and conclude w V i w V i ′ = ( q ) ∞ ( q ) 2 u ( q ) u − h − z ( q ) u + h − z ′ if − d + z ′ ≤ h ≤ d − z and 0 otherwise. □ 7.10. Anomalous self-dual represen tations. Lemma 7.47. L et L , W , and H b e as at the start of Se ction 7.3, and further assume H Γ = 1 . Assume that V i c ontains a nontrivial ve ctor ﬁxe d by at le ast one element of U . L et V i b e self-dual, anomalous, and with char( V i ) | n . L et u b e such that q u i = V · U ′ i . and z = dim κ i H 2 ( H ⋊ Γ , V i ) L ,s H . Ther e is a p articular class ω − 1 ∗ (Φ) ∈ H 2 ( H ⋊ Γ , V i ) deﬁne d in Pr op osition 6.15. If ω − 1 ∗ (Φ) ∈ H 2 ( H ⋊ Γ , V i ) L ,s H , w V i = ∞ Y k =0 (1 + q − k − u i ) − 1 z − 1 Y k =0 (1 − q k − u i ) If ω − 1 ∗ (Φ) ∈ H 2 ( H ⋊ Γ , V i ) L ,s H , w V i = ∞ Y k =0 (1 + q − k − 1 − u i ) − 1 z − 1 Y k =0 (1 − q k − u i ) . In either c ase, ˜ w V i is ﬁnite, and if z ≥ u + 1 , then w V i = 0 and otherwise w V i is p ositive. Pr o of. Let q = q i . By deﬁnition, M i ( e i ) = 1 | GL e i ( q i ) | ( V · U ′ i ) e i X ( G,π ) ∈ E L ( H ,V e i i ) | E 0 , 2 3 | and w V i = X e i ≥ 0 ( − 1) e i q ( e i 2 ) i M i ( e i ) , where E L ( H , V e i i ) is the set of ( G, π ) ∈ Ext Γ ( H , V e i i ) suc h that G ∈ L , and G Γ = 1 , and s H ∈ im π ∗ . By Lemma 5.10, we hav e G Γ = 1 since V i is non trivial. So b y Lemma 7.29, the classes in E L ( H , V e i i ) correspond to the classes in H 2 ( H ⋊ Γ , V e i i ) L ,s H = ( H 2 ( H ⋊ Γ , V i ) L ,s H ) e i . 125 W e can write the elements of ( H 2 ( H ⋊ Γ , V i ) L ,s H ) e i as e i × z matrices ov er κ i , where im α is the column space of the asso ciated matrix and span α is the row space of the asso ciated matrix. Prop osition 6.15 tells us that | E 0 , 2 3 | is 2 q ( e i − r )( e i − r +1) 2 when ω − 1 ∗ (Φ) ∈ span α and q ( e i − r )( e i − r − 1) 2 when ω − 1 ∗ (Φ) ∈ span α . Let r α := dim κ i im α . W e ha ve ∞ X e i =0 ( − 1) e i q ( e i 2 ) | GL e i ( q ) | ( V · U ′ i ) e i X α ∈ ( H 2 ( H ⋊ Γ ,V i ) L ,s H ) e i q ( e i − r α )( e i − r α − 1) 2 = ∞ Y k =0 1 1 + q − k − 1 − u z − 1 Y k =0 (1 − q k − u ) (7.48) (and that the sum con v erges absolutely) exactly as in Lemma 7.39. If ω − 1 ∗ (Φ) ∈ H 2 ( H ⋊ Γ , V i ) L ,s H , then the ab o ve sum is w V i and ˜ w V i is ﬁnite by the same argumen t as in Lemma 7.39. No w w e assume ω − 1 ∗ (Φ) ∈ H 2 ( H ⋊ Γ , V i ) . The argument for absolute conv ergence from Lemma 7.39 also w orks if the terms q ( e i − r )( e i − r − 1) 2 ab o ve are replaced by 2 q ( e i − r )( e i − r +1) 2 . This implies that ˜ w V i is ﬁnite and gives absolute con vergence of the sums we will consider b elow. By Lemma 6.9, w e ha ve that the ω − 1 ∗ (Φ) ∈ H 2 ( H ⋊ Γ , V i ) of Prop osition 6.15 is non-zero. T o coun t α of rank r where ω − 1 ∗ (Φ) ∈ span α we ﬁrst coun t e i × z matrices o ver κ i of rank r whose row space contains a particular non-zero vector. There are q ( z − r )( r − 1) ( q ) z − 1 ( q ) z − r ( q ) r − 1 rank r subspaces of κ z con taining a particular non-zero vector for each 1 ≤ r ≤ z . F or each of these p ossible row spaces, there are q e i r ( q ) e i ( q ) e i − r matrices of dimensions e i × z with that row space if e i ≥ r , and no such matrices if e i < r . So for 1 ≤ r ≤ min( e i , z ) there are q ( z − r )( r − 1)+ e i r ( q ) z − 1 ( q ) e i ( q ) z − r ( q ) r − 1 ( q ) e i − r α of rank r where ω − 1 ∗ (Φ) ∈ span α , and for other r there are 0 suc h α . W e compute ∞ X e i =0 ( − 1) e i q ( e i 2 ) | GL e i ( q ) | ( V · U ′ i ) e i X α ∈ ( H 2 ( H ⋊ Γ ,V i ) L ,s H ) e Φ ∈ span α (2 q ( e i − r α )( e i − r α +1) 2 − q ( e i − r α )( e i − r α − 1) 2 ) = ∞ X e i =0 ( − 1) e i q ( e i 2 ) q e 2 i ( q ) e i q e i u min( e i ,z ) X r =1 q ( z − r )( r − 1)+ e i r ( q ) z − 1 ( q ) e i ( q ) z − r ( q ) r − 1 ( q ) e i − r (2 q ( e i − r )( e i − r +1) 2 − q ( e i − r )( e i − r − 1) 2 ) . 126 Since H 2 ( H ⋊ Γ , V i ) L ,s H con tains a non-zero v ector, z ≥ 1 . W e let e = e i − r and ha ve that the ab o ve sum is = ∞ X e =0 z X r =1 ( − 1) e + r q ( e + r 2 ) − ( e + r ) 2 − ( e + r ) u +( z − r )( r − 1)+( e + r ) r ( q ) z − 1 ( q ) e + r ( q ) e + r ( q ) z − r ( q ) r − 1 ( q ) e (2 q e ( e +1) 2 − q e ( e − 1) 2 ) = ∞ X e =0 z X r =1 ( − 1) e + r q ( r 2 ) + er − ( e + r ) 2 − ( e + r ) u +( z − r )( r − 1)+( e + r ) r + ( e 2 ) ( q ) z − 1 ( q ) z − r ( q ) r − 1 ( q ) e (2 q e − 1) = ∞ X e =0 ( − 1) e +1 q − e − eu − u (2 q e − 1) ( q ) e z X r =1 ( − 1) r − 1 q ( r − 1 2 ) − ( r − 1)( u − 1)+( z − 1)( r − 1) − ( r − 1) 2 ( q ) z − 1 ( q ) z − r ( q ) r − 1 = ∞ X e =0 ( − 1) e q − e ( u +1) − u (1 − 2 q e ) ( q ) e z − 2 Y k =0 (1 − q k − u +1 ) b y the q -binomial theorem (7.35) = q − u ∞ Y k =0 1 1 + q − k − u − 1 − 2 ∞ Y k =0 1 1 + q − k − u ! z − 2 Y k =0 (1 − q k − u +1 ) b y the inﬁnite q -binomial theorem (7.40) = q − u  1 + q − u − 2  ∞ Y k =0 1 1 + q − k − u z − 2 Y k =0 (1 − q k − u +1 ) = − q − u ∞ Y k =0 1 1 + q − k − u z − 1 Y k =0 (1 − q k − u ) . W e obtain w V i b y summing the ab o ve with (7.48), which giv es w V i = ∞ Y k =0 1 1 + q − k − 1 − u z − 1 Y k =0 (1 − q k − u ) − q − u ∞ Y k =0 1 1 + q − k − u z − 1 Y k =0 (1 − q k − u ) =  (1 + q − u ) − q − u  ∞ Y k =0 1 1 + q − k − u z − 1 Y k =0 (1 − q k − u ) . Note that our assumption on U implies that V · U ′ i > 1 so that u > 0 , whic h implies the ﬁnal claim ab out when w V i is 0 or p ositive. □ 7.11. Pro of of Theorem 7.5. Corollary 7.25 red uced the ﬁniteness of ˜ v W , H to that of the ˜ w V i and ˜ w N i . These later ﬁniteness statemen ts are pro ven in Lemmas 7.27, 7.31, 7.34, 7.37, 7.39, 7.42, 7.44, 7.47. The assumption that eac h representation of Γ of c haracteristic dividing n has a nontrivial v ector inv arian t under at least one element of U is used in some of these lemmas, and immediately implies that U is nonempt y , whic h is used in Lemma 7.34. Corollary 7.25 reduced the non-negativity of v W , H to that of the w V i and w N i . These later non-negativity statements follo w from the lemmas listed ab ov e, with a few additional observ ations. When using Lemma 7.31, w e hav e that w V i = ∞ Y j =1  1 − q − j i | H 2 ( H ⋊ Γ , V i ) L | | H 1 ( H ⋊ Γ , V i ) | V · U ′ i  . 127 W e note that H 2 ( H ⋊ Γ , V i ) L , H 1 ( H ⋊ Γ , V i ) , and ( V · U ′ i ) are End H ⋊ Γ ( V i ) -v ector spaces, and th us every term in the pro duct is 1 − q m i for some integer m . These exp onen ts m are either alw ays negative, in which case the pro duct is p ositiv e, or one factor has m = 0 , which makes the pro duct 0 . Similar reasoning applies to the expression for w V i giv en in Lemma 7.37. 7.12. Criteria for nonzero probability. Prop osition 7.49. L et L b e a level of the c ate gory of ﬁnite Γ -gr oups. L et W b e the set of isomorphism classes of ﬁnite n -oriente d Γ -gr oups whose underlying Γ -gr oup is in L . L et U b e a multiset of elements of Γ . Assume al l nonzer o r epr esentations of Γ of char acteristic dividing n c ontain some nontrivial ve ctor ﬁxe d by at le ast one element of U . Fix H ∈ W with H Γ trivial. Then we have v W , H > 0 if and only if, for e ach ﬁnite simple ab elian H ⋊ Γ -gr oup V i of char acteristic prime to n that c an b e the kernel of a simple morphism G → H with G ∈ W we have | H 2 ( H ⋊ Γ , V i ) L | ≤ | H 1 ( H ⋊ Γ , V i ) | V · U ′ i and for e ach ﬁnite simple ab elian H ⋊ Γ -gr oup V i of char acteristic dividing n that c an b e the kernel of a simple morphism G → H with G ∈ W we have | H 1 ( H ⋊ Γ , V ∨ i ) || H 2 ( H ⋊ Γ , V i ) L ,s H | ≤ | H 1 ( H ⋊ Γ , V i ) | V · U ′ i . Pr o of. By Corollary 7.25 we hav e v W , H  = 0 if and only if M H is nonzero and all the w V i and w N i are nonzero. That M H is nonzero follo ws from the assumption that H Γ is trivial. It follows from Lemma 7.27 that the w N i are all nonzero, so it suﬃces to chec k for V i solo that w V i is nonzero if and only if the stated criteria are satisﬁed for V i and for V i paired that w V i w V ′ i is nonzero if and only if the stated criteria are satisifed for V i and V i ′ . In the case of Lemma 7.31, each term q − j i | H 2 ( H ⋊ Γ ,V i ) L | | H 1 ( H ⋊ Γ ,V i ) | V · U ′ i has the form q m i for some m and the pro duct of the 1 − q m i is nonzero if and only if no m is zero, which happ ens and only if eac h m is ≤ − 1 , whic h o ccurs if and only if | H 2 ( H ⋊ Γ , V i ) L | ≤ | H 1 ( H ⋊ Γ , V i ) | V · U ′ i . Lemma 7.34 states that for V i trivial of c haracteristic prime to n , w V i  = 0 if and only if z ≤ u , in other words if | H 2 ( H ⋊ Γ , V i ) L | ≤ V · U ′ i . Since | H 1 ( H ⋊ Γ , V i ) | = 1 in this case by the assumption H Γ = 1 , this is equiv alent to the criterion | H 2 ( H ⋊ Γ , V i ) L | ≤ | H 1 ( H ⋊ Γ , V i ) | V · U ′ i . In the case of Lemma 7.37, w V i is given b y a pro duct with all terms of the form 1 − q m i where the pro duct is nonzero if and only if all m ≤ − 1 , whic h o ccurs if and only if | H 1 ( H ⋊ Γ , V ∨ i ) || H 2 ( H ⋊ Γ , V i ) L ,s H | ≤ | H 1 ( H ⋊ Γ , V i ) | V · U ′ i . Lemma 7.44 states that for V i , V i ′ dual but nonisomorphic of c haracteristic dividing n , we ha ve w V i w V i ′  = 0 if and only if − d + z ′ ≤ h ≤ d − z , which taking exp onen tials with base q and applying the deﬁnitions of d, z , z ′ , h o ccurs if and only if | H 2 ( H ⋊ Γ , V i ′ ) L ,s H V · U ′ i ≤ | H 1 ( H ⋊ Γ , V i ′ ) | | H 1 ( H ⋊ Γ , V i ) | ≤ V · U ′ i | H 2 ( H ⋊ Γ , V i ) L ,s H . The ﬁrst inequalit y is equiv alen t to the criterion | H 1 ( H ⋊ Γ , V ∨ i ′ ) || H 2 ( H ⋊ Γ , V i ′ ) L ,s H | ≤ | H 1 ( H ⋊ Γ , V i ′ ) | V · U ′ i ′ for V ′ i since V · U ′ i ′ = V · U ′ i and V ∨ i ′ ∼ = V i , while the second inequalit y is equiv alent to the criterion | H 1 ( H ⋊ Γ , V ∨ i ) || H 2 ( H ⋊ Γ , V i ) L ,s H | ≤ | H 1 ( H ⋊ Γ , V i ) | V · U ′ i for V i . In the case that V i is selfdual of c haracteristic dividing n , w e hav e H 1 ( H ⋊ Γ , V ∨ i ) ∼ = H 1 ( H ⋊ Γ , V i ) and thus the stated criterion simpliﬁes to | H 2 ( H ⋊ Γ , V i ) L ,s H | ≤ V · U ′ i . This matc hes the condition for w V i  = 0 giv en in Lemmas 7.39, 7.42, and 7.47. □ 128 7.13. The main statemen ts. Pr o of of The or em 1.8. The statemen t of Theorem 1.8 follo ws immediately from Theorem 7.6 and Corollary 7.25 once w e chec k that the form ulas for w V i and w N i giv en in §1.6 match the deﬁnitions of w V i and w N i . W e chec k this by observing that the form ulas giv en in §1.6 match the formulas for w V i and w N i pro ven, in v arious cases, in Lemmas 7.27, 7.31, 7.34, 7.37, 7.39, 7.42, 7.44, 7.47. □ Pr o of of The or em 1.5. Let U b e the multiset consisting of γ v for eac h v ∈ Sp ec k ⊗ R . Let us ﬁrst chec k that U satisﬁes the h yp othesis of Theorem 1.8 that each nonzero representation of Γ of c haracteristic dividing n con tains a nonzero vector ﬁxed by some elemen t of U . If n = 1 then there are no nonzero representations of c haracteristic dividing n and the hypothesis is v acuously satisﬁed. If n = 2 then, by deﬁnition of n , | Γ | is o dd, so γ v m ust b e 1 for all v . If n > 2 then, by deﬁnition of n , k contains the n th ro ots of unit y , which since n > 2 implies that all inﬁnite places of v are complex, so again γ v is 1 for all v . In either case, since there is at least one inﬁnite place v , and w e ha ve γ v = 1 for this place, γ v ﬁxes ev ery nonzero vector and the h yp othesis is again satisﬁed. Theorem 1.5 now follo ws from sp ecializing Theorem 1.8 to this v alue of U , except for the ﬁnal claim ab out con vergence conditional on Conjecture 1.3. This follo ws from the last part of Theorem 1.8 ab out the limit of a sequence of measures. W e deﬁne a sequence of measures ν B indexed b y B by summing o ver D ≤ B and K ∈ E Γ ( D , k , γ ) a delta measure supp orted on Gal ( K un , | Γ | ′ /K ) and dividing by P D ≤ B | E Γ ( D , k , γ ) | = ν Γ ,n,γ . Conjecture 1.3 implies that the momen ts of the ν B con verge to the moments b H , which b y Theorem 7.6 implies that the ν B con verges to ν Γ ,n,γ . Because the set { X | X W ≃ H } is open and closed, it follows that ν B ( { X | X W ≃ H } ) con verges to ν Γ ,n,γ ( { X | X W ≃ H } ) , whic h gives the desired statemen t. □ Pr o of of The or em 1.4. This follows from sp ecializing Theorem 1.8 to U = { 1 } , except for the ﬁnal claim ab out conv ergence. Observe that for a function F of parameters b, q and limit ℓ , w e hav e lim q →∞ lim sup b →∞ F ( b, q ) = ℓ and lim q →∞ lim inf b →∞ F ( b, q ) = ℓ if and only if, for each sequence of pairs b i , q i , as long as b i gro ws suﬃcien tly fast with resp ect to q i , w e ha ve lim i →∞ F ( b i , q i ) = ℓ . W e therefore ﬁx a lev el L and a group H , and a sequence of pairs b i , q i , where w e ma y assume that q i gro ws arbitrarily fast with resp ect to b i , and that ( q i , | Γ | M ) = 1 and ( q i − 1 , nM ) = n . It suﬃces to prov e that (7.50) lim i →∞ P m ≤ b i P K ∈ E Γ ( q m i , F q i ( t ))    { K ∈ E Γ ( q m i , F q i ( t ) | Gal ( K un , | Γ | ′ /K ) W ≃ H }    P m ≤ b i | E Γ ( q m i , F q i ( t )) | = ν Γ ,n, { 1 } ( { X | X W ≃ H } ) . W e now deﬁne a sequence of measures ν i b y summing ov er m ≤ b i and K ∈ E Γ ( q m i , F q i ( t )) a delta measure supp orted on Gal ( K un , | Γ | ′ /K ) W and dividing by P m ≤ b i | E Γ ( q m i , F q i ( t )) | . Equation (7.50) follo ws from the claim that ν i con verges to the pro jection of ν Γ ,n, { 1 } on to W . By [SW22, Theorem 1.6], this follows if we chec k that G -momen t of ν i , for each G ∈ W , con verges to the G -momen t of ν Γ ,n, { 1 } , whic h is ( | H Γ || H 2 ( H ⋊ Γ , Z /n ) | | H || H 3 ( H ⋊ Γ , Z /n ) | if H Γ = 1 0 if H Γ  = 1 . 129 The G -momen t of ν i is giv en by P m ≤ b i P K ∈ E Γ ( q m i , F q i ( t )) Sur( Gal ( K un , | Γ | ′ /K ) , H ) P m ≤ b i | E Γ ( q m i , F q i ( t )) | so it suﬃces to prov e that (7.51) lim i →∞ P m ≤ b i P K ∈ E Γ ( q m i , F q i ( t )) Sur( Gal ( K un , | Γ | ′ /K ) , H ) P m ≤ b i | E Γ ( q m i , F q i ( t )) | = ( | H Γ || H 2 ( H ⋊ Γ , Z /n ) | | H || H 3 ( H ⋊ Γ , Z /n ) | if H Γ = 1 0 if H Γ  = 1 . Let us now c heck that we hav e ( q i , | Γ || G | ) = 1 and q i ≡ 1 mo d n and ( q i − 1 , | G | ) = ( n, | G | ) . The ﬁrst claim follo ws from ( q 1 , | Γ | M ) = 1 and the fact that | G | divides a p ow er of M by deﬁnition of M . The second claim follows from ( q i − 1 , nM ) = n . The third claim fails if there is a prime p where the p -adic v aluation of q i − 1 and the p -adic v aluation of | G | are b oth greater than the p -adic v aluation of n . In particular, this requires p to divide M , and so implies that the p -adic v aluation of ( q i − 1 , nM ) is greater than the p -adic v aluation of n , con tardicting the assumption that ( q i − 1 , nM ) = n . (7.51) now follows, for b i gro wing suﬃciently fast with resp ect to q i , from Theorem 4.1 and the ab o v e observ ation on the relationship b etw een double limits and single limits. (Theorem 4.1 states an identit y that holds for q suﬃciently large dep ending on H , Γ , and th us alw ays holds in the limit as q tends to inﬁnity .) □ Pr o of of The or em 1.1. W e sp ecialize Theorem 1.4 to the setting where Γ = Z / 3 , n = 2 , and L is a level consisting of 2 -groups with an action of Γ . W e ha ve M = 2 so the condition ( q , | Γ | M ) = 1 reads ( q , 6) = 1 and is equiv alent to q b eing o dd and not divisible b y 3 , while the cond ition ( q − 1 , nM ) = n reads ( q − 1 , 4) = 2 and is equiv alent to q b eing congruent to 3 mo dulo 4 . So these conditions matc h the conditions q ≡ 3 mo d 4 and 3 ∤ q from the statemen t of Theorem 1.1. Gal( K ur , 2 /K ) is the maximal 2 -group quotien t of Gal( K ur , | Γ | ′ /K ) . The set of proﬁnite groups whose maximal 2 -group quotien t is isomorphic to H is the ﬁnite disjoin t union of, for eac h Γ -group H ′ whose underlying group is isomorphic to H , the set of proﬁnite groups whose maximal 2 -group quotien t is isomorphic as a Γ -group to H ′ . By Lemma 3.15 this set has the form { X | X L ∼ = H ′ } for L a lev el in the category of Γ -groups consisting only of p -groups with an action of Γ , including H ′ and every extension of H ′ b y V i for V i an y group of the form F d p with an irreducible action of Γ . Th us the probability that Gal( K ur , | Γ | ′ /K ) is the sum o ver H ′ of the probabilit y that Gal( K ur , | Γ | ′ /K ) lies in { X | X L ∼ = H ′ } . By Theorem 1.4, the double limit (with limsup or liminf ) of the probability that Gal( K ur , | Γ | ′ /K ) lies in { X | X L ∼ = H ′ } is equal to ν Z / 3 , 2 , { 1 } ( { X | X L ∼ = H ′ } ) (b ecause the set { X | X L ∼ = H ′ } is itself a disjoint union of { X | X L ∼ = H ′ } for oriented Γ -groups H ′ whose underlying group is H ′ ). Hence if we take p H = X H ′ ν Z / 3 , 2 , { 1 } ( { X | X L ∼ = H ′ } ) w e obtain the ﬁrst part of Theorem 1.1. W e ha ve ν Z / 3 , 2 , { 1 } ( { X | X L ∼ = H ′ } ) = 0 if ( H ′ ) Γ  = 1 b y Theorem 1.8. 130 F or H the Klein four group or the eight-elemen t quaternion group, there is a unique non- trivial Γ -action on H up to isomorphism, and ν Z / 3 , 2 , { 1 } ( { X | X L ∼ = H ′ } ) w ere calculated resp ectiv ely in Lemmas 3.21 and 3.23, and these agree with the formula for p H claimed in Theorem 1.1. Clearly p H is zero if H is not a 2 -group. The remaining 2 -groups of order at most 8 are the trivial group, Z / 2 , Z / 4 , ( Z / 2) 3 , Z / 4 × Z / 2 , Z / 8 , and the dihedral group D 4 . The groups Z / 2 , Z / 4 , Z / 4 × Z / 2 , Z / 8 , and the dihedral group D 4 do not ha v e a nontrivial automorphism of order 3 , so the only Γ -group structure is the trivial Γ -action. The group ( Z / 2) 3 has one non-trivial Γ -action up to automorphism (the regular represen tation of Γ ), but that action still has non-trivial coin v arian ts. W e ha v e p H = 0 if H Γ is nontrivial since the moment is already trivial for these groups, so w e get that p H = 0 for the remaining H . Finally , for the trivial group, w e apply Equation (3.16) to H the trivial group and L the lev el generated by , in the notation of §3.3, the groups V 1 and V 2 . All the m ultiplicativ e factors outside the sum are 1 , the group H 3 ( H ⋊ Γ , F 2 ) being summed o ver has a single element s H = 0 , the factor w V 1 ( s H ) inside the sum gives 1 b y Equation (3.17), and the factor w V 2 ( s H ) inside the sum giv es Q ∞ k =0 (1 + 4 − k − 1 2 − 1 ) − 1 b y Equation (3.18). Hence the probability in this case is Q ∞ k =0 (1 + 4 − k − 1 2 − 1 ) − 1 . □ 8. Non-existence resul ts Let k b e a num b er ﬁeld, Γ a ﬁnite group, K an extension of k with Galois group Γ . Let V b e a ﬁnite ab elian group of order prime to | Γ | with an action of Gal( K un , | Γ | ′ /k ) . Let H i c (Sp ec O K , V ) be the compactly supported étale cohomology groups in the sense of [Mil06, p. 165]. Let H i par (Sp ec O K , V ) b e the image of the natural map H i c (Sp ec O K , V ) → H i (Sp ec O K , V ) (the p ar ab olic c ohomolo gy gr oups .) Lemma 8.1. With notation as in the start of Se ction 8, we have | H 0 (Sp ec O K , V ) Γ || H 2 c (Sp ec O K , V ) Γ | | H 1 p ar (Sp ec O K , V ) Γ || H 3 c (Sp ec O K , V ) Γ | = Y v ar chime dean plac e of k | V Gal( k sep v /k v ) | , wher e the action of the absolute Galois gr oup Gal( k sep v /k v ) on V is thr ough the map Gal( k sep v /k v ) → Gal( K un , | Γ | ′ /k ) (only deﬁne d up to c onjugacy). Pr o of. Let U b e an op en subset of Sp ec O k a voiding all the primes that ramify in K or divide the order of V . By [Mil06, I I, Theorem 2.13(b)] the righ t-hand side is equal to | H 0 c ( U, V ) || H 2 c ( U, V ) | | H 1 c ( U, V ) || H 3 c ( U, V ) | where H i c ( U, V ) arises from the mapping cone of the natural map from H i ( U, V ) to the sum o ver places v of F not in U of H i ( k v , V ) , where H i ( k v , V ) is understo o d as Galois cohomology for non-arc himedean places and T ate cohomology for arc himedean places. Let ˜ U b e the inv erse image of U in Sp ec O K . W e ha ve H i ( U, V ) = H i ( ˜ U , V ) Γ b y the Ho c hschild-Serre sp ectral sequence [Mil80, Ch. II I, Theorem 2.20] because V has order prime to Γ and hence so do all its cohomology groups. W e can upgrade this to H i c ( U, V ) = H i c ( ˜ U , V ) Γ using the long exact sequence for compactly supp orted cohomology [Mil06, Ch. 2, Prop. 2.3(a)], and the similar fact that H i (Gal( k sep v /k v )) , V ) → H i (Gal( K sep w /K w )) , V ) Γ v for an y place v of k , place w of K abov e v , and stabilizer Γ v of w in Γ . 131 A long exact sequence relates H i c ( ˜ U , V ) → H i c ( O K , V ) and the sum ov er non-archimedean places w of K not in ˜ U of H i ( R w , V ) where R w is the residue ﬁeld [Mil06, Ch. 2, Prop. 2.3(d)].The cohomology of the residue ﬁeld is concen trated in degrees 0 , 1 so nonzero maps H i c ( O K , V ) → Q w ∈ ˜ U H i ( R w , V ) o ccur only for i = 0 , 1 and nonzero maps Q w ∈ ˜ U H i − 1 ( R w , V ) → H i c ( ˜ U , V ) o ccur only for i = 1 , 2 . In particular these only o ccur in degrees b etw een 0 and 3 , giving | H 0 c ( ˜ U , V ) Γ || H 2 c ( ˜ U , V ) Γ | | H 1 c ( ˜ U , V ) Γ || H 3 c ( ˜ U , V ) Γ | = | H 0 c (Sp ec O K , V ) Γ || H 2 c (Sp ec O K , V ) Γ | | H 1 c (Sp ec O K , V ) Γ || H 3 c (Sp ec O K , V ) Γ | | ( Q w ∈ ˜ U H 0 ( R w , V )) Γ | | ( Q w ∈ ˜ U H 1 ( R w , V )) Γ | . The inv ariants ( Q w ∈ ˜ U H 0 ( R w , V )) Γ split as a product o v er one representativ e w of each Γ -orbit of H 0 ( R w , V ) Γ w , where Γ w ⊂ Γ is the stabilizer of w , and the same is true for ( Q w ∈ ˜ U H 1 ( R w , V )) Γ . Thus to sho w | ( Q w ∈ ˜ U H 0 ( R w V )) Γ | | ( Q w ∈ ˜ U H 1 ( R w ,V )) Γ | = 1 it suﬃces to show H 0 ( R w , V ) Γ w and H 1 ( R w , V ) Γ w ha ve the same cardinality for all w , which follo ws from the Γ w -equiv ariant long exact sequence 0 → H 0 ( R w , V ) → V F rob w − 1 → V → H 1 ( R w , V ) → 0 . This giv es | H 0 c (Sp ec O K , V ) Γ || H 2 c (Sp ec O K , V ) Γ | | H 1 c (Sp ec O K , V ) Γ || H 3 c (Sp ec O K , V ) Γ | = Y v archimedean place of k | V Gal( k sep v /k v ) | . T o obtain the statement, w e use the long exact sequence [Mil06, Ch. 2, Prop. 2.3(a)], whic h together with the v anishing of H − 1 (Sp ec O K , V ) giv es a long exact sequence 0 → ( Y w arch H − 1 ( K w , V )) Γ → H 0 c (Sp ec O K , V ) Γ → H 0 (Sp ec O K , V ) Γ → ( Y w arch H 0 ( K w , V )) Γ → H 1 c (Sp ec O K , V ) Γ → H 1 (Sp ec O K , V ) Γ and th us 0 → ( Y w arch H − 1 ( K w , V )) Γ → H 0 c (Sp ec O K , V ) Γ → H 0 (Sp ec O K , V ) Γ → ( Y w arch H 0 ( K w , V )) Γ → H 1 c (Sp ec O K , V ) Γ → H 1 par (Sp ec O K , V ) Γ → 0 . Since the Galois group of every archimidean place is Z / 2 or 0 , the T ate cohomology groups are isomorphic in each degree and thus ( Q w H − 1 ( K w , V )) Γ and ( Q w H 0 ( K w , V )) Γ ha ve the same cardinalit y . This gives | H 0 (Sp ec O K , V ) Γ | | H 1 par (Sp ec O K , V ) Γ | = | H 0 c (Sp ec O K , V ) Γ | | H 1 c (Sp ec O K , V ) Γ | and the statement. □ W e now compare these cohomology groups to the Galois cohomology groups H i (Gal( K un , | Γ | ′ /K ) , V ) . The next tw o lemmas are num b er ﬁeld analogs of standard facts in top ology (e.g. see [SW24, Lemma 2.1].) 132 Lemma 8.2. With notation as in the start of Se ction 8, ther e is a Γ -e quivariant isomorphism H 1 (Gal( K un , | Γ | ′ /K ) , V ) → H 1 p ar (Sp ec O K , V ) . Pr o of. By the long exact sequence [Mil06, Ch. 2, Prop. 2.3(a)], we ha ve that H 1 par (Sp ec O K , V ) is precisely the subgroup of elemen ts of H 1 (Sp ec O K , V ) that are trivial in all maps to H 1 ( K v , V ) for v an arc himedian place of K . Remark 2.3 gives a map H 1 (Gal( K un , | Γ | ′ /K ) , V ) → H 1 (Sp ec O K , V ) . An y cohomology class arising from this map must restrict to a trivial class at each archimedean place, since K un , | Γ | ′ /K is split completely at each archimedean place, and so the image of the map is con tained in H 1 par (Sp ec O K , V ) . Let K unf b e the maximal algebraic extension of K that is unramiﬁed at all ﬁnite places. By the deﬁnition, one can see that the map H 1 (Gal( K un , | Γ | ′ /K ) , V ) → H 1 (Sp ec O K , V ) then factors through H 1 (Gal( K unf /K ) , V ) . The resulting map H 1 (Gal( K unf /K ) , V ) → H 1 (Sp ec O K , V ) is an isomorphism (see, e.g., [Y ou15, Theorem 15]). Note that b y the inﬂation-restriction exact sequence H 1 (Gal( K un , | Γ | ′ /K ) , V ) → H 1 (Gal( K unf /K ) , V ) is injec- tiv e. Th us it remains to sho w that H 1 (Gal( K un , | Γ | ′ /K ) , V ) is the subgroup of H 1 (Gal( K unf /K ) , V ) that is trivial in all maps to H 1 ( K v , V ) for v arc himedian. Let G = Gal( K unf /K ) and N = Gal( K unf /K un , | Γ | ′ ) . Consider an elemen t α of H 1 (Gal( K unf /K ) , V ) that is trivial in all maps to H 1 ( K v , V ) for v arc himedian. It is represented by a co cycle G → V that gives a section s of V ⋊ G → G . W e comp ose s with the quotien t to obtain G → V ⋊ G → V ⋊ G/ N . The comp osite map has image whose order is prime to | Γ | . Moreo ver if G v ⊂ G is the decomp osition group for an archimedian place, since G v acts trivially on V , we hav e that s G v is a homomorphism and is trivial b y our assumption on α . So we hav e G → V ⋊ G/ N whic h is trivial on all G v for v arc himedian and has image of order prime to | Γ | . Thus this map factors through G/ N , and we see our co cycle is the image of a co cycle G/ N → V , as desired. □ Lemma 8.3. With notation as in the start of Se ction 8, ther e is a Γ -invariant inje ction H 2 (Gal( K un , | Γ | ′ /K ) , V ) → H 2 c ( O K , V ) . Pr o of. The homomorphism is giv en by Lemma 2.4, and the Γ -in v ariance can b e seen from the deﬁnition. It remains to c heck injectivity . Fix α ∈ H 2 (Gal( K un , | Γ | ′ /K ) , V ) whose image in H 2 c ( O K , V ) is zero. In particular, the image of α in H 2 ( O K , V ) is zero. Fix a ﬁnite unramiﬁed | Γ | ′ Galois extension K ′ of K suc h that α arises from a class in H 2 (Gal( K ′ /K ) , V ) . Then the induced class in H 2 ( O K , V ) arises from a Cech co cycle of the co v ering Sp ec O K ′ → Sp ec O K . The map from the Cec h cohomology of this cov ering to the sheaf cohomology is the edge map of the Cec h-to-sheaf sp ectral sequence of the cov ering Sp ec O K ′ → Sp ec O K . Since α is sent to 0 , the asso ciated co cyle must b e in the image of a diﬀerential of this sp ectral sequence, which m ust b e d 0 , 1 2 : E 0 , 1 2 → E 2 , 0 2 . The source of this diﬀeren tial is a subgroup of H 1 (Sp ec O K ′ , V ) . The class β in H 1 (Sp ec O K ′ , V ) whose image under d 0 , 1 2 is α corresponds to a V -torsor on O K ′ . Let K ∗ b e a Galois extension of K o v er which the torsor V splits. Since V is unramiﬁed o ver K , w e can tak e K ∗ to be unramiﬁed at all ﬁnite places, but not necessarily at the inﬁnite places. Since this torsor splits ov er K ∗ , the pullback of β to H 1 (Sp ec O K ∗ , V ) v anishes. Since the diﬀeren tials of this sp ectral sequence are compatible with pullbacks, the pullbac k of α to the Cec h cohomology of Sp ec O K ∗ → Sp ec O K is the image of d 0 , 1 2 applied to the pullback of β , which is 0 , and th us must b e 0 . 133 The Cech complex of Sp ec O K ∗ → Sp ec O K is isomorphic to the group cohomology com- plex of Gal( K ∗ /K ) , and our map from group cohomology co cycles to Cec h cohomology uses this isomorphism, so the pullback of α to C 2 (Gal( K ∗ /K ) , V ) is a cob oundary dγ for some γ ∈ C 1 (Gal( K ∗ /K ) , V ) . The image of α under the homomorphism of Remark 2.3 is represented b y a pair consisting of the Cech co cycle on the cov ering Sp ec O K ∗ → Sp ec O K asso ciated to α and, for each inﬁnite place v of K , a cochain δ v for the Cec h co ver Sp ec K ′ ⊗ K K v → K v of K v whose diﬀeren tial is the pullback of α . Since α = dγ , this co cycle is the cob oundary ( dγ , ( γ ) v |∞ ) plus the coycle (0 , ( δ v − γ ) v |∞ ) . The co cycle (0 , ( δ v − γ ) v |∞ ) is the image of the co cycle ( δ v − γ ) v |∞ ∈ Q v |∞ H 1 ( K v , V ) under the connecting homomorphism Q v |∞ H 1 ( K v , V ) → H 2 c ( O K , V ) . It follows that ( δ v − γ ) v |∞ is in the image of H 1 ( O K , V ) → Q v |∞ H 1 ( K v , V ) . A class in H 1 ( O K , V ) whose image is ( δ v − γ ) v |∞ corresp onds to a V -torsor on Sp ec O K . By a p ossible mo diﬁcation of K ∗ , we may assume that this torsor is also split ov er K ∗ , so that the class in H 1 ( O K , V ) is representated b y a group cohomology co cycle ϵ ∈ C 1 (Gal( K ∗ ) /K, V ) . F or eac h v , we conclude that the pullbac k of ϵ to H 1 ( K v , V ) agrees with δ v − γ . Explicitly , the co cycle α endo ws Gal( K ′ /K ) × V with the structure of a group extension of Gal( K ′ /K ) with V . Then γ is a function Gal( K ∗ /K ) → V suc h that g 7→ ( g Gal( K ∗ /K ′ ) , γ ( g )) is a group homomorphism. Without loss of generalit y , we may assume α (1 , 1) = 0 . F or any inﬁnite place v that ramiﬁes in K ∗ , w e hav e K v = R and K ∗ ⊗ K K v is a pro duct of copies of C . W e can compute Cec h cohomology classes in H ∗ ( K v , V ) by pulling back from the co ver Sp ec K ∗ ⊗ K K v → Sp ec K v to a single copy of C . W e can compute the pullbacks of α and δ v b y ﬁrst pulling bac k to Spec K ′ ⊗ K K v → Sp ec K v , whic h is a pro duct of copies of K v since K ′ is split at v , and the pullback of α to one of these copies is zero since α (1 , 1) = 0 , so the pullbac k of δ v to one of these copies is zero as w ell. Th us, for each inﬁnite place v , the pullbac k of ϵ to H 1 (Gal( K v ) , v ) agrees with the pullback of − γ to H 1 (Gal( K v , V ) . Since Gal( K v ) acts trivially on V , H 1 (Gal( K v , V ) is simply the set of homomorphisms from Gal( K v ) to V , so γ + ϵ is the trivial homomorphism Gal( K v ) → V . Hence γ ( g ) + ϵ ( g ) = 0 for g ∈ Gal( K v ) . Th us the function f : Gal( K ∗ /K ) → Gal( K ′ /K ) × V sending g 7→ ( g Gal( K ∗ /K ′ ) , γ ( g ) + ϵ ( g )) is trivial on Gal( K v ) for eac h inﬁnite place v . The co cycle condition for ϵ implies that f is a group homomorphism. Since f is trivial on the Galois group of eac h inﬁnite place, f is unramiﬁed. Because the image of f has order prime to Γ , f factors through Gal( K un , | Γ | ′ /K ) so α splits o ver Gal( K un , | Γ | ′ /K ) and thus α = 0 , as desired. □ Lemma 8.4. With notation as in the start of Se ction 8, assume for some p ositive inte ger n that V has char acteristic dividing n and k c ontains the n th r o ots of unity. L et V ∨ b e the dual r epr esentation of V and let D V b e the Cartier dual of the gr oup scheme asso ciate d to V . Then ther e is a natur al map V ∨ → D V of gr oup schemes over O K and the induc e d Γ -e quivariant map H 1 ( O K , V ∨ ) → H 1 ( O K , D V ) is an inje ction. Pr o of. W e hav e V ∨ ∼ = Hom( V , Z /n ) and D V ∼ = Hom( V , µ n ) but ﬁxing a generator of the n th ro ots of unity in k gives a map Z /n → µ n and th us V ∨ → DV . The fact that the ﬁxed generator lies in k and not just in K insures that the induced map is Γ -equiv ariant. This map 134 is an isomorphism on the op en set deleting the primes dividing n . A class in H 1 ( O K , V ∨ ) in the kernel of the map H 1 ( O K , V ∨ ) → H 1 ( O K , D V ) represen ts a torsor that b ecomes a trivial torsor when comp osed with V ∨ → D V , hence trivial up on restriction to the op en set, hence trivial ev erywhere since O K is in tegrally closed, sho wing the map is injective. □ Lemma 8.5. With notation as in the start of Se ction 8, assume V is a simple Gal( K un , | Γ | ′ /k ) - gr oup. L et p b e the char acteristic of V . If k c ontains the p th r o ots of unity then | H 3 c (Sp ec O K , V ) Γ | = | H 0 (Sp ec O K , V ) Γ | . If K un , | Γ | ′ do es not c ontain the p th r o ots of unity, | H 3 c (Sp ec O K , V ) Γ | = 1 W e will later see in Theorem 8.14 that if k do es not contain the p th roots of unity and has no nontrivial unramiﬁed extensions of order prime to | Γ | then K un , | Γ | ′ con tains the p th ro ots of unity only if K con tains the p th ro ots of unity . Pr o of. The group H 0 (Sp ec O K , V ) Γ is equal to the Gal( K un , | Γ | ′ /k ) -in v ariants of V . Since V is a smo oth group scheme, H 3 c (Sp ec O K , V ) Γ = H 3 c, fppf (Sp ec O K , V ) Γ . By Artin-V erdier dualit y for ﬂat cohomology , H 3 c, fppf (Sp ec O K , V ) Γ is dual to H 0 fppf (Sp ec O K , D V ) Γ where D V is the Cartier dual of V . Then H 0 fppf (Sp ec O K , D V ) Γ is isomorphic to the Galois inv arian ts of D V , hence dual to the Galois coin v ariants of the T ate twist V ( − 1) of V . If k contains the p th ro ots of unit y then V ∼ = V ( − 1) as Gal( K un , | Γ | ′ /k ) -represen tations and, b ecause V is simple, its in v ariants and coin v ariants are equal, so these are equal. F or the second statemen t, if H 3 c (Sp ec O K , V ) Γ w ere nontrivial, the Galois action on V ( − 1) w ould b e trivial, so V would b e isomorphic as a Galois mo dule to F p (1) , showing the Galois group of k ( µ p ) is a quotient of Gal( K un , | Γ | ′ /k ) and hence that the p th ro ots of unity lie in K un , | Γ | ′ . □ W e are no w ready to prov e our results giving constrain ts on the Galois cohomology of Gal( K un , | Γ | ′ /K ) ⋊ Γ . Theorem 8.6. With notation as in the start of Se ction 8, let V b e a simple Gal( K un , | Γ | ′ /k ) - gr oup. Assume that K un , | Γ | ′ do es not c ontain the p th r o ots of unity. Then | H 2 (Gal( K un , | Γ | ′ /k ) , V ) | ≤ | H 1 (Gal( K un , | Γ | ′ /k ) , V ) | Q v ar chime dean plac e of k | V Gal( k sep v /k v ) | | V Gal( K un , | Γ | ′ /k ) | . Pr o of. This follows immediately from combining Lemmas 8.1, 8.2, 8.3, and 8.5, together with the observ ation that | H 0 (Sp ec O K , V ) Γ | = | V Gal( K un , | Γ | ′ /k ) | , to show. □ Let n b e the num b er of ro ots of unit y in k of order prime to | Γ | . F or V a simple Gal( K un , | Γ | ′ /k ) -group of c haracteristic p where p | n , let H 2 (Gal( K un , | Γ | ′ /k ) , V ) s K b e the sub- group of classes in H 2 (Gal( K un , | Γ | ′ /k ) , V ) whose cup pro duct with any elemen t of H 1 (Gal( K un , | Γ | ′ /k ) , V ∨ ) has zero Artin-V erdier trace, and, if V ∼ = F p , whose image under the Bo ckstein homomor- phism H 2 (Gal( K un , | Γ | ′ /k ) , F p ) → H 3 (Gal( K un , | Γ | ′ /k ) , Z /n ) has zero Artin-V erdier trace. 135 Theorem 8.7. With notation as in the start of Se ction 8, let V b e a simple Gal( K un , | Γ | ′ /k ) - gr oup of char acteristic p . Assume k c ontains the p th r o ots of unity and k ( µ pn ) is a r amiﬁe d extension of k . Then | H 1 (Gal( K un , | Γ | ′ /k ) , V ∨ ) || H 2 (Gal( K un , | Γ | ′ /k ) , V ) s K | ≤ | H 1 (Gal( K un , | Γ | ′ /k ) , V ) | Q v ar chime dean plac e of k | V Gal( k sep v /k v ) | | V Gal( K un , | Γ | ′ /k ) | . Pr o of. It follows from Lemmas 8.1, 8.2, and 8.5 that | H 2 c ( O K , V ) | = | H 1 (Gal( K un , | Γ | ′ /k ) , V ) | Y v archimedean place of k | V Gal( k sep v /k v ) | so it suﬃces to show | H 1 (Gal( K un , | Γ | ′ /k ) , V ∨ ) || H 2 (Gal( K un , | Γ | ′ /k ) , V ) s K | ≤ | H 2 c ( O K , V ) | | V Gal( K un , | Γ | ′ /k ) | . Lemma 8.3 gives an injective map (8.8) H 2 (Gal( K un , | Γ | ′ /k ) , V ) s K → H 2 c ( O K , V ) while Lemmas 8.2, 8.4, and Artin-V erdier duality give a comp osition of injectiv e maps (8.9) H 1 (Gal( K un , | Γ | ′ /k ) , V ∨ ) → H 1 par ( O K , V ∨ ) → H 1 ( O K , V ∨ ) → H 1 ( O K , D V ) → ( H 2 c ( O K , V )) ∨ and by deﬁnition of H 2 (Gal( K un , | Γ | ′ /k ) , V ) s K the images of (8.8) and (8.9) are orthogonal and hence the pro duct of their cardinalities is at most the cardinality of H 2 c ( O K , V ) . This handles the case where V is non trivial and thus | V Gal( K un , | Γ | ′ /k ) | = 1 . In the complemen tary case where V is trivial, i.e. V ∼ = F p , it suﬃces to sho w that the images of (8.8) and (8.9) are not orthognal complemen ts of each other as then the pro duct of their cardinalities will b e at most | H 2 c ( O K , V ) | /p , as desired. T o do this, observ e that the image of (8.8) consists of classes in H 2 c ( O K , F p ) whose image under the Bo c kstein homomorphism H 2 c ( O K , F p ) → H 2 c ( O K , Z /n ) has zero Artin-V erdier trace. By Eq. (3.2), the trace of the image of a class α under the Bo ckstein morphism is the Artin-V erdier pairing of α with the image under the dual Bo c kstein morphism H 0 ( O K , µ n ) → H 1 ( O K , µ p ) of the class in H 0 ( O K , µ n ) asso ciated to a generator in µ n . This gives a class in H 1 ( O K , µ p ) = H 1 ( O K , D V ) and induces a linear form in ( H 2 c ( O K , V )) ∨ . T o show the images of (8.8) and (8.9) are not orthogonal complemen ts it suﬃces to show that this linear form is not in the image of (8.9). Since the last map in the construction of (8.9) is injective, it suﬃces to show that the Bo c kstein class in H 1 ( O K , µ p ) does not lie in the image of H 1 par ( O K , F p ) . If it was, it would v anish o v er some everywhere unramiﬁed extension K ′ of K and hence the map H 0 ( O K ′ , µ np ) → H 0 ( O K ′ , µ n ) w ould b e surjective. (Note if p = 2 then “everywhere unramiﬁed" includes the inﬁnite place, since the torsors parameterized by parab olic H 1 are those split at the inﬁnite place.) Since K ′ con tains the n th ro ots of unity and p | n this implies that K ′ con tains the pn ’th ro ots of unity . Thus K ( µ pn ) is unramiﬁed o ver K . Thus the order of the inertia group of k ( µ pn ) /k divides the degree of K/k and thus divides | Γ | , but on the other hand divides the degree of Q ( µ pn ) / Q ( µ n ) whic h is p , and p is the cardinality of V which is prime to Γ , so this is 1 and k ( µ pn ) is an unramiﬁed extension of k , con tradicting our assumption. □ 136 Lemma 8.10. L et L b e a level in the c ate gory of ﬁnite Γ -gr oups, H the maximal quotient of Gal( K un , | Γ | ′ /K ) in L , and s H the induc e d orientation on H . F or V a H ⋊ Γ -gr oup, the natur al map H 1 ( H ⋊ Γ , V ) → H 1 (Gal( K un , | Γ | ′ /K ) ⋊ Γ , V ) is inje ctive. If V ⋊ H ∈ L , this map is an isomorphism. Pr o of. Classes in H 1 ( G, V ) represent lifts of G → Aut( V ) → G → V ⋊ ( H ⋊ Γ) , so the induced map on H 1 is alw a ys injectiv e for a surjective homomorphism of groups. If V ⋊ H ∈ L , for an y lift Gal( K un , | Γ | ′ /K ) ⋊ Γ → V ⋊ ( H ⋊ Γ) the image of Gal( K un , | Γ | ′ /K ) is a Γ -in v ariant subgroup of V ⋊ H whose pro jection to H is surjectiv e, hence either V ⋊ H or H , so in either case in L , and thus m ust b e a quotien t of H , sho wing the map is surjectiv e in this case. □ Lemma 8.11. L et L b e a level in the c ate gory of ﬁnite Γ -gr oups, H the maximal quotient of Gal( K un , | Γ | ′ /K ) in L , and s H the induc e d orientation on H . F or V an admissible ﬁnite simple Γ -gr oup, the map H 2 ( H ⋊ Γ , V ) L → H 2 (Gal( K un , | Γ | ′ /K ) ⋊ Γ , V ) is inje ctive. Pr o of. Each class in H 2 ( H ⋊ Γ , V ) represents an extension of H ⋊ Γ b y V , equiv alently , a Γ -equiv ariant extension of H b y V , and its pullbac k to H 2 (Gal( K un , | Γ | ′ /K ) ⋊ Γ , V ) is trivial if and only if the extension splits Γ -equiv arian tly ov er Gal( K un , | Γ | ′ /K ) , equiv alently , the map Gal( K un , | Γ | ′ /K ) → H lifts Γ -equiv ariantly to the extension. If the extension class lies in H 2 ( H ⋊ Γ , V ) L then the associated extension is in L , so an y lift m ust factor through H , sho wing the extension splits already o ver H and its class is trivial. □ Lemma 8.12. L et L b e a level in the c ate gory of ﬁnite Γ -gr oups, H the maximal quotient of Gal( K un , | Γ | ′ /K ) in L , and s H the induc e d orientation on H . L et V b e an admissible simple H ⋊ Γ -gr oup of char acteristic p . Assume that K un , | Γ | ′ do es not c ontain the p th r o ots of unity. Then | H 2 ( H ⋊ Γ , V ) L | ≤ | H 1 ( H ⋊ Γ , V ) | Q v ar chime dean plac e of k | V Gal( k v ) | | V H ⋊ Γ | . Pr o of. This follows immediately from com bining Lemmas 8.6, 8.10, and 8.11. □ Lemma 8.13. L et L b e a level in the c ate gory of ﬁnite Γ -gr oups, H the maximal quotient of Gal( K un , | Γ | ′ /K ) in L , and s H the induc e d orientation on H . L et V b e an admissible simple H ⋊ Γ -gr oup of char acteristic p . Assume k c ontains the p th r o ots of unity. Then | H 1 ( H ⋊ Γ , V ∨ ) || H 2 ( H ⋊ Γ , V ) L ,s H | ≤ | H 1 ( H ⋊ Γ , V ) | Q v ar chime dean plac e of k | V Gal( k v ) | | V H ⋊ Γ | . Pr o of. In view of Theorem 8.7 and Lemma 8.10 it suﬃces to pro ve | H 1 ( H ⋊ Γ , V ∨ ) || H 2 ( H ⋊ Γ , V ) L ,s H | ≤ | H 1 (Gal( K un , | Γ | ′ /K ) ⋊ Γ , V ∨ ) || H 2 (Gal( K un , | Γ | ′ /K ) ⋊ Γ , V ) s K | . W e ﬁrst handle the case where V is nontriv ial. F or p the characteristic of V , the righ t-hand side can b e viewed as p raised to the p o wer of dim H 1 (Gal( K un , | Γ | ′ /K ) ⋊ Γ , V ∨ ) plus dim H 2 (Gal( K un , | Γ | ′ /K ) ⋊ Γ , V ) min us the rank of the bilinear form H 1 (Gal( K un , | Γ | ′ /K ) ⋊ Γ , V ∨ ) × H 2 (Gal( K un , | Γ | ′ /K ) ⋊ Γ , V ) → F p arising from cup pro duct and Artin-V erdier duality . 137 Similarly , the left-hand side is at most p raised to the p o wer of dim H 1 ( H ⋊ Γ , V ∨ ) plus dim H 2 ( H ⋊ Γ , V ) L min us the rank of the pullback bilinear form H 1 ( H ⋊ Γ , V ∨ ) → dim H 2 ( H ⋊ Γ , V ) L → F p . By Lemma 8.10 this is the pullback along injectiv e maps H 1 ( H ⋊ Γ , V ) → H 1 (Gal( K un , | Γ | ′ /K ) ⋊ Γ , V ) and H 2 ( H ⋊ Γ , V ) L → H 2 (Gal( K un , | Γ | ′ /K ) ⋊ Γ , V ) of the previous bilinear form. It is straigh tforw ard to c heck b y linear algebra that the sum of the dimensions of t w o v ector spaces min us the rank of a bilinear form on them can only decrease when w e pull bac k the bilinear form along injectiv e maps. In the case when V is nontrivial, the same claims are true except replacing H 2 (Gal( K un , | Γ | ′ /K ) ⋊ Γ , V ) b y the subspace of classes whose image under the Bo c kstein homomorphism has trivial Artin-V erdier trace, and doing the same for H 2 ( H ⋊ Γ , V ) L . The same argument applies. □ Theorem 8.14. L et k b e a numb er ﬁeld, Γ a ﬁnite gr oup, and p a prime. Assume that k lacks nontrivial unr amiﬁe d extensions of de gr e e prime to | Γ | . Then K un , | Γ | ′ c ontains the p th r o ots of unity if and only if K c ontains the p th r o ots of unity. Pr o of. The “if ” direction is clear so we establish “only if ” and th us assume that K un , | Γ | ′ con tains the p th ro ots of unit y . Let ℓ be a prime dividing the order of Gal( k ( µ p ) /k ) . W e sho w that ℓ | | Γ | . Since Gal( k ( µ p ) /k ) is ab elian, it has some quotient of order ℓ , corresp onding to a degree ℓ ex- tension of k . If this extension is unramiﬁed then b y assumption ℓ | | Γ | so we ma y assume this extension is ramiﬁed at some place v and th us the inertia group of k ( µ p ) /k at v has order a multiple of ℓ . If K un , | Γ | ′ con tains k ( µ p ) then the inertia group of K un , | Γ | ′ /k at v surjects on to the inertia group of k ( µ p ) /k at v . Since K un , | Γ | ′ is an everywhere unramiﬁed extension of K , the inertia group of K un , | Γ | ′ /k at v is isomorphic to the inertia group of K /k at v . Hence ℓ divides the order of the inertia group of k ( µ p ) /k at v which divides the order of the inertia group of K /k at v whic h divides | Γ | , and in particular ℓ | | Γ | , as desired. If K un , | Γ | ′ con tains k ( µ p ) then the extension K ( µ p ) /K must hav e degree relativ ely prime to | Γ | but on the other hand its degree divides the order of Gal( k ( µ p ) /k ) . Since eac h prime divding the order of Gal( k ( µ p ) /k ) divides | Γ | , it follows that K ( µ p ) /K has degree 1 and thus µ p ∈ K , as desired. □ W e are now ready to verify the criteria for nonzero probability in Prop osition 7.49 holds for n umber ﬁelds under a mild assumption on ro ots of unit y . Prop osition 8.15. L et k b e a numb er ﬁeld and Γ a ﬁnite gr oup. L et n b e the numb er of r o ots of unity in k of or der prime to | Γ | . Assume that k lacks nontrivial unr amiﬁe d extensions of de gr e e prime to | Γ | . L et K b e an extension of k with Galois gr oup Γ . L et U b e a multiset of elements of Γ , c onsisting of, for e ach inﬁnite plac e v of k , a gener ator of Gal( k v ) . Assume that K do es not c ontain any r o ots of unity that ar e not c ontaine d in k . L et L b e a level of the c ate gory of ﬁnite Γ -gr oups. L et W b e the set of isomorphism classes of ﬁnite n -oriente d Γ -gr oups whose underlying Γ -gr oup is in L . L et H the maximal quotient of Gal( K un , | Γ | ′ /K ) in L , and s H the orientation on H . 138 Then for e ach ﬁnite simple ab elian H ⋊ Γ -gr oup V i of char acteristic prime to n that c an b e the kernel of a simple morphism G → H with G ∈ W we have (8.16) | H 2 ( H ⋊ Γ , V i ) L | ≤ | H 1 ( H ⋊ Γ , V i ) | V · U ′ i and for e ach ﬁnite simple ab elian H ⋊ Γ -gr oup V i of char acteristic dividing n that c an b e the kernel of a simple morphism G → H with G ∈ W we have (8.17) | H 1 ( H ⋊ Γ , V ∨ i ) || H 2 ( H ⋊ Γ , V i ) L ,s H | ≤ | H 1 ( H ⋊ Γ , V i ) | V · U ′ i . The assumption that k lac ks any nontrivial everywhere unramiﬁed ﬁeld extension of degree prime to | Γ | is (weak er than) one of the assumptions in the main conjecture Conjecture 1.3, so it is not surprising that w e need it to verify a prediction of that conjecture. The assumption that K do es not con tain an y ro ots of u nit y that are not contained in k is more subtle, but should alw a ys hold outside a set of density zero. Indeed, when a veraging o ver n umber ﬁelds K in the con text of the Cohen-Lenstra-Martinet heuristics, one ideally w ants to order n umber ﬁelds so as to av oid the subﬁeld pr oblem , as coined by Ko ymans and Pagano [KP23]. In other words, w e would lik e to order ﬁelds in such a wa y that each non trivial sub exension o ccurs in density 0 of the ﬁelds. When this do es not happ en, w e kno w b y work of Bartel and Lenstra [BL20] that the heuristics can give incorrect p redictions. The ﬁeld extensions with Galois group Γ , con taining additional ro ots of unity all contain one of ﬁnitely man y non trivial sub extensions, that b eing the cyclotomic extensions of degree at most | Γ | , so indeed should hav e densit y zero. Pr o of. T ak e V = V i and let p b e the c haracteristic of V i . Since every elemen t of W has order prime to | Γ | , the assumption that V i can b e the k ernel of a simple morphism implies that p is prime to | Γ | . In the case that the characteristic p of V i is prime to n , it follows b y deﬁnition of n that k do es not con tain the p th ro ots of unity . By assumption, K also do es not con tain the p th ro ots of unit y . Theorem 8.14, K un , | Γ | ′ also do es not con tain the p th ro ots of unit y . Thus we can apply Lemma 8.12, which gives | H 2 ( H ⋊ Γ , V i ) L | ≤ | H 1 ( H ⋊ Γ , V i ) | Q v archimedean place of k | V Gal( k v ) i | | V H ⋊ Γ i | . W e ha v e (8.18) V · U ′ i = Q v archimedean place of k | V Gal( F v ) i | | V H ⋊ Γ i | b y deﬁniton of U and V · U ′ i , and thus we obtain (8.16). In the case that the the c haracteristic p of V i divides n , by deﬁnition of n , k contains the p th ro ots of unity . Hence we can apply Lemma 8.13 whic h gives | H 1 ( H ⋊ Γ , V ∨ i ) || H 2 ( H ⋊ Γ , V i ) L ,s H | ≤ | H 1 ( H ⋊ Γ , V i ) | Q v archimedean place of k | V Gal( k v ) i | | V H ⋊ Γ i | whic h by (8.18) implies (8.17). □ 139 References [A c h06] Jeﬀrey D. Ac hter. The distribution of class groups of function ﬁelds. Journal of Pur e and Applie d A lgebr a , 204(2):316–333, F ebruary 2006. [A GV72] M. Artin, A. Grothendieck, and J.-L. V erdier. Thé orie des top os et c ohomolo gie étale des schémes. Séminair e de Gé ométrie Algébrique Du Bois Marie 1963/63 (SGA 4) vol. 3 , volume 305 of L e ctur e Notes in Mathematics . Springer-V erlag, Berlin-New Y ork, 1972. [AM15] Mic hael Adam and Gunter Malle. 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Dep ar tment of Ma thema tics, Princeton University, Fine Hall, W ashington Ro ad, Prince- ton, NJ 08540 USA Email addr ess : wsawin@math.princeton.edu Dep ar tment of Ma thema tics, Har v ard University, Science Center Room 325, 1 Oxf ord Street, Cambridge, MA 02138 USA Email addr ess : mmwood@math.harvard.edu 142

Distributions of unramified extensions of global fields

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