Functoriality of the canonical fractional Galois ideal

F unctorialit y of the canonical fra ctional Galois ideal P aul Buc kingham Victor Snaith Abstract The fractional Galois ideal of [Victor P . Snaith, Stark’s conjecture and new Stick elberger p h enomena, Canad. J. Math. 58 (2) (2006) 419–448] is a conjectural impro vemen t on the higher Stickel b erger id eals defined at negative i ntegers , and is exp ected to provide non-trivial annihilators for higher K -groups of rings of intege rs of number fields. In th is article, we extend the definition of the fractional Galois id eal to arbitrary (p ossibly infinite and non-abelian) Galois extensions of n um b er fields under th e assumption of Stark’s conjectures, and p ro ve naturality prop erties under canonical c hanges of extension. W e discuss applications of this to the construction of ideals in non-commutativ e Iwasa w a algebras. 1 In tro duction Let E /F be a Galo is extension of num b er fields with Galois gr oup G . I n seeking annihilators in Z [ G ] of the K -groups K 2 n ( O E ,S ) ( S a finite set of places of E containing the infinite o nes), Stickelberger elements have long b een a sour c e of interest. This bega n with the cla ssical Stick elb erger theorem, s howing that for ab elian e x tensions E / Q , annihila to rs of T or s( K 0 ( O E ,S )) can be constructed from Stic kelberger elements. Coates and Sinnott la ter conjectured in [12] that the ana logous pheno meno n would o ccur for higher K -groups. Ho w ever, defined in terms of v alues of L -functions at negative integers, these elements do no t provide all t he a nnihila tors, b ecause of the pr ev alen t v a nishing of the L -function v alues. This difficult y is hop ed to be overcome b y considering the “fractional Ga- lois ideal” introduced by the seco nd a uthor in [33, 34] and defined in terms of le ading c o efficients o f L - functions at negative integers under the assump- tion of the hig her Stark conjectures. A version more suitable for the case of T ors( K 0 ( O E ,S )) = Cl( O E ,S ) was defined in [5] by the first author. Ev- idence that the frac tional Galois ideal annihilates the appr opriate K -gr o ups (resp. class-gr oups) ca n be found in [34] (resp. [5 ]). In the first ca se, it is ´ etale cohomolog y that is annihilated, but this is exp ected to give K -theory by the Lich tenbaum–Quillen co njecture (see [34, Section 1] for details). With a view to r elating the fractional Galois ideal to characteris tic ideals in Iwasa wa theory , we would like to des crib e how it b ehav es in towers of num b er 1 fields. That it exhibits naturality in certain c hanges of extension w as o bserved in particula r cases in [5], and part of the a im of this pap er is to explain these phenomena generally . Passage to subextensions corr esp onding to quo tient s of Galois gr oups will b e of par ticular interest in the situation of non-ab elia n exten- sions, because o f the relativ ely recent emergence of no n- commutativ e Iwasaw a theory in, for example, [11, 15]. Consequently , the aims of this pap er a re (i) to pro ve formal prop e rties of the fractiona l Galois ideal with re- sp ect to changes o f extension, in the commutativ e setting first ( § 3.3 to § 3 .6) (ii) to extend the definition of the fractional Galois idea l to non- ab elian Galois ex tensions ( § 5), having previously defined it only for ab elia n extensions (iii) to show that it be haves w ell under pa ssing to sub ex tens ions in the non-commutativ e s etting also (Pro p osition 5.3) (iv) to show that in order for the non-co mmut ative fra ctional Galois ideals to annihilate the appropr iate ´ etale co homology groups, it is sufficient that the commutativ e o nes do ( § 7). W e will a lso provide an explicit example (in the co mmu tative case) in § 6.2 .1 illustrating how a limit o f fra ctional Galois idea ls gives the Fitting ideal for a n inv erse limit Cl ∞ of ℓ -pa rts of class-gr oups. This sho uld make clear the imp or - tance of taking leading co efficients of L -functions rather than just v alues, since it will b e the par t of the fra ctional Galois idea l corr esp onding to L -functions with first-or der v anishing a t 0 which provides the Fitting ideal for the plus-part of Cl ∞ . In § 8, we will conclude with a discussio n of how the c o nstructions of this pap er fit into no n-commutativ e Iwasa wa theory . In par ticular, under s ome as- sumptions which, compar e d with the many conjectures p ermea ting this a rea, are r elatively weak, we will be able to give a partia l answer to a que s tion of Ardako v–Brown in [1 ] on constr ucting ideals in Iwasaw a algebras . 1.1 Er r or in Prop osition 3.6 Since the a c c eptance of the pap er , the autho rs were made aw ar e of a prob- lem in P rop osition 3.6. It has to do with the fact that the induction map on representations is an additive homo morphism of r epresentation r ings, while the functoriality of L -functions refers to m ultiplication. Co ntrary to our expecta - tions at the time, we hav e not b een able to reso lve this issue. W e thank Andreas Nic kel for br inging this problem to our attention. 2 Notation and the Stark conjectures In wha t follows, by a Galois repre s entation of a num b er field F w e shall mean a contin uo us , finite-dimensiona l complex representation of the absolute Galois group of F , which amounts to sa ying that the represe ntation factors through the Galois group Gal( E /F ) of a finite Galois extension E /F . W e b eg in with 2 the Stark conjecture (at s = 0) and its g eneraliza tio ns to s = − 1 , − 2 , − 3 , . . . which were in tro duced in [16] a nd [34] indep endently . Let Σ( E ) denote the set of e mbedding s o f E into the complex n umbers. F o r r = 0 , − 1 , − 2 , − 3 , . . . set Y r ( E ) = Y Σ( E ) (2 π i ) − r Z = Map(Σ( E ) , (2 π i ) − r Z ) endow ed with the G ( C / R )-action diagonally on Σ( E ) and on (2 π i ) − r . If c 0 denotes complex co njugation, the action of c 0 and G co mmute so that the fixed po ints of Y r ( E ) under c 0 , denoted by Y r ( E ) + , form a G -module. It is easy to see that the r ank of Y r ( E ) + is given b y rk Z ( Y r ( E ) + ) =  r 2 if r is odd , r 1 + r 2 if r ≥ 0 is even . where | Σ( E ) | = r 1 + 2 r 2 and r 1 is the num ber of real embeddings o f E . 2.1 Stark regulators W e be g in with a slight mo dification of the or iginal Star k reg ula tor [36]. Now let G denote the Galois gro up of an extension of n um b er fields E / F . W e extend the Dirichlet reg ulator homomorphis m to the Laurent p olyno mials with co efficients in O E to give an R [ G ]-mo dule isomo rphism of the form R 0 E : K 1 ( O E h t ± 1 i ) ⊗ R = O E h t ± 1 i × ⊗ R ∼ = → Y 0 ( E ) + ⊗ R ∼ = R r 1 + r 2 by the for mu lae, for u ∈ O × E , R 0 E ( u ) = X σ ∈ Σ( E ) log( | σ ( u ) | ) · σ and R 0 E ( t ) = X σ ∈ Σ( E ) σ . The ex istence of this is omorphism implies (see [30, Section 12.1] and [36, p.26]) that there exists a t least one Q [ G ]-mo dule isomorphism of the for m f 0 E : O E h t ± 1 i × ⊗ Q ∼ = → Y 0 ( E ) + ⊗ Q . F or any c hoice of f 0 E Stark forms the c o mp osition R 0 E · ( f 0 E ) − 1 : Y 0 ( E ) + ⊗ C ∼ = → Y 0 ( E ) + ⊗ C which is an isomorphism of complex r epresentations of G . Let V be a finite- dimensional complex repre sentation of G whose contragredient is deno ted by 3 V ∨ . The Stark r egulator is defined to be the exp o nential homomo rphism V 7→ R ( V , f 0 E ), from r epresentations to no n-zero complex num b er s , given by R ( V , f 0 E ) = det(( R 0 E · ( f 0 E ) − 1 ) ∗ ∈ Aut C (Hom G ( V ∨ , Y 0 ( E ) + ⊗ C ))) where ( R 0 E · ( f 0 E ) − 1 ) ∗ is comp osition with R 0 E · ( f 0 E ) − 1 . F or r = − 1 , − 2 , − 3 , . . . there is a n iso morphism of the for m [26] K 1 − 2 r ( O E h t ± 1 i ) ⊗ Q ∼ = K 1 − 2 r ( O E ) ⊗ Q bec ause K − 2 r ( O E ) is finite. Therefore the Bo rel regulator homomor phism de- fines an R [ G ]-mo dule isomorphism of the for m R r E : K 1 − 2 r ( O E h t ± 1 i ) ⊗ R = K 1 − 2 r ( O E ) ⊗ R ∼ = → Y r ( E ) + ⊗ R . Cho ose a Q [ G ]-mo dule isomorphism of the for m f r E : K 1 − 2 r ( O E h t ± 1 i ) ⊗ Q ∼ = → Y r ( E ) + ⊗ Q and form the analog ous Stark re gulator, ( V 7→ R ( V , f r E )), from representations to non-zero complex num b er s given by R ( V , f r E ) = det(( R r E · ( f r E ) − 1 ) ∗ ∈ Aut C (Hom G ( V ∨ , Y r ( E ) + ⊗ C ))) . 2.2 Stark’s conjectures Let R ( G ) denote the co mplex repre sentation ring of the finite group G ; tha t is, R ( G ) = K 0 ( C [ G ]). Since V determines a Galois repr esentation of F , we hav e a non-zero complex num b er L ∗ F ( r , V ) given by the lea ding co efficient of the T aylor series at s = r of the Ar tin L -function asso ciated to V ([2 3], [36, p.23]). W e may modify R ( V , f r E ) to give another exp onential ho momorphism R f r E ∈ Hom( R ( G ) , C × ) defined by R f r E ( V ) = R ( V , f r E ) L ∗ F ( r , V ) . Let Q denote the algebra ic c lo sure o f the ratio nals in the complex n umbers and let Ω Q denote the abso lute Galois group of the rationals, w hich acts con- tin uously o n R ( G ) and Q × . The Star k conjecture as serts that for each r = 0 , − 1 , − 2 , − 3 , . . . R f r E ∈ Hom Ω Q ( R ( G ) , Q × ) ⊆ Hom( R ( G ) , C × ) . In other words, R f r E ( V ) is a n algebr a ic num b er for each V and for all z ∈ Ω Q we have z ( R f r E ( V )) = R f r E ( z ( V )). Since any t wo c hoices of f r E differ by multi- plication by a Q [ G ]-automorphism, the truth of the conjecture is independent of the choice of f r E ([36] pp.28-3 0). 4 When s = 0 the conjecture whic h w e ha ve just form ulated apparently dif- fers from the classical Stark conjecture of [36], therefore we s hall pause to show that the tw o conjectures are equiv ale nt. F or the cla ssical Stark conjecture one replaces Y 0 ( E ) + by X 0 ( E ) + where X 0 ( E ) is the k ernel of the augmentation ho- momorphism Y 0 ( E ) → Z , which adds together all the c o ordinates. The Dirichlet regulator gives a n R [ G ]-mo dule isomo rphism ˜ R 0 E : O × E ⊗ R ∼ = → X 0 ( E ) + ⊗ R and choosing a Q [ G ]-mo dule isomorphis m ˜ f 0 E : O × E ⊗ Q ∼ = → X 0 ( E ) + ⊗ Q we may form ˜ R 0 E · ( ˜ f 0 E ) − 1 : X 0 ( E ) + ⊗ C ∼ = → X 0 ( E ) + ⊗ C . T aking its Stark de ter minant w e o btain ˜ R ( V , ˜ f 0 E ) and finally ˜ R ˜ f 0 E ( V ) = ˜ R ( V , ˜ f 0 E ) L ∗ F (0 , V ) . Prop ositi on 2.1 In § 2. 2 R f 0 E ∈ Hom Ω Q ( R ( G ) , Q × ) ⊆ Hom( R ( G ) , C × ) if and only if ˜ R ˜ f 0 E ∈ Hom Ω Q ( R ( G ) , Q × ) ⊆ Hom( R ( G ) , C × ) indep endently of the choic e of f 0 E or ˜ f 0 E . Pr o of . Given a ny Q [ G ]-isomo rphism ˜ f 0 E we may fill in the following co mmu- tative diagr am by Q [ G ]- is omorphisms f 0 E and f 0 E . Conv er sely , given a ny Q [ G ]- isomorphisms f 0 E and f 0 E we may fill in the diag r am with a Q [ G ]-is o morphism ˜ f 0 E . O × E ⊗ Z Q / / ˜ f 0 E   O E [ t ± 1 ] × ⊗ Z Q / / f 0 E   Q ¯ f 0 E   X 0 ( E ) + ⊗ Z Q / / Y 0 ( E ) + ⊗ Z Q / / Q Similarly there is a co mmutative dia gram in whic h the vertical arrows are reversed, Q is replaced b y R and ˜ f E , f E and f E by ˜ R 0 E , R 0 E and R 0 E , respe ctively . F urthermore R 0 E is multiplication by a rationa l n umber. The result now follows from the multiplicativit y of the determinant in short exact sequences . W e sha ll b e particularly in terested in the cas e when G is abelia n, in which case the follo wing obs erv ation is important. Let b G = Hom( G, Q × ) deno te the 5 set o f c haracter s on G a nd let Q ( χ ) denote the field gener ated by the character v alues of a representation χ . W e may iden tify Hom Ω Q ( R ( G ) , Q )with the ring Map Ω Q ( b G, Q ). Prop ositi on 2.2 L et G b e a fi nite ab elian gr oup. Then t her e exists an isomor- phism of rings λ G : Map Ω Q ( b G, Q ) = Hom Ω Q ( R ( G ) , Q ) ∼ = → Q [ G ] given by λ G ( h ) = X χ ∈ b G h ( χ ) e χ wher e e χ = | G | − 1 X g ∈ G χ ( g ) g − 1 ∈ Q ( χ )[ G ] . In p articular ther e is an isomorphism of unit gr oups λ G : Hom Ω Q ( R ( G ) , Q × ) ∼ = → Q [ G ] × . Pr o of . There is a w ell-known isomo rphism of rings ([22] p.6 48) ψ : Q [ G ] → Y χ ∈ b G Q = Map( b G, Q ) given by ψ ( P g ∈ G λ g g )( χ ) = P g ∈ G λ g χ ( g ). If Ω Q acts on Q and b G in the canonical manner , then ψ is Galois equiv a riant and induces an isomor phism of Ω Q -fixed p oints of the form Q [ G ] = ( Q [ G ]) Ω Q ∼ = Map Ω Q ( b G, Q ) ∼ = Hom Ω Q ( R ( G ) , Q ) . It is straig ht forward to verify that this isomo r phism is the inv er se of λ G . 3 The canonical fractional Galois ideal J r E /F in the ab elian case 3.1 Definition of J r E /F In this sectio n we r ecall the canonical fractional Galo is ideal intro duced in [3 4] (see also [5], [31] a nd [33]). In [34] this was denoted merely b y J r E but in this pap er we sha ll need to keep track of the base field. As in § 2.2, let E /F b e a Galois extension of n umber fields . Throug ho ut this section we shall a s sume that the Stark co njecture o f § 2.2 is true for all E / F 6 and that G = Gal( E /F ) is ab elian. Therefor e, by Prop o sition 2.2, for each r = 0 , − 1 , − 2 , − 3 , . . . we have an element R f r E ∈ Hom Ω Q ( R ( G ) , Q × ) ∼ = Q [ G ] × which dep ends up on the choice of a Q [ G ]-isomo rphism f r E in § 2.2. Let α ∈ End Q [ G ] ( Y r ( E ) + ⊗ Q ) a nd extend this b y the ide ntit y on the ( − 1)- eigenspace of complex c o njugation Y r ( E ) − ⊗ Q to give α ⊕ 1 ∈ End Q [ G ] ( Y r ( E ) ⊗ Q ) . Since Y r ( E ) ⊗ Q is free ov er Q [ G ], we may form the de ter minant det Q [ G ] ( α ⊕ 1) ∈ Q [ G ] . In terms o f the isomorphism o f Pr op osition 2.2, det Q [ G ] ( α ⊕ 1) cor resp onds to the function which sends χ ∈ b G to the determina nt of the endo morphism of e χ Y r ( E ) ⊗ Q induced by α ⊕ 1. F ollowing [34, Section 4.2] (see also [33, 3 1]), define I f r E to b e the (finitely generated) Z [1 / 2][ G ]-submo dule of Q [ G ] generated b y all the elemen ts det Q [ G ] ( α ⊕ 1) satisfying the integrality condition α · f r E ( K 1 − 2 r ( O E [ t ± 1 ])) ⊆ Y r ( E ) . Define J r E /F to b e the finitely generated Z [1 / 2 ][ G ]-submo dule of Q [ G ] given by J r E /F = I f r E · τ ( R − 1 f r E ) where τ is the automor phism of the group-r ing induced by sending each g ∈ G to its inv er se. Prop ositi on 3.1 ([34 , Pr op.4.5] ) L et E /F b e a Galois exten s ion of numb er fields with ab elian Galois gr oup G . Then, assuming t hat the Stark c onje cture of § 2.2 holds for E /F for r = 0 , − 1 , − 2 , − 3 , . . . , the fin itely gener ate d Z [1 / 2][ G ] - submo dule J r E /F of Q [ G ] just define d is indep endent of the choic e of f r E . 3.2 Naturality examples Given an extension E /F o f num b er fields satisfying the Stark conjecture a t s = 0 and a finite s et of places S of F containing the infinite places, let J ( E /F , S ) denote the fractional Galois ideal as defined in [5], a slight modification of the one just defined so that w e can take into account finite places. Le t us consider the following situation: ℓ is an odd prime, E n = Q ( ζ ℓ n +1 ) fo r a primitiv e ℓ n +1 th ro ot of unit y ζ ℓ n +1 ( n ≥ 0), and S = {∞ , ℓ } . The descriptions b elow of J ( E n / Q , S ) and J ( E + n / Q , S ) are provided in [5, Section 4]: J ( E n / Q , S ) = 1 2 e + ann Z [ G n ] ( O × E + n ,S / E + n ) ⊕ Z [ G n ] θ E n / Q ,S (3.1) J ( E + n / Q , S ) = 1 2 ann Z [ G + n ] ( O × E + n ,S / E + n ) (3.2) 7 where G n = Gal( E n / Q ), G + n = Gal( E + n / Q ), E + n is the Z [ G + n ]-submo dule of O × E + n ,S generated by − 1 and (1 − ζ ℓ n +1 )(1 − ζ − 1 ℓ n +1 ), a nd θ E n / Q ,S is the Stick el- ber ger ele ment at s = 0. Also, e + = 1 2 (1 + c ) is the plus-idemp o tent for complex conjugation c ∈ G n . It is immediate from these descriptions tha t the natura l maps Q [ G n ] → Q [ G + n ], Q [ G n ] → Q [ G n − 1 ] and Q [ G + n ] → Q [ G + n − 1 ] give rise to a commutativ e diagram J ( E n / Q , S ) / /   J ( E + n / Q , S )   J ( E n − 1 / Q , S ) / / J ( E + n − 1 / Q , S ) . (3.3) ( O × E + n − 1 ,S / E + n − 1 embeds in to O × E + n ,S / E + n , and Stick elb erger e lement s ar e well known (e.g. [18]) to map to each other in this wa y .) Now supp os e that ℓ ≡ 3 mo d 4 , so that E n contains the imaginary quadratic field F = Q ( √ − ℓ ). Ag a in, letting S F consist of the infinite place of F a nd the unique pla ce a b ov e ℓ , J ( E n /F , S F ) has a simple desc ription. Indeed, if H n = Gal( E n /F ), then J ( E n /F , S F ) = 1 µ n ann Z [ H n ] ( O × E n ,S / E n ) (3.4) where E n is genera ted ov er Z [ H n ] by ζ ℓ n +1 and (1 − ζ ℓ n +1 ) µ n ˜ θ n . Here, µ n = | µ ( E n ) | and ˜ θ n = P σ ∈ H n ζ E n / Q ,S (0 , σ − 1 ) σ ∈ Q [ H n ], a sort of “half Stick elb erger element” o btained by keeping only those terms corresp o nding to elements in the index tw o subgro up H n of G n . (Note that µ n ˜ θ n ∈ Z [ H n ].) Compar ing (3.2) and (3.4), we see without to o muc h difficult y that Prop ositi on 3.2 The isomorphism Φ n : Q [ H n ] → Q [ G + n ] identifies J ( E n /F , S F ) with 2Φ n ( ˜ θ n ) J ( E + n / Q , S ) . W e now explain the ab ov e phenomena by proving some genera l relationships betw een the J r E /F under natura l changes of extensio n. 3.3 Behav iour under quotien t maps G al( L/F ) → Gal( K /F ) Suppo se that F ⊆ K ⊆ L is a to wer of num b er fields with L/F ab elian. The inclusion of K into L induces a homomo rphism K 1 − 2 r ( O K [ t ± 1 ]) → K 1 − 2 r ( O L [ t ± 1 ]) . When r = 0 K 1 ( O K [ t ± 1 ]) T orsio n ∼ = O × K / ( µ ( K )) ⊕ Z h t i maps injectively to the Galois in v ariants of O × L / ( µ ( L )) ⊕ Z h t i se nding t to itself. F or strictly neg ative r , K 1 − 2 r ( O K [ t ± 1 ]) T orsio n ∼ = K 1 − 2 r ( O K ) T orsio n 8 embeds in to the Gal( L/K )-inv ariants of K 1 − 2 r ( O L [ t ± 1 ]) T orsion . There is a homomo r - phism Y r ( K ) → Y r ( L ) which sends n σ · σ to n σ · ( P ( σ ′ | F )= σ σ ′ ) which is a n iso- morphism onto the Gal( L/ K )-inv aria nt s Y r ( L ) Gal( L/K ) . F or r = 0 , − 1 , − 2 , − 3 , . . . there is a co mm utative diag ram of regulator s in § 2 .1 K 1 − 2 r ( O K [ t ± 1 ]) ⊗ Z R R r K / /   Y r ( K ) + ⊗ Z R   K 1 − 2 r ( O L [ t ± 1 ]) ⊗ Z R R r L / / Y r ( L ) + ⊗ Z R W e may choo s e f r K and f r L as in § 2.1 to make the cor resp onding diagra m of Q -vector spaces co mmut e K 1 − 2 r ( O K [ t ± 1 ]) ⊗ Z Q f r K / /   Y r ( K ) + ⊗ Z Q   K 1 − 2 r ( O L [ t ± 1 ]) ⊗ Z Q f r L / / Y r ( L ) + ⊗ Z Q (3.5) Let V b e a o ne-dimensional complex represe nt ation of Gal( K /F ) and let W = Inf Gal( L/F ) Gal( K/F ) ( V ) denote the infla tion of V . Then Hom Gal( L/F ) ( W ∨ , Y r ( L ) + ⊗ C ) = Hom Gal( L/F ) ( W ∨ , ( Y r ( L ) Gal( L/K ) ) + ⊗ C ) = Hom Gal( K/F ) ( V ∨ , Y r ( K ) + ⊗ C ) and thes e isomorphisms transp ort ( R r L · ( f r L ) − 1 ) ∗ int o ( R r K · ( f r K ) − 1 ) ∗ by virtue of the ab ov e co mmu tative diagrams. F urthermore, since the Artin L -function is inv ariant under infla tion, L ∗ F ( r , V ) = L ∗ F ( r , W ). On the other hand, the inflation homomorphism Inf Gal( L/F ) Gal( K/F ) : R (Gal( K/F )) → R (Gal( L /F )) induces the canonica l quotient map π L/K : Q [Gal( L/F )] × → Q [Gal( K /F )] × via the isomo rphism of Prop o sition 2.2. Hence π L/K ( R f r L ) = R f r K . Let α ∈ End Q [Gal( L/F )] ( Y r ( L ) + ⊗ Q ) satisfy the integrality condition of § 3.1 α · f r L ( K 1 − 2 r ( O L [ t ± 1 ])) ⊆ Y r ( L ) . Extend this by the identit y on the ( − 1)-eige nspace of complex conjugation Y r ( L ) − ⊗ Q to give α ⊕ 1 ∈ End Q [Gal( L/F )] ( Y r ( L ) ⊗ Q ) . 9 The endomorphism α comm utes with the action by Ga l( L/K ) so there is ˆ α ∈ End Q [Gal( K/F )] ( Y r ( K ) + ⊗ Q ) making the following diagram commute Y r ( K ) + ⊗ Z Q ˆ α / /   Y r ( K ) + ⊗ Z Q   Y r ( L ) + ⊗ Z Q α / / Y r ( L ) + ⊗ Z Q . Therefore ˆ α satisfies the integrality condition of § 3.1 ˆ α · f r K ( K 1 − 2 r ( O K [ t ± 1 ])) ⊆ Y r ( K ) . W e ma y choo se a Z [1 / 2][Gal( K/F )] basis for Y r ( K ) ⊗ Z [1 / 2] co ns isting of em- bedding s σ i : K → C for 1 ≤ i ≤ m . Let σ ′ i be an embedding o f L which extends σ i for 1 ≤ i ≤ m . Then a Z [1 / 2 ][Gal( L/F )] bas is for Y r ( L ) ⊗ Z [1 / 2 ] is given by { σ ′ 1 , σ ′ 2 , . . . , σ ′ m } . The embedding of Y r ( K ) into Y r ( L ) is given by σ i 7→ P g ∈ Gal( L /K ) g ( σ ′ i ) which implies that the m × m matrix for ˆ α with respe c t to the Z [1 / 2][Gal( K/ F )] basis of σ i ’s is the image of the m × m matrix for α with resp ect to the Z [1 / 2][Gal( L/F )] basis of σ ′ i ’s under the cano nic a l sur jection Q [Gal( L/F )] → Q [Gal( K /F )] . This discussio n has established the following result. Prop ositi on 3.3 Su pp ose that F ⊆ K ⊆ L is a t ower of numb er fields with L/F ab elian. Then, in the notation of § 3.1, the c anonic al s urje ction π L/K : Q [Gal( L/F )] → Q [Gal( K/ F )] satisfies π L/K ( J r L/F ) ⊆ J r K/F . Prop os itio n 3 .3 explains the e x istence of the ma ps in (3.3). 3.4 Behav iour under inclusion maps Gal( L/K ) → Gal( L/F ) As in § 3.3, supp ose that F ⊆ K ⊆ L is a tow er of num b er fields with L/ F ab elian. The inclusion of Gal( L /K ) int o Gal( L/F ) induces an inclusion o f group-r ings Q [Ga l( L/K )] int o Q [Ga l( L/F )]. In terms o f the is o morphism of Prop os itio n 2.2, a s is eas ily seen by the form ula, this homomo rphism is induced by the re striction of repres entations Res Gal( L/F ) Gal( L/K ) : R (Gal( L/F )) → R (Gal( L/K )) . 10 If V is a complex repr esentation of Gal( L/ F ) then R f r L (Res Gal( L/F ) Gal( L/K ) ( V )) = R (Res Gal( L/F ) Gal( L/K ) ( V ) , f r L ) L ∗ K ( r , Res Gal( L/F ) Gal( L/K ) ( V )) = R (Res Gal( L/F ) Gal( L/K ) ( V ) , f r L ) L ∗ F ( r , Ind Gal( L/F ) Gal( L/K ) (Res Gal( L/F ) Gal( L/K ) ( V ))) = R (Res Gal( L/F ) Gal( L/K ) ( V ) , f r L ) L ∗ F ( r , V ⊗ Ind Gal( L/F ) Gal( L/K ) (1)) . If W i ∈ d Gal( L/F ) for 1 ≤ i ≤ [ K : F ] is the set of one-dimensional repr esenta- tions whic h restrict to the trivial representation on Gal( L /K ) then Ind Gal( L/F ) Gal( L/K ) (1)) = ⊕ i W i . By F rob enius recipro c ity Hom Gal( L/K ) (Res Gal( L/F ) Gal( L/K ) ( V ) ∨ , Y r ( L ) + ⊗ C )) = Hom Gal( L/F ) ( ⊕ i ( V ⊗ W i ) ∨ , Y r ( L ) + ⊗ C )) so that R (Res Gal( L/F ) Gal( L/K ) ( V ) , f r L ) = Y i R ( V ⊗ W i , f r L ) and R f r L (Res Gal( L/F ) Gal( L/K ) ( V )) = Y i R f r L ( V ⊗ W i ) . Let H ⊆ G be finite groups with G abelia n. It will suffice to consider the case in which G/H is cyclic of order n generated by g H . Let W ⊗ Q b e a free Q [ G ]-mo dule with ba sis v 1 , . . . , v r . Then W ⊗ Q is a fre e Q [ H ]-mo dule with basis { g a v i | 0 ≤ a ≤ n − 1 , 1 ≤ i ≤ r } . Set S = { 0 , . . . , n − 1 } × { 1 , . . . , r } ; then for u = ( a, i ) ∈ S , we set e u = g a v i . If ˜ α ∈ E nd Q [ H ] ( W ⊗ Q ) we may wr ite ˜ α ( e w ) = X u A u .w e u so that A is a n nr × nr matrix with entries in Q [ H ]. Now co ns ider the induced Q [ G ]-mo dule Ind G H ( W ⊗ Q ), whic h is a free Q [ G ]- mo dule on the basis { 1 ⊗ H e u | u ∈ S } . Hence the nr × nr matrix, with entries in Q [ G ], for 1 ⊗ H ˜ α with resp ec t to this ba s is is the imag e of A under the ca nonical inclusion of φ H,G : Q [ H ] → Q [ G ]. In par ticula r φ H,G (det Q [ H ] ( ˜ α )) = det Q [ G ] ( Q [ G ] ⊗ Q [ H ] ˜ α ) and, by induction on [ G : H ], this relation is true for an a rbitrar y inclusion H ⊆ G of finite ab elian gro ups. This discussio n yields the following result: 11 Prop ositi on 3.4 Su pp ose that F ⊆ K ⊆ L is a t ower of numb er fields with L/F ab elian. Then, in the notation of § 3.1, the c anonic al inclu s ion φ K/F : Q [Gal( L/K )] → Q [Gal( L/F )] maps J r L/K onto the Z [1 / 2][Gal( L/K )] -submo dule Z [1 / 2][Gal( L/K )] h det Q [Gal( L/F )] ( Q [Gal( L/F )] ⊗ Q [Gal( L/K )] ( α ⊕ 1)) τ ( ˆ R f r L ) − 1 i . Her e, in terms of Pr op osition 2.2, ˆ R f r L ∈ Q [Gal( L/F )] × is given by ˆ R f r L ( V ) = R f r L ( V ⊗ Ind Gal( L/F ) Gal( L/K ) (1)) and α ∈ End Q [Gal( L/K )] ( Y r ( L ) + ⊗ Q ) ru ns thr ough endomorphisms satisfying the inte gr ality c ondition of § 3.1. 3.5 Behav iour under fixed-point maps As in § 3.3, supp ose that F ⊆ K ⊆ L is a tow er of num b er fields with L/ F ab elian. Let e L/K = [ L : K ] − 1 ( P y ∈ Gal( L/K ) y ) denote the idemp otent asso ci- ated with the subgr oup Gal( L/K ). There is a homomo rphism of unital rings of the form λ K/F : Q [Gal( K /F )] → Q [Gal( L/F )] given, for z ∈ Ga l( L/F ), b y the fo rmula λ K/F ( z Ga l( L/K )) = (1 − e L/K ) + z · e L/K ∈ Q [Gal( L/F )] . F rom Pro p osition 2.2 it is eas y to see that in terms of gro up characters Map( d Gal( K/ F ) , Q ) → Map( d Gal( L/F ) , Q ) this sends a function h on d Gal( K/ F ) to the function h ′ given by h ′ ( χ ) = ( h ( χ 1 ) if Inf Gal( L/F ) Gal( K/F ) ( χ 1 ) = χ, 1 otherwise . Sending a complex re pr esentation V of Gal( L/F ) to its Ga l( L/K )-fixed po ints V Gal( L/K ) gives a homomo rphism Fix : R (Ga l( L/F )) → R (Ga l( K /F )) . In ter ms of one-dimensiona l respr esentations (i.e. characters) the a b ov e condi- tion Inf Gal( L/F ) Gal( K/F ) ( χ 1 ) = χ is equiv alent to Fix( χ ) = χ 1 . Let V b e a one-dimensio nal complex repres e nt ation of Gal( L /F ) fixed by Gal( L/K ). Then we have iso morphisms of the for m Hom Gal( L/F ) (( V Gal( L/K ) ) ∨ , Y r ( L ) + ⊗ C ) = Hom Gal( K/F ) ( V ∨ , ( Y r ( L ) Gal( L/K ) ) + ⊗ C ) = Hom Gal( K/F ) ( V ∨ , Y r ( K ) + ⊗ C ) 12 and, by inv ar iance of L -functions under inflation, L ∗ F ( r , V ) = L ∗ F ( r , V Gal( L/K ) ). Therefore, by the dis c ussion of § 3 .3, R f r L ( V ) = R f r K ( V Gal( L/K ) ) . On the other ha nd, if V Gal( L/K ) = 0 then R f r K ( V Gal( L/K ) ) = 1 since b oth L ∗ F ( r , 0) and the determinan t of the iden tit y ma p o f the trivial vector space are equal to one. This establishes the formula λ K/F ( R f r K ) = (1 − e L/K ) + R f r L · e L/K . Now consider an endomorphism α ∈ End Q [Gal( K/F )] ( Y r ( K ) + ⊗ Q ) satisfying the integrality condition of § 3.1 αf r,K ( K 1 − 2 r ( O K [ t ± 1 ])) ⊆ Y r ( K ) + ∼ = ( Y r ( L ) + ) Gal( L/K ) . Let v 1 , v 2 , . . . , v d be a Z [1 / 2][Gal( L/F )]-basis of Y r ( L )[1 / 2] so that a Z [1 / 2][Gal( K/ F )]-basis of the subspace ( Y r ( L ) + ) Gal( L/K ) [1 / 2] ∼ = Y r ( K )[1 / 2] is given by { ( P y ∈ Gal( L/K ) y ) v i | 1 ≤ i ≤ d } . T o construct the gener ators of J r K/F , as in § 3.1, we must calculate the deter minant of α ⊕ 1 on Y r ( K ) + ⊗ Q ⊕ Y r ( K ) − ⊗ Q = Y r ( K ) ⊗ Q with res pe c t to the bas is { ( P y ∈ Gal( L/K ) y ) v i } and divide by τ ( R f r K ). Let ˆ α ∈ End Q [Gal( L/F )] ( Y r ( L ) ⊗ Q ) b e g iven by α on Y r ( L ) Gal( L/F ) ⊗ Q and the identit y on (1 − e L/K ) Y r ( L ) ⊗ Q . Hence ˆ α satisfies the in tegrality co ndition ˆ α · f r L ( K 1 − 2 r ( O L [ t ± 1 ])) Gal( L/F ) ⊆ Y r ( L ) Gal( L/F ) , bec ause, as in § 3.3, f r K may b e a ssumed to extend to f r L . Ther efore e L/K det( ˆ α ) τ ( R f r L ) ∈ e L/K J r L/F ⊂ Q [Gal( L/F )] . On the other hand it is clear that λ K/F (det( α ⊕ 1)) = det( ˆ α ). This discussio n has established the following result. Prop ositi on 3.5 Su pp ose that F ⊆ K ⊆ L is a t ower of numb er fields with L/F ab elian and let λ K/F : Q [Gal( K /F )] → Q [Gal( L/F )] denote the unital ring homomo rphism of § 3.5. Then λ K/F ( J r K/F ) ⊆ (1 − e L/K ) Q [Gal( L/F )] + e L/K J r L/F . 13 3.6 Behav iour under corestriction maps As in § 3.3, supp ose that F ⊆ K ⊆ L is a tow er of num b er fields with L/ F ab elian. There is an additive ho momorphism of the for m ι K/F : Q [Gal( L/F )] → Q [Gal( L/K )] called the transfer or corestriction map. In terms of Prop ositio n 2.2 it is induced by the induction of repres entations Ind Gal( L/F ) Gal( L/K ) : R (Gal( L/K )) → R (Gal( L/F )) . That is, the imag e ι K/F ( h ) of h ∈ Hom Ω Q ( R (Gal( L/F )) , Q ) is given b y ι K/F ( h )( V ) = h (Ind Gal( L/F ) Gal( L/K ) ( V )) . By F ro b enius recipro city , fo r each V ∈ R (Gal)( L /K )) there is a n isomor - phism Hom Gal( L/F ) ((Ind Gal( L/F ) Gal( L/K ) ) ∨ , Y r ( L ) + ⊗ C ) = Hom Gal( L/K ) ( V ∨ , Y r ( L ) + ⊗ C ) . Also L ∗ F ( r , Ind Gal( L/F ) Gal( L/K ) ( V )) = L ∗ K ( r , V ) s o that ι K/F ( R f r L ) = R f r L . Now consider an endomorphism α ∈ End Q [Gal( L/F )] ( Y r ( L ) + ⊗ Q ) satisfying the integrality condition of § 3.1 αf r,L ( K 1 − 2 r ( O L [ t ± 1 ])) ⊆ Y r ( L ) + . Then it is straightforw ard to see from P rop ositio n 2.2 that the determinant of α ⊕ 1 as a map of Q [Ga l( L/F )]-modules det Q [Gal( L/F )] ( α ⊕ 1) is mappe d to det Q [Gal( L/K )] ( α ⊕ 1), the determinant of α ⊕ 1 as Q [Gal( L/ K )]- mo dules. This discussio n has es ta blished the following result. (This result has a prob- lem. See Section 1.1.) Prop ositi on 3.6 Su pp ose that F ⊆ K ⊆ L is a t ower of numb er fields with L/F ab elian, and let ι K/F : Q [Gal( L/F )] → Q [Gal( L/K )] denote the additiv e homo morphism of § 3.6. Then ι K/F ( J r L/F ) ⊆ J r L/K . 14 3.7 W e can now ex plain the second example in § 3.2, i.e . P r op osition 3.2. Let us work more generally to beg in with. E a nd F can b e any num b er fields, and we suppo se we have a diag ram E C ⑦ ⑦ ⑦ ⑦ ⑦ ⑦ ⑦ ⑦ H ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ L G ′ ❅ ❅ ❅ ❅ ❅ ❅ ❅ F ⑦ ⑦ ⑦ ⑦ ⑦ ⑦ ⑦ ⑦ K satisfying the following: E /K is Galois (though not necess arily abelia n), LF = E , L ∩ F = K , the extension L/K is ab elian (and hence so is E /F ), and L/K and E /F satisfy the Stark co njectur e. W e let G = Gal( E /K ), a nd the Galois groups of the other Ga lo is extensions are marked in the diag ram. W e observe that C need not b e ab elian here. Owing to the na tur al is omorphism G/C → H , each character ψ ∈ b H extends to a unique one-dimensional representation b ψ : G → C × which is trivial on C . Denote by ch( G ) the set of irreducible c haracter s of G . Then ha v ing c hosen a Q [ G ]-mo dule isomo r phism f as in § 2.2, we can define an element Ω f ∈ C [ H ] × by Ω f = Y χ ∈ ch( G ) r { 1 }   X ψ ∈ b H R f E /K ( χ b ψ ) d χ e ψ   , where for a character χ of G , d χ is the multiplicit y of the triv ial c haracter of H in Res G H ( χ ). W e have opted to denote by R f E /K the gr oup-ring ele ment R f E defined in § 3 .1, to emphasize which extensio n is b eing cons idered. The following lemma shows that the gr oup-ring elemen t R f E /F for the ex- tension E /F is related, v ia Ω f , to the co rresp o nding e lement for the extension L/K . Lemma 3. 7 Ω f has r ational c o efficients, and the image of R f E /F under the isomorphi sm Φ : Q [ H ] → Q [ G ′ ] is R f ′ L/K Φ(Ω f ) , wher e f ′ is the Q [ G ′ ] -mo dule isomorphism making diagr am (3.5) c ommute. The pro o f of the lemma is little more than a combination of § 3.3 and § 3.6. In the situa tion of Pr op osition 3.2 (with L = E + and K = Q now) we find that the element 2 ˜ θ o ccurring there is just τ (Ω f ) − 1 (for any choice of f in this 15 case). Indeed, let ρ ∈ b G be the unique non-trivia l character ex tending the trivial character of H . Then the only χ ∈ ch ( G ) r { 1 } with d χ 6 = 0 is ρ , and d ρ = 1, so Ω f = X ψ ∈ b H R f E / Q ( ρ b ψ ) e ψ = X ψ ∈ b G ψ even R f E / Q ( ρψ ) e ψ | H . How ever, for ψ even, ρψ is o dd so that R f ( ρψ ) = L E / Q ,S (0 , ρψ ) − 1 . Using the easily verified fact that (1 − c ) ˜ θ = θ E / Q ,S , where c ∈ G is complex conjugation, we see that L E / Q ,S (0 , ρψ ) = 2 ψ | H ( τ ˜ θ ), fro m whic h the a ssertion follows. Applying Lemma 3.7 now justifies the app ear ance of 2Φ n ( ˜ θ n ) in P r op osition 3.2. 4 The passage to non-ab elian groups 4.1 In this s ection w e shall use the E xplicit Brauer Inductio n constructions of [32, pp.138–1 47] to pa ss from finite ab elia n Galois gro ups to the non-a b elia n c ase. Let G b e a finite group and co nsider the additive homomorphism X H ⊆ G Ind G H Inf H H ab : ⊕ H ⊆ G R ( H ab ) → R ( G ) . Let N ⊳ G b e a norma l subgr o up a nd let π : G → G/ N denote the quotien t homomorphism. Define a ho momorphism α G,N : ⊕ J ⊆ G/ N R ( J ab ) → ⊕ H ⊆ G R ( H ab ) to b e the homomo rphism which s ends the J -comp o nent R ( J ab ) to the H = π − 1 ( J )-comp onent R ( π − 1 ( J ) ab ) via the map Inf π − 1 ( J ) ab J ab ( R ( J ab )) → R ( π − 1 ( J ) ab ) . Lemma 4. 1 In the notation of § 4.1 the fol lowing diagr am c ommu tes: L J ⊆ G/ N R ( J ab ) / / α G,N   R ( G/ N ) Inf G G/ N   L H ⊆ G R ( H ab ) / / R ( G ) . 16 Pr o of . Since the kernel of π − 1 ( J ) → J and that of π : G → G/ N coincide, bo th b eing equal to N , we have Inf G G/ N Ind G/ N J = Ind G π − 1 ( J ) Inf π − 1 ( J ) J . Therefore, given a c haracter φ : J ab → Q × in the J -co or dinate, w e have Ind G π − 1 ( J ) Inf π − 1 ( J ) π − 1 ( J ) ab α G,N ( φ ) = Ind G π − 1 ( J ) Inf π − 1 ( J ) π − 1 ( J ) ab Inf π − 1 ( J ) ab J ab ( φ ) = Ind G π − 1 ( J ) Inf π − 1 ( J ) J Inf J J ab ( φ ) = Inf G G/ N Ind G/ N J Inf J J ab ( φ ) , as required. 4.2 The homomor phism of § 4.1 is inv ariant under group conjuga tion and therefore induces an additive homomorphism of the for m B G : ( ⊕ H ⊆ G R ( H ab )) G → R ( G ) where X G denotes the coinv ariants of the co njugation G -action. This homomor - phism is a split surjection whose rig ht inv ers e is given by the Explicit Brauer Induction homomor phism A G : R ( G ) → ( ⊕ H ⊆ G R ( H ab )) G constructed in [32, Section 4.5.1 6 ]. W e shall b e interested in the dual homomo r- phisms ([32, Section 4 .5.20]) B ∗ G : Hom Ω Q ( R ( G ) , Q ) → ( ⊕ H ⊆ G Hom Ω Q ( R ( H ab ) , Q )) G and A ∗ G : ( ⊕ H ⊆ G Hom Ω Q ( R ( H ab ) , Q )) G → Hom Ω Q ( R ( G ) , Q ) where X G denotes the subgr oup of G -inv aria nts. As in [32, Def.4.5.4], denote by Q { G } the rational vector space whose basis consists of the conjugacy classes of G . There is an isomorphism ([32, Pro p.4 .5.14]) ψ : Q { G } ∼ = → Hom Ω Q ( R ( G ) , Q ) given by the formula ψ ( P γ m γ γ )( ρ ) = P γ m γ T race( ρ ( γ )). When G is abelia n, we hav e Q { G } = Q [ G ] and under the ident ification Hom Ω Q ( R ( G ) , Q ) = Map Ω Q ( b G, Q ) of P rop ositio n 2 .2 we hav e ψ ( g ) = ( χ 7→ χ ( g )), which is a ring isomorphism inv erse to λ G . 17 5 J r E /F in general Let G denote the Galois gr o up of a finite Galois extension E /F of num b er fields . Hence each subgroup of G has the form H = Gal( E /E H ), whos e ab elianiza tion is H ab = Gal( E [ H,H ] /E H ) wher e [ H, H ] is the commut ator subgroup of H . F or each integer r = 0 , − 1 , − 2 , − 3 , . . . , we have the canonical fractiona l Galois ideal J r E [ H,H ] /E H ⊆ Q [ H ab ] as defined in § 3.1. Definition 5.1 In the notation of § 5, define a sub gr oup J r E /F of Q { G } by J r E /F = ( B ∗ G ) − 1 ( ⊕ H ⊆ G J r E [ H,H ] /E H ) . Lemma 5. 2 In § 5 and Definition 5.1, when G = Gal( E /F ) is ab elian then J r E /F c oincides with the c anonic al fr actional Galo is ide al of § 3.1 . Pr o of . The H -co mp o nent of B ∗ G has the form Q [Gal( E /F )] i E H /F → Q [Gal( E /E H )] π E /E [ H,H ] → Q [Gal( E [ H,H ] /E H )] which maps J r E /F to J r E [ H,H ] /E H by Prop o sition 3 .3 a nd Prop ositio n 3.6 so that J r E /F ⊆ ( B ∗ G ) − 1 ( ⊕ H ⊆ G J r E [ H,H ] /E H ) . On the o ther hand, the G -comp onent o f B ∗ G is the identit y map fr om Q [ G ] to itself. Therefore if z ∈ Q [ G ] r J r E /F then B ∗ G ( z ) 6∈ ⊕ H ⊆ G J r E [ H,H ] /E H , as required. Prop ositi on 5.3 Su pp ose that F ⊆ K ⊆ L is a tower of fi nite extens ions of numb er fields with L /F and K/F Galois. Then, for r = 0 , − 1 , − 2 , − 3 , . . . , the c anonic al homomorphism π L/K : Q { Ga l( L/F ) } → Q { Ga l( K/F ) } satisfies π L/K ( J r E /F ) ⊆ J r K/F . Pr o of . This follows immediately from Prop os ition 3.3, Lemma 4 .1 and Defi- nition 5.1. Definition 5.4 L et F b e a n umb er fi eld and L/F a (p ossibly infin ite) Galois extension with Galois gr oup G = Gal( L/F ) . F or r = 0 , − 1 , − 2 , − 3 , . . . define J r E /F to b e the ab elian gr oup J r E /F = lim ← H J r L H /F , wher e H ru n s thr ough the op en normal sub gr oups of G . 18 6 J r E /F and the annihilation of H 2 ´ et (Sp ec( O L,S ) , Z ℓ (1 − r )) 6.1 Let ℓ b e an o dd pr ime. W e contin ue to assume the Stark c o njecture as stated in § 2 .2 for r = 0 , − 1 , − 2 , − 3 , . . . . Replacing Q by Q ℓ in § 3.1 and Definition 5.1 we ma y a sso ciate a finitely gener a ted Z ℓ -submo dule of Q ℓ { Gal( E /F ) } , again denoted by J r E /F , to any finite extension E /F of num b er fields. In this section we are going to explain a conjectura l pr o cedure to pass from J r E /F to the constructio n o f elemen ts in the annihilator ideal of the ´ eta le c o ho- mology of the r ing of S -integers of E , ann Z ℓ [ G ( E /F )] ( H 2 ´ et (Spec ( O E ,S ( E ) ) , Z ℓ (1 − r ))) , where S deno tes a finite se t of primes of F including a ll archimedean primes and all finite primes which ramify in E /F , a nd S ( E ) deno tes a ll the primes of E ov er those in S . This c o njectural pro cedure was first describ ed in [34, Thm.8.1]. W e shall r estrict ourselves to the ca se when r = − 1 , − 2 , − 3 , . . . . I n sev- eral ways this is a simplification ov er the ca se when r = 0. In this case H 1 ´ et (Spec ( O E ,S ( E ) ) , Z ℓ (1 − r )) is indep endent of S ( E ), while it is related to the group of S ( E )-units when r = 0. Also, when r ≤ − 1, H 2 ´ et (Spec( O E ,S ( E ) ) , Z ℓ (1 − r )) is a s ubgroup of the corr e sp onding cohomolog y g roup when S ( E ) is enlar ged to S ′ ( E ), but when r = 0 the class-group of O E ,S ′ ( E ) is a quotien t o f that o f O E ,S ( E ) . F ur thermore (see [5], [36]), there are subtleties co nc e rning whether or not to us e the S -mo dified L -function in § 2 when r = 0 , w hile for r ≤ − 1 this is immaterial. When r = 0 the annihila tor pr o cedure is similar to the other cases but the additional complicatio ns hav e prompted us to omit this case. W rite G = Ga l( E /F ), and for each subgr oup H = Gal( E /E H ) ⊆ G let S ( E H ) denote the set of primes of E H ab ov e those of S . Then H ab = Gal( E [ H,H ] /E H ) where [ H , H ] deno tes the commutator subgr o up of H . The following conjecture or iginated in [3 1, 33, 34]. Conjecture 6.1 In the notation of § 6.1, when r = − 1 , − 2 , − 3 , . . . , (i) In tegrali t y: J r E [ H,H ] /E H · ann Z ℓ [ H ab ] (T ors H 1 ´ et (Spec( O E [ H,H ] ,S ) , Z ℓ (1 − r ))) ⊆ Z ℓ [ H ab ] . (ii) Annihilation: J r E [ H,H ] /E H · ann Z ℓ [ H ab ] (T ors H 1 ´ et (Spec ( O E [ H,H ] ,S ) , Z ℓ (1 − r ))) ⊆ a nn Z ℓ [ H ab ] ( H 2 ´ et (Spec ( O E [ H,H ] ,S ) , Z ℓ (1 − r ))) . (W e have adopted the sho rthand: O E [ H,H ] ,S = O E [ H,H ] ,S ( E [ H,H ] ) .) 19 6.2 Evidence Part (i) of C o njecture 6.1 is analo g ous to the Stick elb er ger integrality , which is descr ib ed in [3 4, Section 2.2]. Stic kelberger in tegrality w as prov en in cer tain totally real cases in [21, 9, 8, 14], for r = 0. In g eneral, when r = 0 , it is part of the Brumer conjecture [4 ]. The no velt y of part (ii) of Co njecture 6.1, when it was in tro duced in [33] a nd [34 ], was the annihilator prediction when the L -function v anishes at s = r . F or the pa rt of the fractional ide a l corres p o nding to characters whose L -functions a r e non-zero at s = r , g enerated by the higher Stic kelberger element a t s = r , pa r t (ii) is the c onjecture of [12]. Let us co nsider the cyclotomic example J r L/ Q ( r < 0) when L = Q ( ζ ) for some ro ot of unit y ζ , and supp ose ℓ is an o dd prime dividing the order o f ζ . In this case, J r L/ Q splits into plus and minus parts for complex conjuga tio n, i.e. J r L/ Q = e r + J r L/ Q ⊕ e r − J r L/ Q , where e r + = 1 2 (1 + ( − 1) r c ), e r − = 1 2 (1 − ( − 1) r c ) and c ∈ G = Gal( L/ Q ) is complex conjuga tio n. By the pro o f of [3 4, Theorem 6.1 ], e r − J r L/ Q is genera ted by the Stic kelberger element θ L/ Q ,S ( r ) defined in terms of L - function v alues at s = r . Howev er, by [14], ann Z ℓ [ G ] (T ors( H 1 ´ et (Spec O L,S , Z ℓ (1 − r )))) θ L/ Q ,S ( r ) ⊆ Z ℓ [ G ] . F urther, the pro of of [34, Theorem 7.6] shows that e r + J r L/ Q ⊆ Z ℓ [ G ]. In fact, [34, Theo rem 6.1] a lso shows that par t (ii) of Conjecture 6.1 ho lds in this ca s e (with E = Q and H = G ), the intersection “ ∩ Z ℓ [ G ]” found in the s tatement of that theorem b eing unnecess ary . T urning now to the case r = 0, with the fie ld E n as in § 3.2, we hav e a s imilar scenario for J ( E n / Q , S ), where S = {∞ , ℓ } . Indeed, we see from (3.1) that J ( E n / Q , S ) ag ain splits into plus and minus parts, with the minus part b eing generated by the Stic kelberger element θ E n / Q ,S defined at s = 0. Stick elb er ger’s theorem then implies that ann Z ℓ [ G n ] ( µ ( E n )) e − J ( E n / Q , S ) ⊆ Z ℓ [ G n ] , and e + J ( E n / Q , S ) is already in Z ℓ [ G n ]. The ro les of the plus and minus pa rts of J ( E n / Q , S ) will b e come clea r in § 6.2.1 b elow. 6.2.1 An Iw asa w a-theoretic example (3.1) ca n b e used to pr ovide a n exa mple of the r elationship of J ( E n / Q , S ) to Iwasa wa theory , with a n inv ers e limit of the J ( E n / Q , S ) ov er n giving rise, in a suitable way , to Fitting ideals of b oth the plus and minus parts of a n inv e rse limit of class-g roups (Pro p osition 6 .2). Given n ≥ 0, let Q ( n ) / Q b e the degr ee ℓ n sub e xtension of the (unique) Z ℓ -extension Q ( ∞ ) of Q . W e then have the fie ld 20 diagram E n ∆ n ③ ③ ③ ③ ③ ③ ③ ③ Γ n ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ Q ( n ) ✷ ✷ ✷ ✷ ✷ ✷ ✷ ✷ ✷ ✷ ✷ ✷ ✷ ✷ ✷ E 0 ∆ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ Q in which Q ( n ) ∩ E 0 = Q and Q ( n ) E 0 = E n , so that the Galo is g roup G n = Gal( E n / Q ) is the in terna l direct pro duct of ∆ n and Γ n . S will deno te the set of places {∞ , ℓ } of Q . By virtue o f the na tural is omorphism ∆ n → ∆, characters of ∆ n corres p o nd to characters of ∆. If δ ∈ b ∆, we let δ n denote the corres po nding ch aracter in b ∆ n . Now, the idea is to view the group-ring C [ G n ] a s C [Γ n ][∆ n ]. In doing this, we can define a pro jection π n ( δ ) : C [ G n ] → C [Γ n ] by extending δ n linearly (ov er C [Γ n ]). Finally , fix an isomor phism ν : C ℓ → C and let ω : ∆ → C × be the com- po sition of the T eichm¨ uller character ∆ → C × ℓ with ν : C × ℓ → C × . Then g iven δ ∈ b ∆, δ ∗ will deno te ω δ − 1 . O bserve that since ω is o dd, δ is e ven if a nd o nly if δ ∗ is o dd. Prop ositi on 6.2 L et δ ∈ b ∆ . ( δ may b e even or o dd.) Fitt Z ℓ [ [Γ ∞ ] ] ( e δ ∗ Cl ∞ ) = ( lim ← n Z ℓ π n ( δ ∗ )( J ( E n / Q , S )) if δ 6 = 1 lim ← n Z ℓ π n ( δ ∗ )((1 − (1 + ℓ ) σ − 1 n ) J ( E n / Q , S )) if δ = 1 wher e σ n = (1 + ℓ, E n / Q ) . Pr o of . This stems fro m (3.1), which we repro duce for conv enie nce : J ( E n / Q , S ) = 1 2 e + ann Z [ G n ] ( O × E + n ,S / E + n ) ⊕ Z [ G n ] θ E n / Q ,S . Let us deal with even characters δ ∈ b ∆ fir s t. F or simplicit y , we will assume that δ 6 = 1, though in fact the cas e δ = 1 is similar. (3.1) tells us that fo r each n ≥ 0, Z ℓ π n ( δ ∗ )( J ( E n / Q , S )) = Z ℓ [Γ n ] π n ( δ ∗ )( θ E n / Q ,S ). How ever, Iwasaw a ’s construction o f ℓ -adic L - functions (see [1 9] and [40, Cha pter 7 ]) shows tha t this lies in Z ℓ [Γ n ] a nd that the in verse limit of these ideals is generated by the algebraic ℓ - adic L -function corresp onding to the even c haracter δ . Mazur a nd Wiles’ pro of (see [24]) of the Main Conjecture of Iwasaw a theory , and later Wiles’ genera lization of this (see [41]), show that this in turn is equal to the Fitting ideal app ear ing in the statement of the pro p o sition. 21 Now we tur n to o dd characters δ ∈ b ∆. Referring to (3.1) a gain, we find that Z ℓ π n ( δ ∗ )( J ( E n / Q , S )) = π n ( δ ∗ )(Fitt Z ℓ [ G n ] (( O × E + n ,S / E + n ) ⊗ Z Z ℓ )) . This uses that ( O × E + n ,S / E + n ) ⊗ Z Z ℓ is co cyc lic as a Z ℓ [ G n ]-mo dule so that, since G n is cyc lic , the Fitting and annihilator ideals of ( O × E + n ,S / E + n ) ⊗ Z Z ℓ agree. [13, Theorem 1 ] says in particular that this Fitting ideal is equal to tha t of Cl( E + n ) ⊗ Z Z ℓ . Combining the above a nd pa ssing to limits completes the pr o of. W e o bserve the impor tance here o f taking leading co efficients o f L -functions at s = 0 ra ther than just v alues. F o r δ even (i.e. δ ∗ o dd), π n ( δ ∗ )( J ( E n / Q , S )) concerns L -functions which ar e non- zero at 0, and we g et the usual Stic kelberger elements which a re related to minus parts of class-g roups via ℓ - adic L -functions. How ever when δ is o dd (i.e. δ ∗ is e ven), π n ( δ ∗ )( J ( E n / Q , S )) is concer ned with L -functions having simple z ero es at 0 , which are related to plus parts o f cla ss- groups via cycloto mic units. 7 J r E /F and annihilation Let ℓ b e a n o dd prime. Given α ∈ J r E /F and H ⊆ G = Gal( E /F ), c ho ose any β ∈ a nn Z ℓ [ H ab ] (T ors H 1 ´ et (Spec ( O E [ H,H ] ,S ) , Z ℓ (1 − r ))) . Then the H - comp onent B ∗ G ( α ) H lies in Q ℓ [ H ab ] N G H , the fixed p oints under the conjugation ac tio n by N G H , the norma lizer of H in G . Assuming Co njecture 6.1(i), B ∗ G ( α ) H · β ∈ Z ℓ [ H ab ] N G H . Cho o se z H,α,β ∈ Z ℓ [ H ] such tha t π ( z H,α,β ) = B ∗ G ( α ) H · β . Consider the co mpo sition H 2 ´ et (Spec( O E ,S ( E ) ) , Z ℓ (1 − r )) T r E /E [ H,H ] → H 2 ´ et (Spec( O E [ H,H ] ,S ) , Z ℓ (1 − r )) B ∗ G ( α ) H · β → H 2 ´ et (Spec ( O E [ H,H ] ,S ) , Z ℓ (1 − r )) j → H 2 ´ et (Spec ( O E ,S ( E ) ) , Z ℓ (1 − r )) in which j is induced by the inclusion o f fields and T r E /E [ H,H ] denotes the transfer homomorphism. Assuming Conjecture 6.1(ii), this compos itio n is zero. How ever, by F ro b e- nius recipro city for the co ho mology transfer, for all a ∈ H 2 ´ et (Spec ( O E ,S ( E ) ) , Z ℓ (1 − r )) 0 = j ( π ( z H,α,β )T r E /E [ H,H ] ( a )) = j · T r E /E [ H,H ] ( z H,α,β · a ) = ( P h ∈ Gal( E /E [ H,H ] ) h ) z H,α,β · a. 22 Definition 7.1 In the situation of § 6.1 and § 7, let I ( E /F , r ) ⊆ Z ℓ [ G ] denote the left ide al gener ate d by the elements ( P h ∈ Gal( E /E [ H,H ] ) h ) z H,α,β as α , H and β vary thr ough al l the p ossibilities ab ove. Theorem 7. 2 If Conje cture 6.1 is tru e for al l ab elian interme diate extens ions E [ H,H ] /E H of E /F then the left action of the left ide al I ( E /F , r ) annihilates H 2 ´ et (Spec ( O E ,S ( E ) ) , Z ℓ (1 − r )) . R emark . If G is ab elia n in Definition 7.1 a nd Theorem 7.2, then I ( E /F, r ) = J r E /F · ann Z ℓ [ G ] (T ors H 1 ´ et (Spec ( O E ,S ( E ) ) , Z ℓ (1 − r ))) . That is, I ( E /F , r ) equals the left hand side of C o njecture 6.1(ii). Prop ositi on 7.3 In Definition 7.1, I ( E /F , r ) is a two-side d ide al in Z ℓ [ G ] . Pr o of . In the no ta tion of § 7, it suffices to show that w   X h ∈ Gal( E /E [ H,H ] ) h   z H,α,β w − 1 lies in I ( E /F , r ). Consider w   X h ∈ Gal( E ,E [ H,H ] ) h   w − 1 = X h ∈ Gal( E /E [ wH w − 1 ,wH w − 1 ] ) h and wz H,α,β w − 1 . Since z H,α,β lies in Z ℓ [ H ] and maps to B ∗ G ( α ) β in Z ℓ [ H ab ], we see that wz H,α,β w − 1 lies in Z ℓ [ wH w − 1 ] and maps to wB ∗ G ( α ) H w − 1 wβ w − 1 in Z ℓ [ H ab ]. How ever, wB ∗ G ( α ) H w − 1 = B ∗ G ( α ) wH w − 1 and w β w − 1 lies in ann Z ℓ [( w H w − 1 ) ab ] (T ors H 1 ´ et (Spec ( O E [ wH w − 1 ,wH w − 1 ] ,S ) , Z ℓ (1 − r ))) , completing the pro o f. Prop ositi on 7.4 Su pp ose that F ⊆ K ⊆ E is a tower of numb er fields with E /F and K /F Galois. Then for r = − 1 , − 2 , − 3 , . . . , t he c anonic al homomor- phism π E /K : Z ℓ [Gal( E /F )] → Z ℓ [Gal( K/F )] satisfies π E /K ( I ( E / F, r )) ⊆ I ( K/F , r ) . Pr o of . This follows easily from Lemma 4 .1 and Prop ositio n 5 .3. 23 8 Relation to Iw asa w a theory As discusse d in the Intro duction, the motiv ation fo r exa mining the b ehaviour o f the fractiona l Galois idea l under changes of extension is to set up in vestigating a po ssible role in Iwasaw a theo ry . Via the rela tionship of the fractional ideal with Stark-type elements (eg cyclotomic units in the case r = 0 a nd Beilinson elements in the case r < 0 , discuss ed in [5] and [3 3] res p.), o ne might hop e that an approach inv olving E uler systems would b e fr uitful here. A gener al connection of the fractional Galois ideal to Stark elements o f ar bitrary rank was demo nstrated in [6], and the link of Star k elements with cla s s-gro ups using the theory of Eule r sys tems is discussed in [29, 25], so that a stra tegy as above would seem pro mising. W e conclude the pap e r with so me sp ecula tion co ncerning wha t the no n- commutativ e Iw asaw a theory of F uk ay a–Kato [15], Ka to [20] and Ritter– W eiss [27] sugge s ts a b out J r E /F of Definition 5.4 and I ( E / F, r ) of Definition 7.1 . It is worth p ointing out, b efor e we b egin the reca pitulation prop er, that [1 5, 20, 2 7] often r estrict to the situatio n wher e the ex tension fields ar e totally r eal, which tends to involv e only one o f the eigenspa ces o f complex conjuga tion acting on J r E /F and I ( E / F, r ). W e have tried to give some exa mples (for example, § 6.2.1) which illustrate the exp ected role a nd pro pe r ties of the other eig enspace. F urther, in this area ther e is an immense litany of conjectures (see [15, 7]) of which Stark’s co njecture is approximately the weak est. All the c o nstructions we hav e made a r e contingen t only on the truth of Stark’s co njectur e, which is crucial for us but a lso seems fundamental; it is assumed, for e xample, in [28 ]. F ollowing [20], let ℓ b e an o dd prime (denoted p there), F a totally real nu mber field and F ∞ a to ta lly r eal Lie extension of F con taining Q ( ζ ℓ ∞ ) + . Here, Q ( ζ ℓ ∞ ) + is the union of the totally rea l fields Q ( ζ ℓ n ) + = Q ( ζ ℓ n + ζ − 1 ℓ n ) ov er all n ≥ 1. Let G = Gal( F ∞ /F ), and assume tha t only finitely ma ny primes of F r a mify in F ∞ . Fix a finite set Σ of primes of F containing the o nes which r amify in F ∞ /F . Define Λ( G ) to be the Iwasa wa algebra of G , g iven by Λ( G ) = Z ℓ [ [ G ] ] = lim ← U Z ℓ [ G/U ], where the limit runs ov er a ll o pe n nor mal subgroups of G . Let C denote the c o chain complex o f Λ( G )-mo dules g iven by RHom(RΓ ´ et ( O F ∞ [1 / Σ] , Q ℓ / Z ℓ ) , Q ℓ / Z ℓ ) , so that H 0 ( C ) = Z ℓ with trivia l G -action and H − 1 ( C ) = Gal( M /F ∞ ), the Galois g roup of the ma ximal pro - ℓ ab elian extension of F ∞ unramified outside Σ. The o ther H i ( C )’s are zer o and Gal( M /F ∞ ) is a finitely generated tors io n (left) Λ( G )-mo dule. Let F cy c ⊆ F ∞ denote the cyclotomic Z ℓ -extension and set H = Gal( F ∞ /F cy c ) ⊆ G so that G/H ∼ = Z ℓ . As in [11], let S = { f ∈ Λ ( G ) | Λ( G ) / Λ( G ) f is finitely gener ated as a Λ( H )-mo dule } . Then S is an Ore set, whic h mea ns that its ele ments may b e inv erted to form the lo calized ring Λ( G ) S , and there is a n exact loca lization sequence of algebr aic 24 K-gro ups K 1 (Λ( G )) → K 1 (Λ( G ) S ) ∂ → K 0 (Λ( G ) , Λ( G ) S ) → K 0 (Λ( G )) → K 0 (Λ( G ) S ) . By [17], Iwasaw a’s co njecture concer ning the v anishing of the µ - inv aria nt im- plies that the cohomolog y of the p erfect complex C v anishes when S -lo calized. This gives rise to a class [ C ] ∈ K 0 (Λ( G ) , Λ( G ) S ). In the case of finite Galois extensions the class [ C ] accounts for the Stick e lbe r ger phenomena (c.f. [34]) but on the other hand so do v alues of Artin L -functions. The main conjecture of no n-commutativ e Iw asaw a theory , describ ed b elow following [20], makes this relation clear in ter ms of Λ( G ) S -mo dules. There is an ℓ -adic deter minantal v aluation which ass igns to f ∈ K 1 (Λ( G ) S ) and a contin uo us Artin r epresentation ρ a v a lue f ( ρ ) ∈ Q ℓ ∪ {∞} . The main conjecture of no n- commutativ e Iwasaw a theory as s erts that there exists ξ ∈ K 1 (Λ( G ) S ) such that (i) ∂ ( ξ ) = − [ C ] and (ii) ξ ( ρκ r ) = L Σ (1 − r, ρ ) for any even r ≥ 2 where κ is the ℓ -adic cyclotomic character and L Σ ( s, ρ ) is the Artin L -function of ρ with the Euler factors a t Σ removed. The main conjecture of Iw a saw a theory was form ulated in [2 8] and studied in the s eries of pap ers [27] when the Lie gr oup G has rank zero or one. The case of G = GL 2 ( Z ℓ ) is of particula r interest in the study of elliptic curves E / Q without complex multiplication [11] a nd is pr oven for the ℓ -a dic Heisenber g g roup in [2 0]. F or a compr ehensive survey s e e [15]. Motiv ated by the main conjecture of Iwasaw a theory , and mo re genera lly by the r ole of Λ ( G ) in the arithmetic geo metry of elliptic curves and their Selmer groups, there ha s b een considera ble ring-theo retic activity concerning Λ ( G ) and Ω( G ) = Λ( G ) /ℓ Λ( G ) (see [1, 2, 3, 37, 3 8, 39]). The rings Λ( G ) and Ω ( G ) are examples of “just-infinite rings” whic h b oth satisfy the Auslander– Gorenstein condition and ar e th us amenable to Lie theo retic analys is . In the survey article [1 ], a num b er of questio ns a re p osed. In par ticular the constructions of § 7 are directly related to [1, Ques tion G]: “Is there a mecha- nism for constructing idea ls of Iw asaw a alg ebras which inv olves neither central elements nor clo sed normal subgr oups?” Prop ositi on 8.1 If F ∞ /F is any ℓ -adic Lie ex tension of a n umb er fi eld F with Galois gr oup G then, under t he assumption of § 7 for the fin ite interme diate sub exten s ions E / F for r = − 1 , − 2 , − 3 , . . . we may define a two-side d ide al I ( F ∞ /F , r ) = lim ← E I ( E /F, r ) in Λ( G ) , wher e the limit is taken over finite Galois sub exten s ions E /F of F ∞ /F . In view of the annihilation discussio n o f § 7, Prop o sition 8.1 suggests the following: Question 8 . 2 What is the int erse ction of the c anonic al Or e set S of [11, 20 ] with I ( F ∞ /F , r ) ? 25 In many wa y s the most interesting ca se is when G = GL 2 ( Z ℓ ) ( ℓ ≥ 7) arising from the tower of ℓ -primary torsion p oints on a n elliptic c urve ov er Q without complex multiplication [10, 11]. In this cas e one has particula rly stro ng information co ncerning tw o-sided primes ideals o f Λ( G ) – see [3]. There is a po ssibly alter native approa ch to the construction of fractiona l Galois idea ls in Q ℓ [Gal( K/ Q )] based o n assuming that a type of Stark co njecture holds for the Hasse–W eil L -function of the elliptic curve [35]. It would be interesting to know whether this leads to the same tw o -sided ideal as Pro po sition 8.1. References [1] K. Ardak ov and K. A. Brown. Ring -theoretic prop erties of Iwasaw a alge- bras: a s urvey . Do c. 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