The rigid analytical regulator and K_2 of Drinfeld modular curves

THE RIGID ANAL YTICAL REGULA TOR AND K 2 OF DRINFELD MODULAR CUR VES Ambrus P ´ al May 31, 2009. A bs tr act . W e ev aluate a r igid analytical analogue of the Beilinson-Blo c h-Deligne regulator on certain explicit elemen ts i n the K 2 of Drinfeld mo dular curves, co n- structed from analogues of mo dular units, and relate its v alue to sp ecial v alues of L -series using the Rankin-Selb erg method. 1. Introduction Motiv at ion 1.1. In the pap er [10 ] which is no w cla s sical B. Gross formulated a generaliza tion of his or iginal p - adic analogue of Stark ’s conjecture in a form which makes g o o d sense b oth over num ber ﬁelds and function ﬁelds. This conjecture was prov ed by D. Hay es for function ﬁelds in [12]. In this pap er Hayes gav e an explicit rigid analy tical co nstruction o f Stark units and expres sed them in terms o f sp ecial v a lues o f L -functions using this explicit constructio n. This pap er is part of the pro ject to formulate and prove results which gener alize Hayes’s theorem the same wa y a s B eilinson’s conjectures g eneralize Stark’s. In a previous pap er ([22]) w e constructed a rigid analy tical reg ula tor ana logous to the cla ssical Beilinson-B lo ch- Deligne regula tor reﬁning the tame regulator in case of Mumford cur v es. In our current work we ex press the v alue of this regulator on certain explicit elements of the K 2 group o f Drinfeld mo dular curves, which are a nalogues o f A. Beilinson’s construction using mo dular units, in ter ms of sp ecial v alues of L -functions. Using the function ﬁeld analo gue of the Shimura-T aniyama-W eil conjecture we derive a formula for every elliptic curve deﬁned over the rational function ﬁeld o f tr a nscen- dence degr ee o ne a ﬁnite ﬁeld having split m ultiplicative reduction at the p oint a t inﬁnit y analog ous to the classica l theorem of Beilinso n o n the K 2 of elliptic curves deﬁned ov er the ra tio nal num ber ﬁeld. In the rest o f this in tro ductory chapter we ﬁrst describ e the rigid analy tic reg u- lator for T a te elliptic cur v es then deﬁne the ∞ - adic L -function o f elliptic cur ves o f the type men tioned ab ov e and for m ulate our main theo r em. Notation 1.2 . Let F ∞ be a ﬁeld complete with res pect to a dis c rete v a luation and let O ∞ be its v a lua tion ring. There is a cano nical wa y to extend the abso lute v a lue of F ∞ induced b y its v aluation to its algebraic closure. Let C ∞ denote the completion of the algebraic clo sure of F ∞ with respect to this absolute v alue and 2000 Mathematics Subj ect Classiﬁc ation. 11G05 (primary), 11G40 (secondary). Ty p eset b y A M S -T E X 1 2 AMBRUS P ´ AL let | · | denote the absolute v alue induced b y the completion pr oces s. Let | C ∞ | denote the set o f v alues of the latter. L e t P 1 denote the pro jective line o ver C ∞ . W e call a set D ⊂ P 1 an o pen disc if it is the imag e of the set { z ∈ C ∞ || z | < 1 } under a M¨ obius transformation. Recall that a subset U of P 1 is a connected rational sub do main, if it is non-empty and it is the c omplemen t of the union of ﬁnitely man y pair-wise dis jo in t op en discs. Let ∂ U denote the set of these complementary o pen discs. Let O ( U ) and O ∗ ( U ) deno te the alge br a of ho lomorphic functions on U and the group o f inv ertible elemen ts of this algebra, respec tively . F or each f ∈ O ( U ) let k f k denote sup z ∈ U | f ( z ) | . T his is a ﬁnit e n um ber , and mak es O ( U ) a Ba nach algebra ov er C ∞ . The latter is the closure of the subalgebr a o f restrictions of rational functions with resp ect to the supremum nor m k · k by deﬁnition. F o r every real num ber 0 < ǫ < 1 we deﬁne the sets O ǫ ( U ) = { f ∈ O ( U ) |k 1 − f k ≤ ǫ } , and U ǫ = { z ∈ C ∞ || 1 − z | ≤ ǫ } . Recall that a function f : C ∗ ∞ → C ∞ is holo morphic if its r estriction f | U is ho lomorphic for every connected rational sub domain U ⊂ C ∗ ∞ . F or ev ery x ∈ P 1 and every pair of rational non-zero functions f , g ∈ C ∞ (( t )) on the pro jective line let { f , g } x denote the ta me symbol of the pair ( f , g ) a t x . Let M ( C ∗ ∞ ) denote the ﬁeld of meromor phic functions of C ∗ ∞ . F or every ﬁeld L let K 2 ( L ) denote the Milnor K 2 of the ﬁeld L . Finally for ev ery x ∈ C ∞ and p ositive nu mber ρ ∈ | C ∞ | let D ( x, ρ ) denote the o p en disc { z ∈ C ∞ || z − x | < ρ } . The following re s ult is a n immediate co ns equence of the results o f [22 ]. Theorem 1 .3. F o r every 0 < r ∈ | C ∞ | ther e is a u nique homomorphism: {·} r : K 2 ( M ( C ∗ ∞ )) → C ∗ ∞ with the fol lowing pr op erties: ( i ) for every p ai r of r ational functions f , g ∈ M ( C ∗ ∞ ) ∗ we have: { f ⊗ g } r = Y x ∈ D (0 ,r ) { f , g } x , ( ii ) for every r e al numb er 0 < ǫ < 1 and fun ctions f ∈ M ( C ∗ ∞ ) ∩ O ǫ ( U ) and g ∈ M ( C ∗ ∞ ) ∩ O ∗ ( U ) we have { f , g } r ∈ U ǫ wher e U is a c onne ct e d r ational sub doma in U ⊂ C ∗ ∞ such that D (0 , r ) ∈ ∂ U .  Notation 1. 4. F or every ﬁeld K , for any v ariety V deﬁned o ver K and for an y extension L o f K let V L denote the ba s e change o f V to L . F or every ﬁeld K and regular irr educible pro jective curve C deﬁned ov er K let F ( C ) denote the function ﬁeld o f the curve C ov er K . F or every clo sed p oint x o f C there is a tame symbol at x which is a homomor phism from K 2 ( F ( C )) into the multiplicativ e group of the residue ﬁeld at x . W e deﬁne the group K 2 ( C ) as the intersection of the kernels of all tame symbols. (In this pap er we will sometimes use the so mewhat incorr ect notation K 2 ( X ) to denote H 2 M ( X, Z (2)) for v a r ious types of s pa ces X as the latter is ra ther awkw ard.) Let E b e an elliptic curve de ﬁned over F ∞ which has a r igid- analytic T ate uniformization o ver F ∞ . The latter is equiv alen t to the pro perty that the sp ecial ﬁb er of the N´ eron mo del of E ov er the spectrum o f O ∞ is split m ultiplicative. Let θ : C ∗ ∞ → E ( C ∞ ) b e the T ate uniformiza tion (ov er C ∞ ). The latter induces a homomorphism θ ∗ : F ( E F ∞ ) → M ( C ∗ ∞ ) by pull-back which in turn induces a ho mo morphism K 2 ( F ( E F ∞ )) → K 2 ( M ( C ∗ ∞ )) which will b e deno ted by the sa me s y m bol b y slig ht a buse of notatio n. THE RIGID ANAL YTICAL REGULA TO R AND DRINFELD MODULAR CUR VES 3 Prop osition 1.5. F or every k ∈ K 2 ( E F ∞ ) and 0 < r ∈ | C ∞ | we ha ve { θ ∗ ( k ) } r ∈ F ∗ ∞ and the latter is indep ende nt of the choic e of r .  Let {·} : K 2 ( E F ∞ ) → F ∗ ∞ denote the ho momorphism deﬁned by the common v a lue of the reg ulators { θ ∗ ( · ) } r . Deﬁnition 1.6. F or every ﬁeld K let K denote its sepa rable closur e. Let F de- note the function ﬁeld o f X , where the latter is a geometric a lly c onnected smo oth pro jective curve deﬁned ov er the ﬁnite ﬁeld F q of characteristic p . Fix a closed po in t ∞ of the curve X and let E be an elliptic curve deﬁned ov er F which has split multiplicativ e reduction a t ∞ . F or ev ery closed p oin t x of X let deg ( x ) and L x ( E , t ) denote the degree of x and the loc a l factor of the Hasse-W eil L -function of E at x , resp ectively . The latter is a n element of ∈ Z [[ t ]]. L et ψ ∗ E ( x n ) ∈ Z deno te the unique n um b er such that L x ( E , t ) = ∞ X n =0 ψ ∗ E ( x n ) t n deg( x ) . Let K b e a num ber ﬁeld and let ∆ deno te its ring o f integers. Let χ : Gal( F | F ) → K ∗ be a K -v a lued o ne-dimensional Galois repr esen tation of F which has ﬁnite image. Note that χ is automa tica lly almost everywhere unr amiﬁed and its image lies in ∆ ∗ . Let Γ denote the q uotien t of Gal( F | F ) by the kernel o f χ . Assume that χ splits a t ∞ a nd let m b e an eﬀective diviso r whose supp ort do es not contain ∞ and the co nductor of χ and E divides m and m ∞ , resp ectively . (Note that such an m exis ts b ecause we assumed tha t E has split multiplicativ e reduction at ∞ .) F or every Galois gro up G of a ﬁnite ab elian extension of F and for every closed po in t x o f X where G is unramiﬁed let φ G x denote the imag e of a g eometric F ro benius at x in G . The element φ G x ∈ G is well-deﬁned as G is ab elian. Assume now that G is the Galo is g roup of a ﬁnite abelia n extension of F which only ramiﬁes at ∞ . W e deﬁne the L -function L G m ( E , χ, t ) as the Euler pr o duct: L G m ( E , χ, t ) = Y x / ∈ supp ( m ∞ ) ∞ X n =0 ψ ∗ E ( x n ) χ ( φ Γ x )( φ G x ) n t n deg( x ) ! ∈ ∆[ G ][[ t ]] , where sup p ( d ) denotes the supp ort of any eﬀectiv e divisor d on X . The inﬁnite pro duct L G m ( E , χ, t ) is well-deﬁned, as the constant term of every facto r app earing in the pro duct is 1, and there a re only ﬁnitely many fa c tors with a term of degr ee less than m for any p ositive integer m . Actually even more is true: Prop osition 1 .7. The p ower series L G m ( E , χ, t ) is an element of ∆[ G ][ t ] . Deﬁnition 1.8. An imp ortant consequence of the prop osition ab ov e is that the po lynomial L G m ( E , χ, t ) can b e ev alua ted at 1, i.e. the element L G m ( E , χ, 1) ∈ ∆[ G ] is well-deﬁned. Let G ∞ denote the Galois gr oup of the maxima l ab elian extension of F unramiﬁed at every closed p oin t x of X diﬀerent fr om ∞ . It is a pro ﬁnite gro up. Also note that if H denotes the Galois gro up o f the maxima l a belian extension of F unramiﬁed at every closed p oint x of X and tota lly split at ∞ , then the kernel of the natural pro jection G ∞ → H is canonica lly isomor phic to t he proﬁnite completion of F ∗ ∞ / F ∗ q , the multiplicative group o f the completion F ∞ of F with with resp ect to the v aluation at ∞ divided out by the multiplicative group of the constant ﬁeld of 4 AMBRUS P ´ AL X . (Note that this notation is compa tible with what we hav e introduced in 1.2 and 1.4.) F or any ring R and ab elian pr oﬁnite group M let R [[ M ]] denote the R -dual of the ring of c on tin uous functions f : M → R , where f is contin uous with r espect to the discrete topolo gy on R a nd the Krull topology on M . The ring R [[ M ]] is also the pro jective limit of R -co eﬃcient gr oup rings of the ﬁnite quotien ts of M . The elements L G m ( E , χ, 1) satisfy the obvious compatibility , so their limit deﬁnes an element L m ( E , χ ) in ∆[[ G ∞ ]], which we will call the ∞ -adic L -function of E t wisted with χ . F or every M as ab ov e let I M ⊳ ∆[[ M ]] denote the kernel of the natural augmentation map ∆[[ M ]] → ∆. W e will usually drop the subscript M to ease notation. It is known that the group I M /I 2 M is naturally isomo r phic to M ⊗ ∆. Finally let θ ′ ∈ M ⊗ ∆ denote the class o f any θ ∈ I M in I M /I 2 M . Prop osition 1 .9. We have L m ( E , χ ) ∈ I and L m ( E , χ ) ′ ∈ F ∗ ∞ / F ∗ q ⊗ ∆ . Let L denote the Galois extension of F whose Galo is group is Γ. By our a ssump- tions the ﬁeld L has an imbedding in to F ∞ which extends the cano nical inclusion F ⊂ F ∞ . Fix once and for all such a n imbedding. By sligh t abuse of notatio n let {·} : K 2 ( E L ) ⊗ K → F ∗ ∞ ⊗ K denote also the comp osition o f the homomorphis m K 2 ( E L ) ⊗ K → K 2 ( E F ∞ ) ⊗ K induced by the imbedding ab o ve and the unique K - linear extension of the homomorphis m {·} . Assume that F = F q ( T ) is the rational function ﬁeld of trans cendence degree one o ver F q , where T is an indeter minate, and ∞ is the po in t a t inﬁnity on X = P 1 F q . Also a s sume that χ is no n- trivial. Now we are a ble to state our ma in re sult: Theorem 1 .10. Ther e is an element κ E ( χ ) ∈ K 2 ( E L ) ⊗ K such that { κ E ( χ ) } = L ( E , q − 1 ) L m ( E , χ ) ′ in F ∗ ∞ ⊗ K . It is easy to deduce that the v alua tion of L m ( E , χ ) ′ with r espect to ∞ ⊗ id K is equal to − L m ( E , χ, 1) from the interpo la tion pro perty . (F o r the explanatio n of this notation see the next chapter.) Deligne’s purity theorem implies that the latter is non-zer o under mild, purely lo cal conditions on χ and m . If the sp ecial v alue L ( E , q − 1 ) als o happens to b e non-zer o we get that the element κ E ( χ ) ∈ K 2 ( E L ) ⊗ K is not tor sion hence our main result is non- v acuous . Con ten ts 1.11. In the nex t c hapter we prov e the basic pro perties o f the L - function L m ( E , χ ) by simple cohomolo gical means. W e int ro duce our mail to ol, which we call double E isenstein ser ies, in the third chapter. They are r eally a nalogous to the pro duct of tw o Eisens tein se ries in the class ical setting, but they canno t b e written as such due to the lac k of lo garithm in po sitiv e characteristic. Here we als o es ta blish their basic prop erties, among them Prop osition 3.5, which is a nalogous to analytic contin uation. The link b et ween double Eisens tein ser ies and the r ig id analytic regulator of element s in K 2 analogo us to Beilinson’s constr uction is provided b y the Krokeck er limit for m ula 4.1 0 o f the fourth chapter. The ﬁfth chapter is somew ha t techn ical: it iden tiﬁes function ﬁeld a nalogues of modular units with the rigid analytic functions app earing in the prev io us chapter and studies the action of the Heck e algebra on the sour ce and target groups of the r igid analytic regulato r. W e execute the principal calculation o f the pa per in the sixth c hapter. Perhaps the crucial rea son why the Rankin-Selb erg computation can b e car ried out is that the double Eisenstein ser ies do b ecome a pr oduct o f t w o series after the ﬁrst step of the calculation. In the seventh chapter w e use the function ﬁeld analogue of the THE RIGID ANAL YTICAL REGULA TO R AND DRINFELD MODULAR CUR VES 5 Shim ura-T aniyama-W eil conjecture as well as its explicit description due to Gekeler and Reversat to conclude the pro of of our ma in r esult. The aim o f the last chapter is to prov e a useful lemma on the action of cor r espo ndences on motivic cohomolo gy groups which is used in the ﬁfth chapter. Ac kno wledgmen t 1.12. I wish to thank the CRM and the IH ´ ES for their warm hospitality and the plea san t environment they cre ated fo r productive research, where this a rticle was written. 2. The ∞ -adic L -functions of elliptic cur ves Deﬁnition 2.1. Note that fo r a ﬁnite gr o up G we hav e ∆[[ G ]] = ∆[ G ] na turally . Let M be an abe lia n proﬁnite group, let H b e a ﬁnite quotient of M and let K denote the kernel of the quotient map M → H . W e let I M H denote the ideal of the quotient map ∆[[ M ]] → ∆[ H ]. It is obvious that the augmentation ideal I = I M { 1 } and I M H ⊆ I for any H . Lemma 2.2. We have θ ′ ∈ K ⊗ ∆ for any θ ∈ I M H . Pro of. The same as the pro of of Lemma 3.9 of [21] whose claim is just slightly diﬀerent.  Notation 2.3. Let E b e an elliptic cur ve deﬁned o ver F which has split multi- plicative reduction at ∞ as in the intro duction whose notation we a re go ing to use without further notice. Let G b e the Galo is g r oup o f a ﬁnite ab elian extension of F which only r amiﬁes at ∞ and let H ( G ) denote the maximal q uotien t o f G s uc h that the corresp onding ab elian extension of F is unramiﬁed at every closed p oint x of X and totally split at ∞ . Prop osition 2 .4. The fol lowing ho lds: ( i ) the p ow er series L G m ( E , χ, t ) is an element of ∆[ G ][ t ] , ( ii ) we have L G m ( E , χ, 1) ∈ I G H ( G ) . Pro of. Let l b e a prime diﬀeren t fro m p . The Gal( F | F )-mo dule H 1 ( E F , Q l ) is absolutely ir reducible b ecause the curve E is not iso trivial. Let ρ denote the cor - resp onding l -a dic Galois representation. F or every character φ : G → Q ∗ l let the same symbol denote the corresp onding homo morphism Q l [ G ][[ t ]] → Q l [[ t ]] and the corres p onding l -adic Galois representation by the usual abuse of notation. F or ev - ery l -adic Ga lois re pr esent ation ψ which is unramiﬁed at almost all closed p oints of X will use the same symbol to deno te the constructible l -adic shea f on X which is the direct image o f ψ with res p ect to the g eneric point Spec( F ) → X . F ix an im bedding of K in to Q l . This wa y the Galo is re presentation χ beco mes an l -adic r e pr esent ation, too. The series L G m ( E , χ, t ) ∈ Q l [[ t ]] is characterized by the prop erty: φ ( L G m ( E , χ, t )) = L ( X ( m ∞ ) , ρ ⊗ χφ, t ) for every character φ : G → Q ∗ l where wher e X ( d ) denotes complement of the suppo rt of any eﬀective diviso r d in X and L ( U, ψ, t ) denotes the Grothendieck L -function of an y constr uctible l -adic sheaf ψ on a v a riet y U over F q . The l - adic Ga lois repres e n tation ρ ⊗ χφ is a bs olutely irreducible, therefor e the t wisted L -function L ( X ( m ∞ ) , ρ ⊗ χ φ, t ) is a p olynomia l for every character φ : G → Q ∗ l by the Grothendieck-V erdier fo rm ula. Hence so do es L G m ( E , χ, t ) as c la im ( i ) says. F o r 6 AMBRUS P ´ AL every character φ : H ( G ) → Q ∗ l let the same symbol denote the co mposition of the quotient map G → H ( G ) and the character φ as well. In this cas e the restriction of the l -adic Galois representation corre s ponding to φ to the decomp osition g r oup at ∞ is trivial. The sa me holds for χ by assumption. Moreov er E has split mu ltiplicative reduction at ∞ s o we have: φ ( L G m ( E , χ, t )) = (1 − t deg( ∞ ) ) L ( X ( m ) , ρ ⊗ χφ, t ) for ev ery such character. As the t wisted L -function L ( X ( m ) , ρ ⊗ χ φ, t ) is a p oly- nomial by the Gro thendiec k-V e r dier for m ula, we hav e φ ( L G m ( E , χ, 1)) = 0 for every such character as well. The latter is e q uiv alent to the prop erty that L H ( G ) m ( E , χ, 1) is zero a s cla im ( ii ) says.  As we explained in Deﬁnition 1.8 part ( i ) of the pr o pos ition ab ov e implies that the o b ject L m ( E , χ ) is well-deﬁned. F or every group M let c M denote its proﬁnite completion a nd let ∞ : \ F ∗ ∞ / F ∗ q ⊗ ∆ → b ∆ = b Z ⊗ ∆ deno te the pro ﬁnite completio n of the v aluation ∞ as well. The following pr opo s ition takes ca r e of Pr opo sition 1.9 and the remar k after Theorem 1.10. F or the sa k e o f simple notation let L m ( E , χ, t ) denote L ( X ( m ) , ρ ⊗ χ, t ). Prop osition 2 .5. The fol lowing ho lds: ( i ) we have L m ( E , χ ) ∈ I and L m ( E , χ ) ′ ∈ F ∗ ∞ / F ∗ q ⊗ ∆ . ( ii ) we have ∞ ( L m ( E , χ ) ′ ) = − L m ( E , χ, 1) . Pro of. The ﬁrst half of claim ( i ) and the fact that L m ( E , χ ) ′ ∈ \ F ∗ ∞ / F ∗ q ⊗ ∆ follows at once from claim ( ii ) of Prop osition 2 .4 and Lemma 2.2 by taking the limit. On the hand note that F ∗ ∞ / F ∗ q ⊗ ∆ is the pr e-image of ∆ with resp ect to ∞ in \ F ∗ ∞ / F ∗ q ⊗ ∆ hence the second half of claim ( i ) is an immediate consequence of claim ( ii ). No w we only have to show the latter. The pr oﬁnite g roup G ∞ surjects onto the Galois group of the ma ximal consta n t ﬁeld extension of F which is isomorphic to b Z . This induces a sur jection ∆[[ G ∞ ]] → ∆[[ b Z ]]. T he choice of a topo logical generato r of b Z , or equiv alently the choice of a system o f generators o f the ﬁnite quotients o f b Z compatible with the pro jections furnishes an injection ∆[ t ] → ∆[[ b Z ]] such that the image of t is the generator. In case of the natur al choice of the global geometric F r o benius as a top ologica l generato r, the image φ x of a geometric F r obe nius at x in G ∞ maps to t deg( x ) for every c lo sed p oint x on X under the map ab ov e. Hence the image o f L m ( E , χ ) under this map is e L m ( E , χ, t ) = (1 − t deg( ∞ ) ) L m ( E , χ, t ) as we saw in the pro of of Pro positio n 2 .4. The idea l I ⊳ ∆[[ G ∞ ]] maps into the aug men tation ideal J ⊳ ∆[[ b Z ]] cor resp onding to the trivial quo tien t of b Z , and the induced map I /I 2 → J /J 2 is the tensor pro duct o f the surjection G ∞ → b Z int ro duced ab ov e and the iden tit y of ∆. Since the int ersection J ∩ ∆[ t ] is the ideal gener ated by t − 1, the image of L m ( E , χ ) ′ under the map I /I 2 → J/J 2 is just the deriv ativ e e L ( E , 1) ′ ∈ ∆ ⊂ b ∆. On the other hand the restr iction o f the surjection G ∞ → b Z to F ∗ ∞ is deg ( ∞ ) times the v a luation map ∞ : F ∗ ∞ → Z , so : deg( ∞ ) ∞ ( L m ( E , χ ) ′ ) = ((1 − t deg( ∞ ) ) L m ( E , χ, t )) ′     t =1 = − deg( ∞ ) L m ( E , χ, 1) as we claimed.  THE RIGID ANAL YTICAL REGULA TO R AND DRINFELD MODULAR CUR VES 7 3. Double E isenstein series Notation 3.1. Let | X | , A , O denote set of closed po in ts of X , the ring of a deles of F and its maximal co mpact s ubring of A , respectively . As in the introduction we will ﬁx a clo sed point ∞ in the set | X | . F or every divisor m of X let m also denote the O -mo dule in the ring A gener ated by the ideles whose divisor is m , by abuse of notation. F or every idele m ∈ A ∗ let the same symbol als o denote the diviso r of m if this notation do es no t cause c onfusion. F o r any closed p oint v in | X | w e will let F v , f v and O v denote the cor resp o nding completion of F , its constant ﬁeld, a nd its discrete v aluation ring, resp ectively . F or every v ∈ | X | let v : F ∗ v → Z denote the v alua tion normalized suc h that v ( π v ) = deg( v ) for every uniformizer π v ∈ F v . F or any idele, adele, adele-v alued matrix or function deﬁned on the ab ov e which decompo ses as an inﬁnit e pro duct of functions deﬁned on the individual comp o nen ts the subscr ipt v will denote the v -th comp onent. Let A f , O f denote the restricted direct product Q ′ x 6 = ∞ F x and the direct product Q x 6 = ∞ O x , resp ectively . The former is also called the ring of ﬁnite adeles of F and the latter is its maximal compac t subring. F or every g ∈ GL 2 ( A ) (o r g ∈ A , etc .) let g f denote its ﬁnite comp onent in GL 2 ( A f ). W e will consider A ∗ f as well as F ∗ v (for every place v ∈ | X | ) as a subgr o up of A ∗ in the natur al way . Similarly we w ill consider A f and F v as a subring of A and GL 2 ( A ) a nd GL 2 ( F v ) a s a subgroup of GL 2 ( A ). Let | · | denote the nor malized absolute v alue on the ring A and for any idele or divisor y let deg( y ) denote its degree related to the normalized abs olute v alue by the formula | y | = q − deg( y ) . In a ccordance with o ur co n ven tion | · | will denote the absolute v alue with res pect to ∞ if its argument is in F ∞ . F or each ( u, v ) ∈ F 2 ∞ let k ( u, v ) k , ∞ ( u, v ) denote max( | u | , | v | ) and min( ∞ ( u ) , ∞ ( v )), resp ectively . Le t Z denote the center of the group scheme GL 2 , let Γ ∞ =  a b c d  ∈ GL 2 ( O ∞ ) |∞ ( c ) > 0  be the Iwahori subgroup of GL 2 ( F ∞ ) and let K ( m ) = { g ∈ GL 2 ( O ) | g ≡ I mo d m } , for every eﬀective diviso r m wher e I is the iden tit y ma trix. W e will adopt the conv en tion which assigns 0 o r 1 a s v alue to the empty sum o r pro duct, resp ectively . Deﬁnition 3.2. Let F 2 < denote the s e t: F 2 < = { ( a, b ) ∈ F 2 ∞ || a | < | b |} . Let m be an eﬀective divisor o n X whos e supp ort do es not con tain ∞ . Let the same symbol a lso denote the ideal m ∩ O f by abus e of no tation. F or every g ∈ GL 2 ( A ), ( α, β ) ∈ ( O f / m ) 2 , and n integer let W m ( α, β , g , n ) = { 0 6 = f ∈ F 2 | f g f ∈ ( α, β ) + m O 2 f , − n = ∞ ( f g ∞ ) } , V m ( α, β , g , n ) = { f ∈ W m ( α, β , g , n ) | f g ∞ ∈ F 2 < } and U m ( α, β , g , n ) = W m ( α, β , g , n ) − V m ( α, β , g , n ) . Also let W m ( α, β , g f ) = [ n ∈ Z W m ( α, β , g , n ) , 8 AMBRUS P ´ AL U m ( α, β , g ) = [ n ∈ Z U m ( α, β , g , n ) and V m ( α, β , g ) = [ n ∈ Z V m ( α, β , g , n ) . Obviously the ﬁrst set is well-deﬁned. F or every ﬁnite quotient G of F ∗ \ A ∗ / O ∗ f let · G : A ∗ → G denote the quotient map. Let E G m ( α, β , γ , δ, g , x, y ) denote the Z [ G ][[ x, y ]]( x − 1 , y − 1 )-v a lued function E G m ( α, β , γ , δ, g, x, y ) = det( g − 1 f ) G ( xy ) deg(d et( g )) · X ( a,b ) ∈ U m ( α,β ,g ) ( c,d ) ∈ V m ( γ ,δ,g ) det  a b c d  G ∞ x 2 ∞ (( a,b ) g ∞ ) y ∞ (2( c,d ) g ∞ ) , for e very g ∈ GL 2 ( A ), v ariables x , y , and pair s ( α, β ) and ( γ , δ ) as ab ov e. In or de r to see tha t this function is indee d well-deﬁned ﬁrst note that  a b c d  =  a 1 b 1 c 1 d 1  · det( g ∞ ) − 1 = ( a 1 d 1 − b 1 c 1 ) · det( g ∞ ) − 1 is no n-zero wher e ( a 1 , b 1 ) = ( a, b ) g ∞ and ( c 1 , d 1 ) = ( c, d ) g ∞ bec ause | a 1 | ≥ | b 1 | a nd | c 1 | < | d 1 | by the deﬁnition of the sets U m ( α, β , g ) and V m ( γ , δ, g ) therefor e | a 1 d 1 − b 1 c 1 | = | a 1 d 1 | 6 = 0 . Hence the terms of the inﬁnite sum ab ov e a re deﬁned. The sum itself is well- deﬁned and Z [ G ][[ x, y ]]( x − 1 , y − 1 )-v a lued a s the cardinality o f the sets U m ( α, β , g ) and V m ( γ , δ, g ) ar e ﬁnite for all n and zero for n suﬃciently sma ll. Prop osition 3 .3. The fol lowing ho lds: ( i ) the function E G m ( α, β , γ , δ, g , x, y ) is left-invariant with re sp e ct t o GL 2 ( F ) and right-invariant with r esp e ct to K ( m ∞ )Γ ∞ Z ( F ∞ ) , ( ii ) the C [ G ] -value d inﬁnite sum E G m ( α, β , γ , δ, g , q − s , q − t ) c onver ges absolutely, if Re( s ) > 1 and Re( t ) > 1 , for ev ery g . Pro of. W e a re go ing to prove claim ( i ) ﬁrst. Since for every ρ ∈ GL 2 ( F ) and n ∈ Z we have: U m ( α, β , ρg , n ) = U m ( α, β , g , n ) ρ − 1 and V m ( γ , δ, ρg , n ) = V m ( γ , δ, g , n ) ρ − 1 , we get tha t E G m ( α, β , γ , δ, ρg , x, y ) = det( ρ − 1 f ) G det( g − 1 f ) G ( xy ) deg(d et( ρ ))+deg(det( g )) · X ( a,b ) ∈ U m ( α,β ,g ) ( c,d ) ∈ V m ( γ ,δ,g ) det  a b c d  G ∞ det( ρ − 1 ∞ ) G x 2 ∞ (( a,b ) ρ − 1 ρg ∞ ) y 2 ∞ (( c,d ) ρ − 1 ρg ∞ ) = E G m ( α, β , γ , δ, g , x, y ) , bec ause det( ρ − 1 ) ∈ F ∗ and deg(det( ρ )) = 0 as the deg r ee of e very principal divis or is zer o. O n the other hand for every λ ∈ GL 2 ( F ∞ ) the set { f ∈ F 2 ∞ | f λ ∈ F 2 < } is obviously left inv ar ian t by Γ ∞ Z ( F ∞ ) hence U m ( α, β , g ρ ) = U m ( α, β , g ) a nd V m ( γ , δ, g ρ ) = V m ( γ , δ, g ) THE RIGID ANAL YTICAL REGULA TO R AND DRINFELD MODULAR CUR VES 9 for every ρ ∈ K ( m ∞ )Γ ∞ Z ( F ∞ ) and g ∈ GL 2 ( A ). Therefore E G m ( α, β , γ , δ, g ρ, x, y ) = det( ρ − 1 f ) G det( g − 1 f ) G ( xy ) deg(d et( z ))+de g(det( g )) · X ( a,b ) ∈ U m ( α,β ,g ) ( c,d ) ∈ V m ( γ ,δ,g ) det  a b c d  G ∞ x 2 ∞ (( a,b ) g ∞ )+ ∞ (det( z )) y 2 ∞ (( c,d ) g ∞ )+ ∞ (det( z )) = E G m ( α, β , γ , δ, g , x, y ) , where ρ = κz with κ ∈ K ( m ∞ )Γ ∞ and z ∈ Z ( F ∞ ) b ecause κ ∞ is a n isometry with resp ect to the norm k · k , deg(det( κ )) = 0 and det( κ f ) G = 1 by deﬁnition. Our pro of of claim ( i i ) is the s a me as the ar gumen t that may be found in [20]. The co eﬃcien t of each element of G in the s e ries E G m ( α, β , γ , δ, g , q − s , q − t ) is ma jorated by the pro duct E ( g , s ) E ( g , t ) where: E ( g , s ) = | det( g ) | s X f ∈ F 2 −{ 0 } f g ∈O 2 f k ( f g ) ∞ k − 2 s , so it will b e suﬃcient to pr o ve that E ( g , s ) conv erges abs olutely for each g ∈ GL 2 ( A ) if Re( s ) > 1. F or every g ∈ GL 2 ( A ) let E ( g ) deno te the sheaf on X whos e gro up o f sections is for every op en subset U ⊆ X is E ( g )( U ) = { f ∈ F 2 | f g ∈ O 2 v , ∀ v ∈ | U |} , where we denote the set of closed p oints of U by | U | . The sheaf E ( g ) is a coher en t lo cally free s heaf of rank tw o. If F n denote the sheaf F ⊗ O X ( ∞ ) n for every coherent sheaf F on X and integer n , then fo r every g ∈ GL 2 ( A ) and s ∈ C the s e ries ab ov e can b e rewritten a s E ( g , s ) = X n ∈ Z | H 0 ( X, E ( g ) n ) − H 0 ( X, E ( g ) n − 1 ) | q − s deg( E ( g ) n ) . By the Riemann-Ro ch theo r em for curves: dim H 0 ( X, F ) − dim H 0 ( X, K X ⊗ F ∨ ) = 2 − 2 g ( X ) + deg ( F ) for any co herent lo cally free shea f of ra nk tw o F on X , where K X , F ∨ and g ( X ) is the ca nonical bundle o n X , the dual of F , and the g en us of X , respec tively . Because dim H 0 ( X, F − n ) = 0 for n suﬃciently large dep ending o n F , we hav e that | H 0 ( X, E ( g ) n ) | = q 2 − 2 g ( X )+deg ( E ( g ))+2 n deg( ∞ ) and | H 0 ( X, E ( g ) − n ) | = 1 , if n is a suﬃciently la rge p ositive num b er. Hence E ( g , s ) = p ( q − s ) + q 2 − 2 g ( X )+(1 − s ) de g( E ( g )) (1 − q − deg( ∞ ) ) ∞ X n =0 q 2 n (1 − s ) deg( ∞ ) , where p is a polyno mial. The claim now follows fr o m the conv ergence of the g eo- metric ser ies.  10 AMBRUS P ´ AL Deﬁnition 3.4. F or every abe lia n group M and for every ﬁnite set S let M [ S ] and M [ S ] 0 denote the group of funct ions f : S → M and its subgroup co nsisting of functions f ∈ M [ S ] with the pro perty X α ∈ S f ( α ) = 0 , resp ectively . Let V m denote the set ( O f / m ) 2 − { 0 , 0 } and for every C ∈ R [ V m ] and D ∈ R [ V m ] let E G m ( C, D , g , x, y ) denote the function: E G m ( C, D , g , x, y ) = X ( α,β ) ∈V m ( γ ,δ ) ∈V m C ( α, β ) D ( γ , δ ) E G m ( α, β , γ , δ, g , x, y ) , where R ⊇ Z is an arbitra r y commutativ e ring. Prop osition 3.5. F or every C , D ∈ R [ V m ] 0 the function E G m ( C, D , g , x, y ) t akes values in R [ G ][ x, y , x − 1 , y − 1 ] . Pro of. W e may assume by bilinear it y tha t C = ( α, β ) − ( γ , δ ) and D = ( ǫ, ι ) − ( κ, λ ) for some pairs ( α, β ), ( γ , δ ), ( ǫ, ι ) and ( κ, λ ) ∈ V m . Pick t wo elements ( r , s ) ∈ U m ( α − γ , β − δ, g ) a nd ( u , v ) ∈ V m ( ǫ − κ, ι − λ, g ). Then for every suﬃcient ly la rge natural num ber n we have: U m ( α, β , g , n ) = { ( a + r , b + s ) | ( a, b ) ∈ U m ( γ , δ, g , n ) } and V m ( ǫ, ι, g , n ) = { ( a + u, b + v ) | ( a, b ) ∈ V m ( κ, λ, g , n ) } . Therefore E G m ( C, D , g , x, y ) = P ( x, y ) + det( g − 1 f ) G ( xy ) deg(d et( g )) X ( a,b ) ∈ U m ( γ ,δ,g f ) ( c,d ) ∈ V m ( κ,λ,g )  a b c d  x 2 ∞ (( a,b ) g ∞ ) y 2 ∞ (( c,d ) g ∞ ) , where P ( x, y ) ∈ Z [ G ][ x, y , x − 1 , y − 1 ] and  a b c d  = det  a + r b + s c + u d + v  G ∞ − det  a + r b + s c d  G ∞ − det  a b c + u d + v  G ∞ + det  a b c d  G ∞ . In order to ﬁnish the pro of it is enough to show that the deter minan ts in the expres - sion ab ov e can b e paired in such a way that in every pair the determinants hav e dif- ferent signs a nd they represent the sa me element in G if max( k ( a, b ) g ∞ k , k ( c, d ) g ∞ k ) is suﬃciently lar g e. This follows the lemma below o r its pair which we get by switch- ing the rows of the matrices de p ending on whether k ( a, b ) g ∞ k or k ( c, d ) g ∞ k is the larger one a mong the t wo, resp ectively .  THE RIGID ANAL YTICAL REGULA TO R AND DRINFELD MODULAR CUR VES 11 Lemma 3. 6. F or e ach k ∈ GL 2 ( F ∞ ) let k k k denote the maximum of absolute values of the entries of k . Then for every g ∈ GL 2 ( F ∞ ) and ( r, s ) , ( a, b ) , ( c, d ) ∈ F 2 ∞ such that ( a, b ) g / ∈ F 2 < and ( c, d ) g ∈ F 2 < we have:      1 − det(  a + r b + s c d  ·  a b c d  − 1 )      ≤ k ( r , s ) kk g − 1 k| det( g ) | k ( a, b ) g k . Pro of. Using Cramer ’s rule w e get that  a + r b + s c d  ·  a b c d  − 1 =  1 + r d − sc ad − bc − r b + sa ad − bc 0 1  , so      1 − det(  a + r b + s c d  ·  a b c d  − 1 )      =     rd − sc ad − bc     ≤ k ( r , s ) k · k ( c, d ) k | ad − bc | . On the other hand let ( a 1 , b 1 ) = ( a, b ) g a nd ( c 1 , d 1 ) = ( c, d ) g . Then | a 1 | ≥ | b 1 | and | c 1 | < | d 1 | similarly a s w e noted a t the end of Deﬁnition 3.2 therefo r e | ad − bc | =     det  a 1 b 1 c 1 d 1  · det( g − 1 )     = | a 1 d 1 − b 1 c 1 | · | det( g ) | − 1 = | a 1 d 1 | · | det( g ) | − 1 = k ( a 1 , b 1 ) k · k ( c 1 , d 1 ) k · | det( g ) | − 1 ≥k ( a, b ) g k · k ( c, d ) k · k g − 1 k − 1 · | det( g ) | − 1 .  Deﬁnition 3.7. As a consequence of P rop osition 3.5 the function E G m ( C, D , g , x, y ) can b e ev aluated at x = y = 1 . Let E G m ( C, D , g ) = E G m ( C, D , g , 1 , 1) ∈ R [ G ] for every g ∈ GL 2 ( A ). In accordance with the previously introduced notation for every ﬁnite ab elian gro up G we let I G ⊳ Z [ G ] denote the aug mentation ideal of Z [ G ] that is t he kernel of the augment ation map Z [ G ] → Z . There is an isomo rphism I G /I 2 G = G induced by the ma p given by the rule g 7→ 1 − g ∈ I G for every g ∈ G . Prop osition 3.8. Assume t hat R = Z . Then we have E G m ( C, D , g ) ∈ I G for every g ∈ GL 2 ( A ) . Pro of. It will be suﬃcient to prove that E { 1 } m ( C, D , g ) = 0 wher e { 1 } is the trivia l group. W e ma y as sume again by bilinearity that C = ( α, β ) − ( γ , δ ) and D = ( ǫ, ι ) − ( κ, λ ) for some pairs ( α, β ), ( γ , δ ), ( ǫ, ι ) and ( κ, λ ) ∈ V m . Pick a g ain t w o elements ( r, s ) ∈ U m ( α − γ , β − δ, g ) a nd ( u , v ) ∈ V m ( ǫ − κ, ι − λ, g ). Then for every suﬃciently la r ge natur a l num be r n w e have: [ m ≤ n U m ( α, β , g , m ) = [ m ≤ n { ( a + r, b + s ) | ( a, b ) ∈ U m ( γ , δ, g , m ) } 12 AMBRUS P ´ AL and [ m ≤ n V m ( ǫ, ι, g , m ) = [ m ≤ n { ( a + u, b + v ) | ( a, b ) ∈ V m ( κ, λ, g , m ) } . Hence we have: | [ m ≤ n U m ( α, β , g , m ) | = | [ m ≤ n U m ( γ , δ, g , m ) | , | U m ( α, β , g , n ) | = | U m ( γ , δ, g , n ) | and | [ m ≤ n V m ( ǫ, ι, g , m ) | = | [ m ≤ n V m ( κ, λ, g , m ) | , | V m ( ǫ, ι, g , n ) | = | V m ( κ, λ, g , n ) | for every suﬃciently larg e natural num b er n . Therefor e E { 1 } m ( C, D , g ) = X m,n ∈ Z  ( | U m ( α, β , g , m ) | − | U m ( γ , δ, g , m ) | ) · ( | V m ( ǫ, ι, g , n ) | − | V m ( κ, λ, g , n ) | )  = lim n →∞  ( | [ m ≤ n U m ( α, β , g , m ) | − | [ m ≤ n U m ( γ , δ, g , m ) | ) · ( | [ k ≤ n V m ( ǫ, ι, g , k ) | − | [ k ≤ n V m ( κ, λ, g , k ) | )  =0.  Deﬁnition 3. 9. In a ccordance with the notatio n w e introduced in Deﬁnition 1.8 let θ ′ ∈ G denote the class of any θ ∈ I G . F or every C , D ∈ Z [ V m ] 0 and N ∈ Z le t E m ( C, D , g , n ) denote the F ∗ ∞ -v a lued function: E m ( C, D , g , N ) = Y m,n ≤ n ( α,β ) ∈V m ( γ ,δ ) ∈V m Y ( a,b ) ∈ U m ( α,β ,g ,m ) ( c,d ) ∈ V m ( γ ,δ,g ,n ) det  a b c d  C ( α,β ) D ( γ , δ ) . Finally let E m ( C, D , g ) denote the limit E m ( C, D , g ) = lim N →∞ E m ( C, D , g , N ) if the latter exists. The following cla im is an immediate corollary to Lemma 3.6 and Prop osition 3.8 using the same argument we used in the pro of of Pr opo sition 3.5. Prop osition 3 .10. The limit ab ove exits and E G m ( C, D , g ) ′ = E m ( C, D , g ) G .  THE RIGID ANAL YTICAL REGULA TO R AND DRINFELD MODULAR CUR VES 13 4. The Kronecker limit f ormula Notation 4.1. W e a re g oing to use the notatio n we intro duced in 1.2. F or every connected r ational sub domain U of P 1 the element s of ∂ U are called the bo undary comp onen ts of U , b y slight abuse of lang ua ge. Let R ( U ) ⊂ O ( U ) denote the subalgebra of r e strictions o f ratio na l functions holomor phic on U and R ∗ ( U ) denote the gr oup of invertible elemen ts of this algebra . The g roup R ∗ ( U ) co nsists of rational functions w hich do not hav e p oles or z e r os lying in U . Theorem 4 .2. Ther e is a u nique map {· , ·} D : O ∗ ( U ) × O ∗ ( U ) → C ∗ ∞ for every D ∈ ∂ U , c al le d the rigid analytic re gulator, with the fol lowing pr op erties: (i) F or any two f , g ∈ R ∗ ( U ) their r e gulator is: { f , g } D = Y x ∈ D { f , g } x , (ii) the r e gulator {· , ·} D is biline ar in b oth variables, (iii) the r e gulator { · , ·} D is alternating: { f , g } D · { g , f } D = 1 , (iv) if f , 1 − f ∈ O ( U ) ∗ , then { f , 1 − f } D is 1 , (v) for e ac h f ∈ O ǫ ( U ) and g ∈ O ∗ ( U ) we have { f , g } D ∈ U ǫ . Pro of. This is Theorem 2 .2 of [22].  Deﬁnition 4.3 . If U is still a connected ra tio nal sub domain of P 1 , and f , g are t wo meromorphic functions on U , then fo r a ll x ∈ U the functions f and g ha v e a power series expansion around x , so in particular their tame sym bo l { f , g } x at x is deﬁned. Let M ( U ) denote the ﬁeld of meromorphic functions o f U . The tame symbols extends to a homomorphism {· , ·} x : K 2 ( M ( U )) → C ∗ ∞ . W e deﬁne the group K 2 ( U ) as the kernel o f the direct sum of tame symbols : M x ∈ U {· , ·} x : K 2 ( M ( U )) → M x ∈ U C ∗ ∞ . Let k = P i f i ⊗ g i ∈ K 2 ( U ), where f i , g i ∈ M ( U ), and le t D ∈ ∂ U . Let mor eov er Y be a connected ra tional s ubdomain of U such that f i , g i ∈ O ∗ ( Y ) for all i and ∂ U ⊆ ∂ Y . Deﬁne the r igid analytica l r egulator { k } D by the formula: { k } D = Y i { f i | Y , g i | Y } D . Theorem 1.3 is based on the previous result a nd the following theor em: Theorem 4.4. ( i ) F or e ach k ∈ K 2 ( U ) the rigid analytic al r e gulator { k } D is wel l- deﬁne d, and it is a homomorphism {·} D : K 2 ( U ) → C ∗ ∞ , ( ii ) for any two functions f , g ∈ O ∗ ( U ) we have { f ⊗ g } D = { f , g } D , ( iii ) for every k ∈ K 2 ( U ) the pr o duct of al l r e gulators on t he b o undary c omp onents of U is e qu al t o 1: Y D ∈ ∂ U { k } D = 1 , ( iv ) for every c onne cte d sub domain Y ⊆ U , b oundary c omp onent D ∈ ∂ Y ∩ ∂ U and k ∈ K 2 ( M ( U )) we have: { k | Y } D = { k } D . 14 AMBRUS P ´ AL Pro of. This is Theorem 3 .2 of [22].  Deﬁnition 4.5. F or every ρ ∈ GL 2 ( F ∞ ) and z ∈ P 1 let ρ ( z ) denote the image of z under the M¨ obius transfor mation corres ponding to ρ . Let mor eov er D ( ρ ) denote the op en disc D ( ρ ) = { z ∈ P 1 ( C ∞ ) | 1 < | ρ − 1 ( z ) |} . Let D denote the set of op en discs of the form D ( ρ ) where ρ ∈ GL 2 ( F ∞ ). F or each D ∈ D let D ( F ∞ ) denote D ∩ P 1 ( F ∞ ). Let P deno te those subs e ts S of D such that the sets D ( F ∞ ), D ∈ S form a pa ir-wise disjoin t partition of P 1 ( F ∞ ). F or each S ∈ P let Ω( S ) deno te the unique connected rational sub domain deﬁned ov er F ∞ with the prop erty ∂ Ω( S ) = S . Let Ω denote the rigid analytic upp er half plane, or Drinfeld’s upp er half plane ov er F ∞ . The set of p oints of Ω is C ∞ − F ∞ , denoted a lso b y Ω by abuse of no ta tion. Recall that a function f : Ω → C ∞ is holomorphic if the res tr iction of f onto Ω( S ) is ho lomorphic for every S ∈ P . Let O (Ω) and M (Ω) denote the C ∞ -algebra of holomo r phic functions and the ﬁeld of mer o morphic function of Ω, resp ectively . The la tter is of co urse the quotient ﬁeld of the former. W e deﬁne K 2 (Ω) as the intersection o f the kernels of all the tame symbols {· , ·} x inside K 2 ( M (Ω)) where x runs through the set Ω. By pa rt ( iv ) of Theo rem 4.4 for each k ∈ K 2 (Ω) the v alue { k } ( ρ ) = { k | Ω( S ) } D ( ρ ) , where ρ ∈ GL 2 ( F ∞ ) and D ( ρ ) ∈ S ∈ P , is indep endent of the c hoice of S . W e deﬁne the regulator { k } : GL 2 ( F ∞ ) → C ∗ ∞ of k as the function given by this r ule. Lemma 4 .6. L et ρ =  x y 0 1  wher e x ∈ F ∗ ∞ and y ∈ F ∞ . Then for every 0 6 = ( a, b ) ∈ F 2 ∞ and 0 6 = ( c, d ) ∈ F 2 ∞ the fol lowing holds: ( i ) if ( a, b ) ρ ∈ F 2 < and ( c, d ) ρ ∈ F 2 < then { ( az + b ) ⊗ ( cz + d ) } D ( ρ ) = 1 , ( ii ) if ( a, b ) ρ / ∈ F 2 < and ( c, d ) ρ / ∈ F 2 < then a 6 = 0 , b 6 = 0 and { ( az + b ) ⊗ ( cz + d ) } D ( ρ ) = b/a , ( iii ) if ( a, b ) ρ / ∈ F 2 < and ( c, d ) ρ ∈ F 2 < then a 6 = 0 and { ( az + b ) ⊗ ( cz + d ) } D ( ρ ) = 1 a det  a b c d  , ( iv ) if ( a, b ) ρ ∈ F 2 < and ( c, d ) ρ / ∈ F 2 < then b 6 = 0 and { ( az + b ) ⊗ ( cz + d ) } D ( ρ ) = c det  a b c d  − 1 . Pro of. Let D ( ρ ) c denote the complement of D ( ρ ) in P 1 . Obviously D ( ρ ) c = { z ∈ C ∞ || z − y | ≤ | x |} . Hence ( a, b ) ρ ∈ F 2 < if a nd only if the p olynomial az + b ha s no zeros in D ( ρ ) c and ( a, b ) ρ / ∈ F 2 < if and only if a 6 = 0 and the polyno mial az + b do es hav e a z e r o in D ( ρ ) c . By W eil’s recipro city law: { ( az + b ) ⊗ ( cz + d ) } − 1 D ( ρ ) = Y t ∈ D ( ρ ) − 1 { az + b, cz + d } t . THE RIGID ANAL YTICAL REGULA TO R AND DRINFELD MODULAR CUR VES 15 Since the tame symbo l o f az + b and cz + d at t ∈ C ∞ is 1 if neither az + b nor cz + d has a zero at t claim ( i ) is c lear. In the second cas e b oth az + b and c z + d has a single pole at ∞ but has no zero in D ( ρ ) therefor e { ( az + b ) ⊗ ( cz + d ) } D ( ρ ) = { az + b, cz + d } ∞ = b/a. Claim ( iv ) follo ws from claim ( iii ) b y the a ntisymmetry of the r egulator. In the latter case az + b has a s ing le zero in D ( ρ ) c and cz + d has no zer o s in D ( ρ ) c hence { ( az + b ) ⊗ ( cz + d ) } D ( ρ ) = { az + b, cz + d } − 1 − b/a = 1 a det  a b c d  .  Deﬁnition 4.7. W e are going to need a mild extension of the regulato r we in- tro duced in Deﬁnition 4.5. Let K 2 ( GL 2 ( A f ) × Ω) denote the set of functions k : GL 2 ( A f ) → K 2 (Ω). W e deﬁne the reg ulator of an element k ∈ K 2 ( GL 2 ( A f ) × Ω) as th e function { k } : GL 2 ( A ) → C ∗ ∞ given by the rule { k } ( g ) = { k ( g f ) } ( g ∞ ) f or every g ∈ GL 2 ( A ). Since the set K 2 ( GL 2 ( A f ) × Ω) consists of functions ta king v a lues in the group K 2 (Ω) it is eq uipped with a group structure who se op eration will b e denoted by addition. Let O ∗ ( GL 2 ( A f ) × Ω) denote the set of functions u : GL 2 ( A f ) × Ω → C ∗ ∞ which are holomor phic in the second v a riable. Then there is a bilinea r map: ⊗ : O ∗ ( GL 2 ( A f ) × Ω) × O ∗ ( GL 2 ( A f ) × Ω) → K 2 ( GL 2 ( A f ) × Ω) given by the rule ( u ⊗ v )( g ) = u | g ×{· } ⊗ v | g ×{· } for every g ∈ GL 2 ( A f ). F o r each ( α, β ) ∈ ( O f / m ) 2 , and N p ositive in teger let ǫ m ( α, β , N )( g , z ) denote the function: ǫ m ( α, β , N )( g , z ) = Y n ≤ N   Y ( a,b ) ∈ W m ( α,β ,g ,n ) ( az + b ) · Y ( c,d ) ∈ W m (0 , 0 ,g ,n ) ( cz + d ) − 1   . on the set GL 2 ( A f ) × Ω. The latter is clea r ly holomorphic in the second v ariable. Lemma 4.8. The limit ǫ m ( α, β )( g , z ) = lim N →∞ ǫ m ( α, β , N )( g , z ) c onver ges uniformly in z on every admissible o p en sub domain o f Ω for eve ry ﬁxe d g and deﬁnes a fun ction holomo rphic in the se c ond variable. Pro of. See Lemma 4.5 o f [20 ] o n pages 145 -146.  Deﬁnition 4.9. F or every C ∈ Z [ V m ] 0 let ǫ m ( C, g , z ) de no te the function: Y ( α,β ) ∈V m ǫ m ( α, β )( g , z ) C ( α,β ) on the set GL 2 ( A f ) × Ω. F or every C , D ∈ Z [ V m ] 0 let κ m ( C, D ) denote the element : ǫ m ( C, g , z ) ⊗ ǫ m ( D , g , z ) of the set K 2 ( GL 2 ( A f ) × Ω). 16 AMBRUS P ´ AL Kronec k er L imit F orm ula 4.10. F or al l g ∈ GL 2 ( A ) we have: { κ m ( C, D ) } ( g ) G = ( E G m ( C, D , g ) − E G m ( D , C , g )) ′ . Pro of. Assume ﬁrst that g ∞ =  x y 0 1  for some x ∈ F ∗ ∞ and y ∈ F ∞ . By Pr opos i- tion 3.1 0 it will b e suﬃcient to prov e that { Y ( α,β ) ∈V m ǫ m ( α, β , N )( g , z ) C ( α,β ) ⊗ Y ( γ ,δ ) ∈V m ǫ m ( γ , δ, N )( g , z ) D ( γ ,δ ) } D ( g ∞ ) is equal to E m ( C, D , g , N ) · E m ( D , C , g , N ) − 1 for ev ery suﬃciently lar g e N . By bilinea r it y and Lemma 4.6 the r egulator in the left hand s ide o f the equation tha t we wish to prov e is equal to = Y m,n ≤ N ( α,β ) ∈V m ( γ ,δ ) ∈V m  Y ( a 1 ,b 1 ) ∈ U m ( α,β ,g ,m ) ( c 1 ,d 1 ) ∈ V m ( γ ,δ,g ,n ) a − 1 1 det  a 1 b 1 c 1 d 1  · Y ( a 2 ,b 2 ) ∈ V m ( α,β ,g ,m ) ( c 2 ,d 2 ) ∈ U m ( γ ,δ,g ,n ) c 2 det  a 2 b 2 c 2 d 2  − 1 · Y ( a 3 ,b 3 ) ∈ U m ( α,β ,g ,m ) ( c 3 ,d 3 ) ∈ U m ( γ ,δ,g ,n ) a − 1 3 c 3  C ( α,β ) D ( γ , δ ) . Therefore what w e need to s how is: Y m,n ≤ N ( α,β ) ∈V m ( γ ,δ ) ∈V m  Y ( a,b ) ∈ U m ( α,β ,g ,m ) a −| W m ( γ ,δ,g ,n ) | · Y ( c,d ) ∈ U m ( γ ,δ,g ,n ) c | W m ( α,β ,g ,m ) |  C ( α,β ) D ( γ , δ ) = 1 . The la tter follows from the fact that for every C ∈ Z [ V m ] 0 and for every suﬃciently large N the equation: X n ≤ N ( α,β ) ∈V m C ( α, β ) | W m ( α, β , g , n ) | = 0 holds. On the o ther hand the la tter has be en a lr eady shown in the co ur se of the pro of of Prop osition 3.8 (at lea st in the sp ecial case when C = ( α, β ) − ( γ , δ ) for some ( α, β ), ( γ , δ ) ∈ V m but the g eneral cas e follows at once from this one b y linearity). Let us co ns ider now the general case. First note that b oth sides of the equation in the theor e m a bov e are rig ht-inv a riant with resp ect to Z ( F ∞ )Γ ∞ hence if the claim is tr ue for g then it is true for g z as well for ev ery z ∈ Z ( F ∞ )Γ ∞ . Let Π ∈ GL 2 ( F ∞ ) be the matrix whos e diago nal entries are zero , a nd its low er left and upper right entry is π and 1, re s pectively , where π is a uniformizer of F ∞ . Then THE RIGID ANAL YTICAL REGULA TO R AND DRINFELD MODULAR CUR VES 17 for every ρ ∈ GL 2 ( F ∞ ) the op en disks D ( ρ ) and D ( ρ Π) are complementary in P 1 hence claim ( i ii ) of Theor em 4.4 (W eil’s recipro city law) implies that { κ m ( C, D ) } ( g ) G · { κ m ( C, D ) } ( g Π) G = 1 for ev ery g ∈ GL 2 ( A ). A matrix ρ ∈ GL 2 ( F ∞ ) can be written as a product ρ =  x y 0 1  z wher e x ∈ F ∗ ∞ , y ∈ F ∞ and z ∈ Z ( F ∞ )Γ ∞ if a nd only if ∞ ∈ D ( ρ ). Since either D ( ρ ) or D ( ρ Π) contains the p oin t ∞ it will b e suﬃcient to prove that the identit y ab ov e holds for E G m ( C, D , g ) ′ /E G m ( D , C , g ) ′ as well. But U m ( α, β , g Π) = V m ( α, β , g ) and V m ( α, β , g Π) = U m ( α, β , g ) , therefore E G m ( C, D , g ) = ( − 1) G E G m ( D , C , g Π) for any g ∈ GL 2 ( A ), so the latter is obvious.  5. Modular units and Hecke opera tor s Notation 5.1. Let A = O f ∩ F : it is a Dedekind domain. The ideals of A a nd the eﬀective divisor s on X with s upport awa y from ∞ ar e in a bijective cor resp ondence. These tw o sets will be iden tiﬁed in all that follows. F or any non- zero ideal m ⊳ A let Y ( m ) denote the coar se moduli for par ameterizing Drinfeld mo dules of r ank tw o ov er A of gener al characteristic with full level m -structure. It is a n a ﬃne a lgebraic curve deﬁned ov er F . F or ev ery Drinfeld mo dule φ : A → C { τ } of r ank tw o equipp e d with a full level m -structure ι : ( A/ m ) 2 → C , where C is a n F -alg ebra, let u φ,ι : Y ( m ) → Sp ec( C ) b e the universal map. Lemma 5.2. Ther e is a unique element ǫ m ( D ) ∈ Γ( Y ( m ) , O ∗ ) for every D ∈ Z [ V m ] 0 such that u ∗ φ,ι ( ǫ m ( D )) = Y ( α,β ) ∈V m ι ( α, β ) D ( α,β ) ∈ C for every C , φ and ι as ab ove. Pro of. W e may a ssume that m is a pro per ideal without the loss of generality . F or any non- z ero ideal m ⊳ A let H ( m ) denote Γ( Y ( m ) , O ). W e may assume b y linearity that D = ( α, β ) − ( γ , δ ) for some ( α, β ), ( γ , δ ) ∈ V m . Let ( φ, ι ) and ( ψ , κ ) be order ed pairs of t wo Drinfeld mo dules φ and ψ o f rank t wo over C equipp e d with a full level m -structure ι a nd κ , resp ectively . Recall that ( φ, ι ) and ( ψ , κ ) are isomor phic if there is an isomor phis m j : G a → G a betw een φ and ψ such that the co mpositio n j ◦ ι is equal to κ . As j is just a multiplication by a scala r we g e t that the element ι ( α, β ) /ι ( γ , δ ) dep ends only on the isomorphism class of the pair ( φ, ι ). In particula r the cla im is obvious when the mo duli scheme Y ( m ) is ﬁne b ecause w e have ǫ m ( D ) = ι m ( α, β )) /ι m ( γ , δ ) in this case where the map ι m : ( A/ m ) 2 → H ( m ) is the universal full level m -str ucture for the universal Dr infeld mo dule φ m : A → H ( m ) { τ } over Y ( m ). The latter holds if m ha s at leas t tw o prime divisors. In general the universal map H ( m ) → L p ∤ m H ( mp ) is an ´ etale injection, so it is faithfully ﬂat. Therefore the sequence 0 → H ( m ) → M p ∤ m H ( mp ) ⇒   M p ∤ m H ( mp )   ⊗ H ( m )   M p ∤ m H ( mp )   18 AMBRUS P ´ AL is exact by P rop osition 2.18 o f [18 ], pag es 1 6-17. F or every ( κ, λ ) ∈ V m and prime ideal p ∤ m let ( κ, λ, p ) denote the unique element of V mp such that ( κ, λ, p ) ≡ ( κ, λ ) mo d m and ( κ, λ, p ) ≡ (0 , 0) mo d p . Moreov er for ev ery prime ideal p ∤ m let D ( p ) ∈ Z [ V mp ] 0 denote the element ( α, β , p ) − ( γ , β , p ). Then the element M p ∤ m ǫ mp ( D ( p )) ∈ M p ∤ m H ( mp ) is in the k ernel o f the second map in the exact sequence abov e, therefore it is the image of a unique element ǫ m ( D ) ∈ H ( m ) which satisﬁes the required prop erty .  Deﬁnition 5. 3. The g roup GL 2 ( F ) acts on the pro duct GL 2 ( A f ) × Ω on the left by a cting on the ﬁr st fa c to r via the natural embedding a nd on Drinfeld’s upp er half plane v ia M¨ obius transformations. The group K f ( m ) = K ( m ) ∩ GL 2 ( O f ) acts o n the r igh t of this pr oduct by a c ting on the ﬁrs t factor via the regular a ction. Since the quotient set GL 2 ( F ) \ GL 2 ( A f ) / K f ( m ) is ﬁnite, the set GL 2 ( F ) \ GL 2 ( A f ) × Ω / K f ( m ) is the disjoint union of ﬁnitely many sets o f the form Γ \ Ω, where Γ is a subgr oup of GL 2 ( F ) of the form GL 2 ( F ) ∩ g K f ( m ) g − 1 for s ome g ∈ GL 2 ( A f ). As these gr oups act on Ω discr etely , the set ab ov e naturally has the structure of a r ig id analytic curve. L e t Y ( m ) F ∞ also denote the underlying rigid analytical space o f the base change of Y ( m ) to F ∞ by abus e of no tation. Theorem 5 .4. Ther e is a rigid-analytic al isomorp hism: Y ( m ) F ∞ ∼ = GL 2 ( F ) \ GL 2 ( A f ) × Ω / K f ( m ) . Pro of. See [4], Theore m 6 .6.  Prop osition 5.5. F or every D ∈ Z [ V m ] 0 the function c orr esp onding t o ǫ m ( D ) under the isomorphism of The or em 5.4 ab ove is t he fu n ction ǫ m ( D , g , z ) intr o d uc e d in Deﬁnition 4.9. Pro of. Fir s t we are going to recall the map underlying the isomorphism of Theo - rem 5 .4 on C ∞ -v a lued p oint s. F or every ( g , z ) ∈ GL 2 ( A f ) × Ω let e ( g,z ) ( w ) denote the cor r espo nding exp onential function: e ( g,z ) ( w ) = z Y ( a,b ) ∈ W m (0 , 0 ,g f ) (1 − w az + b ) . The inﬁnite pro duct ab ov e is con verging abso lutely and deﬁnes an entire func- tion. The exp onential e ( g,z ) uniformizes a Drinfeld mo dule φ ( g,z ) ov er C ∞ which is equipp e d with a full level m - structure ι given by the formula: ι ( α, β ) = e ( g,z ) ( az + b ) where ( a, b ) ∈ W m ( α, β , g f ) for every ( α, β ) ∈ ( O f / m ) 2 independent of the choice of ( a, b ). Since obviously we hav e ι ( α, β ) = ǫ m ( α, β , g , z ) the claim is now clear.  THE RIGID ANAL YTICAL REGULA TO R AND DRINFELD MODULAR CUR VES 19 Deﬁnition 5 .6. Le t M b e a n ab elian group a nd let n b e any eﬀective diviso r on X . By an M -v alued automorphic fo r m over F of level n (and trivial ce ntral character) we mean a lo cally c onstant function φ : GL 2 ( A ) → M satisfying φ ( γ g k z ) = φ ( g ) for all γ ∈ GL 2 ( F ), z ∈ Z ( A ), and k ∈ K 0 ( n ), where K 0 ( n ) = {  a b c d  ∈ GL 2 ( O ) | c ≡ 0 mo d n } . Let A ( n , M ) denote the Z -mo dule o f M -v alued a uto morphic for ms of level n . Now let n b e an eﬀective divisor on X whose supp ort do es not contain ∞ . Let H ( n , M ) denote the Z -mo dule of automorphic forms f of level n ∞ satisfying the follo wing t wo identities: φ ( g  0 1 π 0  ) = − φ ( g ) , ( ∀ g ∈ GL 2 ( A )) , and φ ( g  0 1 1 0  ) + X ǫ ∈ f ∞ φ ( g  1 0 ǫ 1  ) = 0 , ( ∀ g ∈ GL 2 ( A )) , where π is a uniformizer in F ∞ and we consider GL 2 ( F ∞ ) as a subg roup of GL 2 ( A ) and w e understand th e pro duct of their elements accordingly . Such automorphic forms are called harmonic. Deﬁnition 5.7. Let m , n be eﬀective divisor s of X . Deﬁne the set: H ( m , n ) = {  a b c d  ∈ GL 2 ( A ) | a, b, c, d ∈ O , ( ad − cb ) = m , n ⊇ ( c ) , ( d ) + n = O } . The set H ( m , n ) is co mpact and it is a do uble K 0 ( n )-coset, so it is a disjo in t unio n of ﬁnitely many r igh t K 0 ( n )-cosets. Let R ( m , n ) b e a set of repr esent atives of these cosets. F or any φ ∈ A ( n , R ) deﬁne the function T m ( φ ) by the formula: T m ( φ )( g ) = X h ∈ R ( m , n ) φ ( g h ) . It is easy to chec k that T m ( φ ) is indep enden t of the choice of R ( m , n ) and T m ( φ ) ∈ A ( n , M ) as well. So we have an Z -linear o p era to r T m : A ( n , M ) → A ( n , M ). Deﬁnition 5.8. Let X b e a Hausdo rﬀ to p olog ical space. F or any R commutativ e ring let M ( X , R ) denote the s et o f R -v alued ﬁnitely additive measures on the o pen and compact subsets o f X . F or every M ab elian group let C 0 ( X , M ) denote the group of compa ctly suppo rted lo cally constant functions f : X → M . F or every f ∈ C 0 ( X , M ) and µ ∈ M ( X , R ) we deﬁne the mo dulus µ ( f ) of f with r e s pect to µ as the Z -submo dule of R g e nerated by the elements µ ( f − 1 ( g )), where 0 6 = g ∈ M . W e also deﬁne the in tegral o f f on X with re spect to µ as the sum: Z X f ( x ) dµ ( x ) = X g ∈ M g ⊗ µ ( f − 1 ( g )) ∈ M ⊗ µ ( f ) . Of course this deﬁnition is nothing more than a con venien t formalism. F or its elementary pr oper ties see Lemma 5.2 of [2 2]. Let M be a Q - v ector spa c e and let φ be an element of A ( n , M ). If for all g ∈ GL 2 ( A ): Z F \ A φ (  1 x 0 1  g ) dµ ( x ) = 0 , 20 AMBRUS P ´ AL where µ is the normalized Ha a r measure on F \ A such that µ ( F \ A ) = 1 w e call φ a cusp form. Let A 0 ( n , M ) (resp e ctiv ely H 0 ( n , M )) deno te the Q -mo dule o f M - v a lued cuspidal automor phic for ms (r espectively cuspidal harmonic forms) o f level n (resp. o f level n ∞ ). Notation 5.9. F or any idea l n ⊳ A let Y 0 ( n ) denote the co arse mo duli sc heme for rank t wo Drinfeld mo dules of general characteristic eq uipped with a Hec ke level- n structure. It is an aﬃne alg ebraic cur v e deﬁned o ver F . Let X 0 ( n ) denote the unique irreducible smo oth pro jective cur v e ov er F whic h contains Y 0 ( n ) as an op en subv ariety . F or every prop er ideal m ⊳ A there is a m -th Heck e co rresp ondence on the Drinfeld mo dular curv e X 0 ( n ) which in turn induces an endomorphism of the Jacobian J 0 ( n ) of the curve X 0 ( n ), called the Heck e o pera tor T m (for a detailed description s e e for example [6] o r [7].) The m -th Hecke c o rresp ondence also induces a pair o f compatible homomor phisms: T m : H 2 M ( X 0 ( n ) L , K (2)) → H 2 M ( X 0 ( n ) L , K (2)) and T m : H 2 M ( Y 0 ( n ) L , K (2)) → H 2 M ( Y 0 ( n ) L , K (2)) for every num ber ﬁeld K and for every L ⊇ F extension. These o p era to rs are denoted by the same sy m bol w e use fo r the op erators introduced in Deﬁnition 5.7, but this will not cause confusion as we will see. F or the momen t it is s uﬃcien t to remark that they act on diﬀerent ob jects. Notation 5.10. Let π ( n ) : Y ( n ) → Y 0 ( n ) b e the map induced by the for g etful func- tor which assigns to every Drinfeld mo dule φ : A → C { τ } of r ank tw o equipp ed with a full level m -structure ι : ( A/ m ) 2 → C , where C is an F - a lgebra, the Drinfeld mod- ule φ equipp ed with the Hec ke level- n str ucture gener a ted by ι (0 , 1). Hence Y 0 ( n ) also ha s a rigid a nalytic uniformiza tion of the kind desc ribed in Theorem 5 .4 where the ro le of the gro up K ( n ) is played by K 0 ( n ). Hence we ma y ev aluate the reg ula tor int ro duced in Deﬁnition 4.7 on the pull-ba c k of the elements of H 2 M ( Y 0 ( n ) F ∞ , Z (2)) with resp ect to this uniformizatio n. Let { ·} denote also the unique K -linear exten- sion to H 2 M ( Y 0 ( n ) F ∞ , K (2)) of this regulato r for every num b er ﬁeld K by abuse of notation. F or the rest of the pap er we assume that F = F q ( T ) is the ra tional function ﬁeld of transcendence degree one o ver F q , where T is an indeterminate, and ∞ is the po in t at inﬁnity on X = P 1 F q . Prop osition 5 .11. F or every k ∈ H 2 M ( Y 0 ( n ) F ∞ , K (2)) fol lowing holds: ( i ) we have { k } ∈ H ( n , F ∗ ∞ ⊗ K ) , ( ii ) we have { k } ∈ H 0 ( n , F ∗ ∞ ⊗ K ) when k ∈ H 2 M ( X 0 ( n ) F ∞ , K (2)) , ( iii ) we have { T m ( k ) } = T m { k } for every m ⊳ A pr op er id e al . Pro of. By deﬁnition and the inv a riance theorem of [22] the reg ulator { k } is le ft GL 2 ( F )-inv ariant and right K 0 ( n ∞ ) Z ( F ∞ )-inv a riant. By our a ssumptions o n F and ∞ w e hav e F ∗ O ∗ f = A ∗ f hence { k } is also Z ( A )-inv ar ian t. Therefo r e it is an element of A ( n ∞ , F ∗ ∞ ⊗ K ). By claim ( ii i ) of Theorem 4 .4 the additional conditions of Deﬁnition 5.6 also hold for { k } as the following tw o sets of disks: D ( ρ ) , D ( ρ  0 1 π 0  ) and D ( ρ  0 1 1 0  ) , { D ( ρ  1 0 ǫ 1  ) | ǫ ∈ f ∞ } THE RIGID ANAL YTICAL REGULA TO R AND DRINFELD MODULAR CUR VES 21 give a pa ir-wise disjoint covering of the set P 1 ( F ∞ ) for every ρ ∈ GL 2 ( F ∞ ). Claim ( i ) is prov ed. The seco nd claim is an immediate co nsequence of Theor em 6.3 of [22]. Fina lly let us concern ourselves with the pr oo f of claim ( iii ). F o r every h ∈ GL 2 ( A f ) let h : GL 2 ( A f ) × Ω → GL 2 ( A f ) × Ω s imply denote the map given by the rule ( g , z ) 7→ ( g h, z ) for every g ∈ GL 2 ( A f ) and z ∈ Ω. By slight abuse of nota tion let the same symbol deno te unique the map h : Y ( n ) F ∞ → Y ( n ) F ∞ which satisﬁes the relation h ◦ π ( n ) = π ( n ) ◦ h . Let R ( m , n ) ⊂ H ( m , n ) b e a set of representatives which is also a subset of GL 2 ( A f ). Then we hav e: π ( n ) ∗ ( T m ( k )) = X h ∈ R ( m , n ) h ∗ ( k ) in H 2 M ( Y ( n ) F ∞ , K (2)). Hence b y the inv ar iance theorem (Theorem 3.1 1 of [22]) for every g ∈ GL 2 ( A ) we hav e: { T m ( k ) } ( g ) = X h ∈ R ( m , n ) { h ∗ ( k ) } ( g ) = X h ∈ R ( m , n ) { k } ( g h ) = T m { k } ( g ) as we claimed.  Let L ⊂ F ∞ be a ﬁnite ex tension of F and let {·} : H 2 M ( X 0 ( n ) L , K (2)) → F ∗ ∞ ⊗ K denote als o the c o mpos itio n of the homomo r phism H 2 M ( X 0 ( n ) L , K (2)) → H 2 M ( X 0 ( n ) F ∞ , K (2)) induced b y the functoriality of motivic cohomolo gy and the ho momorphism { ·} . Let C ⊂ X 0 ( n ) × X 0 ( n ) b e a co rresp ondence and let C ∗ : H 2 M ( X 0 ( n ) L , K (2)) → H 2 M ( X 0 ( n ) L , K (2)) denote the homo morphism induced by C . Lemma 5.12. We have { C ∗ ( k ) } = 0 for every k ∈ H 2 M ( X 0 ( n ) L , K (2)) if the endomorphi sm J ( C ) : J 0 ( n ) → J 0 ( n ) induc e d by C is zer o. Pro of. W e may assume without the loss of g enerality that k ∈ H 2 M ( X 0 ( n ) L , Z (2)). Let Y b e a smo oth, pro jective curve ov er F q whose function ﬁeld is L and let X b e a regular ﬂat pr o jective mo del of X 0 ( n ) L ov er Y . By passing to a ﬁnite extension of L , if it is nec essary , we may also assume that X is semi-sta ble. By P rop o sition 8.6 there is a p ositive integer j such that the element j C ∗ ( k ) ∈ H 2 M ( X 0 ( n ) L , Z (2)) lies in the image o f the natura l map H 2 M ( X , Z (2)) → H 2 M ( X 0 ( n ) L , Z (2)). By Prop osition 6.5 of [23] the group H 2 M ( X , Z (2)) is the extensio n of a torsion gro up b y a p - divisible s ubgroup. The image of the restriction of the r egulator of Notatio n 5.10 to H 2 M ( X 0 ( n ) L , Z (2)) lies in H ( n , F ∗ ∞ ) so its image has a torsion p -divis ible part. Therefore the image of the restrictio n of this reg ulator to H 2 M ( X , Z (2)) is torsion. The claim is now clear.  22 AMBRUS P ´ AL Remarks 5.13. Let T ( n ) denote the algebr a with unity generated b y the e ndo - morphisms T m of the Ja cobian J 0 ( n ), where m ⊳ A is any pro per ideal. The algebr a T ( n ) is known to b e co mm utative. By cla im ( iii ) of P rop osition 5.11 and the lemma ab o ve the alg ebra of corre s pondences generated b y the Heck e co rresp ondences leav es the kernel of the regula tor of Nota tio n 5.10 r estricted to H 2 M ( X 0 ( n ) L , K (2)) inv ar i- ant and its action on the image o f this ho momorphism factors through t he Hecke algebra T ( n ) ⊗ Q . Mo reov er the Hec ke o per a tor T m acts on this image via the op erator T m given b y the formula in Deﬁnition 5.7 by claim ( iii ) of P rop osition 5.11. Deﬁnition 5.14. Le t µ G be the unique left-inv ariant Haar measure on the lo cally compact abelian topolog ical group GL 2 ( A ) / Z ( A ) such that µ G ( GL 2 ( O ) / Z ( O )) is equal to 1 . Since this measure is left-inv ar ian t with r espect to the discrete actio n of the group GL 2 ( F ) / Z ( F ), it induces a measur e Z ( A ) GL 2 ( F ) \ GL 2 ( A ) which will be denoted by the same symbol b y abuse of notatio n. Let V , W are vector spaces ov er Q , and let φ , ψ be a V -v alued and a W -v alued, lo cally constant function on Z ( A ) GL 2 ( F ) \ GL 2 ( A ), resp ectively . Also as sume that ψ has compa ct supp ort, for example ψ ∈ H 0 ( n , W ). Then the in tegral Z Z ( A ) GL 2 ( F ) \ GL 2 ( A ) φ ( g ) ⊗ ψ ( g ) dµ G ( g ) ∈ V ⊗ Q W is well-deﬁned. It will be denoted by h φ, ψ i , and will b e called the P etersson pro duct of φ a nd ψ . Lemma 5 .15. F or every k ∈ H 2 M ( Y 0 ( n ) L , K (2)) ther e is a k ′ ∈ H 2 M ( X 0 ( n ) L , K (2)) such that h{ k } , ψ i = h{ k ′ } , ψ i for eve ry ψ ∈ H 0 ( n , Q ) . Pro of. Let U denote the group H 0 ( Y 0 ( n ) L , O ∗ ) and let V denote the K -vector s ub- space of H 2 M ( Y 0 ( n ) L , K (2)) generated b y the pro duct L ∗ ⊗ U ⊆ H 2 M ( Y 0 ( n ) L , Z (2)). By our assumptions on F and ∞ the c ur v e X 0 ( n ) is geometrically irreducible. Moreov er the group gener a ted by the linear equiv alence class of degre e ze r o divis o rs deﬁned ov er L supp orted on the co mplemen t of Y 0 ( n ) L in the Jacobian of X 0 ( n ) L is ﬁnite by the main theorem of [8]. Hence there is a u ∈ V suc h that { k } x = { u } x for every closed p oint x in the complement o f Y 0 ( n ) L where {·} x denotes the K -linea r extension of the tame symbol at x (this fact is refer red to as Blo c h’s lemma in [24]). Therefore k ′ = k − u lies in H 2 M ( X 0 ( n ) L , K (2)) by the lo caliza tion sequence. Hence it w ill b e suﬃcient to prove that h{ u } , ψ i = 0 for every u ∈ V and for ev ery ψ ∈ H 0 ( n , Q ). In fact w e will show the same claim for every ψ ∈ H 0 ( n , Q ). The op erators T m act s emisimply on the ﬁnite-dimensiona l vector space H 0 ( n , Q ) ther e- fore the latter decompo ses as the direct sum of Heck e-eigenspaces. Hence we may assume tha t ψ above is a Hec ke-eigenform by linea rit y . By the pro jection formula for the norm map in Milnor K -theor y we hav e T m ( u 1 ⊗ u 2 ) = u 1 ⊗ T m ( u 2 ) for every u 1 ∈ L ∗ , u 2 ∈ U and Hecke corr e s pondence T m . Let q 6 | n b e a non-zero prime ideal which has a genera to r π ∈ A suc h that π ≡ 1 mo d n . Then the Hec k e cor- resp ondence T q maps the cusps, the geometric po in ts in the complement of Y 0 ( n ), int o themselves with multiplicit y 1 + q deg( q ) according to the pro of o f Propo s ition 3.1 of [8] on pag e 3 65. (Strictly sp eaking this claim is pr oved for the cusps of the Drinfeld mo dular curve X ( n ) ther e but the fo r mer claim immedia tely follows fr om the latter.) Hence we hav e: (1 + q deg( q ) ) h{ u } , ψ i = h T q { u } , ψ i = h{ u } , T q ( ψ ) i = ψ ∗ ( q ) h{ u } , ψ i THE RIGID ANAL YTICAL REGULA TO R AND DRINFELD MODULAR CUR VES 23 using the self-adjointness of the op erator T q with resp ect to the Petersson pro duct where ψ ∗ ( q ) ∈ Q is the q -th Hec ke eigenv alue of ψ . By the Ra man ujan-Petersson conjecture (proved in [4] ﬁr st in this c ase) the la tter is not equal to 1 + q deg( q ) when deg( q ) is suﬃciently la rge hence h{ u } , ψ i m ust b e zer o.  6. The Rankin-Selberg method Notation 6.1. Let m ⊳ A be a prop er ideal. Reca ll that a Dirichlet character of conductor m is a contin uous ho momorphism χ : A ∗ f → C ∗ which is trivial on F ∗ O m where O m = { u ∈ O ∗ f | u ≡ 1 mo d m } . Then there is a unique homomorphism from ( O f / m ) ∗ int o C ∗ , which will be denoted by χ a s well b y abuse o f notation, such that the la tter is tr iv ial on the class of constants F ∗ q ⊂ O ∗ f and we have χ ( z ) = χ ( z ) for ev ery z ∈ O ∗ f where z denote the cla ss of z in the quotient group ( O f / m ) ∗ . Moreov er w e let χ denote also the unique ex tension of these t w o ho momorphisms onto A f and O f / m which is zer o on the complement of A ∗ f and ( O f / m ) ∗ , respec- tively . W e are go ing to assume that the homomorphism χ is non-trivial. In this case P α ∈O f / m χ ( α ) is zer o . Let χ 1 , χ 2 ∈ C [ V m ] 0 denote the functions given by the rules: χ 1 ( α, β ) = χ ( α ) ( ∀ ( α, β ) ∈ V m ), and χ 2 ( α, β ) = χ ( β ) when α = 0 , and χ 2 ( α, β ) = 0, other wise, res pectively . Let E G m ( χ, g , x, y ) denote the function χ (det( g f )) − 1 E G m ( χ 1 , χ 2 , g , x, y ) for every ﬁnite quotient G of F ∗ \ A ∗ / O ∗ f . Lemma 6.2. The function E G m ( χ, g , x, y ) is left-invariant with r esp e ct to GL 2 ( F ) and right-invariant with r esp e ct to K ( m ∞ )Γ ∞ Z ( A ) . Pro of. By cla im ( i ) o f Prop osition 3.3 we only need to sho w that E G m ( χ, g , x, y ) is right-in v aria n t with r espect to Z ( A f ). But Z ( A f ) = Z ( F ) Z ( O f ) hence w e o nly hav e to s ho w that E G m ( χ, g , x, y ) is rig h t-inv a riant with r espect to Z ( O f ). In o r der to do so we will introduce so me co n venien t notation which we will also use later on without further notice. By o ur usual abuse of nota tio n fo r i = 1 , 2 let χ i : O 2 f → C denote the function such tha t χ i ( f ) = χ i ( f ) for every f ∈ O 2 f where f denotes the class of f in the quotient group ( O f / m ) 2 . F or every g ∈ GL 2 ( A ) let W ( g ) = { 0 6 = f ∈ F 2 | f g f ∈ O 2 f } , V ( g ) = { f ∈ W ( g ) | f g ∞ ∈ F 2 < } and U ( g ) = W ( g ) − V ( g ). F or every g ∈ GL 2 ( A ) and z ∈ Z ( O f ) = O ∗ f we hav e U ( g z ) = U ( g ) and V ( g z ) = V ( g ). Moreov er we have χ i ( f z ) = χ i ( f ) χ ( z ) for every f ∈ O 2 f and i = 1, 2. Therefo r e E G m ( χ, g z , x, y ) = χ ( z ) − 2 χ (det( g f )) − 1 det( z − 1 ) G det( g − 1 f ) G ( xy ) deg(d et( z ))+de g(det( g )) · X ( a,b ) ∈ U ( g ) ( c,d ) ∈ V ( g )  χ 1 (( a, b ) g f z ) χ 2 (( c, d ) g f z ) · det  a b c d  G ∞ x 2 ∞ (( a,b ) g ∞ ) y 2 ∞ (( c,d ) g ∞ )  = E G m ( χ, g , x, y ) , bec ause deg(det( z )) = 0 and det( z ) G = 1 by deﬁnition.  24 AMBRUS P ´ AL Deﬁnition 6.3. Let χ 0 : O f → C denote the function such that χ 0 ( u ) = χ ( u ) for every u ∈ O f where u denotes again the class of u in the quotient gr oup O f / m . Let G b e a ﬁnite quotient group o f F ∗ \ A ∗ / O ∗ f . Note tha t for ev ery non- z e ro q ⊳ A the v alue y G ∈ G depends only o n q for every y ∈ A ∗ f where the divisor o f y is q . Let q G denote this common v alue. Similarly note that for every no n-zero idea l q ⊳ A relatively prime to m the v alue χ 0 ( a ) dep ends only on q for every a ∈ A which generates the idea l q . W e let χ ( q ) deno te this common v alue. F or every G as ab ove let L G m ( χ, x ) b e the inﬁnite series: X ( q , m )=1 χ ( q )( q G ) − 1 x deg( q ) ∈ C [ G ][[ x ]] . Note that for ev ery complex n um b er s the C [ G ]-v alued series L G m ( χ, q − s ) is abso- lutely convergen t when Re( s ) > 1. F or every z ∈ A ∗ f let L G m ( χ, z , s ) denote the C [ G ]-v alued serie s: L G m ( χ, z , s ) = ( z − 1 ) G | z | s X u ∈ F ∗ uz ∈ O f χ 0 ( uz ) u G ∞ | u | − s ∞ if the la tter is absolutely co n vergen t. Lemma 6.4. F or every z ∈ A ∗ f we have L G m ( χ, z , s ) = χ ( z )( q − 1) L G m ( χ, q − s ) . Pro of. Fir s t we are going to show that the function χ ( z ) L G m ( χ, s ) is inv ariant with resp ect to O ∗ f . F or every v ∈ O ∗ f we hav e χ ( z v ) L G m ( χ, η z v , s ) = χ ( v )( v − 1 ) G | v | s χ ( z )( z − 1 ) G | z | s · X u ∈ F ∗ uz v ∈O f χ 0 ( uz ) χ 0 ( v ) u G ∞ | u | − s ∞ = L G m ( χ, z , s ) , bec ause χ ( v ) = χ 0 ( v ), v G = 1 and | v | = 1 . Now we may assume that z ∈ F ∗ bec ause A ∗ F = F ∗ O ∗ f . In this case χ ( z ) L G m ( χ, z , s ) = L G m ( χ, z , s ) = X 0 6 = u ∈ A χ 0 ( u )( u G f ) − 1 | u | − s ∞ =( q − 1) L G m ( χ, q − s ) bec ause we hav e ( u G f ) − 1 = u G ∞ for every u ∈ F ∗ , and becaus e χ ( z ) = 1 and | z f | = | z | − 1 ∞ as the deg ree of every principal divis or is zero.  Deﬁnition 6.5. Let B deno te the group scheme o f inv ertible upp er tria ngular tw o by t wo matrices. F or every ﬁnite quotient G as ab o ve and g ∈ B ( A ) let K G m ( χ, g , s ) denote the C [ G ]-v alued function: K G m ( χ, g , s ) = χ (( xz ) f )(( xz ) − 1 f ) G | xz 2 | s X ( v, w ) ∈ U ( g ) χ 0 (( v xz ) f ) v G ∞ | ( v xz ) ∞ | − 2 s THE RIGID ANAL YTICAL REGULA TO R AND DRINFELD MODULAR CUR VES 25 where g =  xz y z 0 z  ∈ B ( A ) and s is a complex n um b e r w hen this inﬁnite sum is absolutely conv ergent. The latter ho lds if s with Re( s ) > 1 b ecause the series ab ov e is ma jorated by the s eries E ( g , s ). Finally for ev ery pair of complex num be rs s , t with Re( s ) > 1, Re( t ) > 1 we let H G m ( χ, g , s, t ) denote the C [ G ]-v a lued function on GL 2 ( A ) given by the formula: H G m ( χ, g , s, t ) = L G m ( χ, q − 2 t ) | x | t K G m ( χ,  xz y z 0 z  , s ) if g =  xz y z 0 z  k where k ∈ K ( m ∞ )Γ ∞ , and H G m ( χ, g , s, t ) = 0, other wise. Lemma 6.6. The fol lowing ho lds: ( i ) the function K G m ( χ, g , s ) is left-invariant with r esp e ct t o B ( F ) and right- invariant with r esp e ct t o ( B ( O ) ∩ K ( m )) Z ( A ) , ( iii ) the function H G m ( χ, g , s, t ) is wel l-deﬁne d and it is left-invariant with r esp e ct to B ( F ) and right-invariant wi th r esp e ct to K ( m ∞ )Γ ∞ Z ( A ) . Pro of. The pro of o f the ﬁrst claim is the same as the pro ofs of claim ( i ) o f P rop o - sition 3 .3 and Lemma 6.4. In order to prov e that H G m ( χ, g , s, t ) is well-deﬁned we need to show that | x | t = | a | t and K G m ( χ,  xz y z 0 z  , s ) = K G m ( χ,  ac bc 0 c  , s ) where  xz y z 0 z  ,  ac bc 0 c  ∈ B ( A ) and  xz y z 0 z  ·  ac bc 0 c  − 1 ∈ K ( m ∞ )Γ ∞ . The latter is an immediate cons e quence of cla im ( i ). Similarly the inv aria nce pro p- erties o f H G m ( χ, g , s, t ) claimed ab o ve are ob vious fro m claim ( i ) and its deﬁnition.  Prop osition 6.7. F or every g ∈ GL 2 ( A ) the sum on t he right hand side b elow is absolutely c onver gent and we have: E G m ( χ, g , q − s , q − t ) = ( q − 1) X ρ ∈ B ( F ) \ GL 2 ( F ) H G m ( χ, ρg , s, t ) , when Re( s ) > 1 and Re( t ) > 1 . Pro of. F or every ρ ∈ GL 2 ( F ) the v a lue of H G m ( χ, ρg , s, t ) dep ends only on the left B ( F )-cos et of ρ be cause H G m ( χ, g , s, t ) is left-inv ar ia n t with r espect to B ( F ). Hence the inﬁnite sum on the right hand side ab ov e is well-deﬁned. By gro uping the terms of the absolutely conv ergent s eries o n the left ha nd side we get: E G m ( χ, g , q − s , q − t ) = X ρ ∈ B ( F ) \ GL 2 ( F ) χ (det(( ρg ) f )) − 1 det(( ρg ) − 1 f ) G | det( ρg ) | s + t ·  X ( v, w ) ∈ U ( ρg ) u ∈ F ∗ , (0 ,u ) ∈ V ( ρg ) χ 1 (( v , w ) ρg f ) χ 2 ((0 , u ) ρg f )( v u ) G ∞ · k ( v, w ) ρg ∞ k − 2 s · k (0 , u ) ρg ∞ k − 2 t  . 26 AMBRUS P ´ AL By the Iwasaw a decomp osition we may write g as g = pk , where p =  a b 0 c  ∈ B ( A ) and k =  k 11 k 12 k 21 k 22  ∈ GL 2 ( O ) . Because k ∞ is an is ometry we only have to show that H G m ( χ, g , s, t ) =  χ ( c f )( c − 1 f ) G | ac | t X u ∈ F ∗ (0 ,u ) ∈ V ( g ) χ 2 ((0 , u ) g f ) u G ∞ k (0 , u ) p ∞ k − 2 t  ·  χ ( a f )( a − 1 f ) G | ac | s X ( v, w ) ∈ U ( g ) χ 1 (( v , w ) g f ) v G ∞ k ( v , w ) p ∞ k ) − 2 s  by the ab ov e. T he ﬁr st inﬁnite sum is zer o unless there is a d ∈ A f such that d ( k 21 , k 22 ) f ∈ O 2 f and the la tter is congr uen t to (0 , α ) modulo m O f for some α ∈ ( O f / m ) ∗ . The latter is po ssible exa ctly when k f is in K 0 ( m ). W e may even a ssume that k f is in K ( m ) by changing p , if neces sary . By the deﬁnition o f the s et V ( g ) we also need tha t | ( k 22 ) ∞ | > | ( k 21 ) ∞ | for the ﬁrst sum to b e non-zero. Since ( k 22 ) ∞ ∈ O ∞ we hav e ∞ (( k 21 ) ∞ ) > 0 so k ∞ ∈ Γ ∞ . In this cas e we have (0 , u ) g ∞ ∈ F 2 < for every u ∈ F ∗ automatically so { u ∈ F ∗ | (0 , u ) ∈ V ( g ) } = { u ∈ F ∗ | uc ∈ O f } . Hence the ﬁr st term of the pro duct above is | a/c | t χ ( c f ) L G ( χ, c f , 2 t ). By Lemma 6.4 we know that the latter is | a/c | t ( q − 1) L ( χ, q − 2 t ). On the other ha nd the second term is visibly K G m ( χ, g , s ) beca use U ( g ) = U ( p ) and χ 1 ( f k f ) = χ 1 ( f ) for every f ∈ O 2 f since k f ∈ K ( m ).  Deﬁnition 6.8. Let µ B be the unique left -inv ariant Haa r measur e on the lo cally compact ab elian top ological gro up Z ( A ) \ B ( A ) such that µ B ( Z ( O ) \ B ( O )) is equal to 1. Since this measure is left-in v ar ian t with res pect to the discrete action o f the group Z ( F ) \ B ( F ), it induces a measure on Z ( A ) B ( F ) \ B ( A ), whic h will b e denoted by the same sym b ol by abuse of no tation. The measure µ B has a s imple description. Let µ and µ ∗ be the unique Haar measur e on the loca lly c ompact ab elian top ological g roup A and A ∗ , res pectively , suc h that µ ( O ) and µ ∗ ( O ∗ ) ar e bo th equal to 1 . Since the measures µ and µ ∗ are left-inv ar ia n t with resp ect to the discrete s ubgroups F ⊂ A , and F ∗ ⊂ A ∗ , resp ectively , b y deﬁnition, they induce a measure o n F \ A a nd F ∗ \ A ∗ , resp ectively , which will b e deno ted by the same letter by abuse of notation. Then we hav e Z Z ( A ) B ( F ) \ B ( A ) f  x y 0 1  µ B (  x y 0 1  ) = Z F ∗ \ A ∗ µ ∗ ( x ) Z F \ A f  x y 0 1  µ ( y ) | x | for every Leb esgue-mea surable function f : Z ( A ) B ( F ) \ B ( A ) → C . Lemma 6.9. F or every ψ ∈ A 0 ( m ∞ , C ) t he inte gr ands of the two inte gr als b elow ar e absolutely L eb esgue-inte gr able and Z Z ( A ) GL 2 ( F ) \ GL 2 ( A ) E G m ( χ, g , q − s , q − t ) ψ ( g ) dµ G ( g ) = µ ( m ) Z Z ( A ) B ( F ) \ B ( A ) H G m ( χ, b, s, t ) ψ ( b ) dµ B ( b ) wher e µ ( m ) = ( q − 1) µ G ( Z ( O ) \ K ( m ∞ )Γ ∞ Z ( O )) when Re( s ) > 1 and Re( t ) > 1 . THE RIGID ANAL YTICAL REGULA TO R AND DRINFELD MODULAR CUR VES 27 Pro of. W e may ta lk ab out the Leb esgue-integrability of the in tegrands a bove b e- cause they ar e C [ G ]-v alued functions. By Theor em 2.2.1 in [11 ], pages 25 5 -256, we know that an y c us pidal automor phic form which is inv ariant with resp ect to Z ( A ) has compact suppor t as a function on Z ( A ) GL 2 ( F ) \ GL 2 ( A ). Hence the integral on the left in the equa tion a bov e is absolutely conv ergent a nd we may interc hange the integration a nd the summation in Prop osition 6.7 to get that Z Z ( A ) GL 2 ( F ) \ GL 2 ( A ) E G m ( χ, g , q − s , q − t ) ψ ( g ) dµ G ( g ) = ( q − 1) Z Z ( A ) B ( F ) \ GL 2 ( A ) H G m ( χ, b, s, t ) ψ ( b ) dµ G ( b ) where the measure on Z ( A ) B ( F ) \ GL 2 ( A ) induced b y µ G will be denoted b y the same symbol by the usual abuse of notation. The ma p: π : Z ( A ) B ( F ) \ B ( A ) × Z ( O ) \ K ( m ∞ )Γ ∞ Z ( O ) → Z ( A ) B ( F ) \ GL 2 ( A ) given by the rule ( b, k ) 7→ b k is con tinuous, hence for every Borel- measurable set B ⊆ Z ( A ) B ( F ) \ GL 2 ( A ) the pre-imag e π − 1 ( B ) is a lso Bore l- measurable. Let µ B × µ G denote the direct pro duct of the mea sures µ B and µ G on the direct pro duct Z ( A ) B ( F ) \ B ( A ) × Z ( O ) \ K ( m ∞ )Γ ∞ Z ( O ). Then w e have µ B × µ G ( π − 1 ( B )) = µ G ( B ) for every B ab ov e. Mo r eov er the ma p π maps surjectively onto the s uppor t of H G m ( χ, b, s, t ) as a function o n GL 2 ( A ) as we s a w in Deﬁnition 6.4 so the integral ab o ve is equa l to: ( q − 1) Z Z ( A ) B ( F ) \ B ( A ) dµ B ( b ) Z Z ( O ) \ K ( m ∞ )Γ ∞ Z ( O ) H G m ( χ, bk , s, t ) ψ ( bk ) dµ G ( k ) by F ubini’s theo r em. B y deﬁnition the integrand of the in terior integral is constant on the doma in of in tegratio n. The cla im is now obvious.  Deﬁnition 6. 10. Let τ : F \ A → C ∗ be a non-tr ivial contin uous character and le t d b e an idele such that D = d O , where D is the O -mo dule deﬁned a s D = { x ∈ A | τ ( x O ) = 1 } . It is well-kno wn the linear equiv alence class of the diviso r of d is the anti-canonical class. More over for every η ∈ F ∗ the map x 7→ τ ( η x ) is ano ther non-triv ial contin- uous ho momorphism. Therefore b y choos ing an a ppropriate character τ , we may assume that d is any idele o f deg ree tw o, as every such div is or is linea rly equiv alent to the anti-canonical class. In particular we ma y assume that d = π 2 where π ∈ F ∞ is a unifor mizer. F or every r ⊳ A non-zero ideal let S ( m , r ) deno te the set: S ( m , r ) = { 0 6 = q ⊳ A | ( m , q ) = 1 , q | r } . Moreov er for every G as in Deﬁnition 6.3 let σ G m ( χ, r , x ) ∈ C [ G ][ x ] denote the po lynomial given by the formula: σ G m ( χ, r , x ) = X q ∈ S ( m , r ) χ ( q )( q G ) − 1 x deg( q ) . 28 AMBRUS P ´ AL Prop osition 6 .11. F or e ach c ompl ex s with Re( s ) > 1 we have: Z F \ A K G m ( χ,  x d y 0 1  , s ) τ ( − y ) dµ ( y ) = ( q − 1) | x d | 1 − s σ G m ( χ, x f , q 1 − 2 s ) , if the divisor of x is eﬀe ctive, and it is zer o, otherwise. Pro of. The int egra l above is well-deﬁned b ecause the in tegrand is F -inv ariant by claim ( i ) of Lemma 6.6. Note that we have u 6 = 0 for every 0 6 = ( u, v ) ∈ F 2 such that χ 0 (( ux d ) f ) 6 = 0. Therefor e by grouping the terms in the inﬁnite sum of Deﬁnition 6.5 we ge t the following identit y: K G m ( χ,  x d y 0 1  , s ) = χ (( x d ) f )(( x d ) − 1 f ) G | x d | s X v ∈ F X u ∈ F ∗ ( u, 0) ∈ U ( „ x d ( y + v ) 0 1 « ) χ 0 (( ux d ) f ) u G ∞ | ( ux d ) ∞ | − 2 s . Hence Z F \ A K G m ( χ,  x d y 0 1  , s ) τ ( − y ) dµ ( y ) = χ (( x d ) f )(( x d ) − 1 f ) G · | x d | s X u ∈ F ∗ u ( x d ) f ∈O f χ 0 (( ux d ) f ) u G ∞ | ux d | − 2 s ∞ Z uy f ∈O f | y ∞ |≤| x d | ∞ τ ( − y ) dµ ( y ) by in terchanging summation and integration. F or e very u ∈ F ∗ the doma in of int egratio n of t he in tegral above is a direct pro duct of the sets u − 1 f O f ⊂ A f and x d O ∞ ⊂ F ∞ . The integral itself is non-zero if and o nly if the pro duct set ab o ve lies in the kernel of τ . The latter is equiv a len t to the conditions ( u d ) − 1 f = u − 1 f ∈ O f and ∞ ( x ) ≥ 0. In this case the integral is equa l to: µ ( u − 1 f O f × x d O ∞ ) = | u | − 1 µ ( O f × ux d O ∞ ) = | u x d | ∞ . Let T ( m , x ) denote the set: T ( m , x ) = { u ∈ F ∗ | ( ux ) f ∈ O f , u − 1 f ∈ O f } . By the ab ov e the left hand s ide of the equa tion in the claim ab ov e is equal to: χ ( x f ) | x d | 1 − s X u ∈ T ( m ,x ) χ 0 (( ux ) f )(( ux ) − 1 f ) G | ( ux ) f | 2 s − 1 when ∞ ( x ) ≥ 0, a nd it is zero , otherwise. The set T ( m , x ) is empty when x f is not an element of O f . Therefor e the expression a bove is zero unles s the divis o r of x is eﬀective. Note that in the latter case for every u ∈ T ( m , x ) the num b e r χ 0 (( ux ) f ) is zero unless the divis o r of ( ux ) f is a n e le men t of S ( m , x f ). On the other hand every element of S ( m , x f ) is the divisor o f an idele of the form ( ux ) f for some u ∈ T ( m , x ) and u is unique up to fac tor in F ∗ q . Note that the sum a bove is inv aria n t in the v a riable x with resp ect to the action of O ∗ f . Hence we may assume that x f = η f for some η ∈ F ∗ . In this case w e ha ve χ ( x f ) = 1 and χ ( q ) = χ 0 (( ux ) f ) for ev ery u ∈ T ( m , x ) where q is the divis o r of ( ux ) f . The claim is now clear .  THE RIGID ANAL YTICAL REGULA TO R AND DRINFELD MODULAR CUR VES 29 Notation 6.1 2. Recall that w e ca ll tw o divisor s r and s on X relatively prime if their s upp ort is disjoin t. F or ev ery ψ ∈ A 0 ( m , C ) let ψ ∗ : Div( X ) → C denote the F our ie r co eﬃcients of ψ whos e existence was established in Prop osition 1 of Chapter I II in [25], page 21, prov ed on pages 19-20 of [2 5 ]. Recall that a function f : Div( X ) → R is called multiplicativ e, where R is a commutativ e ring with unity , if it is zero on no n-eﬀectiv e divisors , f (1) = 1 and for every pair of rela tiv ely pr ime divisors r and s we hav e f ( rs ) = f ( r ) f ( s ). (Similarly an R -v alued function on the set of non- zero ideals of A is called mult iplicative if it satisﬁes the last tw o prop erties of the previous deﬁnition.) Let us recall the situation considered in the int ro duction. Let E b e an elliptic curve deﬁned ov er F which has split multiplicativ e reduction at ∞ . By assumption the co nductor of E is o f the for m n ∞ wher e n is a n eﬀective divisor which is supp orted in the complement of ∞ in X . Let ψ ∗ E denote the unique multiplicativ e function in to the multiplicativ e se mig roup o f Q such that ψ ∗ E ( x n ) is the same as in 1.6 for each natur a l num ber n and ea c h closed po in t x on X . A c us pidal harmonic form φ E ∈ H 0 ( n , Q ) is called a normalized Heck e eigenform a tta ched to E is if its F ourier coe ﬃcie nt φ ∗ E ( q ) is equal to | q | ψ ∗ E ( q ) for every eﬀective divisor q . The following prop osition is an easy conseque nce of the La nglands corresp on- dence: Prop osition 6 .13. Ther e is a unique normalize d He cke eigenform attache d to E . Pro of. The o nly no t entirely obvious fact is that the nor malized Heck e eigenform has v alues in Q , see for example the pro of o f Pr opo sition 3.3 in [21].  Theorem 6.1 4. Assume that n divides m . Then for every Re( s ) > 1 and Re( t ) > 1 we have: Z Z ( A ) GL 2 ( F ) \ GL 2 ( A ) E G m ( χ, g , q − s , q − t ) φ E ( g ) dµ G ( g ) = ( q − 1) µ ( m ) L G m ( χ, q − 2 t ) | d | t − s 1 − q s − t − 1 X 0 6 = r ⊳A | r | 1+ t − s σ G m ( χ, r , q 1 − 2 s ) ψ ∗ E ( r ) . Pro of. By Lemma 6.9 and the description o f the measure µ B as a do uble integral at the end of Deﬁnition 6.8 we know that the int egral on the left hand side o f the equation ab ov e is equal to: µ ( m ) L G m ( χ, q − 2 t ) Z F ∗ \ A ∗ µ ∗ ( x ) Z F \ A | x | t K G m ( χ,  x y 0 1  , s ) φ E (  x y 0 1  ) d µ ( y ) | x | . 6 . 1 4 . 1 Using the F ourier expansion of φ E = φ E we get from P rop o sition 6.1 1 that Z F \ A K G m ( χ,  x d y 0 1  , s ) φ E (  x d y 0 1  ) dµ ( y ) = ( q − 1) X η ∈ F ∗ | η x d | 1 − s σ G m ( χ, ( ηx ) f , q 1 − 2 s ) φ ∗ E ( η x ) . 30 AMBRUS P ´ AL By plugging the eq ua tion a bov e into the double integral in 6.14.1 we get that the latter is equal to: ( q − 1) Z A ∗ | x d | t − s σ G m ( χ, x f , q 1 − 2 s ) φ ∗ E ( x ) dµ ∗ ( x ) if we also in terchange the summation in the index η and the in tegration. The int egrand ab ov e is constant on the co sets of the subg roup O ∗ ⊂ A ∗ hence the int egral is equal to the inﬁnite sum: | d | t − s X 0 6 = r ⊳A k ∈ N | r ∞ k | t − s σ G m ( χ, r , q 1 − 2 s ) φ ∗ E ( r ∞ k ) = | d | t − s 1 − q s − t − 1 X 0 6 = r ⊳A | r | 1+ t − s σ G m ( χ, r , q 1 − 2 s ) ψ ∗ E ( r ) , where we also used that the function ψ ∗ E is multiplicativ e and ψ ∗ E ( ∞ k ) = 1 for every k ∈ N .  7. An ∞ -adic analogue of Beilinson’s theorem Let R be a n arbitra ry commutativ e r ing with unity . Let ∗ : R [[ t ]] × R [[ t ]] → R [[ t ]] denote the map given by the rule: ( X n ∈ N a n t n ) ∗ ( X n ∈ N b n t n ) = X n ∈ N a n b n t n . Lemma 7.1. We have: 1 (1 − α 1 t )(1 − β 1 t ) ∗ 1 (1 − α 2 t )(1 − β 2 t ) = 1 − α 1 β 1 α 2 β 2 t 2 (1 − α 1 α 2 t )(1 − α 1 β 2 t )(1 − β 1 α 2 t )(1 − β 1 β 2 t ) for every α 1 , β 1 ∈ R and α 2 , β 2 ∈ R . Pro of. W e may a ssume that R = Z [ x 1 , y 1 , x 2 , y 2 ] and α i = x i , β i = y i for i = 1 , 2 without the lo ss of genera lity . Note that α i − β i (1 − α i t )(1 − β i t ) = α i (1 − α i t ) − β i (1 − β i t ) for i = 1 and i = 2. Also note that 1 1 − γ 1 t ∗ 1 1 − γ 2 t = 1 1 − γ 1 γ 2 t for every γ 1 , γ 2 ∈ R by deﬁnition. Bec a use the map ∗ is R -bilinea r we have ( α 1 − β 1 )( α 2 − β 2 ) 1 (1 − α 1 t )(1 − β 1 t ) ∗ 1 (1 − α 2 t )(1 − β 2 t ) =  α 1 1 − α 1 t − β 1 1 − β 1 t  ∗  α 2 1 − α 2 t − β 2 1 − β 2 t  = ( α 1 − β 1 )( α 2 − β 2 )(1 − α 1 β 1 α 2 β 2 t 2 ) (1 − α 1 α 2 t )(1 − α 1 β 2 t )(1 − β 1 α 2 t )(1 − β 1 β 2 t ) by the a bove. Since our assumption a b ov e implies that α i − β i is not a ze ro divis or in R for i = 1, 2 the claim is now clear.  THE RIGID ANAL YTICAL REGULA TO R AND DRINFELD MODULAR CUR VES 31 Notation 7.2. Let us consider the situation describ ed in Deﬁnition 1.6 . The Galois representation χ co rresp onds to a Dirichlet character of conductor m desc r ibed in Deﬁnition 6.1 b y class ﬁeld theory if a n embedding of K in to the ﬁeld of co mplex nu mbers is also provided. W e let χ deno te this Dirichlet character, to o. Mo reov er the proﬁnite completion of the group F ∗ \ A ∗ / O ∗ f and G ∞ are c anonically is o morphic by cla s s ﬁeld theory . In particular there is a bijectiv e corres p ondence b etw een the ﬁnite quotients of these gro ups. These tw o sets ar e going to b e identiﬁed in all tha t follows. F or every eﬀective divis or d on X let L d ( E , x ) be the L -function: L d ( E , t ) = L ( X ( d ) , ρ, t ) ∈ C [ t ] where we contin ue to use the notation intro duced in the pro of o f Pr opo sition 2.4 . Prop osition 7 .3. We have: L m ∞ ( E , t ) L G m ( E , χ, xt ) L G m ( χ, q xt 2 ) = X 0 6 = q ⊳A ( q , m )=1 ψ ∗ E ( q ) σ G m ( χ, q , x ) t deg( q ) . Pro of. Note that the l -adic representation ρ is unramiﬁed at every prime ideal q ⊳ A which do es not divide m ther efore the lo cal factor L q ( E , t ) o f the Hasse-W eil L -function of E at q ca n b e written as L q ( E , t ) = 1 (1 − α ( q ) t deg( q ) )(1 − β ( q ) t deg( q ) ) , where α ( q ) and β ( q ) a re complex n um ber s such that α ( q ) + β ( q ) = ψ ∗ E ( q ) a nd α ( q ) · β ( q ) = q deg( q ) . On the other hand it is clear from the deﬁnition of σ G m ( χ, q , x ) that the la tter is a K [ G ][ x ]-v alued m ultiplicative function on the set o f non-zero ideals of A . Therefore the pow er series in both sides of the equation in the claim ab o ve ar e Euler pro ducts, that is, the left hand s ide and the right hand side o f the equation ab ov e is equal to: Y q ∈| X | q 6∈ supp ( m ∞ ) A q ( x, y ) and Y q ∈| X | q 6∈ supp ( m ∞ ) B q ( x, y ) , resp ectively , where A q ( x, y ) = 1 − χ ( q )( q G ) − 1 ( q xt 2 ) deg( q ) (1 − α ( q ) t deg( q ) )(1 − β ( q ) t deg( q ) ) · 1 (1 − α ( q ) χ ( q )( q G ) − 1 ( xt ) deg( q ) )(1 − β ( q ) χ ( q )( q G ) − 1 ( xt ) deg( q ) ) and B q ( x, y ) = ∞ X n =0 ψ ∗ E ( q n ) σ G m ( χ, q n , x ) t deg( q ) n for every q ∈ | X | such that q 6∈ supp ( m ∞ ) b y the a bov e. Clear ly it is suﬃcien t to prov e that for every q the factor s o f these E uler pro ducts at q are equa l. But the latter follows at once from Lemma 7.1 a nd the fact that ∞ X n =0 σ G m ( χ, q n , x ) t deg( q ) n = 1 (1 − t deg( q ) )(1 − χ ( q )( q G ) − 1 ( xt ) deg( q ) ) .  32 AMBRUS P ´ AL Theorem 7 .4. We have: h E G m ( χ, g , x, y ) , φ E i = ( q − 1) µ ( m )  x y  2 L m ( E , y q x ) L G m ( E , χ, xy ) . Pro of. By deﬁnition b oth sides of the equation a bov e are elements of the ring C [ G ][[ x, y ]][ x − 1 , y − 1 ]. But in fa ct the left and the right hand sides are elements of the ring C [ G ][ x, y , x − 1 , y − 1 ] b y P r opo sitions 3.5 and 2.4, res p ectively . W e also know that after we substitute q − s and q − t int o x and y , res pectively , b oth sides of the equation a b ov e b ecome abs olutely c o n vergen t when Re( s ) > 1 a nd Re( t ) > 1 . Therefore it will b e suﬃcient to prov e that they are eq ual after these substitutions by the unique c on tin uation of holomo rphic functions. Since d is the anticanonical class, its deg ree is tw o, so the int egral h E G m ( χ, g , q − s , q − t ) , φ E i ca n b e r ewritten a s the inﬁnite sum: ( q − 1) µ ( m ) L G m ( χ, q − 2 t ) | d | t − s 1 − q s − t − 1 X 0 6 = r ⊳A | r | 1+ t − s σ G m ( χ, r , q 1 − 2 s ) ψ ∗ E ( r ) = ( q − 1) µ ( m ) q 2 s − 2 t L m ( E , q 1+ s − t ) L G m ( E , χ, q − s − t ) by Theorem 6 .14 and Prop osition 7.3. The cla im is now clea r.  Notation 7.5. It is clea r from Theorem 5.4 that the irreducible co mponents of the curve Y ( m ) F ∞ are in a bijective corr espondenc e with the set: GL 2 ( F ) \ GL 2 ( A f ) / K f ( m ) of double cosets. In fact for a double coset represented by an element g ∈ GL 2 ( A f ) the corresp onding co nnected co mponent is the imag e of { g } × Ω under the uni- formization map of Theorem 5.4. There fore the rule which asso ciates χ (deg ( g )) − 1 to the irreducible comp onent corresp onding to the double coset repre sen ted by the element g ∈ GL 2 ( A f ) gives rise to a well-deﬁned K -v alued function o n the irr e - ducible co mponents of the curve Y ( m ) F ∞ . Actually this function is in v a riant under the action o f the abso lute Galois gro up of the extension L of F we in tro duced after Prop osition 1.9 hence the function ab ov e is an algebra ic cycle on Y ( m ) L of co-dimension zero with co eﬃcien ts in K . F or every irre ducible comp onent C of Y ( m ) L we let χ − 1 ( C ) denote the co eﬃcient of C in this algebr aic cycle. Deﬁnition 7.6. F or every C , D ∈ Z [ V m ] 0 let κ m ( C, D ) denote the element: ǫ m ( C ) ⊗ ǫ m ( D ) ∈ H 2 M ( Y ( m ) , Z (2)) where w e use the notation of Lemma 5.2 . By Prop osition 5.5 the pull-bac k of κ m ( C, D ) with r e spect to the uniformiza tion map of Theorem 5.4 is the element of K 2 ( GL 2 ( A f ) × Ω) introduced in Deﬁnition 4.9 which is denoted by the s ame symbol hence our new no ta tion will not cause any confusion. Clearly κ m ( C, D ) is linear in the v ariables C and D . Le t the s a me s y m bo l denote by abuse o f nota tio n the unique ∆-bilinear extension: κ m ( · , · ) : ∆[ V m ] 0 × ∆[ V m ] 0 → H 2 M ( Y ( m ) , ∆(2)) of this pairing. Let κ m ( χ ) denote the unique element of H 2 M ( Y ( m ) L , ∆(2)) whose restriction to ev ery irreducible co mponent C of Y ( m ) L is χ − 1 ( C ) κ m ( χ 1 , χ 2 ) | C . THE RIGID ANAL YTICAL REGULA TO R AND DRINFELD MODULAR CUR VES 33 Theorem 7 .7. We have h{ κ m ( χ ) } , φ E i = b ( E , m ) L ( E , q − 1 ) L m ( E , χ ) ′ in F ∗ ∞ ⊗ K wher e b ( E , m ) ∈ K ∗ . Pro of. The Hecke eigenfor m φ E is lo cally consta n t and has compact suppo rt as a function on GL 2 ( F ) \ GL 2 ( A ) hence it takes only ﬁnitely ma n y v alues. In par ticular there is a p ositive n ∈ N s uch that nφ E takes integer v alues. Let C and D ∈ ∆[ V m ] 0 be t w o functions such that the function χ (det( g f )) − 1 E G m ( C, D , · , x, y ) is right Z ( A ) K ( m ∞ )-inv ar ian t. Then the integral: P G E ( C, D , x, y ) = n h χ (det( g f )) − 1 E G m ( C, D , · , x, y ) , φ E i ∈ ∆[[ x, y ]]( x − 1 , y − 1 ) is well-deﬁned and it is in fact an elemen t of ∆[ x, y , x − 1 , y − 1 ] acco rding to P rop o- sition 3.5 . Therefore we may ev aluate P G E ( C, D , x, y ) at x = y = 1. As we alr eady noted at the end of the pro of o f the limit fo r m ula 4.10 we have: E G m ( C, D , g ) = ( − 1) G E G m ( D , C , g Π) for ev ery g ∈ GL 2 ( A ) (using the no tation of lo c. cit. ). Ther efore we get the equality P G E ( C, D , 1 , 1) = − ( − 1) G P G E ( D , C , 1 , 1) b ecause φ E is harmo nic and the Petersson- pro duct is tr a nslation-inv ariant. The elements P G E ( C, D , 1 , 1) satisfy the o b vious compatibility: let P E ( C, D ) denote their limit. Then P E ( C, D ) ∈ ∆[[ G ∞ ]] lies in I by Prop osition 3.8. Mo reov er we hav e: 2 P E ( χ 1 , χ 2 ) ′ = P E ( χ 1 , χ 2 ) ′ /P E ( χ 2 , χ 1 ) ′ = h{ κ m ( χ ) } , φ E i n ∈ G ∞ ⊗ K by the K roneck er limit formula 4.1 0 using the nota tio n we introduced in Deﬁnition 6.1. Therefor e we get that h{ κ m ( χ ) } , φ E i = q − 1 2 µ ( m ) L m ( E , q − 1 ) L m ( E , χ ) ′ using Theor em 7.4. Since L m ( E , q − 1 ) = a ( E , m ) L ( E , q − 1 ) for some a ( E , m ) ∈ Q ∗ the claim follows.  The function ﬁeld a nalogue of the Shimura-T a niyama-W eil conjecture claims the following: Theorem 7 .8. Ther e is a non-trivial map π : X 0 ( n ) → E deﬁne d over F . Pro of. Although this theorem is certainly very well known and have bee n stated in the literature sev eral tim es already , in some cases with an indication of pro of, its co mplete pro of hav e not been written y et, which w e will prese nt now for the sake of rec o rd. Let l b e a prime diﬀer en t from p and le t V F denote the ba se change of an y algebraic v ariet y V o v er F to the separable closure F of the ﬁeld F . The Gal( F | F )-mo dule H 1 ( E F , Q l ) is abs olutely irr e ducible, b ecause the curve E is not isotrivial. By the global Langlands corresp ondence for function ﬁelds (see [17], prov ed in this c a se in [3] already) there is a corresp onding cuspidal auto morphic representation π of GL 2 ( A ). Let ω denote the g r¨ o ssencharacter of F which assig ns to each id ´ ele its no rmalized absolute v alue. By the co mpatibilit y of the loca l and 34 AMBRUS P ´ AL global Langlands corres pondences the ∞ -adic comp onent of π ⊗ ω − 1 is isomorphic to the Steinberg representation. Also the conducto r of π is n ∞ , so there is a non-zero automorphic form φ o f lev el n ∞ and trivial central character which is an element of π ⊗ ω − 1 . By the a b ov e φ is also harmonic, so b y the ma in theorem of [4] there is a n abso lutely irreducible Gal( F | F ) submo dule o f H 1 ( X 0 ( n ) F , Q l ) corresp onding to the repr e s en tation π . This representation must be isomorphic to H 1 ( E F , Q l ), bec ause the Langlands co rresp ondence is a bijection. By Zarhin’s theore m (see [26] a nd [27 ]) there is a homo morphism from the Jacobian of X 0 ( n ) o n to E which induces this isomorphism. W e get the map of the cla im by comp osing the map ab o ve with a ﬁnite-to-one map fro m X 0 ( n ) into its Jacobian.  Our next goal is to give an explicit descriptio n of the r elation betw een the mo du- lar parameter ization of the elliptic curve E in the theo rem a bov e and the normalized Heck e eigenfo r m a ttac hed to E , due to Gekeler and Reversat [9]. Deﬁnition 7.9. Let deg( u ) : GL 2 ( F ∞ ) → Z denote the unique function for every holomorphic function u : Ω → C ∗ ∞ such that the reg ulator { c ⊗ f } introduced in Deﬁnition 4.5 is equal to c deg( u ) for every c ∈ C ∗ ∞ . Then deg( u ) is just the v an der P ut logarithmic der iv ative of u introduced in [5]. Similar ly to the notation w e int ro duced in Deﬁnition 4.7 let deg( u ) : GL 2 ( A ) → Z b e the function given by the formula deg ( g f , g ∞ ) = deg ( u ( g f , · ))( g ∞ ) for each g ∈ GL 2 ( A f ) if u : GL 2 ( A f ) × Ω → C ∗ ∞ is holomorphic in the sec o nd v ariable. Recall that θ : C ∗ ∞ → E ( C ∞ ) denotes the T ate uniformization of E . A theta function attached to E (and the mo dular parameteriza tio n π ) is a function u E : GL 2 ( A f ) × Ω → C ∗ ∞ holomorphic in the s econd v a r iable for each g ∈ GL 2 ( A f ) if it satisﬁes the following prop erties: ( a ) w e hav e u E ( g k , z ) = u E ( g , z ) for each g ∈ GL 2 ( A f ), z ∈ Ω and k ∈ K 0 ( n ) ∩ GL 2 ( A f ), ( b ) the harmonic co chain deg ( u E ) is c E φ E , where c E is a p ositive in teger. ( c ) the dia gram: GL 2 ( A f ) × Ω − − − − → u E C ∗ ∞   y   y θ Y 0 ( n ) − − − − → π E ( C ∞ ) is commutativ e where the vertical ma p o n the left is the uniformization map men tioned in Nota tion 5.10 . Theorem (Gekeler-Rev ersat) 7.10. Ther e is a theta function attache d to E . Pro of. See [9], Section 9.5, pages 86 - 88.  Pro of o f Theorem 1.1 0 . Let κ m , n ( χ ) ∈ H 2 M ( Y 0 ( n ) L , K (2)) denote the push- forward of the elemen t κ m ( χ ) with r espect to the map Y ( m ) → Y 0 ( m ) induced the the forgetful ma p b etw een the functors r e presented by these mo duli curves. Then we have h{ κ m ( χ ) } , ψ i = h{ κ m , n ( χ ) } , ψ i for every ψ ∈ H 0 ( n , Q ) b y the in v a riance theorem (Theorem 3.11 of [22]). Moreover there is a κ m , n ( χ ) ′ ∈ H 2 M ( X 0 ( n ) L , K (2)) such that h{ κ m , n ( χ ) } , ψ i = h{ κ m , n ( χ ) ′ } , ψ i for ev ery ψ as ab ov e by Lemma 5.15. Let C ⊂ X 0 ( n ) × X 0 ( n ) denote the corr espo ndence which is the comp osition of the uniformization map π : X 0 ( n ) → E o f Theorem 7.8 and its gr aph Γ( π ) ⊂ E × X 0 ( n ) considered as a corr espo ndence fro m E to X 0 ( n ). Then the endomorphism J ( C ) : J 0 ( n ) → J 0 ( n ) induced by C is equal to d ( E ) T E where d ( E ) is a non-zero ratio na l THE RIGID ANAL YTICAL REGULA TO R AND DRINFELD MODULAR CUR VES 35 nu mber a nd T E ∈ T ( n ) ⊗ Q is a pro jection o pera tor. Moreov er we hav e T E ( φ E ) = φ E therefore b ( E , m ) L ( E , q − 1 ) L m ( E , χ ) ′ = h{ κ m ( χ ) } , φ E i = h{ κ m , n ( χ ) ′ } , φ E i = h{ κ m , n ( χ ) ′ } , T E ( φ E ) i = h T E { κ m , n ( χ ) ′ } , φ E i = d ( E ) − 1 h{ C ∗ ( κ m , n ( χ ) ′ ) } , φ E i using the self-adjoin tness of the op era to r T E with respe ct to the Petersson pr od- uct in the fourth e q uation and Lemma 5 .12 in the last equation. By the inv ari- ance theo rem (Theorem 3.11 o f [22]) the har monic form { C ∗ ( κ m , n ( χ ) ′ ) } is equal to { π ∗ ( κ m , n ( χ ) ′ ) } deg( u E ) where π ∗ ( κ m , n ( χ ) ′ ) ∈ H 2 M ( E L , K (2)) is the push-for w ard of κ m , n ( χ ) ′ with resp ect to the unifomization π : X 0 ( n ) → E . Hence we hav e L ( E , q − 1 ) L m ( E , χ ) ′ = c ( E ) b ( E , m ) d ( E ) { π ∗ ( κ m , n ( χ ) ′ ) }h φ E , φ E i by the deﬁnition of theta functions. As the Petersson pro duct is po sitiv e deﬁnite restricted to H 0 ( n , Q ) the claim is now obvious.  8. The action of cer t ain correspondences on K 2 In this chapter the no tation used will b e somewha t independent o f the one used in the r est o f the pap er. Notation 8.1. Let l be a prime num b e r and for e very scheme S on which l is inv ertible let c 2 , 2 : H 2 M ( S, Q (2)) → H 2 et ( S, Q l (2)) denote the ´ etale Chern class ma p. Let L b e a ﬁeld complete with respect to a discrete v alua tion and let O denote its v aluation ring. Ass ume that the residue ﬁeld o f O is a ﬁnite ﬁeld of c haracter istic p 6 = l . Let X → Sp e c ( O ) b e a ﬂat, regular , prop er a nd semi-stable scheme ov er Spe c( O ) such that its generic ﬁbe r X is a smo oth, geo metrically irreducible curve o ver Sp ec( L ). Let Y denote the sp ecial ﬁber of X and let ∂ : H 2 M ( X, Q (2 )) → H 1 M ( Y , Q (1)) denote the b oundary map furnished by the lo calization sequence for the pair ( X , Y ). Lemma 8.2. F or every element k ∈ H 2 M ( X, Q (2 )) such that c 2 , 2 ( x ) = 0 we have ∂ ( k ) = 0 . Pro of. Let R denote the r ing of g lobal sections of the sheaf of total quotient rings of O Y . Since the residue ﬁeld of any closed po in t y of Y is a ﬁnite ﬁeld, the homomorphism j ∗ : H 1 M ( Y , Q (1)) → H 1 M (Spec ( R ) , Q (1)) = R ∗ ⊗ Q induced by the natural map j : Sp ec( R ) → Y is injective. Let c 1 , 1 : R ∗ ⊗ Q → H 1 et (Spec ( R ) , Q (1)) 36 AMBRUS P ´ AL be the c o nnecting homo morphism o f the long coho mological exac t se quence at- tached to Kummer’s shor t exa ct seq uence. Let π : e Y → Y be the normaliza tion of Y , and let Div( e Y ) denote the group of diviso rs on e Y . Since the homo morphism R ∗ → Div( e Y ) which a ssigns to every r ∈ R ∗ the divisor of π ∗ ( r ) has a ﬁnite kernel and the group Div( e Y ) is a free ab elian gr oup, the intersection T n ∈ N ( R ∗ ) l n is ﬁnite. Hence the homomorphism c 1 , 1 is injective. Therefor e for every k ∈ H 2 M ( X, Q (2 )) we have ∂ ( k ) = 0 if the equation c 1 , 1 ◦ j ∗ ◦ ∂ ( k ) = 0 holds. L e t K b e the function ﬁeld of the curve X and let i : Sp ec( K ) → X b e the g eneric p o in t. The cla im now follows form the fact that the diagrams: H 2 M ( X, Q (2 )) i ∗ − − − − → H 2 M (Spec ( K ) , Q (2)) ∂ − − − − → R ∗ ⊗ Q c 2 , 2   y c 2 , 2   y c 1 , 1   y H 2 et ( X, Q l (2)) i ∗ − − − − → H 2 et (Spec( K ) , Q l (2)) ∂ − − − − → H 1 et (Spec( R ) , Q l (1)) and H 2 M ( X, Q (2 )) i ∗ − − − − → H 2 M (Spec ( K ) , Q (2)) ∂   y ∂   y H 1 M ( Y , Q (1)) j ∗ − − − − → R ∗ ⊗ Q are comm utative, where the sym bo l ∂ denote the respec tiv e loca lisation ma p ev- erywhere in the diagrams .  Notation 8.3. F o r every smo oth, pro jective, geometrica lly ir reducible cur v e Z deﬁned ov er a ﬁeld K le t Jac( Z ) denote the J acobian o f Z , a s us ual. Mo reov er for every corr espo ndenc e C ⊂ Z × Z let C ∗ : H 2 M ( Z, Q (2)) → H 2 M ( Z, Q (2)) and J ( C ) : Ja c( Z ) → Jac( Z ) denote the endomorphisms induced b y C on H 2 M ( Z, Q (2)) and Jac( Z ), res pectively . Let L a nd X be as in Nota tion 8 .1 and let C ⊂ X × X b e a corres pondence. Lemma 8.4. We have c 2 , 2 ( C ∗ ( k )) = 0 for every k ∈ H 2 M ( X, Q (2 )) if t he endo- morphism J ( C ) is zer o. Pro of. Let X deno te the base change o f X to the sepa rable clo sure L o f L . Note that there is a Hoschsch ield-Serre sp ectral sequence: H i (Gal( L | L ) , H j et ( X , Q l (2))) ⇒ H i + j et ( X, Q l (2)) . Because H 0 et ( X , Q l (2)) = Q l (2) and H 2 et ( X , Q l (2)) = Q l (1) we have H 2 (Gal( L | L ) , H 0 et ( X , Q l (2))) = 0 = H 0 (Gal( L | L ) , H 2 et ( X , Q l (2))) by lo cal cla s s ﬁeld theory . In particular E 2 , 0 ∞ = E 0 , 2 ∞ = 0 for the sp ectral seq ue nce men tioned above. Moreover H 3 (Gal( L | L ) , M ) = 0 for ev ery Gal( L | L )-mo dule M THE RIGID ANAL YTICAL REGULA TO R AND DRINFELD MODULAR CUR VES 37 hence E 3 , 0 2 = 0. T he r efore we hav e E 1 , 1 ∞ = E 1 , 1 2 = H 1 (Gal( L | L ) , H 1 ( X , Q l (2))) and so there is an isomor phism: ι X : H 2 et ( X, Q l (2)) → H 1 (Gal( L | L ) , H 1 ( X , Q l (2))) . Let T l ( C ) : H 1 ( X , Q l (2))) → H 1 ( X , Q l (2))) be the endomorphism of H 1 ( X , Q l (2))) induced by C . The map T l ( C ) induces a homomorpism on cohomology : T l ( C ) ∗ : H 1 (Gal( L | L ) , H 1 ( X , Q l (2))) → H 1 (Gal( L | L ) , H 1 ( X , Q l (2))) by functoriality . The n we have the following co mmutative dia g ram: H 2 M ( X, Q (2 )) c 2 , 2 − − − − → H 2 et ( X, Q l (2)) ι X − − − − → H 1 (Gal( L | L ) , H 1 ( X , Q l (2))) C ∗   y C ∗   y T l ( C ) ∗   y H 2 M ( X, Q (2 )) c 2 , 2 − − − − → H 2 et ( X, Q l (2)) ι X − − − − → H 1 (Gal( L | L ) , H 1 ( X , Q l (2))) . Since w e have H 1 ( X , Q l (2)) = H 1 (Jac( X ) , Q l (2)) the map T l ( C ) is zero when the endomorphism J ( C ) is. In this case T l ( C ) ∗ is also z ero, so the claim is now clear .  Notation 8. 5. As in the introductio n, let F denote the function ﬁeld of X , where the latter is a geometrically co nnected smo oth pro jective curv e deﬁned over the ﬁnite ﬁeld F q of character is tic p . Let Z be a smo oth, pro jective, geometrically irreducible curve deﬁned ov er F and le t C ⊂ Z × Z be a corresp ondence. Assume that Z has a ﬂat, regula r, pro per a nd semi- stable mo del Z → X ov er X . Prop osition 8.6. Assume that the endomorphism J ( C ) is zero . Then for every k ∈ H 2 M ( Z, Q (2)) the element C ∗ ( k ) ∈ H 2 M ( Z, Q (2)) lies in the image of the n atur a l map H 2 M ( Z , Q (2)) → H 2 M ( Z, Q (2)) . Pro of. F or every closed point x of X let Z x denote the ﬁber of Z at x . By the exactness of the lo calisation s equence it will b e suﬃcient to show that the image of C ∗ ( k ) under the b oundar y map: ∂ : H 2 M ( Z, Q (2)) → H 1 M ( Z x , Q (1)) is zero for every x as ab ov e. 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P´ al, On the k e rnel and the image of the rig id analytic r e gula tor in p ositi v e char acte ristic , to app ear, Publ. Res. Inst. Math. Sci. (2009). 24. N. Schapp ac her and A. J. Scholl, Bei linson ’s the or e m on mo dular curves , Beilinson’s conjec- ture on sp ecial v alues of L -functions, Academic Pr ess, 1988, pp. 273–304. 25. A. W eil , Dirichlet series and automorphic forms , Springer, Berlin-H eidelberg-New Y ork, 1971. 26. J. G. Zarhin, Iso g e nies of ab e lian varieties over ﬁelds of ﬁnite char acteristi cs , M ath U SSR-Sb. 24 (1974), 451–461. 27. J. G. Zar hi n, A rema rk on endomorphisms of ab elian varieties over function ﬁelds of ﬁnite char act e ristic , Math USSR-Izv. 8 (1974), 477-480. D epar tme nt of M athe matic s, 18 0 Q ue en’ s G ate, Im p eri al C ol leg e, L on do n S W 7 2 A Z, U n i te d K i ngd o m E-mail addr e ss : a.pal @imperial .ac.uk

The rigid analytical regulator and K_2 of Drinfeld modular curves

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