Introduction to Number Theory
Introduction to Number Theory MATH 110
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Annals of Mathematics7 163 2006 3017346 Elliptic units for real quadratic elds By HENRI DARMON and SAMIT DASGUPTA Contents 1 A review of the classical setting 2 Elliptic units for real quadratic elds 21 p adic measures 22 Double integrals 23 Splitting a two cocycle 24 The main conjecture 25 Modular symbols and Dedekind sums 26 Measures and the Bruhat Tits tree 27 Inde nite integrals 28 The action of complex conjugation and of Up 3 Special values of zeta functions 31 The zeta function 32 Values at negative integers 33 The p adic valuation 34 The Brumer Stark conjecture 35 Connection with the Gross Stark conjecture 4 A Kronecker limit formula Measures associated to Eisenstein series 42 Construction of the p adic L function 43 An explicit splitting of a two cocycle 44 Generalized Dedekind sums 45 Measures on Zp x Zp 46 A partial modular symbol of measures on Zp x Zp 47 From 2 x 2 to X 48 The measures In and P invariance Introduction Elliptic units7 which are obtained by evaluating modular units at quadratic imaginary arguments of the Poincar upper half plane7 provide us with a rich source of arithmetic questions and insights They allow the analytic construc tion of abelian extensions of imaginary quadratic elds7 encode special values 302 HENRI DARMON AND SAMIT DASGUPTA of zeta functions through the Kronecker limit formula and are a prototype for Stark s conjectural construction of units in abelian extensions of number elds Elliptic units have also played a key role in the study of elliptic curves with complex multiplication through the work of Coates and Wiles This article is motivated by the desire to transpose the theory of elliptic units to the context of real quadratic elds The classical construction of elliptic units does not give units in abelian extensions of such elds1 Naively one could try to evaluate modular units at real quadratic irrationalities but these do not belong to the Poincare upper half plane H We are led to replace H by a p adic analogue Hp P1Cp ilP lQp equipped with its structure of a rigid analytic space Unlike its archimedean counterpart Hp does contain real quadratic irrationalities generating quadratic extensions in which the rational prime p is either inert or rami ed Fix such a real quadratic eld K C C17 and denote by K1 its completion at the unique prime above p Chapter 2 describes an analytic recipe which to a modular unit 04 and to 739 6 HP K associates an element ua 739 6 K5 and conjectures that this element is a p unit in a speci c narrow ring class eld of K depending on 739 and denoted H7 The construction of uoz739 is obtained by replacing in the de nition of Stark Heegner points77 given in Dar1 the weight 2 cusp form attached to a modular elliptic curve by the logarithmic derivative of a an Eisenstein series of weight 2 Conjecture 214 of Chapter 2 which formulates a Shimura reciprocity law for the p units uoz739 suggests that these elements display the same behavior as classical elliptic units in many key respects Assuming Conjecture 214 Chapter 3 relates the ideal factorization of the p unit ua 739 to the Brumer Stickelberger element attached to H Thanks to this relation Conjecture 214 is shown to imply the primeto 2 part of the Brumer Stark conjectures for the abelian extension H K7an implication which lends some evidence for Conjecture 214 and leads to the conclusion that the p units uoz739 are essentially the p adic Gross Stark units which enter in Gross s p adic variant Gr1 of the Stark conjectures in the context of ring class elds of real quadratic elds Motivated by Gross s conjecture Chapter 4 evaluates the p adic logarithm ofthe norm from Kp to Q17 of ua 739 and relates this quantity to the rst deriva tive of a partial p adic zeta function attached to K at s 0 The resulting formula stated in Theorem 41 can be viewed as an analogue of the Kronecker limit formula for real quadratic elds In contrast with the analogue given in Ch ll 3 of Sie1 see also Za Theorem 41 involves non archimedean in 1Ebrcept when the extension in question is contained in a ring class eld of an auxiliary imaginary quadratic eld an exception which is the basis for Kronecker s solution to Pell s equation in terms of Values of the Dedekind nfunction ELLIPTIC UNITS FOR REAL QUADRATIC FIELDS 303 tegration and p adic rather than complex zeta values Yet in some ways it is closer to the spirit of the original Kronecker limit formula because it in volves the logarithm of an expression which belongs at least conjecturally to an abelian extension of K Theorem 41 makes it possible to deduce Gross s p adic analogue of the Stark conjectures for HTK from Conjecture 214 It should be stressed that Conjecture 214 leads to a genuine strengthening of the Gross Stark conjectures of Gr1 in the setting of ring class elds of real quadratic elds and also of the re nement of these conjectures proposed in Gr2 Indeed the latter exploits the special values at s 0 of abelian L series attached to K as well as derivatives of the corresponding p adic zeta functions to recover the images of Gross Stark units in K X where 5 is the topological closure in K of the unit group of K Conjecture 214 of Chapter 2 proposes an explicit formula for the GrossStark units themselves It would be interesting to see whether other instances of the Stark conjectures both classical and p adic are susceptible to similar re nements2 1 A review of the classical setting Let H be the Poincare upper half plane and let P0N denote the standard Hecke congruence group acting on H by Mobius transformations Write Y0N and X0N for the modular curves over Q whose complex points are identi ed with HP0N and HP0N respectively where H H U lP lQ is the extended upper half plane A modular unit is a holomorphic nowherevanishing function on HP0N which extends to a meromorphic function on the compact Riemann surface X0NC A typical example of such a unit is the modular function ArANr More generally let DN be the free Z module generated by the formal Z linear combinations of the positive divisors of N and let DR be the submodule of linear combinations of degree 0 We associate to each element 6 Z ndd 6 DR the modular unit 1 A5r H AltdTm39 le Fix such a modular unit 04 A5 on P0N lts level N will remain xed from now on Let M0N C M2Z denote the ring of integral 2 x 2 matrices which are upper triangular modulo N Given 739 E H its associated order in M0N denoted 97 is the set of matrices in M0 N which x 739 under Mobius trans 2In a purely archimedean context recent work of Ken and Sczech on the Stark conjectures for a complex cubic eld suggests that the answer to this question should be yes 304 HENRI DARMON AND SAMIT DASGUPTA formations 2 0 ZgtEM0N suchthat a7b072d7 This set of matrices is identi ed with a discrete subring of C by sending the 1 matrix lt f d gt to the complex number 0739 1 Hence 9 is identi ed either with Z or with an order in an imaginary quadratic eld K Let 9 be such an order of discriminant 7D relatively prime to N De ne 7750 739 E H such that O 2 O This set is preserved under the action of P0N by Mobius transformations and the quotient HOP0N is nite lf 739 u 1 iv belongs to H0 then the binary quadratic form 490 y UAW W95 i y of discriminant 74 is proportional to a unique primitive integral quadratic form denoted 3 Q7xy Amz Bzy Cyz with A gt 0 Since D is relatively prime to N we have NlA and B2 7 4A0 7D We introduce the invariant 4 uaT 047 The theory of complex multiplication cf KL Ch 9 Lemma 11 and Ch 11 Th 12 implies that ua 739 belongs to an abelian extension of the imaginary quadratic eld K QT More precisely class eld theory identi es PicO with the Galois group of an abelian extension H of K the so called ring class eld attached to 9 Let OH denote the ring of integers of H lf 739 belongs to H0 then 5 uoz739 belongs to OH1NX and 6 a 71ua739 belongs to 0 for all a E GalHK Let 7 rec PicOgtGalHK denote the reciprocity law map of global class eld theory which for all prime ideals p f D of K sends the class of p O O to the inverse of the F robenius element at p in GalHK One disposes of an explicit description of the action of GalHK on the uoz739 in terms of To formulate this description ELLIPTIC UNITS FOR REAL QUADRATIC FIELDS known as the Shimura reciprocity law7 it is convenient to denote by Q N the set of homothety classes of pairs A17 A2 of lattices in C satisfying 8 A1 3 A2 and A1A2 ZNZ Let x H m denote the nontrivial automorphism of GalKQ There is a natural bijection I from 9N to Hl ON7 de ned by sending z A17 A2 6 9N to the complex number 9 190w1w27 where wl mg is a basis of A1 satisfying 10 lmw1w2 7 wiwg gt 07 and A2 ltNw17w2gt A point r E H O K belongs to 1QNK7 where 11 GNU AhAg EQN with A17A2 CKKX Given an order 9 of K denote by bxO the set of A17 A2 6 GNU such that O is the largest order preserving both A1 and A2 Note that 19N0 HOP0N Any element a E PicO acts naturally on bxO by translation uA1A2 ct1ctl27 and hence also on HOF0N Denote this latter action by 12 a T H aw for u e PicO T e HOP0N Implicit in the de nition of this action is the choice of a level N7 which is usually xed and therefore suppressed from the notation Fix a complex embedding HgtC The following theorem is the main statement that we wish to generalize to real quadratic elds THEOREM 11 If 739 belongs to 7 IfOl 0N7 then 7117 739 belongs to HX7 and a 71uozr belongs to 91 for all a E GalHK Furthermore7 13 1404 cw r recu 1uoz7 r7 for all u E Pic0 Let log RgtOgtR denote the usual logarithm The Kronecker limit for mula expresses log luoz rl2 in terms of derivatives of certain zeta functions The remainder of this chapter is devoted to describing this formula in the shape in which it will be generalized in Chapter 4 To any positive de nite binary quadratic form Q is associated the zeta function 14 625 2 0717 70757 mn7oo 306 HENRI DARMON AND SAMIT DASGUPTA where the prime on the summation symbol indicates that the sum is taken over pairs of integers m n different from 00 lf 739 belongs to H0 de ne 15 93 6273 04 T s andisgms le Note that for any le A 62114907 24 5902 Ba y 103427 so that the terms in the de nition of 04 739 s are zeta functions attached to integral quadratic forms of the same discriminant 7D Note also that 04 739 5 depends only on the F0N orbit of 739 The Kronecker limit formula can be stated as follows THEOREM 12 Suppose that 739 belongs to H0 The function 04739 s is holommphic except for a simple pole at s 1 It vanishes at s 0 and 16 04 70 logNormCRuoz739 1 75 Proof The function 75 is known to be holomorphic everywhere except for a simple pole at s 1 Furthermore the rst Kronecker limit formula cf Sie1 Theorem 1 of Ch I 1 states that for all 739 uiv 6 HO the function 75 admits the following expansion near 8 1 lt17 27139 7 47139 1 98 e 71gt 1 5 o 7 510g2 5v710glnr12gt 0lts 71gt where ClLr 01ilogn is Euler s constant and 00 TNT EMT12 H 1 7 eZWimT m1 is the Dedekind n function satisfying 7W MT The reader should note that Theorem I of Ch I of Sie1 is only written down for D 4 the case for general D given in 17 is readily deduced from this The functional equation satis ed by 75 allows us to write its expansion at s 0 as 78 1 i H 21Og5177T12l 8 0827 ELLIPTIC UNITS FOR REAL QUADRATIC FIELDS 307 where FL is a constant which is unchanged when 739 6 H0 is replaced by 1739 with d dividing N It follows that 04 739 0 O and a direct calculation shows that Kan 0 is given by 16 El 2 Elliptic units for real quadratic elds Let K be a real quadratic eld and x an embedding K C R Also x a prime p which is inert in K and does not divide N as well as an embedding K C Cp Let Hp ED1CP ED1QP denote the p adic upper half plane which is endowed with an action of the group P0N and of the larger p arithmetic group P de ned by 18 r Z Z gt e SL2Z1p such that Nth Given 739 6 HP O K the associated order of 739 in M0N1p denoted 9 is de ned by analogy with 2 as the set of matrices in M0N1p which x 739 under Mobius transformations ie d This set is identi ed with a Z1p order in Kiie a subring of K which is a free Z1p module of rank 2 Conversely let D gt 0 be a positive discriminant which is prime to Np and let 9 be the Z1p order of discriminant D Set Hg T 6 HP such that 0 0 This set is preserved under the action of P by Mobius transformations and the quotient HgF is nite Note that the simplifying assumption that N is prime to D implies that the Z1p order O is in fact equal to the full order associated to 739 in M2Z1p Our goal is to associate to the modular unit 04 and to each 739 6 Hz taken modulo the action of F a canonical invariant uoz739 E K behaving just like77 the elliptic units of the previous chapter in a sense that is made precise in Conjecture 214 To begin it will be essential to make the following restriction on a 19 0 Z 5 gt e M0N1p such that aTb CT2dT Assumption 21 There is an element 5 E llDlQ such that 04 has neither a zero nor a pole at any cusp which is P equivalent to 5 Examples of such modular units are not hard to exhibit For example when N 4 the modular unit 20 Oz Az2A2z 3A4z 308 HENRI DARMON AND SAMIT DASGUPTA satis es assumption 21 with E 00 More generally this is true of the unit A5 of equation 1 provided that 6 satis es 21 and 0 d Remark 22 When N is squarefree two cusps E and 5 Z are F0N equivalent if and only if gcdvN gcd1 N Because p does not divide N it follows that two cusps are P equivalent if and only if they are P0N equivalent Remark 23 Note that as soon as X0N has at least three cusps there is a power 045 of 04 which can be written as as a0a007 where 04739 satis es Assumption 21 with E j This will make it possible to de ne the image of ua r in K X Q by the rule 1 Wow uao7ruo7 F rom now on we will assume that 04 A5 is of the form given in 1 with the ad satisfying 21 The construction of uar proceeds in three stages which are described in Sections 21 22 and 23 21 p adz39c measures Recall that a Zp valued resp integral p adic measure on llDlQp is a nitely additive function Compact open u 39 subsets U C lP lQp gtZp reSP39 Z39 Such a measure can be integrated against any continuous Up valued function h on lP lQp by evaluating the limit of Riemann sums W W hm wiij mop 26 j taken over increasingly ne covers of 11 Q17 by mutually disjoint compact open subsets Uj If a is an integral measure and h is nowhere vanishing one can de ne a multiplicative re nement of the above integral by setting 22 ht d t lim ht39MU7 lt gt 7432 o w mm lt7 A ball in lP lQp is a translate under the action of PGL2QP of the basic compact open subset 2 C llDlQp Let 3 denote the set of balls in P1Qp The following basic facts about balls will be used freely ELLIPTIC UNITS FOR REAL QUADRATIC FIELDS 309 1 A measure a is completely determined by its values on the balls This is because any compact open subset of lP lQp can be written as a disjoint union of elements of B 2 Any ball B 39yZp can be expressed uniquely as a disjoint union of p balls7 23 B B0 U B1 U U B1771 where Bj pr The following gives a simple criterion for a function on B to arise from a measure on llDlQp LEMMA 24 If a is any Zp ualued function on B satisfying MP1 Qp B MB7 MB M30 M31271 for all B 6 B then a extends uniquely to a measure on lP l Qp with total measure 0 Remark 25 The proof of Lemma 24 can be made transparent by us ing the dictionary between measures on lPl Q17 and harmonic cocycles on the Bruhat Tits tree of PGL2QP7 as explained in Section 26 Let 042 denote the modular unit on P0Np de ned by 04 azapz Note that p71 z 7 4 25 0 galt7 l aw mngago 02 auamap 25 Mz W 0412 awmmp 7 where 25 follows from the fact that the weight two Eisenstein series dlog 04 on P0N whose q expansion is given by 59 and 63 below is an eigenvector of Tp with eigenvalue p 1 The following proposition is a key ingredient in the de nition of uoz7 r PROPOSITION 26 There is a unique collection of integral p adic mea sures on lP l Q177 indexed by pairs r73 6 PE X PE and denoted law gt s7 satisfying the following axioms for all r s 6 PE Laveamwmo 310 HENRI DARMON AND SAMIT DASGUPTA 2 My swap isdlogaz 2m 3 P equluan39ance For all 39y E P and all compact open U C llDlQp Nah H YS YU Mal n 8U Proof The key point is that the group P acts almost transitively on 3 There are two distinct P orbits for this action one consisting of the orbit of Zp and the other of its complement lP lQp 7 Zp To construct the system of measures aa r a s satisfying properties 173 above we rst de ne them as functions on B If B is any ball then it can be expressed without loss of generality after possibly replacing it by its complement as 26 B 39yZp for some 39y E P Then properties 2 and 3 force the de nition 1 7 19 27 law gt sB dlogozz i Yil The line integral in 27 converges since both endpoints belong to the set PE P0N ithis is the crucial stage where assumption 21 is usediand it is an integer by the residue theorem Note also that the right hand side of 27 does not depend on the expression of B chosen in 26 This is because the element 39y that appears in 26 is well de ned up to multiplication on the right by an element of P0Np StabpZp Since the integrand dlog 04 is invariant under this group 27 yields a well de ned rule The function aa r a 3 thus de ned on 3 extends by additivity to an integral measure on llDlQp To see thislet p71 BBoomon1Uyg 2 j0 be the decomposition appearing in 23 Setting 7 39y lr and s 39y ls 171 171 54 9 wipm a sB7 23 dlogozz Updlogaz 70 j0 Tl By 25 the differential form dlog 042 is invariant under Up and it follows that avewawwvewamavemm Proposition 26 now follows from Lemma 24 CI Remark 27 It follows from property 2 in Proposition 26 that aheaaeeaavea ELLIPTIC UNITS FOR REAL QUADRATIC FIELDS 311 for all r 572 6 P5 In the terminology introduced in Section 257 pa can thus be viewed as a partial modular symbolwith values in the F module of measures on llDlQp 22 Double integrals Let ordp CSAQ C KP logp cgecp be the p adic ordinal and lwasawa s p adic logarithm respectively7 satisfying lngp O Motivated by De nition 19 of Darl7 we set 28 2dlogoz logZ7 dluar a st 71 39r lP lQP T1 for 717 72 6 HP and r s 6 PE Note that this new integraliwhich is Cp valuedi is completely different from the complex line integral of dlog 04 of equation 40 and so there is some abuse of notation in designating the integrand in the same way However this notation is suggestive7 and should result in no confusion since double integral signs are always used to describe the integral of 28 The expression de ned by 28 is additive in both variables of integration Properties 1 and 3 of Proposition 26 imply that it is also l invariant7 ie7 772 79 72 s dloga leg a for all 7 E P YT1 Y f T1 7 Note that the measures par a 3 involved in the de nition of the double integral in 28 are actually Z valued it is possible to perform the same mul tiplicative re nement as in equation 71 of Darl to de ne the Kg valued multiplicative integral 29 72dioga 7 T2 my a st 71 7 FAQ T1 for 7172 6 Hp KP and as 6 PE 23 Splitting a two cocycle Using the double multiplicative integral of equation 297 we may associate to any 7 6 HP KP and to any choice of base point z 6 PE a Kg valued two cocycle M 6 220 Kg 717 71m 2771772 dloga 71727 m It is instructive to compare the following proposition with Conjecture 5 of Darl by the rule 312 HENRI DARMON AND SAMIT DASGUPTA PROPOSITION 28 The two cocycles ordpm 10mm 6 22mm are two coboundaries Their image in H2PKp does not depend on 739 or m An explicit splitting of ordprlt7 will be given in Section 3 Proposition 34 and of logpm in Section 4 Proposition 47 see Section 27 for the connection between the inde nite integrals appearing in those propositions and the two cocycle 9 Given any integer e gt 0 let K e denote the e torsion subgroup of K5 Proposition 28 implies the existence of an element 0 E 01P KI satisfying 30 lt97 de mod K eaD for some ea dividing p2 7 1 The minimal such integer e0 depends only on 04 and not on 739 It is natural to expect that ea 1 but we have not attempted to show this One strategy to do so would be to apply the techniques of Section 47 in a mod p 7 1 re ned77 context as in the work of deShalit deSl deS2 Remark 29 Let MFA denote the group of p 7 1st roots of unity in CS In many cases one can give a direct proof that the natural image of a in H2l Clt up1 vanishes An element of H2FC corresponds to a homo morphism 1C H2PZ 7 CS By the independence of the cohomology class a on 739 this homomorphism takes values in Q5 Up to 2 and 3 torsion H20 Z H1X0NpC 7 Elfin where the space on the right is the p new subspace of the singular homology group of the modular curve X0Np with the cusps in PE removed The homo morphism gtH is Hecke equivariant where the Hecke action on Q is given by the eigenvalues of dlog 04 Thus if there are no p new modular units of level Np regular on P5 and with the same eigenvalues as this Eisenstein series7for example if N is squarefree or if N 47then it follows that the image of 1 lies in the torsion subgroup of Q5 The onecochain p which splits a is uniquely de ned up to elements in Z1FKz lt HomP Fortunately we have LEMMA 210 The abelianizatz39on ofP is nite Proof See Theorem 2 of Me or Theorem 3 of Se2 El ELLIPTIC UNITS FOR REAL QUADRATIC FIELDS 313 Let er denote the exponent of the abelianization of P and let e lcmem er7 U K e The image of p7 in 011 7 K U depends only on a r and on the base point m not on the choice of onecochain p7 satisfying 30 Assume now that r 6 HP O K Let FT be the stabilizer of r in P LEMMA 211 The rank of P7 is equal to one Proof The group FT is identi ed with the group OTlt of elements of norm 1 in the order OT associated to 739 By the Dirichlet unit theorem this group has rank one7 and in fact the quotient PTlti1gt is isomorphic to Z El LEMMA 212 The restriction of p7 to P7 depends only on 04 and 739 not on the choice of base point m 6 PE that was made to de ne 97 Proof Write mm and paw for m and 077 respectively7 to emphasize the dependence of these invariants on the choice of base point z 6 PE A direct computation cf for example Lemma 84 of Dar2 shows that if y is another choice of base point7 then quot971 7 Hay 0101217 where the onecochain pm 6 011 7 Kg vanishes on P7 The lemma follows El Let 6 be a fundamental unit of OTlt C KX chosen to be greater than 1 or less than 1 according to whether 739 gt r or r lt 7quot respectively7 where r is the Galois conjugate of r The unit 6 is independent of the choice of real embedding of K Let W be the unique element of PT satisfying 739 T 17 1 8 1 39 We de ne 1404 739 by setting lt31 m T pm 6 KgU Note that 1404 739 depends only on the P orbit of 739 Remark 213 It may not be apparent to the reader why the somewhat intricate construction of 1404 739 given above is analogous to the construction of Section 1 leading to elliptic units Some further explanation of the analogy in the context of the Stark Heegner points of Dar1 can be found in Sections 4 and 5 of BDG7 and in the uniformization theory developed in Das17 Das3 314 HENRI DARMON AND SAMIT DASGUPTA 24 The main conjecture The elements Mag 6 K U are expected to behave exactly like the elliptic units 14047739 of Chapter 1 To make this statement more precise we now formulate a conjectural Shimura reciprocity law for these elements A Z1p lattice in K is a Z1p submodule of K which is free of rank 2 Let K denote the multiplicative group of elements of K of positive norm By analogy with 11 we then set A a Z1p lattice in K7 KX AlAZSZNZ Jr In this de nition it is important to take equivalence classes under multipli cation by K rather than KX see also Remark 219 of Section 28 As in Chapter 17 there is a natural bijective map I from QNK to Hp Kl 7 which to z A17 A2 assigns 33 z wlwg where whwg is a Z1p basis of A1 satisfying 32 QNK A1A2 with Lulu2 7 wiwg gt 07 34 ordpw1w 27 wiwg E 0 mod 27 and A2 ltNw17w2 Recall that O is a Z1p order of K of discriminant prime to N and 197 by convention As before denote by bxO the set of pairs A17 A2 6 QNK such that O is the maximal Z1p order of K preserving both A1 and A2 Note that QNO Hgr Let PicO denote the narrow Pleard group of 97 de ned as the group of projective O submodules of K modulo homothety by Class eld theory identi es PicO with the Galois group of an abelian extension H of K7 the narrow rz39ng class eld attached to 9 Let 35 rec Pic0gtGalHK denote the reciprocity law map of global class eld theory The group PicO acts naturally on QNO by translation7 and hence it also acts on 1QNO HgP Adopting the same notation as in equation 12 of Chapter 17 denote this latter action by 36 w H aw for a e Pic0 T e Hfr The following conjecture can be viewed as a natural generalization of Theo rem 11 for real quadratic elds CONJECTURE 214 If 739 belongs to HgP then 7217 739 belongs to 9H1pXU7 and in fac l7 37 1404 cwr reca 1uoz7 739 mod U7 for all u E Pic0 ELLIPTIC UNITS FOR REAL QUADRATIC FIELDS 315 In spite ofits strong analogy with Theorem 11 Conjecture 214 appears to lie deeper its proof would yield an explicit class eld theory for real quadratic elds Chapter 11 of Das1 cf also Das2 describes ef cient algorithms for calculating ua r and uses these algorithms to obtain numerical evidence for Conjecture 214 Evidence of a more theoretical nature will be given in Chapters 3 and 4 by relating the analytically de ned elements ua r to special values of zeta functions in the spirit of Theorem 12 The remainder of this chapter contains some preliminaries of a more tech nical nature which the reader may wish to skip on a rst reading 25 Modular symbols and Dedekind sums We discuss the notion of partial modular symbols and the associated Dedekind sums that will be useful for the calculation of the ua riboth from a computational and a theoretical point of view Partial modular symbols Let Mg denote the module of Z valued func tions in on P5 x PE denoted r s gt gt mr a s and satisfying 38 mr gt s ms gt t mr gt t for all r st 6 PE Functions of this sort will be called partial modular symbols with respect to E and P This terminology is adopted because in satis es all the properties of a modular symbol except that it is not de ned on all of lP lQ but only on a P invariant subset of it More generally if M is any P module write MAM for the group of M valued partial modular symbols equipped with the natural F module structure 39 39ymr gt s y myyilr H V718 To the modular unit 04 is associated the Z valued TO N invariant partial modular symbol 1 S 40 mar a s dloga Dedekind sums The line integrals in 40 de ning the modular symbol ma can be expressed in terms of classical Dedekind sums D 3131 31 for gcdam 1 m gt 0 m1 where B1m 7 12 x7 712 316 HENRI DARMON AND SAMIT DASGUPTA is the rst Bernoulli polynomial made periodic Corresponding to the element 6 used to de ne Oz A5 in 1 one de nes the modi ed Dedekind sum 13 Z ndDdz le Following Maz ll 2 we introduce the modi ed Dedekind Rademacher ho momorphism on P0N 41 I a b 0 if c 0 6 lt NC 1 gt 397 12 signcD6 otherwise as well as the corresponding homomorphism of P0Np ifc0 11 lt a b gt 0 y 5 Npc d 39 12 signc D6 7 D6 otherwise Note that the assumption 21 that was made on 6 created a simpli cation in the behaviour of the Dedekind Rademacher homomorphism making it vanish on the upper triangular matrices and eliminating the extra terms appearing in Equation 21 of Maz when 6 N 7 In particular it is clear that 15 and I take integral values The modi ed Dedekind Rademacher homomorphisms 15 and I attached to 6 encode the periods of dloga and dlog 04 respectively For any choice of base points z E H U PE and 739 E H we have 1 7m 1 lt42 1gt6v dloga mlogam 7 logam 1 7m 1 41w dlogcw mlogaww 7 10mm for all 39y in P0N and P0Np respectively In particular if r s belong to PE we may evaluate the partial modular symbol ma a s by choosing 39y E P0N such that s yr and noting that 1 S 43 dloga 7ltIgt539y 26 Measures and the Bmhat Tz39ts tree Let T denote the Bruhat Tits tree of PGL2QP whose set VT of vertices is in bijection with the ng homothety classes of Zp lattices in 127 two vertices being joined by an edge if the corresponding classes admit representatives which are contained one in the other with index p See Chapter 5 of Dar2 for a detailed discus sion The group f of matrices in PGLZ1p which are upper triangular modulo N acts transitively on VT via its natural left action on 127 and the group P0N is the stabilizer in f of the basic vertex 120 corresponding to the standard lattice 212 ELLIPTIC UNITS FOR REAL QUADRATIC FIELDS 317 The unmmz39 ed upper half plane H is the set of 739 6 HP such that Qp739 generates an unrami ed extension of Qp The Bruhat Tits tree can be viewed as a combinatorial skeleton of Hp and is the target of the reduction map r Hf T This map is compatible with the natural PGL2Qp actions on both source and target and its de nition and main properties can be found for example in Chapter 5 of Dar2 To each u E VT we associate a well de ned partial modular symbol my a s by imposing the rules much gt s m0 gt s mwfyr gt 39ys my gt s for all U E VT 39y E f and r s 6 P5 In addition to the built in P equivariance relation satis ed by the collection my of partial modular symbols the as signment v gt gt my r a 3 satis es the following harmonicity property 44 Z my a s p 41mm 4 s for all u e um d1u1 in which the sum on the left is taken over the p 1 vertices 1 which are adjacent to v The relation 44 follows from the fact that dloga is a weight two Eisenstein series on P0N and hence an eigenvector for the Hecke operator Tp with eigenvalue p 1 Let T denote the set of ordered edges of T ie the set of ordered pairs of adjacent vertices of T If e 12913 is such an edge it is convenient to write 35 Us and 255 U for the source and target vertex of 5 respectively and E 15125 for the edge obtained from 5 by reversing the orientation A Z valued harmonic cocycle on T is a function f TgtZ satisfying 45 2 105 0 for all u e VT se39u as well as f 7fe for all e E T The collection of partial modular symbols my gives rise to a system m5 of partial modular symbols indexed this time by the oriented edges of T by the rule 46 m5r gt s mter gt s 7 m5er gt 3 Note that if r and s 6 P5 are xed the assignment 5 gt gt m5r a s is a Z valued harmonic cocycle on T This follows directly from 44 As explained in Section 12 of Darl or in Chapter 5 of Dar2 to each ordered edge e of T is attached a standard compact open subset of P1Qp denoted U5 Thanks to this assignment the Zp valued harmonic cocycles on 318 HENRI DARMON AND SAMIT DASGUPTA T are in natural bijection with the Zp valued measures on P1Qp by sending a cocycle c to the measure 2 satisfying 47 uU5 Ce7 for all e E T The harmonic cocycles m5r a s of 46 give rise in this way to the p adic measures Ma r a s of Proposition 267 satisfying 48 law gt sUe mer gt s 27 Inde nite integrals The double multiplicative integral of 29 can be used to associate to Oz and 739 an MgK valued one cocycle YT S a e Z1PMgKpX de ned by amp a s j dloga T 7 Let f5 denote the space of Kg valued functions on P5 and denote by d fgKgtMgK the P module homomorphism de ned by the rule de H 8 f8f Finally7 denote by 6 H1PM5KPXgtH2PKPX the connecting homomorphism arising from the P cohomology of the exact sequence OAK mde aMAKQAO One can see cf the discussion in Section 96 of Dar2 that MHWSIWII MWi z Proposition 28 is a consequence of the following more precise statement whose proof will be given in Chapters 3 and 4 PROPOSITION 215 The one cocycles ordpFLT and lngILT are one coboundaries As in the discussion following the statement of Proposition 287 Proposi tion 215 implies the existence of a U C Kimrs such that 49 RT 167 mod U7 for some 57 E tgKzlt7 and the image of 67 in MAX U is unique De ne the inde nite integral involving only one p adic endpoint of inte gration by the rule 7 S dloga 57r a s EKgU T ELLIPTIC UNITS FOR REAL QUADRATIC FIELDS 319 This inde nite integral is completely characterized by the following three prop erties r s r t r t 50 7dlogoz x 7dlogoz7dlogoz7 for all rst 6 PE 71 2 72 SS 717 S 51 jdlogozj dlogaj dlogoz7 for all 7177 2 67117 7 7 72 7 39yr 3979 r s 52 dlogoz7dlogoz7 for all 39y E P YT 7 Letting z 6 PE be the base point that was used to construct 077 we have r 3971 53 p7 y dloga mod U 1 ln particular7 LEMMA 216 The following equality holds in KgU 7 771 uozr 7 dlogoz7 m 28 The action of complex conjugation and of Up The partial modular symbol ma used to de ne uoz7 r is odd in the sense that for any base point m 6 PE ma7x gt 7y 7max a y for all 71 6 PE cf Maz7 Ch ll7 The complex conjugation associated to either of the in nite places 001 or 002 of K is the same in GalHK since H is a ring class eld of K Let r00 6 GalHK denote this element The parity of ma implies the following behaViour of the elements uoz7 739 under the action of 7 00 PROPOSITION 217 Assume conjecture 214 For all r E Hg roouozr uozr 1 Proof The fact that the partial modular symbol ma is odd implies that the sign denoted woo in Proposition 513 of Dar1 satis es woo 71 The proof of Proposition 217 is then identical to the proof of Proposition 513 of Dar1 El Remark 218 In the context of a modular elliptic curve E treated in Dar17 the sign woo can be chosen to be either 1 or 71 by working with either the even or odd modular symbol of E7 corresponding to the choice of the real 320 HENRI DARMON AND SAMIT DASGUPTA or imaginary period attached to E respectively In the situation treated here where E is replaced by the multiplicative group only the odd modular symbol m0 remains available in harmony with the fact that the multiplicative group has a single period 2m which is purely imaginary Remark 219 Suppose that O has a fundamental unit of negative norm Then equivalence of ideals in the strict and usual sense coincide so that the narrow ring class eld H associated to O is equal to the ring class eld taken in the nonstrict sense which is totally real Conjecture 214 predicts that 73900 should act trivially on ua 739 in this case and that the p units ua 739 should be trivial In fact it can be shown independently of any conjectures that ua T 1 for all T 6 H5 This suggests that interesting elements of Hgtlt are obtained only when H is a totally complex extension of K This explains why it is so essential to work with equivalence of ideals in the narrow sense and with narrow ring class elds to obtain useful invariants Similarly to the proof of Proposition 217 the fact that the Eisenstein series dlog 04 is xed by the U17 operator implies that the sign denoted w in Proposition 513 of Dar1 equals 1 Thus the invariance of the inde nite integral given in 52 holds for all 39y E f D Pi1 In particular the element ua 739 depends only on the f orbit of 739 3 Special values of zeta functions It will be assumed for simplicity in this section that p is inert and not rami ed in KQ Recall the p adic ordinal ordp K gtZ mentioned in Section 23 The goal of this section is to give a precise formula for ordpuoz739 when 739 6 HP O K in terms of the special values of certain zeta functions 31 The zeta function Given 739 E Hg the primitive integral binary quadratic form Q associated to 739 can be de ned as in This time Q is non de nite lts discriminant is positive and is ofthe form Dpk for some integer k 2 0 where D is the discriminant of the Z1p order 9 By convention the integer D is taken to be prime to p By replacing 739 by a f translate we may assume without loss of generality that 54 the discriminant of Q is equal to D We will make this assumption from now on In that case the generator y of PTi1gt belongs to P0N Note that the matrix W xes the quadratic form ELLIPTIC UNITS FOR REAL QUADRATIC FIELDS 321 QT under the usual action of SL2Z on the set of binary quadratic forms Furthermore the simplifying assumption that gcdDN 1 implies that W 7 where the latter matrix is taken to be the generator of the stabilizer of the form QT in SL2Z Given any nonde nite binary quadratic form Q whose discriminant is not a perfect square let 39yQ be a generator of its stabilizer in SL2Z Note that Q takes on both positive and negative integral values and that each value in the range of Q is taken on in nitely often since Q is constant on the 39yQ orbits in Z2 The de nition of Qs given in 14 needs to be modi ed accordingly by setting W i Z2 ODWW7 andletting 55 625 2 signQm7nlQm7nl i mm 6W where signm i1 denotes the sign of a nonzero real number x Equivalence classes of binary quadratic forms of discriminant D are in natural bijection with narrow ideal classes of 9 OK ideals by associating to such an ideal class the suitably scaled norm form attached to a representative ideal The partial zeta function attached to the narrow ideal class A is de ned in the usual way by the rule 3 A ZNormI s 16A If A is a narrow ideal class let A be the ideal class corresponding to 04A for some 04 E KX of negative norm and let Q be a quadratic form of discriminant D associated to A A standard calculation cf the beginning of Section 2 of Za for example shows that 56 628 C87 4 C87 4 Note in particular that Qs 0 if 9 contains a unit of negative norm since A A in that case We mimic the de nitions of equation 15 and de ne 57 93 6273 04 T s andsgds le Observe that 3 rather than 78 appears as the exponent of d in the de nition of 04 739 As in 15 the function 04 739 s is a simple linear combination of zeta functions atttached to integral quadratic forms of the same positive discriminant D Note that 04739 5 depends only on the F0N orbit of the element 739 6 Hz normalized to satisfy 54 322 HENRI DARMON AND SAMIT DASGUPTA Let AK denote the ring of adeles of K A nite order idele class character X HX AgWar U is called a ring class character if it is trivial on A6 If X is such a character then its two archimedean components Xgt01 and Xgt02 attached to the two real places of K are either both trivial or both equal to the sign character In the former case X is called even and in the latter it is said to be odd Any ring class character can be interpreted as a character on the narrow Picard group GO PicO of narrow ideal classes attached to a xed order 9 of K whose conductor is equal to the conductor of X Formula 56 shows that the zeta functions 75 with 739 6 HI can be interpreted in terms of partial zeta functions encoding the zeta function of K twisted by odd ring class characters of GO More precisely letting To be any element of 71 which is equivalent to xE under the action of SL2Z we have 7 0 if X is even 58 620 Maxim LKX 3 if X is odd 7 O The main formula of this chapter is THEOREM 31 Suppose that 739 belongs to Hg and is normalized by the action off to satisfy 54 Then 1 a no 5 ordpltultwgtgt 32 Values at negative integers In this section we give a formula for the value of 04730 in terms of complex periods of dloga This formula is a special case of a more general one expressing 04 739 1 7 r in terms of periods of certain Eisenstein series of weight 2r for odd r 2 1 The logarithmic derivatives dloga and dlog 04 can be written as 59 dlogaz 27riF2z dz dlogozz 27riF z dz where F2z and F are the weight two Eisenstein series on P0N and F0N p respectively given by the formulae 60 F2z 724 dndE2dz F2z 7pF2pz w and E2z is the standard Eisenstein series of weight 2 lt61 mew ltlt2gt Z Z ELLIPTIC UNITS FOR REAL QUADRATIC FIELDS 323 We remark that the double series used to de ne E2 is not absolutely con vergent and the resulting expression is not invariant under SL2Z For a discussion of the weight two Eisenstein series see Section 310 of Ap for ex ample The Eisenstein series of 61 and 60 are part of a natural family of Eisenstein series of varying weights For even k 2 2 consider the standard Eisenstein series of weight k 2 k 71 1 1 Bk 62 EW Z W ZUHMW mmiioo n1 De ne likewise as a function of the element 6 2d ndd used to de ne the modular unit 04 the higher weight Eisenstein series 63 F142 44an d Ekdz em 7 48k e 1 1 7 7 27rik 7mg m2 nk dmndd 724 Z ak1n Z nddqnd n1 le The Fk are modular forms of weight k on P0N holomorphic on the upper half plane Note that these Eisenstein series have no constant term and hence are holomorphic at the cusp loo We also de ne for the purposes of p adic interpolation the function Fm Fee 7 pkilFMpz We extend the de nition of and to all k 2 2 by letting Ek Fk 0 for k odd Recall the standard right action of GLR on the space of modular forms of weight k given by FW Fyz when 39y lt j Z gt Now the de nition of F can be written 0 1 The following proposition expresses 04 739 177 in terms of periods of F2 F F epHFklpg where P lt p 0 gt PROPOSITION 32 For all odd integers r gt 0 ea 12 C04T17 r 7 Qz11F2zdz s 324 HENRI DARMON AND SAMIT DASGUPTA Proof Let k 2 2 be a positive integer and let Ek denote the weight k Eisenstein series N Mk i EkEkz 1fkgt2gt By Hilfsatz 1 of Sie27 letting 20 E H be an arbitrary base point7 the following identity holds for all integers r gt 1 7720 Til N Til T 7 Til 7 64 20 Q7 E2TZd2 1 MD 2 gQTWW Suppose that r gt 1 is an odd integer Then 2 cm n ZsigMQdmleQdm701 cm W W so that 65gt Qi l mz dz 20 T7 12 mDT CTUl On the other hand7 it follows from the relation 58 and from the functional equation for LKX7 s for odd characters cf La7 Cor 1 after Th 14 of 8 Ch XlV that 75 satis es the functional equation 139 312 273 2 66 717 3 Wzsill Plt 2 gt 73 Hence if r 2 2 is an even positive integer7 717 T 07 while if r 2 1 is odd7 417 2 67 41 7 WU 1l 70 Combining this functional equation with 657 we obtain 7720 N 27027 Til E d 17 20 Q7 272 2 42r 7 1 T Since V 2 k 7 1 E E W 27W Wt it follows that 68gt W QZ 1E2Tzdz gen 7 r ELLIPTIC UNITS FOR REAL QUADRATIC FIELDS 325 From the de nition of Fk 7720 7720 Q1F2z d2 7242711 Q1E2dz dz zo d N 20 Making the change of variables w 1 2 we obtain 7720 1 dWrzo w 771 d or E2dzdz Q E 1 E2wdw 20 120 Recall that yd denotes the generator of the stabilizer of 1739 in SL2Z chosen in such a way that yd is a positive power of 11 Note that w 1 dwo Mme QT 31 3224121 Hence 3977 20 1 54141120 d or E2dzdz 120 10 1 dril Qd7wT 1E27 w dw The expression on the right is equal to idlgf dTO 7 r by 68 It follows that for all odd r gt 1 10 69 7 Q1F2z d2 12 andHQ u 7 r 12 61317 7 1 le The integrand in the left hand expression involves an Eisenstein series which is holomorphic at 00 hence we may replace the base point 20 E H by the cusp 00 or any other cusp which belongs to the same P0N orbit In the case where r 1 using 59 42 and 41 we see that the expression on the left of 69 is equal to 7720 1 7720 a 70 F22dz i dloga MW 12 signcD6 10 2m 20 7 a 12 V77 NC gtk and D6 is the modi ed Dedekind sum introduced previously Meyer s formula expressing the special values of partial zeta functions attached to real quadratic elds at s 0 in terms of Dedekind sums can be used to derive the identity 71 12C04T0 7ltIgt539y7 Cf Za Eq 41 for a statement of Meyer s formula in the case where D is fundamental the general case can be derived from equation 18 in 5 of CS for example It follows that Proposition 32 holds for r 1 as well in light of the fact that dloga is holomorphic at 5 so that the base point 20 can be replaced by the cusp E in the expression on the left of 70 El where 326 HENRI DARMON AND SAMIT DASGUPTA The evaluation of the right hand side in Theorem 31 is taken up in the next section 33 The p adz39c valuation To compute ordp7lozr7 it will be useful to have at our disposal a formula for the p adic valuation of a p adic mul tiplicative line integral We describe such a formula in the case where the p adic endpoints of integration belong to the unrami ed upper half plane HIE in terms of the reduction map from H to VT introduced in Section 26 LEMMA 33 For all r1772 6 H and for all as 6 PE ordp dlogagt Z m5r 7 s7 errl7lrrg where the sum on the right is taken over the ordered edges in the path of T joining rr1 to rr2 A complete proof of this formula is given7 for example7 in Lemma 25 of BDG El PROPOSITION 34 Let u rr Then ordp ltjTdlogozgt mm a s Proof By Lemma 337 for all 39y E P YT S 72 ordp dlogagt Z m5r a s T T 517vy39u where the sum on the right is taken over the ordered edges in the path joining v to yo By 467 this sum is equal to the telescoping sum 2 mter 7gt s 7 77155 7gt s mwr 7gt s 7 mvr 7gt s 570771 mv39y71r 7gt y 137 mvr 7gt s dmvvr H 8 so that Orde lT dmv It follows from the de ning equation 49 for 67 and from the fact that lgZF 0 that 73 ordp67 my The lemma follows El We may assume without loss of generality that 739 has been normalized to satisfy 547 so that rr 120 where 120 is the vertex of T corresponding to the standard lattice 2127 In this case the matrix W belongs to P0N and generates the stabilizer of r in that group furthermore we have mvo ma ELLIPTIC UNITS FOR REAL QUADRATIC FIELDS 327 COROLLARY 35 Let m be any base point in TE Then 1 W 0rdpu04739 dloga 4pm m Proof By Lemma 216 and Proposition 34 739 77m 74 ordpu04 ordp dlog a mm a W 1 The lemma follows from the de nition of ma given in 40 and from 42 CI The proof of Theorem 31 now follows by combination of 71 and Corol lary 35 34 The Brumer Stdrk conjecture Given 739 E Hg let BS denote the Brumer Stickelberger element in the integral group ring of GO Pic0 de ned by BS Z 00a1 7600 This element is independent of the choice of 739 E Hg up to multiplication by an element of GO Relation 58 implies that BS agrees with the usual Brumer Stickelberger element attached to the extension H K To any modular unit 04 and with 739 6 HI we may also associate the modi ed BrumeT Stichelberger element by setting 75 BSaT Z COiagtk 70071 7600 Let ClH denote the class group of H viewed as a ZG0 module in a natural way Let I denote the augmentation ideal of ZGO The following conjecture is a reformulation of the usual Brumer Stark conjecture for HK generalising Stickelberger s theorem on class groups of abelian extensions of Q CONJECTURE 36 The element BS7 annihilates I ClH X Z12 Conjecture 36 is proved in this case thanks to the work of Wiles We give a more direct proof which is conditional on Conjecture 214 in the spirit of Stickelberger s original proof in the abelian case Because it is only conditional this result is more notable for what it says about Conjecture 214 than about the Brumer Stark conjectures PROPOSITION 37 Assume Conjecture 214 Then the BrumeT Stichelberger element BS7 annihilates I ClH X Z12 Proof For any modular unit 04 we have the relation BSa 739 Ia BST 328 HENRI DARMON AND SAMIT DASGUPTA where JD 6 I is an element which depends on 04 and 739 and is de ned as follows The integral quadratic form Q Amz Bmy Cyz attached to 739 6 Hz determines an O ideal of norm 1 for each le by the rule ad d B 7 V5 Then JD 2nd recud le By the Chebotarev density theorem the elements JD generate I as 04 ranges over the possible modular units Hence it is enough to show that BSaT annihilates ClH X Z12 Let H denote the maximal sub eld of the Hilbert class eld of H which is of odd degree over H Note that H is Galois over K and even over Q Class eld theory identi es GalHH with M CIH Z12 as modules over ZGO12 Since GalHQ is a generalized dihedral group the generator of GalKQ lifts to an involution L E GalHQ Lift L further to an involution in GalHQ This can be done since H is of odd degree over Choose any a E M By the Chebotarev density theorem there exists a rational prime p such that F robpHQ m In particular p is inert in K Note that p as a prime ideal of K splits completely in Choosing a prime p of H above p we have F robpHK F robpHH O39LO39L 00L The factorization of ua 739 and its conjugates given by Theorem 31 implies that BSOLT annihilates F robpHH aw Since a was chosen arbitrarily it follows that 76 BSaT annihilates 1 LM Let 000 E GalHK denote complex conjugation Since L was an arbitrary lift of the generator of GalKQ we could have replaced it by L600 in the preceding argument yielding 77 BSaT annihilates 1 L000M Note furthermore that by de nition 1 000 BSaT 0 so a fortL39orL39 78 BSa 739 annihilates 1 COOM ELLIPTIC UNITS FOR REAL QUADRATIC FIELDS 329 Since the module M has odd order7 it decomposes as a direct sum of simulta neous eigenspaces for the action of the commuting involutions i and 000 Each eigenspace belongs to at least one of the subspaces in 767 777 or 78 The result follows CI 35 Connection with the Gross Stark conjecture A general result of Deligne and Ribet cf the discussion in Gr17 2 implies the existence of a p adic meromorphic function Cpa739 s of the variable 8 E 2 characterized by its values on a dense set of negative integers 79 1704773977117p72nCa7T7n7 for all n g 0 n E 0 mod 2p 71 Let UHp denote the group of p units of H de ned by Gross in Proposition 38 of Gr1 UHp e E Hgtlt 1 for all places 9 which do not divide p Since the places 9 involved in the de nition of UHp include all the archimede an ones7 it follows that UHp is in nite only when H has no real embeddings7 and that images of the elements of UHp under all the complex embeddings of H lie on the unit circle Proposition 217 implies that the p unit iioz7 739 belongs to UHp assuming7 of course7 conjecture 214 Since 0rdpu0477391239C047T707 Conjecture 212 of Gr1 cf the formulation given in Proposition 38 of Gr1 suggests that one should have 80 logp NormKPQpii047 7 712 1047 70 In fact7 the relation 80 is essentially equivalent by varying oz appropriately to the Gross Stark conjecture for HK7 assuming Conjecture 214 The next chapter is devoted to the explicit construction of Cpa739s and to a proof of 80 4 A Kronecker limit formula The rst three sections of this chapter give an explicit construction of the p adic zeta function on 739 s satisfying the interpolation property 79 The following theorem is then proved THEOREM 41 Suppose that 739 belongs to Hg and is normalized by the action off to satisfy 54 Then logp NormKPQpii047 1 Ham 0 5 330 HENRI DARMON AND SAMIT DASGUPTA Note the clear analogy between this formula and the classical Kronecker limit formula stated in Theorem 12 Theorem 41 allows us to deduce the Gross Stark conjecture for HK from Conjecture 214 It should be pointed out that Conjecture 214 is stronger and more precise than Gross s conjecture in that setting7 since it gives a formula for the GrossStark unit uoz7 r itself7 and not just its norm to Q17 41 Measures associated to Eisenstein series Let X Z17 X Zpl C Q17 X Qp 07 considered as column vectors7 where Z1 x Zp denotes the set of primitive uectors z y E Z satisfying gcdzy 1 The space Q127 7 0 is endowed with a natural action of P by left multiplication There is a sz bundle map 7139 X 7 lP lQp given by my 7 The crucial technical ingredient in the construction of Cpar s and in the proof of Theorem 41 is the following result7 which can be viewed as an extension of Proposition 26 to the family of Eisenstein series introduced in the previous section THEOREM 42 Fix oz and E as before There is a unique collection of p adic measures on the space Q12 7 O7 indexed by pairs 7 s 6 PE X PE and denoted ar 7gt s7 satisfying the following properties 1 For euery homogeneous polynomial hx7 y E Zx y of degree k 7 27 231 X Mm cw 7 sum 2 7 Re 17 Shltz1gtFkltzgtdz r 2 P equiuariance For all 39y E P and all compact7 open U C Q12 7 O7 MW H YS YU Mr H 8U 3 Inuariance under multiplication by p M H 5PU M H 5U Furthermore this measure satis es 4 For euery homogeneous polynomial hx7 y E Zx y of degree k 7 27 S ha y cw 7 Shaw 7 Re hlt271gtFzltzgtdzj ZPXZ r Remark 43 The function r73 gt gt ar 7 3 de nes a partial modular symbol with values in the space of measures on Q127 7 Objects of this type appear in Glenn Stevens7 study of two variable p adic L functions attached to Hida and Coleman families of eigenforms More precisely7 when dloga is ELLIPTIC UNITS FOR REAL QUADRATIC FIELDS 331 replaced by a weight two cuspidal eigenform 1 which is ordinary at 197 Stevens attaches to f a measure valued modular symbol via Hida s theory of families of eigenforms7 and uses it to de ne the two variable p adic L function attached to this family There is also a theory in the nonordinary setting7 where it becomes necessary to replace p adic measures by locally analytic distributions in the sense of Stevens The proof of Theorem 42 is postponed to the end of the paper beginning with Section 44 The following lemma shows how the measures Iur a s are related to the measures paT a s of the previous section LEMMA 44 For all compact open U C llD1Qp7 232 w a sltw1Ugt w a 8W Recall that 7r 1U C X by de nition Proof De ne a collection of measures my a s on P1Qp by the rule mm a W w a new Theorem 42 implies that the collection of measures my a 3 satis es all the properties of Ma spelled out in Proposition 26 To see that my a 3 satis es the required P invariance property7 note that mph H YS YU MW n VSW 1VU NW A v8v7r 1U7 where the last equality follows from the fact that both 7r 139yU and 39y7r 1U are fundamental regions for the action of ltpgt on the inverse image of 39yU in 127 7 Hence mm a WWII w a swim mm a 8W Lemma 44 follows from the uniqueness in Proposition 26 I 42 Construction of the p adic L function The special values of 04 r s at certain7 even negative7 integers can be expressed in terms of the measure a described in Section 41 LEMMA 45 For all odd integers r gt 0 121P2772Ca7r71 r A Qltzygt1dis a may Proof This follows directly from Proposition 32 in light of the properties of the measure a spelled out in Theorem 42 CI 332 HENRI DARMON AND SAMIT DASGUPTA Suppose that the integer r in addition to being odd is congruent to 1 modulo p 7 1 Then by Lemma 457 233 1217292 aml r 7 XltQ7m7ygtH M5 4 WW where for z 6 ZS the expression denotes the unique element in 1 pr which differs from x by a p71st root of unity The advantage of the expression 83 is that it interpolates p adically7 expressing 04731 7 r with its Euler factor at p removed7 as a function of the p adic variable r This leads us to de ne cm a s 7 1 12 Ana aw cw 7 may for all s E Zp Note that one recovers the p adic L function introduced in Section 35 which is uniquely characterized by the interpolation property 79 In terms of this explicit de nition of Cpoz r7 s7 we have LEMMA 46 The derivative 04773 3 at s 0 is given by 1 12O 7 T7 0 5 X log QT7 dJ E H VTE7 Proof This is a direct consequence of the de nition CI 43 An explicit splitting of a two cooyole We now turn to the calculation of the onecochain oT7 or7 equivalently7 of the expression 739 S 7dlogoz T A formula for this inde nite integral can be given in terms of the system of p adic measures In of Theorem 42 PROPOSITION 47 Let u be as in Theorem 42 Then logp ltjTdlog a Xlogpz 7 Ty dir 7 sz7 Proof If we de ne jcnoga 410w 7 w cw 7 my then a direct calculation shows that the resulting expression satis es 7 s 7 r t 7 r t 7 84 dloga dloga dlogoz 7 for all rst GTE 7 S 7 71 s 72 s 71 s 85 dloga77 dlogoz7 dlogoz 7 7 7392 7 ELLIPTIC UNITS FOR REAL QUADRATIC FIELDS 333 as well as 39yr 3975 r s 86 dlogoz7dlogoz7 for allyel Y 7 These properties are the additive counterparts of equations 507 51 and 52 of Section 277 which uniquely determine the p adic inde nite multiplicative integral attached to dloga It follows that 739 S 739 S dlog a logp lt7dlogozgt 7 7 as was to be shown El We can now prove Theorem 41 Proof of Theorem 41 By Lemma 2167 we have T 39yrr r 39yrr logpNormKPQpuozr logp dloga x7 dloga 7 7 for any r 6 PE By Proposition 47 and the fact that 6247 y is proportional to m 7 ryz 7 r y and that MX 07 the expression on the right is equal to X meow cw a we 2 The result now follows from Lemma 46 CI The remainder of the paper is devoted to the proof of Theorem 42 44 Generalized Dedekind sums In this section we evaluate the integrals appearing in the right side of 81 in Theorem 427 which characterize the partial modular symbol of measures In The computations of this section are not new7 but we include them for completene and t t Let 1 denote a modular form and let af0 denote the constant term of its q expansion at 00 For any relatively prime integers a and c with c 2 17 the function 00 Afs M emsZail it ac 7 af0ts1 dt 0 is well de ned for Res large enough7 and has a meromorphic continuation to all of C For the Eisenstein series Egk with k gt 17 this is given by 87 7 Us 0 AE2k 5i 0397 C EMS262k 2 2719 Z 0 3 17 2k 110 2 iemmhaC 334 HENRI DARMON AND SAMIT DASGUPTA where 31 denotes the Hurwitz zeta function This is a relatively standard computation carried out for example in Proposition 31 of Let us cal culate the real part of this expression for s an integer 1 g s 2k 7 1 Note that when 3 1 the term for h c in 87 is taken to be lini 8 17 2kCs E R 57 The Hurwitz zeta function has the well known value 1 7 n b 7Bnbn where the Bernoulli polynomials B are de ned by the power series 5th i0 tn7139 n0 5 71 nl Furthermore for any real number m 23371 1 2 H371 1 2 R 7rin 7rin elt 2709 1 mse 227rS mg mse m70 7 45 2 s 7 where 0 if s 1 and x e z 5 l Bsz Bsz7 m otherwise See Section II of Hal for this last equation Hence we obtain 7 5 0 lt88 Re Am 8 a a 7 c2k 2 1 Z B ShC LilaC 2 s h1 We would like to replace the term ng5hc by gk5hc in the sum above Only the term for h c which we now consider may cause dif culty If s is even then B2k751 B2k750 since in general one has Bn17 m 71 Bnz If s is odd then the other term in the product is 1 9hac 0 Thus in either case we may replace the term ng5hc by ngshc This motivates the following de nition 78 De nition 48 Let s t 2 O For a and c relatively prime and c gt O the generalized Dedekind sum D5 16 is de ned by C D9tae CH 2 1301 CBthac h1 Note that the sum may be taken over any complete set of representatives h mod 0 For st 2 1 de ne Damc D5tac ELLIPTIC UNITS FOR REAL QUADRATIC FIELDS Remark 49 When 3 t 17 D11ac D11ac Dac 7 3 4 Equation 88 may be written in terms of the generalized Dedekind sums as 1 s 89 ReAEksac 0971 2 Dkssac This formula continues to hold when k is odd7 since then the Dedekind sum Dk5sae vanishes using the relation like 45341 F rom the de nition of Fk we nd 90 ApksaNc 724anAEksaNcd am We are now ready to evaluate the integrals appearing in 81 Let 0 g n hi2 Using the change of variables 2 it aNe7 we nd 91 Awe dzZn lt gtnilNcilAFkl1aNc N0 0 71 n a 4 fl NC 724Zlt gt NC anAEk lt61a 0 am In view of 897 the real part of 91 is equal to 71 n a 4 a 92 12 71 HD l Z a No l l 7 k 1 l1ltNcdgt 0 w As we now check7 equation 92 remains valid for k 2 In this case the desired formula simpli es to mFd 12E D a 12D5 2 Z 2 nd L1 7 aNc d N Ncd NC which is nothing but equation 43 45 Measures on 2 x Zp Let E 6 Foo7 and assume that p does not divide e In this section we prove the following crucial lemma LEMMA 410 Let E 6 Foo haue denominator not diuisible by p There exists a unique Zp ualued measure 15 on Z1 X Zp such that lt93 Z th7ydvs7yRe1ipk 2 mhzy1Fkzdz PX for euery homogeneous polynomial hmy 6 ley of degree k 7 2 336 HENRI DARMON AND SAMIT DASGUPTA Equation 93 is equivalent to the statement that 94 ioo mnym dug x7 y Re 17 pnm annm2z dz Z27 XZP aNc 121 PW in y 1 g nddianmil111 0 for all integers nm 2 0 Denote the last expression appearing in equation 94 by mm 6 Q Our key tool in showing the existence and uniqueness of Zg is the following result7 which is the two variable version of a classical theorem of Mahler see Theorem 331 of Hida LEMMA 411 Let bmm E Zp be constants indexed by integers nm Z 0 There exists a unique measure 1 on Z1 X Zp such that przp Mg 5mm Thus to prove Lemma 410 we must show that the rational numbers 7L m Jmm Z Z Cni6mj1ij i0 j0 lie in Z17 where the rational numbers c are de ned by the equation 7L ltgt z i0 Our proof of this fact will follow the proof of the existence of p adic Dirichlet L functions7 as in Section 34 of Hida Consider the rightmost term appearing in the de nition 94 of mm here k n m 2 kilig Ncd h ha 1 le z n1 a gt d l Bkililm Bllm 77 NCd d h1 7571 51 7ltNcgtk7172 Ne Bkl1 3 l1 d h1 kitil 61 where 95 follows from the distribution relation for Bernoulli numbers For each h 17 Ne7 write 9 haNe Let x be a formal variable and write u em Then the Bernoulli numbers are given by the power series 1amp9 1 00 Bs1 s 96 u 72Fh z 71 51 7 ELLIPTIC UNITS FOR REAL QUADRATIC FIELDS 337 where Fh 12 when h Nc and Fh 0 otherwise the error term Fh deals with the discrepancy between 1310 and B10 Similarly7 write 3d lidNC7 let y be a formal variable and write 1 59 we then have Utad 0 t1 y t 97 antlLNGVZZW 51 3 7 le t0 le where Gh is a constant in Z Multiplying 96 and 977 and summing over all h we obtain Nc U dd UQ 98 Huv gnd Uwi ah ui1 Fhgt 00 Ne ha hd 7 Bs1mBt1m s y t 99 i Z 2 31 t1l m lt5 39 st0 h1 le Note that the ilm terms from 96 have dropped out in 98 since summing 97 over all h gives the value 0 By the same reasoning7 we may replace uu jl in equation 98 de ning Huv by 7511 this will be useful in later computations Recalling that u em and v 59 we de ne the commuting differential operators 8 8 Duiuaiaandeivmia y Using 95 and 997 we then have lt17 Wquot Z ndH DMWT M g d N nm 1 1 NCVlm qD DTWZHVM Uluv117 where Huv Huv 7 Hupvp We thus nd that 100 7 n n a nquot z nmil z nmil Inm712m lt71gtltNcgt DMDJ Hu7vl11 Nchmmnv 7 Du 12HuUl11 If we de ne a change of variables 74712 247109102417 then Dw NcDu and Dz aDu 7 DE Hence we obtain 1W 12Huvl11 338 HENRI DARMON AND SAMIT DASGUPTA The following lemma will allow us to prove that these rational numbers lie in Zp LEMMA 412 Consider the subset R of Zpu1NCU1Nc de ned by Q Now R is a ring stable under the operators and R 5 where P Q 6 prN ulNc and Q1 1 e zg Proof The proof of this proposition follows exactly as in Lemma 342 of Hida7 except for the subtlety that we must check that Zpa1Ncv1Nc is stable under the given differential operators for this it suffices to check that for example ltDzgtltU1Nc 2wzaNc atAftwzl Nc7 n n 2 n which lies in Zpa1Ncv1Nc because p does not divide Ne similarly for the other cases El Thus to prove that Jmm E Z17 it suffices to prove that Hav is an element of R7 and for this it suffices to prove that Ha7 v E R Writing Ildv 11 Uld Ud71d7 we have U dd 1 7 19 d 101 anUIdilivilzndv a am am am 7 1 2le ndv dd l dW amt 39 UlNC 71 Since the numerator of the rightmost term in 101 is a polynomial in UlNc which vanishes when UlNc 17 the rightmost term itself is a polynomial in UlNc Since we are assuming that p does not divide Ne7 equation 101 then implies that U dd nd 6 R g Uld 7 1 Similarly one shows that 9 1 E R7 and it follows that Hav E B This ail concludes the proof of Lemma 410 46 A partial modular symbol of measures on Zp X Zp In this section7 we use the measures Zg to construct a partial modular symbol of measures on 2 x Zp encoding the periods of Fk Note that 2 x 2 is stable under the action of F0 ELLIPTIC UNITS FOR REAL QUADRATIC FIELDS 339 LEMMA 413 There exists a unique P0N inuariant partial modular sym bol 1 of Zp ualued measures on Z1 X Z17 such that S 102 h7ydVT a Shaw Re lt1 7pm hltz1gtFkltzgtdz ZPXZP r for rs 6 L007 and euery homogeneous polynomial hmy E Zzy of degree k 7 2 Proof Uniqueness follows from Lemma 411 we must show existence Let M denote the F module of degree zero divisors on the set Poo Let M C M be the set of divisors m for which there exists a Zp valued measure 1m on Zp x Zp such that hwy h 7ydl m7y Re 1 Pkizlmh271Fkzdzgt Here fm is de ned by fmim f5 and extended by linearity We must show that M M It is clear that M is a subgroup of M We will show that M is a P0N stable submodule Let m E M and 39y gt E P0N for compact B C D open U C Zp x Zp de ne mmw umltr1Ugt De ne a right action of P0N on the space of polynomials in two variables by Whey W490 By Cm Dy We calculate 103 hltuvgtdumltuvgt hltuvgtdumltr1ltuvgtgt ZPXZP ZPXZP hlwltx7ydvmz7y ZPXZP Relt17pHgt hl7lt21gtFkltzgtdz m Re 17pk2 hu1Fkudugt 39ym where equation 103 uses the change of variables u 39yz and the fact that Fklyil Fk Therefore7 M is a F0N stable submodule of M Lemma 410 shows that aNc 7 00 E M when p does not divide 0 Since the P0N module generated by these elements is all of M7 we indeed have 340 HENRI DARMON AND SAMIT DASGUPTA M M Furthermore7 the F0N invariance of 1 follows from uniqueness and the calculation of 103 above CI 47 From 2 x Zp to X In this section we show that the measures 1z 7 y of Lemma 413 are supported on the set X C Zp x Zp of primitive vectors LEMMA 414 Let as 6 Foo Then7 S ha y cw 7 Shaw Re hltz1gtFltzgtdz ZPXZ r for every homogeneous polynomial hmy E Zzy of degree k 7 2 Proof The characteristic function ofthe open set Zp XZ is limjH00 yp 1p7 For notational simplicity7 let 9 p 7 throughout the remainder of this section Then for 71mm 2 0 and k n m 27 we have 104 ZPXZ mnymdz 7gt 00 my 7 w w 320 120 7W 2 quot 4 X Z Dkg7171l1 NZd le 4 7122 w ed Meanwhile we calculate 105 Re lt 2ng z 12gt N4 13900 13900 Re dz 7 pkiniz ank dz L a N4 N4 12 yil 71 nd a 7 7 pa X E W Dk7l71l1 pk l 2Dk7l71l1 le ELLIPTIC UNITS FOR REAL QUADRATIC FIELDS 341 The following lemma implies that 104 and 105 are equal and nishes the proof El LEMMA 415 Let 375 2 0 For any rational number x in Q17 106 Dsgt me pklbsi nl Proof This essentially follows from the generalized Kummer congruences for Bernoulli polynomials Let x ac and assume rst that p does not divide 0 Let 1 denote an integer such that abp E 1 mod 0 Note that 107 Dstac69 1BS bpCBt c 1 Similarly DsgtaC Cs Z sgbPC tC 1 Dstpaccs 1 BszbCBtzc 1 Write y prc and y Kbc Since 09 a 1 it suffices to prove that 71ng Bsgy BM 7 2994350 For 3 gt 0 this follows from the proof of Theorem 32 of You which applies for our purposes even in the case 3 E 0 mod p E 1 For 3 0 the desired equality follows from the fact that the p adic L function Lps x for a Dirichlet character X is analytic at s 1 unless X 1 in which case Lp has a simple pole with residue 1 E 1p This completes the proof for the case x E Zp We now handle the case z Zp From equation 107 one sees that D9tac CHDMbpc Thus the result proved above is that 108 71320 DnsgbPC 5tsbPC 7 PsilDtsbC whenever p f c By switching indices in a similar fashion equation 106 for z abp becomes 109 jlirgjbpf itbnsw bp bp9 tljtsCbp psilbsitjm b where ac E 1 mod bp We will reduce equation 109 to equation 108 by means of the reciprocity law for these generalized Dedekind sums given in 342 HENRI DARMON AND SAMIT DASGUPTA Theorem 2 of Hal When I gt 0 the reciprocity law states t 3 75 110 bs tDts c b si n c 7 9 quotCHM t7 s c lt gt ltgt gogww 1 b D WW st 54071 9t 111 t Z WPDaba tcswljwsimAO 70 t 7 signc4 ifs t 1 0 otherwise Note that the sum in 110 is taken to be 0 ifs 0 We will call the terms in the sum on line 110 type 177 terms and those on line 111 type ll77 terms Using the Dedekind reciprocity law on each of the terms in 109 one easily checks that the desired limit holds for the type 1 terms by 108 The same is true for each of the type ll terms with a 0 s 25 To conclude the proof one checks that each of the type ll terms for a s t 1 s t g arising from the reciprocity law for bp99 tDtsgcbp has ordp greater than ordpg minus some constant depending only on s and 25 Thus in the limit the sum of these terms vanishes El We can now prove LEMMA 416 The measures 1r gt s are supported on X Proof Let 39y E P0N As in 103 above we calculate for a homogeneous polynomial hm y of degree k 7 2 112 AZPXZhmyd1r a smy Re he 1F11zdz Let m be a set of left coset representatives for P0NP0Np Then p1 U ViltZP X Z 11 is a degree p cover of X Hence from 112 we nd that 113 pmemlr a smy Re ltjhz1FlY1zdzgt Now p1 p1 2111 2 F117 epk2Fkip71gt i1 i1 19 1Fk Tka P Pk71Fk7 ELLIPTIC UNITS FOR REAL QUADRATIC FIELDS 343 since Fk is evidently an eigenform for T1 with eigenvalue 1pk 1 Thus 113 becomes hm y dVr a sxy Re 1 7pk72 s hz1Fkzdz X r Therefore7 the integral on X of any polynomial hzy equals the integral on Zp x Zp of hm7 y this implies that the measure 1r 7 s is supported on X CI 48 The measures a and P inuariance The compact open set X is a fundamental domain for the action of multiplication by p on Q127 7 Hence if we de ne for compact open U C X ar 7gt sU 1r 7gt sU7 then a extends uniquely to a P0N invariant partial modular symbol of Zp valued measures on Q127 7 0 which is invariant under the action of multi plication by p Mr H 8pU Mr H 8U for all compact open U C 127 7 Lemmas 4137 4147 and 416 show that a satis es properties 1 and 4 of Theorem 42 Furthermore7 property 3 is satis ed by construction Thus to complete the proof of Theorem 427 it remains to show that the partial modular symbol of measures a is P invariant LEMMA 417 The partial modular symbol a is invariant under Proof Since f is generated by TO N and P lt 15 it suf ces to 0 1 7 show that a is invariant for the action of P For a homogeneous polynomial hzy of degree k 7 27 we have 114 hm7yduP 1r 7 PlsltP1ltzygtgt 7 X r s hpuv d 7 711 Md mp plt gt Writing P lX as a disjoint union 19 0 71 PM a x mm 0 1 lt2 was and using the invariance of a under multiplication by 197 we see that 114 becomes 344 HENRI DARMON AND SAMIT DASGUPTA By the homogeneity of h one simpli es the above expression 2924147109147 v WED H 5 7w ue o memmdgs kmw Re pzik ipkiz hpz dz Hh hwmamwgt 115 Re 19241 i 1 710927 Upk leW d2 116 Re 1 7pm hu1Fkudugt where 115 uses the de nition of F and 116 uses the change of variables u p2 Since this equals the integral over X of hm y against the measure ar a s we nd that a is indeed invariant for the action of P McGILL UNIVERSITY MONTREAL QUEBEC CA Eemail address damen rnathmcgillca URL httpwwwmathmcgillcaNdannon HARVARD UNIVERSITY CAMBRIDGE MA Eemail address dasgupta mathharvardedu URL httpwwwmathharvardeduNdasgupta REFERENCES Ap T APOSTOL Modular Functions and Dirichlet Series in Number Theory Second edition Grad Texts in Math 41 SpringerVerlag New York 1990 BD1 M BERTOLINI and H DARMON The p adic Lfunctions of modular elliptic curves in Mathematics Unlimitedi2001 and Beyond 1097170 SpringerVerlag New York 2001 BDG M BERTOLINI H DARMON and P GREEN Periods and points attached to quadratic algebras in Heegner points and Rankin Leseries Darmon and S Zhang eds Math Sci Res Inst Publ 49 3237367 Cambridge Univ Press Cambridge 2004 CS J 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