Lecture Notes ni Mathematics Edited by .A Dold and .B nnamkcE Series: Mathematisches Institut der Universit~t Bonn Adviser: .F Hirzebruch 726 Modular Functions fo One Variable lV Proceedings International Conference, University of Bonn, Sonderforschungs- bereich Theoretische Mathematik July 2-14, 1976 Edited by .P-.J Serre dna .D .B Zagier galreV-regnirpS Berlin Heidelberg NewYork 1977 Editors Jean-Pierre Serre College de France 75231 Paris Cedex 05/France Don Bernard Zagier Mathematisches Institut der Universit~t Bonn Wegelerstr. 10 53 Bonn/BRD AMS Subject Classifications (1970): 10C15, 10D05, 10D25, 12A99, 14 H45, 14 K22, 14K25 ISBN 3-540-08530-0 Springer-Verlag Berlin Heidelberg NewYork ISBN 0-387-08530-0 Springer-Verlag NewYork Heidelberg Berlin This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, re- printing, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher. © by Springer-Verlag Berlin Heidelberg 1977 Printed ni Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2140/3140-543210 PREFACE This is the second and final volume of proceedings for the Conference on Modular Forms held in Bonn in July 1976. The first volume appeared as Lecture Notes ° n 601, under the title "Modular Functions of One Variable V" (cf. Lecture Notes ° n 320, 349, 350 and 476). Jean- Pierre Serre Don Zagier CONTENTS .M RAZAR, Values of Dirichlet series at integers in the critical strip I .C MOREN0, Analytic properties of Euler products of automorphic representations 11 J-P. SERRE, H.M. STARK, Modular forms of weight I/2 27 .H COHEN, .J OESTERLE, Dimension des espaces de formes modulaires 69 M-F. VIGNERAS, i Facteurs gamma et &quations fonctionnelles 79 .D ZAGIER, Modular forms whose Fourier coefficients involve zeta-functions of quadratic fields 105 .D ZAGIER, Correction to "The Eichler-Selberg trace formula on SL2(~)" 171 .H COHEN, A lifting of modular forms in one variable to Hilbert modular forms in two variables 175 .M EICHLER, Theta functions over Q and over Q(q~) 197 M-F. VIGNERAS, I S~ries th~ta des formes quadratiques zndeflnles 227 .S GELBART, Automorphic forms and Artin's conjecture 2#1 .S KUDLA, Relations between automorphic forms produced by theta-functions 277 .F HIRZEBRUCH, The ring of Hilbert modular forms for real quadratic fields of small diseriminant 287 A.N. ANDRiANOV, On zeta-functions of Rankin type associated with Siegel modular forms 523 List of adresses of authors 339 Raz-1 Values of Diriohlet Series at Integers in the Critical Strip by Michael J. Razar .i~ Introduction This note is primarily a summary of some recent work on the val- ues at integer points in the critical strip of Dirichlet series assoc- iated to newforms on F (N). The first such results seem to be due O to Shimura [9], who derived them for the Dirichlet series Z m(n)n -s n:l associated to the cusp form A(z) of weight 12 for the full modu- lar group. Somewhat later, Manin [4] extended these results to cusp forms of arbitrary integral weight for the full modular group. In the interim, Birch had introduced the ~'modular symbol" for cusp forms of weight two on F (N) and these were studied and used by Manin o [2] and [3], Mazur and Swinnerton-Dyer [5] and others. Recently, V. Miller in his thesis [6] extended the definition of the modular symbol to Fo(N). Just this year (1976), Shimura [ii], using totally different methods, has extended almost everything to F(N) and has obtained rationality results similar to those describe l below. The main result of the present note is Theorem .4 The proof consists of two main parts. The first is based on Shimura's isomor- phism between cusp forms and Eichler cohomology with real coefficients. In this respect it is similar to the techniques of Manin [4]. The second part is the interpretation of the coefficients of an Eichler cocycle as residues at poles of the Mellin transform of a multiple integral of the corresponding cusp form. This is based on the Hecke correspondence (Preposition i) and on an additive character analog of Weil's theorem (Proposition 2). Well has also developed such a pro- cedure recently in a paper delivered at the Takagi conference (1976). International Summer School on Modular Functions Bonn 1976 Raz-2 2 One advantage of proceeding in this way is that the same methods are applicable in other settings. For example, they work for Eisenstein series. This is discussed briefly in .4§ Detailed proofs of everything discussed here will appear in [7] and [8]. It is convenient in the present note to stick to cusp forms of Hauptypus for Fo(N) since the Shimura isomorphism only makes sense for eusp forms with respect to a real character. However, in [8] a modified version of the Shimura isomorphism is used to prove analogous results for arbitrary cusp forms of Nebentypus. .2§ The Shimura isomorphism We begin with a brief discussion of the Eichler cohomology and its relationship to the space of cusp forms. Fix a Fuchsian group of the first kind G e SL2(IR), an integer k, and a real character v of G. Assume G contains translations. If f is a function on the upper half plane, define (flk,v~)(z) = v(~)(cz + d) -k f\c-~]' ~ = Let K be a subfield of ~ such that G c SL2(K ) and denote by X(K) = Xk_2,v(K) the space of polynomials of degree at most k-2 with coefficients i~ K and G-module structure given by (i), but with k replaced by 2-k. HI(G,X(K)) and ZI(G,X(K)) are respectively the first cohomol- ogy and first oocycle groups of G with values in Xk_2,v(K). Let HI(G,X(K)) be the subspaoe of HI(G,X(K)) consisting of those co- homology classes whose restriction to every (cyclic) parabolic sub- group of G is trivial. Let ZI(G,X(K)) consist of those cocyoles :P a~-~P~(z) such that if ~ ( G is parabolic, then for some Q (X(K), P = llQ 2-k, v~ - Q and such that furthermore, if ~ is a 3 Raz-3 translation, then P = .0 The only coboundaries in ZI(G~X(K)) are a the constant multiples of the eoboundaries a~iJa - I. Thus dim ~i = dim ~i _ .i Let Sk,v(G) be the space of eusp forms and let k be the pos- itive number such that i { k~ generates the group of translations in \ 0 1 I G. Let f 6 Sk,v(G) and suppose that f(z) has the Fourier expan- sion 2~inz k f(z) : [ane (2) n:l Denote by f*(z) the (k-l)-fold integral of f(z) given by 2ninz f*(z) : ~ a n e (3) n;I Let Pa = f* - f*I2_k,v G . , Then Pa E X(¢) for all a ( G and so P (ZI(G,X(~)). Define homomorphisms 6o:Sk,v(G)--. ZI(G,X(¢)) and 5:Sk,v(G)--~ HI(G,X(~)) by 5of = P and 5f = cohomology class of 5of. (4) The maps 5 and 5 are injective and the image of 5 has di- O mension (over ¢) equal to half that of HI(G,X(¢)). To get an iso- morphism, Shimura ([9] and [i0]) defines maps Po and p from Sk(G) to ZI(G,XaR)) and HI(G,XGR)) respectively, by letting po f be the "real part" of the cocycle 6of. ("Real part" here applies only to coefficients of polynomials, not to the variable z.) Then pf is the cohomology class of po f and the map p is an iso- morphism. Furthermore, p commutes with the action of Heeke opera- Raz-h 4 tots (double cosets) on Sk(G) and HI(G,X(K)) and thus is an iso- morphism of Hecke modules. 0- and suppose that e normalizes .G Define the action of E on functions f on the upper half plane by (fls)(z) : f(-Z). (Note that c is only ~-linear, not { linear.) If f(z) has a Fourier expansion (2)~ then fle = f if and only if the Fourier coefficients n a are real and fIs = -f if and only if (if) sI = .f The action of s on a cocycle P is given by (PIE) = Psa -lit and this action induces an automorphism of order two on HI(G,XaR)). Denote the eigenspaces of s corresponding to eigen- values ±I by Z~(G,XGR)) and HI(G,XCIR))+ and let Sk(G)~.) be the space of cusp forms with real coefficients. Since po(fls) = (pof)ls, p restricts to isomorphisms from Sk(G)(R) to H~(G,X(IR)) and iSk(G)~R) to HI(G,X(19))._ If f (Sk(G)(19) , then P : f* - f*la : P+ + iP- (5) I< G where P = po(f) E Z (G,X k 2 (19)) and Pa - : Po (-if) ( ZI(G'Xk _ 2,v aR)). G ,V -- In the case of F (N) we can take advantage of the fact that P O is a Hecke-module isomorphism. The action of the Hecke operators Tp, Uq, Wq (see [I]) on Fo(N) can be used to break Sk(Fo(N)) into spaces of newforms and oldforms and the space of newforms breaks up into one-dimensional eigenspaces with distinct families of eigen- values for the T . This decomposition is carried over by p into P HI(G,X(~)). In fact it is carried over by Po into ~I(G,X(R))~ since the coboundaries lie in the same eigenspaee for the Heeke alge- bra as an Eisenstein series for F (N). Note that ZI(G,XaR)) has O a basis in ZI(G,X(Q)) and this latter space is preserved by the Hecke operators. Finally, by the theory of Atkin-Lehner the newforms 5 Raz-5 are actually cusp forms on G : F *(N), the group generated by O Fo(N) and the involutions Wq, qlN. Thus we get the following theorem. Theorem i: Let f(z) be a newform (of Hauptypus) of weight k on -k+l F (N). Suppose f(z) = Z a e 2~inz , I a = i and f*(z) = (2~i) o n:l n a n -k+l 2~inz e Let K be the (totally real) field generated n:l n + over ~ by the a . There exist real numbers w and w (depend- n ing only on f) such that for all a ( F *(N), there are polynomials O A:(z) and A~(z) of degree at most k-2 with coefficients in K such that (f* - f*I2 ka)(z) : w + +A (z) + iw A (z). (6) 3. Coefficients of the Eiehler cocycles In order to apply Theorem i, we must relate the coefficients of + the polynomials A-(z) to f. The principal tool used is the Hecke correspondence between Fourier series and Dirichlet series via Mellin transform as described in the following Proposition. Proposition i. Let k > 0, f(z) = ~ an e2~inz/k, g(z) = n=O b e 2~inz/x, ~(s) : Z a n -s, ~(s) = ~ b n -s, ~(s) = (2~/k) -s n:0 n n:l n n:l n r(s)9(s), Y(s) : (2~/x)-SF(s)9(s). Assume that for some real num- ber c, the complex numbers an, n b satisfy an, n b : 0(nC). Let y, k ( C. The following are equivalent. A. 9(s) : yZ(k-s) and there is a rational function R(s) such that }(s) - R(s) is entire and bounded in vertical strips (EBV). )( B = Res ) z-Sds ' where the sum is 6 over the poles of R(s). Next, observe that differentiation of f(z) essentially corre- sponds to changing ~(s) to ~(s-l). In general this leads to a more complicated functional equation. However, let k be an integer, k ~ 2 and let Q*(s) : ~(s+k-l) and ~*(s) : ~(s+k-l). In this case, if @*(s) = (2~/k)-sF(s)~*(s) and y*(s) = (2~/k)-sF(s)@*(s), and if @ and Y satisfy condition A of Proposition i, then @*(s) : (-l)k-Iyy*(2-k-s). (7) In addition, there is a rational function R*(s) such that @*(s) - R*(s) is EBV. A residue computation yields the following Theorem. Theorem 2: Let f(z) = [ a e 2~inz/k g(z) = ~ b e 2~inz/k n:O n ' n:O n ' k-1 f*(z) : (k-l)' + a n e , • n:O n k-i ( (k i) boZ -k+l 2~inz/X g*(z) -- (k-l): + ~ b n e and n=O n ~(s) = Z a n -s If k is a positive integer and Y a complex num- n:l n ber such that f(z) = y -k g - ~ i , ~ ±~ then \ z! f*(z) = ¥zk-2g, _ + ~ ~(k _-j) j=0 Let G be a subgroup of SL2(I~) , v a character of G and k a positive integer. Let ~ = (a b) ~ G with c # 0. If f is any function on the upper half plane, define f by Z d) f (z) : f l~l c "