Table Of ContentLecture 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
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© by Springer-Verlag Berlin Heidelberg 1977
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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 "
