Topics in classical automorphic forms

Henryk lwaniec Graduate Studies in Mathematics Volume 17 I@ .J>f';o;tll\t.4;,,,, o; • • ~ American Mathematical Soci...

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Henryk lwaniec

Graduate Studies in Mathematics Volume 17

I@ .J>f';o;tll\t.4;,,,,

o; • • ~

American Mathematical Society

Selected Titles in This Series 17 Henryk Iwanlec, Topics in classical automorphic forms, 1997

16 Richard V. Kadison and John R. Ringrose, Fundamentals of the theory of operator algebras. Volume 11: Advanced theory, 1997

15 Richard V. Kadison and John R. Ringrose, Fundamentals of the theory of operator algebras. Volume I: Elementary theory, 1997

14 Elliott H. Lieb and Michael Loss, Analysis, 1997 13 Paul C. Shields, The ergodic theory of discrete sample paths, 1996 12 N. V. Krylov, Lectures on elliptic and parabolic equations in Holder spaces. 1996 11 Jacques Dixmier, Enveloping algebras, 1996 Printing 10 Barry Simon, Representations of finite and compact groups. 1996 9 Dino Lorensinf, An invitation to arithmetic geometry, 1996 8 Winfrled Just and Martin Weese, Discovering modern set theory. 1: The basics, 1996 7 Gerald J. Janusz, Algebraic number fields, second edition, 1996 6 Jens Carsten Jantsen, Lectures on quantum groups. 1996 5 Rick Miranda, Algebraic curves and Riemann surfaces, 1995 4 Russell A. Gordon, The integrals of Lebesgue, Denjoy, Perron, and Henstock. 1994

3 William W. Adams and Philippe Loustaunau, An introduction to Grobner bases, 1994

2 Jack Graver, Brigitte Servatius, and Herman Servatius, Combinatorial rigidity. 1993 1

Ethan Akin, The general topology of dynamical systems. 1993

Topics in Classical

Automorphic Forms

Topics in Classical

Automorphic Forms Henryk Iwaniec

Graduate Studies in Mathematics Volume 17

American Mathematical Society Providence, Rhode Island

Editorial Board James E. Humphreys (Chair) David J. Saltman David Sattinger Julius L. Shaneson 1991 Mathematics Subject Classification. Primary 32Nxx.

Library of Congress Cataloging-in-Publication Data Iwaniec, Henryk. Topics in classical automorphic forms / Henryk Iwaniec.

p. cm. - (Graduate studies in mathematics ; v. 17) Includes bibliographical references and index. ISBN 0-8218-0777-3 (hardcover : alk. paper) 1. Automorphic forms. I. Title. II. Series. QA243.I95 1997 97-24332

512'.7-dc21

CIP

Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication (including abstracts) is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Assistant to the Publisher, American Mathematical Society, P. 0. Box 6248, Providence, Rhode Island 02940-6248. Requests can also be made by e-mail to reprint -permission®ams.org.

© 1997 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America.

® The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability.

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Contents

Preface

Chapter 0. Introduction Chapter 1. The Classical Modular Forms 1.1. Periodic functions 1.2. Elliptic functions 1.3. Modular functions 1.4. The Fourier expansion of Eisenstein series 1.5. The modular group 1.6. The linear space of modular forms Chapter 2. Automorphic Forms in General 2.1. The hyperbolic plane 2.2. The classification of motions

2.3. Discrete groups - Fuchsian groups 2.4. Congruence groups 2.5. Double coset decomposition 2.6. Multiplier systems 2.7. Automorphic forms 2.8. The eta-function and the theta-function

xi

1

3 3 6 10 13 15

18

23 23

27 29

34 37 40 42

44

vii

viii

Contents

Chapter 3. The Eisenstein and the Poincare Series 3.1. General Poincare series 3.2. Fourier expansion of Poincare series 3.3. The Hilbert space of cusp forms

47

Chapter 4. Kloosterman Sums 4.1. General Kloosterman sums 4.2. Kloosterman sums for congruence groups 4.3. The classical Kloosterman sums 4.4. Power-moments of Kloosterman sums 4.5. Sums of Kloosterman sums 4.6. The Salie sums

57

Chapter 5. Bounds for the Fourier Coefficients of Cusp Forms 5.1. General estimates 5.2. Estimates by Kloosterman sums 5.3. Coefficients of cusp forms with theta multiplier 5.4. Linear forms in Fourier coefficients of cusp forms 5.5. Spectral analysis of the diagonal symbol

69

Chapter 6. Hecke Operators 6.1. Introduction 6.2. Hecke operators Ttz 6.3. The Hecke operators on periodic functions 6.4. The Hecke operators for the modular group 6.5. The Hecke operators with a character 6.6. An overview of newforms 6.7. Hecke eigencuspforms for a primitive character 6.8. Final remarks

91

Chapter 7. Automorphic L-functions 7.1. Introduction 7.2. The Hecke L-functions 7.3. Twisting automorphic forms and L-functions 7.4. Converse theorems

47 49 52

57 58

59 61

65 66

69 72

75 81

83

91

92

94 98 101

107 108 118

119 119 120 124 126

Contents

Chapter 8. Cusp Forms Associated with Elliptic Curves 8.1. The Hasse-Weil L-function 8.2. Elliptic curves Er

8.3. Computing \(p) 8.4. A Hecke Grossencharacter 8.5. A theta series 8.6. The automorphy of f

ix

133

133 136 138 141

142

143

Chapter 9. Spherical Functions 9.1. Positive definite quadratic forms 9.2. Space spherical functions 9.3. The spherical functions reconsidered 9.4. Harmonic analysis on the sphere

147

Chapter 10. Theta Functions 10.1. Introduction 10.2. An inversion formula 10.3. The congruent theta functions 10.4. The automorphy of theta functions

165

10.5. The standard theta function

147 150 155

159

165

166 168 175 176

Chapter 11. Representations by Quadratic Forms 11.1. Introduction 11.2. Siegel's mass formula 11.3. Representations by Eisenstein series and cusp forms 11.4. The circle method after Kloosterman 11.5. The singular series 11.6. Equidistribution of integral points on ellipsoids

179

Chapter 12. Automorphic Forms Associated with Number Fields 12.1. Automorphic forms attached to Dirichlet L-functions 12.2. Hecke L-functions with Grossencharacters 12.3. Automorphic forms associated with quadratic fields 12.4. Class group L-functions reconsidered 12.5. L-functions for genus characters 12.6. Automorphic forms of weight one

203

179 180 185

190 196 199

203 206 211 215

219

224

x

Contents

Chapter 13. Convolution L-functions 13.1. Intro duction 13.2. Ran kin-Selberg integrals 13.3. Selb erg's theory of Eisenstein series 13.4. Stat ement of general results 13.5. The scattering matrix for F0 (N) 13.6. Func tional equations for the convolution L-functions 13.7. Met aplectic Eisenstein series 13.8. Sym metric power L-functions

231

Bibliography

255

Index

257

231 232 235

240 240 243 246 248

Preface

Automorphic forms are present in almost every area of modern number theory. They also appear in other areas of mathematics and in physics. I have lectured on these topics many times at Rutgers University with only slight overlapping of content, and I still have new material to teach that is important. It is indeed a vast territory which cannot be grasped by any one person. While research publications on automorphic forms are rapidly increasing in quantity and quality, the demand for textbooks, particularly on the graduate level, is also growing. There are fine books on this subject such as [Lan], [Miy], [Sh2], but still more are needed, especially those which favor analytic methods. The present book is based on my lecture notes (almost verbatim except for Section 5.5) from a graduate course which I delivered in the Fall of 1994 _

and in the Spring of 1995 at Rutgers. The course was formulated, as the title implies, to acquaint our new students with the subject matter from various perspectives. Thus I have not followed direct or traditional paths, but rather I have frequently ventured into areas where different ideas and methods mix and interact. To cover a lot in a limited time, some material is necessarily presented as a survey. For example, the numerous connections

of automorphic forms with L-functions of number fields are discussed in Chapter 12 without details. However we do provide complete proofs of the most basic results in the earlier sections. An experienced reader will find some of our arguments to be nonstandard. It would be pointless to argue which approach is better, since our choice was made simply for the purpose of showing different possibilities. For example, our presentation of the theory of Hecke operators in Chapter 6

xi

xii

Preface

is completed for primitive characters quickly by establishing the multiplicityone principle using Gauss and Ramanujan sums instead of lengthy considerations of inner products. Of course, this is only a special case (the whole space is spanned by newforms), but it is an important case.

We pay great attention to detail in the subjects of theta functions and representations by quadratic forms (Chapters 10 and 11) because these are not sufficiently covered in textbooks, despite having a long history of research.

Because the original notes where written as the course was progressing, inevitably some redundancy has occurred. Nevertheless we have decided not to eliminate this redundancy, because it offers the option of selective reading. For example, our account of the Shimura-Taniyama conjecture for special curves (the congruent number curves) is self-contained in Chapter 8, even though one could instead appeal to the later chapters on general theta functions.

Sergei Gelfand, Peter Sarnak and others have convinced me that these lecture notes might be useful for a large number of graduate students, and they have urged me to publish them. I would like to thank them for their encouragement. I am grateful to W. A. Gonzalez, C. L. Hamer and C. J. Mozzochi for helping me in the technical preparation of the original notes. Special thanks are expressed to T. Khovanova for corrections and improvements which she contributed when editing these notes for publication. Henryk Iwaniec

Chapter 0

Introduction The concept of an automorphic function (the name was given by F. Klein, 1890) is a natural generalization of a periodic function. Let X be a locally compact space acted on discontinuously by a group IF. Then a function f : X --j C is automorphic with respect to IF if

f(yx)= f(x)

for all ryEI'. In other words, f lives on the quotient space r \ X (the space of orbits). A typical habitat for automorphic functions is the homogeneous space X = G/K of a Lie group G, where K is a closed subgroup. In this case X is a Riemannian manifold, so the differential calculus on X is available. If (0.1)

an autom.orphic function f is a common eigenfunction of the whole °a:Tgebra

D of invariant differential operators on X (D is a commutative algebra; it contains he Laplace-Beltrami operator), (0.2)

Df=)f(D)f forallDED,

then f is called an automorphic form. More generally, one considers automorphic functions for which the transformation rule (0.1) is modified by a differential factor and a suitable multiplier system. Those functions which live on X = SL2 (R) /SO2 (R) and which are complex-analytic (X is identified with the upper-half plane M C C and the differential equations (0.2) are the Cauchy-Riemann equations) are called classical automorphic forms. In this book our main interest will be in the classical automorphic forms with respect to a congruence group, because of their importance for applications to arithmetic. On a few occasions we shall encounter the real-analytic Eisenstein series, and for clarity, we introduce basic concepts in the context of a general group acting discontinuously on the upper-half plane. 1

2

Introduction

There are various books on spectral aspects of automorphic functions, but none covers and treats in detail as much as the expansive volumes by Dennis Hejhal [Hell. I recommend them to anyone who is concerned with research. In these books one also finds a very comprehensive bibliography. Those who wish to learn about the theory of automorphic forms on other symmetric spaces in addition to the hyperbolic plane should read Audrey Terras [Te]. A broad survey with emphasis on new developments is given by A. B. Venkov [Ve].

Chapter 1

The Classical Modular Forms

We begin by giving a glimpse of how automorphic forms emerged on the hyperbolic plane as descendants of those living on flat spaces.

1.1. Periodic functions Take X = R and IF = Z, so R/Z is the circle. A function f : IR -a C is periodic of period 1 if

f (x + n) = f (x)

for all n E Z.

To construct a periodic function one can simply take any f on the segment [0, 1) and extend its values uniquely to R by requiring periodicity. For example {x} = x - [x] (the fractional part of x) is obtained from f (x) = x on [0, 1). Another construction uses the averaging method. Let f : 1[8 --+ C be any function of rapid decay at ±oo so that the series (1.1)

g(x) = > f(x + n) nEZ

converges absolutely. Clearly the function g is periodic of period 1. The exponential function (1.2)

e(x) = e272x = cos 2-7rx + i sin 27rx 3

1. The Classical Modular Forms

4

plays a special role in the linear space of periodic functions. Any periodic and piecewise continuous function f : III -f C has the Fourier series representation (1.3)

f (x) = 1] a,e(nx) nEZ

with coefficients given by

f

an =

(1.4)

f (x)e(-nx)dx.

If f is continuous, the Fourier series (1.3) converges absolutely, hence uniformly.

Exercise. Prove that the Fourier expansion for the fractional part of x is given by the series

{x} =

(1.5)

1

2

°O sin 27rnx

-E n=1

7rn

which is boundedly convergent (the partial sums are uniformly bounded), but not absolutely. Precisely, we have N

{x} = 2 -

sin 27 nx

+

0((1 +

IIxIIN)-1),

ra=1

where jjxjj denotes the distance from x to the nearest integer and the implied constant is absolute.

Developing (1.1) into Fourier series, we find by unfolding the integral (1.4) that an = J

f(x)e(-nx)dx = f(n)

D where R Y) denotes the Fourier transform of f (x) on ]I8; (1.6)

f(y) =

I1f6_xY

)dx.

Therefore the Fourier expansion (1.3) appears as (1.7)

E f (x + n) = > f (n)e(nx). nEZ

nEZG

1.1. Periodic functions

5

If we set x = 0, this becomes the Poisson summation formula

1: f(n)=1: f(n).

(1.8)

nEZ

nEZ

The above argument is valid if f and f have rapid decay at ±oo. As an example, take the function h(x) = e-,x2, which is self-dual with respect to the Fourier transform, namely h(y) = h(y). Therefore, changing variables, for any v > 0 we have the Fourier pair f (x) = e-"x2", f (y) _ -ny2v-' v- 2e . In this case (1.7) yields e-ir(x+n)2v

(1.9)

= v- 2

nEZ

E nEZ

e-7rn2v-'

e(nx).

Exercise. Using (1.8), derive the following formula of Dirichlet: -N 2 e E (9) n (mod N) \

(1.10)

eNV lv,

where EN = 1 1 z

, 2

which is valid for any positive integer N. Here are some suggestions. Assume for simplicity that N is odd. First prove that the sum (1.10), can it G(N), satisfies the asymptotic formula G(N) ' ENVrN as N -- oo, which is easier to do because G(N) can be smoothed before applying the Pois-

son summation. Then show by elementary arguments that G(N2) = N and G(N3) = NG(N). Hence the asymptotic: formula G(N) " ENVW improves itself to the exact formula (1.10) by considering the sequence Nk with k = 1, 2, ... (see [Dav] for a direct derivation).

The theory of periodic functions generalizes with obvious modifications to any given dimension. The space X = Rk is acted on by I' = 7Lk as integral vector translations. The Poisson summation formula asserts that

f(n) _

(1.11) nEZk

f(n) nEVC

where

(y) = ff(x)e(_xY)dx Rk

and xy = xlyl +

+ xkyk is the scalar product. To ensure convergence it suffices that f (x) is in the Schwartz class, i.e., all its partial derivatives

satisfy

f(j)(x) << IxI-A

where 1xI _ (xl + ... + xk) 2

and A > 0 is arbitrary, the implied constant depending on A and (j) _ (j1,

,jk)

1. The Classical Modular Forms

6

Exercise. Show that if f (x) is a distance function, then so is f (y); more precisely, if

f (x) = g(IxI) then

f(y) = h(IyI) where h(y) is a Hankel type transform of g(x) given by (1.12)

h(y) = 27ry1

2 fg(x)J_i(2xY)xdx

and JQ(x) is the Bessel function. Hence the Poisson summation formula becomes cc

(1.13)

oc

E rk(m)g(m) = 1] rk(m)h(m) rn=O

mn.=0

where rk(m) denotes the number of representations of m as the sum of k squares;

rk(m) = I {(ni, ... , nk) E Zk : ni + ... + nk = m}I.

1,.2. Elliptic functions The next generation of periodic functions lives in the space of complex numbers X = C. This is not simply the former case in two dimensions. Here C is considered as a Riemannian manifold with complex structure. Consequently the theory of meromorphic functions can be employed, which is indeed a very powerful tool. For instance, one can establish interesting relations, even of arithmetic type, by examining the location of poles and zeros.

We shall use the standard complex number notation z = x + iy. The group Z acts on C giving periodic functions in the horizontal direction, such as

e(z) = e27rz4 = e(x)e-2'ry.

To produce periodicity in any two described directions, consider a group A = w1Z + w2Z

where W1, w2 are complex numbers linearly independent over 1[8 so that C = w11l + w2lR.

1.2. Elliptic functions

7

W1+ W2

Figure 1. A parallelogram of a lattice

Thus A is a discrete, free abelian subgroup of C of rank 2, i.e., A is a lattice (see Figure 1). A function f : C ---> C is elliptic with respect to A if f is meromorphic on C - f is periodic with periods in A, (1.15)

f (u + w) = f (u)

for all w E A.

Clearly for the latter condition it suffices that f has two periods w1i w2 which generate A. An elliptic function can be regarded as a meromorphic function on the torus C/A. The elliptic functions for a lattice A form a field, say E(A). If f has no poles then it is holomorphic and bounded on C, so by Liouville's theorem f is a constant function. Choose a fundamental parallelogram for the lattice A, say

P = t1w1 + t2w2 + µ,

0 < ti t2 < 1,

with fi E C such that f has neither poles nor zeros on the boundary DP. Integrating along the boundary, we get by Cauchy's theorem (1.16)

E resf(w) = 0 WEP

because the integrals along the opposite sides cancel out by periodicity. Hence there is no elliptic function having exactly one simple pole (mod A). The order of f at w is the integer m = m f (w } such that f(u) = (u - w)mg(u)

where g(u) is meromorphic with g(w) 0, oc. Since f (u) is A-periodic, so is mf(w). Notice that mf(w) 0 for at most a finite number of points in a

1. The Classical Modular Forms

8

given parallelogram. Applying (1.16) to the elliptic function f'(u)/ f (u), we get

E m f(w) = 0.

(1.17)

w (mod A)

Similarly integrating uf'(u)/f(u) along the boundary 8P we get

E mf(w)w - 0 (mod A).

(1.18)

w (mod A)

Define the order of f to be the sum of orders of zeros (mod A) or the negative of the sum of orders of poles (mod A), i.e., (1.19)

rf=

E max(mf(w), 0) = -

E min(mf(w), 0). w (mod A)

w (mod A)

It follows from (1.16) that there is no elliptic function of order 1. From (1.17) applied to f (u) - c it follows that any complex number c is a value of f (u) for exactly r f points (mod A) counted with multiplicity. The simplest elliptic function which is not constant has one pole of order two with zero residue at the origin. Such a function can be constructed by averaging over the lattice; for example, one obtains (1.20)

P(u) = u-2 + E' ((u -

w)-2

- w-2)

wcA

where i indicates that w = 0 is skipped in the summation. Notice that P(u) is even, i.e., yo(u) = p(-u). Since p(u) has order 2 we infer that (1.21)

yo(u) = P (W)

u - ±w (mod A).

Moreover yo(u) - p(w) has a double zero at u = w exactly when w - -w (mod A). There are three such points (mod A), namely wl/2, w2/2 and (J1 + w2) /2 (the point w = 0 is not because it is a pole of p). The numbers (1.22)

rrwlll

_

(W2\l

_

Cwt +W21 2

ll

are all distinct because of (1.21); thus the associated discriminant (1.23)

0 = 16 (el - e2)2 (e2 - e3)2 (e3 - el)2

h, not zero. The function yo(u) is called the Weierstrass function.

1.2. Elliptic functions

9

The Weierstrass function is basic in the sense that it generates the subfield C (p) of all even elliptic functions in E(A). Indeed if f c E(A) is even, then by comparing singularities (the zeros and the poles) we find that

fi

f(u)=c

'(w))"`r(w)

(p(u) -

w (mod A)

where c is a constant. Here the product takes only one singular point of f out of each pair ±w (mod A). Differentiating p(u) we obtain the elliptic function (u) _ -2 E(u - C.J)-3

(1.24)

wEA

which is odd, p'(u) _ -p'(-u). Therefore the field of all elliptic functions for a lattice A is generated by p and p', E(A) = C (p, p')

(1.25)

.

The functions g(u), p'(u) are linearly independent over the constant field C, but they are algebraically related through the Weierstrass equation (1.26)

89 (u)2 = 4 (6o(u) - el) (p(u) - e2) (P('u) - e3)

with el, e2, e3 given by (1.22). To derive this equation recall that P(u) - el has exactly one double zero at u = wi/2, which is also a simple zero of p'(u). The same holds true for the other factors. Therefore the ratio .f (u) =

01

(u)2 (p(u)

- el) 1 (P(u) -- e2)-1 (8J(u) - e3)-1

is an elliptic function whose only possible pole (mod A) is at the origin. But

u = 0 is a regular point of f (u) since

(u)

-

-2u-3 and yo(u) - u-2 as

u --> 0. Therefore f (u) is constant, in fact f (u) = 4 as claimed. From (1.20) one derives easily that oc

(1.27)

P(u) = u-2 + E (m + 1)G7n+2urn rn.=1

where (1.28)

Gk

El L4)-k. wEA

10

1.

Harmonic Analysis on the Hyperbolic Plane

Figure 2.. Geodesics in H.

The hyperbolic circles (loci of points at a fixed distance from a given point in IHI) are represented by the euclidean circles in liii (of course, not with

the same centers). There are various interesting relations in the hyperbolic plane. For instance, the trigonometry for a triangle asserts that sin a sinh a

_

sin /3

_ sing

sinh b

sinh c

and

sina sin,3 coshc= cosa cos,3 +cos7, where a, p, y are the interior angles from which the sides of length a, b, c are seen, respectively. The latter relation reveals that the length of sides depends only on the interior angles. More counter-intuitive features occur with the area. To define area, one needs a measure. The riemannian measure derived from the Poincare differential ds = y-11dzI on IH[ is expressed in terms of the Lebesgue measure simply by (1.12)

duz = y-2 dx dy.

It is easy to show directly by (1.8) and (1.11) that the above measure is G-invariant.

Theorem 1.3 (Gauss defect). The area of a hyperbolic triangle with interior angles a, ,Q, y is equal to (1.13)

7r-a-0 -y.

1.3. Modular functions

11

that two pairs of complex numbers (linearly independent over R) determine the same lattice if they are connected by a iinirnodular transformation wi = awl + bw'2 w2 =cwl+dw2

where a, b, c., d E Z with ad - be = ±1. This is so because the matrix of coefficients } = C a d) is invertible; indeed -y-1 = ± ( d ab) has integer entries. We can interchange the generators w1, w2, if necessary, so that Imwl/w2 > 0, i.e. we can require the point z = w,/w2 to be in the upper half-plane

IHI={z=x+iy:xEIR,yEI18+}.

(1.36)

Then the action of unimodular transformations on lattices A = w1Z + w2Z corresponds to the action of the modular group (1.37)

SL2(Z) _

a

b I

: a, b, c, d E Z, ad - be = 1 I

on IHI by the linear fractional transformations (1.38)

yz = a + d

if y - +

a

d-

Suppose f (u; wl, w2) is an elliptic function in u for a lattice A = w1Z + Suppose w2Z which is homogeneous in r and A of degree -k E Z, i.e.,

f (au; Awl, ,\w2) = A-" f (u; WI , L02) for any A E C, ,\

0. Then we can write /

f (u, wl, w2) = w2

kf

uw w2 w'2

where f (v; z) is a function on C x H. As a function of z it has the property (1.39)

f (yz) = (cz + d)' f (z)

for all -y c SL2(Z).

A function f : IHI -> C satisfying the transformation rule (1.39) is called a modular function of weight k. Note that k must he even, otherwise f (z) vanishes identically. Clearly any modular function is obtained from a homogeneous function of lattices in the above Fashion.

1. The Classical Modular Forms

12

In particular we obtain Gk = Gk (w1, w2) = w2 kGk (w1/W2) where (1.40)

Gk(z) _ E (mz + n)-k (r..n)#(0,0)

provided k > 2 for convergence. This is a modular function of weight k, called the Eisenstein series. Arranging terms in accordance with the greatest common divisor d = (m, n), we can pull out from Gk(z) the following constant factor: (1.41)

((k) = E c'O

d-k

d=1

This is a special value of the Riemann zeta-function. For k even we have (1.42)

((k) =

- (2k!

k

Bk

where Bk are the Bernoulli numbers; they are defined as the coefficients in the power series k

e'x-1

k=0

Then we define the normalized Eisenstein series Ek(z) = Gk(z)/2( (k), which is

(1.43)

Ek(z) = 2 E (mz + n) -k. (rn,n)=1

From the discriminant function (1.34) we get a modular function of weight twelve: (1.44)

A (z) = 92(Z)3 -2793 (Z)

2

where 92(z) = 60G4(z) and 93(z) = 140G6(z). Therefore the function '1.45)

j(z) _

(1292(z))30(z)-1

zas weight zero. The finction j (z) plays an important role in the theory of elliptic curves.

A modular function which is meromorphic on IHI and at oc is called a :Nodular form (later we shall require somewhat stronger analytic conditions). The requirement that f (.z) be meromorphic at z = oo needs a few words

of explanation. The map z H e(z) transforms C onto C* = C \ {0}, and

1.4. The Fourier expansion of Eisenstein series

13

since e(z) is invariant under the translation Tz = z + 1 this map establishes an analytic isomorphism between the strip C/T and the punctured plane C*. Hence a meromorphic function f (z) on IH[ which is T-invariant (periodic of period 1) can be written as f (z) = g(e(z)), where g(q) is meromorphic in

C*. We say that f (z) is meromorphic at z = oc if g(q) is meromorphic at q = 0. Then the power series expansion of g(q) at q = 0, say

g(q) _

angn,

0,

ant

n=m

reads as the Fourier expansion of f (z) at z = oo. cc

f (z) = >] ane(nt). n=m

The smallest integer m = mf(oe) with am 0 is called the order of f at 00, and the complex numbers an with n > m are called the Fourier coefficients of f at oo. Similarly we define the order of f at points w E IHI, say m = m f(w), by examining the Taylor expansion 00

f (z) _ 1: an(z - w)n,

ant

0.

n=m

1.4. The Fourier expansion of Eisenstein series We begin by the well-known product representation for the sine function (which is easy to establish by comparing zeros and applying Liouville's theorem)

x sin7rz=7rz11

z

(1--n ) (1

n=1

z n

Take the logarithmic derivative 7

cos 7rz

sin 7Z

+E( z-n +_z+n ) z 1

1

1

On the other hand, this is equal to 7ri

e(z) + 1 e(z) - 1

= 7ri +

27ri e(z) - 1

00

e(dz).

= 7ri - 27ri d=O

1. The Classical Modular Forms

14

'Ale differentiate these expansions k - 1 times, getting 00

(z - n)-k

(1.46)

00

29ri k

=

n=-oc

(k - 1)1

E dk

le(dz)

d=1

for any k > 2. Hence we derive the Fourier expansion of Gk(z) as follows 00

00

(mz + n)-k = 2((k) + 2 E

Gk(z) = (m,n)54(O,O)

= 2((k) +

(mz + n) -k

-m.=1 n=-o0 (k2-

oo 1)iE k

x

Edk le(dmz),

d,rn.= 1

and by collecting terms with dm = n we arrive at

Gk(z) = 2((k) + 2

(1.47)

(27ri) k

ak-l (n)e(nz).

F (-k-)

n=1

Here a5 (n) is the arithmetic function given by

a,(n) = Eds.

(1.48)

din

For the normalized Eisenstein series Ek(z) = Gk(z)/2((k) we get (by (1.42))

Ek(z) = 1 - 2kBk 1 Eak_1(n)e(nz).

(1.49) Since

o0

[1+240a3(n)e(nz)]

92(x) = (12)

n=1 00

s

1 - 504 93(x)

(216

a5(n)e(nz)

n=1

we find that in the Fourier expansion of 0(z) = g2(z)3 -2793 (Z) 2 the constant terms cancel out, and 0(z) = (2;r)12

(1.50)

T(n)e(nz) n=1

where T(1) = 1. The arithmetic function T(n) is called the Ramanujan function; it possesses fascinating properties, which will be revealed throughout this book.

Exercise. Using the formula (1.46) for k = 2, derive the following Fourier expansion of the Weierstrass function: 8J (u; z) _

(5i

emu) 2

--

2

+ 167x2

00 (dsin2(du)) e(nz). n=1 \ din

1.5. The modular group

15

1.5. The modular group The modular group SL2(Z) is the first discrete subgroup of SL2( R ) which interests arithmeticians. We begin by proving

Theorem 1.1. The modular group is generated by two matrices

i),

T= (1

(1.51)

S=

Proof. The action of S on IHI (the inversion; is involutary, more precisely S2 = -1. We have

(a

S

d) _ (ac

bd)

so S acts by interchanging the rows up to the sign. The matrix T''2 _ (1 acts by translation, a

b

_

(c d)-(

1

)

a+cn b+dn d

c

If c # 0 this has an effect of reducing the left upper entry to 0 < a < Icl by a suitable choice of n E Z. Applying both operations repeatedly, we end up Then applying with a matrix having c = 0, which must be of type T-m we arrive at ±1. Finally we can change -1 to 1 by applying S2. Remark. The procedure described in the proof of Theorem 1.1 follows the steps of the continued fraction expansion of

Theorem 1.2. The set (1.52)

D = { z = x + iy : xj <

2z j > 1 }

is a fundamental domain for the modular group F = SL2(Z), i.e. it has the following properties:

- D is a domain in IH, - every orbit of F has a point in D or on the boundary aD, - distinct points in D are not in the same orbit of F.

Proof. By (1.51) each orbit Fz has a point with largest height (the imaginary part). Such a point, say z, has the property that

Icz+dl >1for all ry= (*

d) EF.

1. The Classical Modular Forms

16

00

P

p,

i

1

-1/2

1/2

Figure 2. A fundamental domain for the modular group

This maximal property is preserved by translations (which are represented by matrices with c = 0), so we can choose a maximal point of the orbit in the strip jxj We shall show that the domain r2.

D'={zEIH[:1xI<2, Icz+dl >1forallc,dwith c$0} coincides with D. Indeed, D' C D by choosing c = 1, d = 0. Conversely, if zEDandc 0 then cz + dl2 = c2 z12 + 2cdx + d2 > c2 - jcdj + d2 , 1, so D C D'. From the construction of D' it follows that distinct points of D' are not equivalent and the closure D' = D' U 3D' contains points of every orbit. Thus D = D' is a fundamental domain. Let D be the standard fundamental polygon (1.52). We shall show that the equivalent points of the boundary 3D are exactly those pairs of points which interchange upon reflection in the line x = 0, and they are identified by the transformations T and S. Indeed, if both z and yz are on 3D then 1 = Icz + d2 = (cx + d)2 + c2y2 '> c2 - jcdj + d2 , 1; hence either c = 0, d = ±1 or c = +1, d = 0, giving T or S respectively. Observe that D has a parabolic vertex at oo, an elliptic vertex at i of order m(i) = 2, and two equivalent elliptic vertices at p = -1 2z ' and p' = 1+2

of order m(p) = m(p') = 3 (see Figure 2 and Section 2.2 for a general definition).

Theorem 1.3. Let f $ 0 be a modular form of weight k , 0. Denote by m f (w) the order off at w. If w is an elliptic fixed point (of the modular grcup) denote its order by m(w), and put m(w) = 1 otherwise. We have

L m(w)

w (mod Pl

12

1.5. The modular group

17

Figure 3. The cut polygon

Proof. Cut off the upper part of the standard fundamental polygon D along a horizontal segment of very large height. Make very small circular cuts at the elliptic vertices and at any singular point of f which lies on the boundary aD in such a way that the equivalent points have congruent cuts. Let P denote the resulting cut polygon (see Figure 3).

Integrate fl f along the boundary aP of the cut polygon, getting by Cauchy's theorem

12,7rif f , (z)dz = 8P

'rnf (w). wEP

Note that at(w) = 1 for any w E P. On the horizontal segment (we can assume that it does not contain singular points) wve see from the Fourier expansions 00

f(z)

ane(nt),

n=m

f'(z) _ n=m

where a,,,, 0 0 for m = m f (oo) that

f,

f (z) _

or bee(Ez) 1=0

with bo = 2lrimf(oo). Hence the integral over the horizontal segment is equal to -mf(oo). The two integrals along equivalent vertical segments with congruent cuts at singular points cancel out by periodicity. At every elliptic vertex w = p, i, p' we have ' (z) = 'nf( U)

f

+ h(2)

1. The Classical Modular Forms

18

where h(z) is holomorphic in a small disc centered at w. Hence the integral over the arc around w tends to p(w)m f(w) as the radius tends to zero, where 21rp.(w) is the arc angle. We have and µ(p) = l1(P) =

1 µ(Z) = 12 = m(i)

1

=

6

1

2m(p)

It remains to evaluate the integral along the arc of the circle Izi = 1 at the bot ,om of P with the elliptic points p, i, p' and other singular points being removed. The middle point i splits the arc into two equivalent arcs, say A and A' = SA. We have

f(Sz) = zkf(z) kzk-l.f f'(Sz)z-2 = (z) + zkf,(z)

f

(SZ)z-2 =

k Z

+

f' (z).

Hence 2iri A'

f

(z)dz =

27ri

f

.f'(Sz)dSz =

-If k A

A

and

kdz _ k

1

1

f (z)dz = 27ri I

27ri AUA'

f'

(z + f (z)) dz,

27ri

z

12

A

because the are A has one-twelfth the length of the whole circle jzI = 1. Gathering together the above evaluations, one obtains the formula (1.53).

Remark. It would have been more appropriate to perform the computat ions in geodesic polar coordinates at elliptic vertices, to avoid taking limits.

1.6. The linear space of modular forms From now on we require any modular form to be holomorphic in H and at 00 (before, at co we required only meromorphy). The Eisenstein series Ek(z)

is a modular form of weight k for every even k > 2, and the discriminant function 0(z) is a modular form of weight 12 in this stronger sense. But j(z) fails to be holomorphic at z = cc, though it is holomorphic in H because A (z) does not vanish. Let Mk denote the linear space of modular forms of weight k > 0 which are holomorphic in H and at cc. Clearly modular forms of different weights

1.6. The linear space of modular forms

19

are linearly independent over C, so the space of all modular forms is the direct sum of the Mk,

M = ®Mk. k>O

The whole space M can also be considered as a, graded algebra with respect to the inclusions MkMe C Mk+P

We shall use the formula (1.53) to examine the structure of M as an algebra over (C. Since f E M is holomorphic, all terms of (1.53) are non-

negative. First let us consider each space Mk for a few small, even values of k separately.

Case k = 0. Then m f(z) = 0 for all z, so f (z i is constant, and (1.54)

M0 = C.

Case k = 2. Then m f(z) = 0 for all z, so f (z''i is constant, and necessarily zero, and (1.55)

M2 = 0.

Case k = 4. Then m1(p) = 1 and mf(z) = 0 for'z

p(mod F). This means f (z) has a simple zero at p and no other zeros mod F. In particular, one knows that G4(z) has a simple zero at p acid no other zeros. Therefore for any f (z) E M. we can find a constant c such that f (z) -cG4(z) vanishes at oc, so it must vanish identically, i.e., f (z) = cG4(z). This proves (1.56)

M4 = G4(z)(C.

Case k = 6. Then m f(i) = 1 and m1(z) = 0 :'or z t i (mod F). The same argument as before shows that f (z) = cG6(z) and (1.57)

M6 = G6(z)C.

Case k = 8. Then m f (p) = 2 and m f (z) = 0 for z f (z) = cG4(z)2 and (1.58)

M8 = G4(z)2(C.

p (mod r). Hence

1. The Classical Modular Forms

20

Case k = 10. Then m f (i) = 1, m f (p) = 1 and m f (z) = 0 for other inequiv-

alent points. Hence we argue as before to show that f (z) = cG4(z)G6(z) and Mlo = G4(z)G6(z)C.

(1.59)

Case k = 12. Recall that 0(z) does not vanish in H and it has a simple zero at z = oc (see (1.23) and (1.50)). Since G12(oo) 0 there exists a constant c such that f (z) - cG12(z) vanishes at oc, so f (z) - cG12(z)

A(z)

E MO = C

and f (z) = b0 (z) + cG12(z). This proves that .M12 = 0(z)C ® G12(z)C.

(1.60)

Similarly by induction we prove that (1.61)

Mk = 0(z)Mk-12 © Gk(z)C,

if k > 12.

From the above results we conclude

Theorem 1.4. The dimension of the space Mk is given by (1.62)

dim Mk =

[kj 12]

if k - 2 (mod 12)

[k/'12] + 1

if k 0- 2 (mod 12).

Next we show that any f E Mk is a polynomial in G4 and G6. This is already proved for k < 10. If k > 12 we subtract from f a monomial in G41 G6 of special type, namely we consider

f-

where 4a + 6b = k,

with a suitable constant c such that the resulting function vanishes at oo. Then divide by 0, getting a function in Mk-12, and the proof is completed by induction. We shall also prove that G4, G6 are algebraically independent. Suppose to the contrary that P (C47 G6) = 0

for some polynomial P 0. Since there is no linear relation between functions of distinct weights, the monomials in P (G4i G6) have the same weight.

1.6. The linear space of modular forms

21

Therefore the above algebraic relation must contain one of the following type:

G4 +G6Q(G4,G6) := 0 Gm

+G4Q(G4,G6)0 Q (G4, G6) - 0

where Q is a polynomial 0 0 of degree smaller than that of P. But the first equation is false because G6(i) = 0 while G4 ;i) # 0 (see the above cases k = 4 and k = 6 respectively). Similarly the -econd equation is false. The third equation yields a new algebraic relation of smaller degree; whence one completes the proof by induction.

Theorem 1.5. The Eisenstein series G4, G6 are algebraically independent and generate the space of all modular forms, that is M = C[G4i G6]

- the polynomial ring.

A modular form f E Mk is called a cusp form if it vanishes at oo, i.e., its Fourier expansion has no constant term,

f (z) _

ane(nz). n=1

For example, A(z) is a cusp form of weight 12. The Eisenstein series Ek(z) is not a cusp form. The linear space Sk of cusp forms together with Ek(z)C gives (1.63)

Mk = Ek(z)C ®SA

if k > 2. It follows from (1.62) that there are no cusp forms of weight k S 10, and for k > 12 the space Sk has dimension one less than Mk.

Since Ek(z)Et(z) - Ek+t(z) vanishes at z = oc, it has to be a cusp form or vanish identically. From this observation we obtain the relations E42(z)

E4(z)E6(z)

E8(z), Elo(z),

E6(z)E8(z) = E4(z)Elo(z)

E'14(z),

E6 (Z)2

c:,(z).

- E12(z)

Comparing the Fourier coefficients (see (1.49) and (1.50)) we infer from the first relation that, (1.64)

cr (n) = o,3 (n) + 120 T 0
0'3('n1)crs(n. -

m).

1. The Classical Modular Forms

22

From the last relation one can infer a formula for the Ramanujan function: (1.65)

5

r(n) = 756ou(n)

756os(n) -

a5(m)a5(n - m).

631 O<m
By a more elaborate method D. Niebur has established another formula: (1.66)

r(n) = n4o(n) - 24 E m2 f (m, n)a(n)o(n - m) O<m
where o(n) = of (n) stands for the sum of divisors of n and f is a binary quadratic form with integral coefficients,

f (m, n) = 35m2 - 52mn + 18n2. In particular, (1.66) shows that r(n) is an integer. There are many formulas of the above type. We give one more example for odd n's which was found by van der Pol,

r(n) = n3o3(n) + 120 E (2n - 3m)o3(m)o3(n - m). O
The formulas of the above type are quite useful for numerical computations of r(n), but they are not good for producing strong estimates.

Chapter 2

Automorphic Forms in General

Of great importance for arithmetic are automorphic forms with respect to a congruence subgroup of SL2 (Z). However a, large part of the theory holds true for more general discrete subgroups of SL2(R); therefore for a better understanding of relevant concepts it is reascnable to deal with the general case as long as possible.

2.1. The hyperbolic plane We begin by recalling the well-known properties of the linear fractional transformations of t = C U {oo} (the Riemann sphere) given by (2.1)

gz =

az + b

cz+d'

a. b, c, d E R. ad - be > 0.

We denote by GL,; (18) the group of 2 2 real matrices of positive determinant. Observe that a linear fractional transformation g determines the

matrix (n d) E GL.; (f8) up to a scalar bec a,use the matrices (a ) with a : 0 give the identity transformation. Dividing by a scalar, we can represent g by a matrix of determinant 1. The group SL2 (JR) of real matrices of determinant 1 contains two elements 1 = (1 } and -1 = (-1 1) which act trivially. and the factor group 1

SL2(R)

is identified

-1}

the linear fractional transformations. 23

2. Automorphic Forms in General

24

The linear fractional transformations are conformal mappings (preserve angles), and they transform a circle into a circle subject to the convention that a straight line is a circle passing through oo. Of course, the center of s. circle may not be mapped onto the center, save for g which represents a translation. For any g = (* d) E SL2(R) we introduce the function (2.2)

j9(z) = cz + d.

This satisfies the chain rule of differentiation (2.3)

igh(z) = jg(hz)jh(z)

We have (2.4)

gz - gw = (z - w)j9(z)-1j9(w)-1;

whence (2.5)

j9(z)-2. clzgz =

Naturally I jg(z) I -2 = I cz + dl -2 is called the deformation factor of g at z.

Suppose g = (a d) E SL2(R) with c 54 0. Putting w = oo in (2.4), we get

gz= a

- c(cz + d)' c 1

(9-a) (+) By (2.4) we also see that I gz - gwl = I z - wI if both points z, w are on the curve (2.6)

CQ = {z E 111

: Ijg(z)I =1},

which is the circle centered at -d/c of radius IcI-1. One can distinguish the exterior from the interior of Cg by considering all points of deformation

< 1 as the exterior. It is easy to see that g maps Cg to Cg-i and reverses the interior of Cg onto the exterior of C9-i (see Figure 4). We shall call Cg the isometric circle of g because g acts on Cg as a euclidean isometry (it preserves the distance). It follows from the formula (2.7)

Ij9(:)I2 In. qz = lm z

2.1. The hyperbolic plane

25

C9

C9-1

Figure 4. Isometric circles

that the complex plane C splits into three invariant subspaces of the group SL2(1R), namely the upper half-plane (2.8)

lE

z=x+iy:xEJ,yEW'},

the lower half-plane IElI (complex conjugate) and the real line IR = IR U {oo} (the common boundary of IEiT and H). Moreover we have (combine (2.5) with (2.7)) (2.9)

(Im gz)--lidgz = (Im z.

dzJ

which shows that the differential form (2.10)

ds2

=

y-2(dx2 + dye)

on TEE is invariant under the group GL2 (IR). With the metric derived from this differential the upper half-plane becomes a Riemannian manifold. This means that the distance function on IHZ is given by

p(z, w) = n n) (x '(t)2 + y,(t)2) y(t)-idt JO

where L = [x(t), y(t)] ranges over smooth curves in IHI joining z with w. More explicitly. we have the relation (2.11)

coshp(z, w) = 1 + 2u(a, w)

with (2.12)

u(z, w) =

wl -_ 41m -- . Im i, z

2. Automorphic Forms in General

26

0

Figure 5. Geodesics in IHI

The upper half-plane I1-1 equipped with the above metric is the Poincare

model of a hyperbolic plane (curvature -1). The invariance of ds2 implies that the linear fractional transformations are hyperbolic isometries of H. In addition to these isometries we have the reflection in the imaginary line (2.13)

which reverses the orientation (thus it is not analytic).

Exercise. Show using the above asserted properties that the whole group of isometries of IH[ is generated by the following ones: z -- z + t (translation). t G R, (dilation), p e- RR', z --* pz z -1/z (inversion'. z -a (reflection). Show that the hyperbolic lines (geodesics in IFI[) are represented by the euclidean semi-circles and half lines orthogonal to I<8 (see Figure 5). The hy-

perbolic circles (loci of points in a, fixed distance from a given point in H) are represented by the eucliclea__ circles in H, of course, not with the same centers (see Figure 0. The hyperbolic measure on 1H[ is expressed in terms of the Lebesque measure by (

n

.14

uz = y _2 dxdy.

2.2. The classification of motions

27

0

Figure 6. Circles in C aLnd H

Figure 7. Hyperbolic triangle

Let T be a hyperbolic triangle with interior angles a, 0, -y (see Figure 7). The hyperbolic area of T is given by (Gauss defect phenomenon) area

(T)=Jdyz=7r-n-3--`y. T

2.2. The classification of motions The linear fractional transformations are rigid motions of the hyperbolic

plane, and they move points in distinct was. Given g E PSL2(R), we denote its conj u;acy class by {9}

= ITgT

1

: T E PSL,2(JR) }.

2. Automorphic Forms in General

28

parabolic

elliptic

hyperbolic

Figure S. Rigid motions in IfII

Conjugate motions act on IEII similarly, so the classification will be invariant under conjugation. The identity motion (every z E IEII is a fixed point) forms a class by itself.

Any other motion g = (a a)

+1 has one or two fixed points in c. There

are three cases: i g has one fixed point on II8, ii g has two distinct fixed points on IEP, iii g has one fixed point in IEII and the complex conjugate one in IR. Accordingly g is called parabolic, hyperbolic, elliptic. By conjugating we can bring g to one of the following types: i z H z + t (translation, fixed point oo), ii z i-- pz (dilation, fixed points 0, oo), iii z p -- k(B)z (rotation, fixed point i), where t E R, p E IR+, 0 E IR and k(B) _

cos B

-sinO

sin 0 cosO

A parabolic motion moves points along horocycles (circles in H tangent to k). A hyperbolic motion moves points along hypercycles (the segments in IEII of circles in t through the fixed points on A). The geodesic line through the fixed points of a hyperbolic motion is mapped to itself, not identically. Of the two fixed points one is repelling, the other is attracting. An elliptic motion moves points along circles centered at its fixed point in ]Bl (see Figure 8).

2.3. Discrete groups - Fuchsian groups

29

The number of fixed points of a rigid motion is invariant under conjugation; therefore the above classification applies naturally to the conjugacy classes. The same classification can be also described in terms of trace (which is an algebraic invariant of conjugation), namely if g = (a b) ±1, then i g is parabolic Ia + dl = 2, ii g i s hyperb o l ic Ia + d l > 2, iii g i s e lli ptic J a + d l < 2.

2.3. Discrete groups - Fuchsian groups The group M2 (IR) of 2 x 2 real matrices is a vector space with a norm given by

IIgII =a2+b2+c2+d2

if(a b) c

d

Besides its usual properties this norm satisfies IlghtI < IIgillihil The norm gives a metric topology to SL2 (R) through the embedding into M2 (R) A subgroup r c SL2 (]R) is discrete if the induced topology in r is discrete, i.e. .

the sets {yEr:flyII< r} are finite for any r> 0. Let X be a topological space (Hausdorff) and r be a group of homeomorphisms of X acting on X (transitively). We say that r acts on X discontinuously if the orbit

rx={'yx:-yEr} has no limit point in X. In particular the stability group

ry={-yer:yx=x} of a point x E X is finite. Moreover, for any x E X there exists a small neighborhood U of x such that

{yEF:yUnU

O}=rx.

If two points z, w are not r-equivalent, i.e. are not on the same orbit, then there are neighborhoods U and V of z, w respectively such that yU n V = 0 for all y E F. We now return to the hyperbolic plane X = IHI. A subgroup IF c PSL2(R) acting discontinuously on H is called a Fuchsian group. In this case Poincare showed that a subgroup of SL2 (lid) is discrete if and only if it acts discontinuously on H (when considered as a subgroup of PSL2(R)).

2. Automorphic Forms in General

30 is

Remark. A discrete subgroup of SL2(R) acts on the whole of C, but not. necessarily discontinuously.

There is a multitude of Fuchsian groups, a lot more by comparision with the discrete groups of motion on R2. The essence of this fact is that PSL2(]R) is non-abelian. Let us mention the following result off. Nielsen: if F C PSL2 (IR) is non-abelian and hyperbolic (it contains only hyperbolic

elements and the identity), then P acts discontinuously on IHI, so it is a discrete group.

Proposition 2.1. Let r be a Fuchsian group. For any z E C the stability group Fz is cyclic (but not necessarily finite if z E R).

A Fuchsian group r is said to be of the first kind if every point on the boundary 8IHI = IR is a limit (in the C-topology) of an orbit Fz for some z E H. Clearly, any subgroup of finite index of a Fuchsian group of the first kind is also a Fuchsian group of the first kind. But a Fuchsian group of the first kind cannot be too small. Obviously, it cannot be cyclic, and a fortiori the stability group Fz of a point z E C is not of the first kind. A Fuchsian group can be visualized by its fundamental domain. A set D C IHI is called a fundamental domain for F if - D is a domain in IHI,

- distinct points in D are not equivalent, - any orbit of r contains a point in the closure D (with respect to the C-topology). There are many ways of constructing fundamental domains. Suppose r

is a Fuchsian group of the first kind and w E IHI is not fixed by any y E F other than the identity motion. Then the set

D = {zEH: p(z,w)
1}

is a fundamental domain for P. This is called a normal polygon. One can show that D is a convex polygon with an even number of sides (subject to the convention that if a side contains a fixed point of an elliptic motion of order

2 from r, then this point is considered as a vertex and the side is divided into two sides). The sides of D can be arranged in pairs of congruent sides so that the side-pairing motions generate the group F. These properties imply the following:

Proposition 2.2. Every Fuchsian group of the first kind has a finite number of generators and a fundamental domain of finite area.

The finiteness of the area of a fundamental domain does not guarantee the compactness (after closure). In fact the most important arithmetic

2.3. Discrete groups - Fbchsian groups

31

groups do not have compact fundamental domains. Suppose that the closed polygon D is not compact. Then D must have a vertex on A, and since the two sides of D which meet at such a vertex are tangent (because they are orthogonal to A), they form a cusp.

For a Fuchsian group of the first kind one can select a fundamental polygon all of whose cuspidal vertices are inequivalent. In such a polygon the two sides joined at a cuspidal vertex are congruent by a motion which fixes the vertex, generates the stability group of the vertex and is parabolic. For this reason a cuspidal vertex is also called a parabolic vertex. Conversely, cusps for r are exactly the fixed points of parabolic motions of F. Therefore we have

Proposition 2.3. A Fuchsian group of the first kind has compact fundamental domain if and only if it has no parabolic elements.

Let F be a Fuchsian group of the first kind. Then the quotient space r \ ]H[ (the set of orbits) is equipped with the topology in which the natural projection -7r : IHI -> F \ H is continuous; it is a connected Hausdorff space, and with properly chosen analytic charts it becomes a Riemann surface. Specifically, given a point z E H, consider discs U,z C H sufficiently small so

that for all 7 E F,'y Fz 'yUz = Uz for all 'y c rz.

-yU,z fl U,z = O

The stability group Fz is cyclic of finite order, say m > 1. Choose Tz E SL2(C) which maps z to 0 and transforms Uz onto the unit disc

U= {zEC:IzI <1}. Let em : U - U be the power map defined by em,(z) = z"`. Then the charts (7rUz7 emrz7r-1) with z ranging over H and Uz (small discs centered at z) form an analytic atlas for IF \ H. If the group F contains only hyperbolic elements and the identity, then F \ H is a compact, smooth surface of genus g > 2. If F has elliptic elements, then the surface F \ H has branch points at the fixed points of the elliptic motions. If F has parabolic motions, then I\ H is not compact. Usually we compactify F \ H by adding cusps with suitable charts. Given a cusp a E A for IF, consider discs Ua C H tangent to A at a sufficiently small so that 7Ua fl U a = o

yUa=Ua

all

forall'yEra.

a

2. Automorphic Forms in General

32

The stability group Pa is cyclic infinite, say generated by rya. There exists Qa E SL2 (R) such that (change rya to -ya if necessary)

Qalyaaa= C1 1). We shall call Qa a scaling matrix of the cusp a; it is determined up to (2.15)

Ora00=a,

composition with a translation from the right side. Thus cr sends a to 00 and transforms the disc Ua (cuspidal zone, so to speak) onto the half-plane IEIIY = {z = x + iy : y > Y}. (2.16) If Y is sufficiently large, the cuspidal zones Ua for distinct cusps are disjoint.

Let e : IHI -> H be the exponential map e(z) = e2,i,. Then for charts at cusp a we take (irUa, The quotient space r \ H equipped with the above charts at points of H and the cusps is a compact Riemann surface. It is convenient to think of the Riemann surface r \ H as being constructed from a fundamental polygon by gluing pairs of congruent sides at equivalent points. eaa1IF-1)

There is a nice construction of a fundamental polygon (due to L.R. Ford) which uses isometric circles (2.6). Suppose r is a Fuchsian group of the first kind which has parabolic motions, so it has a cusp, say a. By conjugation

with the scaling matrix Qa we can assume that a = oo and the stability group r... is generated by the translation z H z + 1 (consider the group Qa 1FQa in place of r). The fundamental domain of r... is the strip

P={z=x+iy:0<x0}. Let E be the set of points in H at which every y E F, y V F... has deformation < 1, i.e.

E = {zEH:Ijy(z)I >1for all yEF,'y

={zeH:Imz>Imyzfor all yeF,yvr,,c}. Thus E is the intersection of exteriors to all isometric circles C,,, ,y E IF, y

Fem. One can show that D = E fl P is a fundamental polygon for F (this is called the standard polygon). Some of the cuspidal vertices of D may be equivalent; nevertheless D has a nice shape which is useful for getting explicit estimates in terms of the group r (see the observations made after Proposition 2.7). If F' C F is a subgroup of finite index, one can construct a fundamental domain for F' out of selected copies of a domain for F. Specifically, letting

yj,1 < j < n, be coset representatives, i.e. F = U F'yj, and D be a fundamental domain for IF, then the union D' = U yjD is a fundamental domain for F' (not necessarily connected). Finally, before giving specific examples of Fuchsian groups, let us mention the following general group-theoretic result.

2.3. Discrete groups - Fuchsian groups

33

Proposition 2.4 (R. Fricke and F. Klein). A Fuchsian group of the first kind is generated by (primitive) motions A1, ... , A9, B17 ... , Bg, El, ... , Ell P1,. .. , Ph satisfying the relations

[All B1]... [Ag,Bg]EI...EQP1...Ph = 1, E73 = 1

1<j
1

Here A3, Bj are hyperbolic motions, [A;, B3] = A; B., A3 1Bj- 1 stands for the commutator, g is the genus of I' \ IEII, E3 are elliptic motions of order m3 >, 2,

Pi are parabolic motions and h is the number of inequivalent cusps.

The symbol (g; ml, ... , ml; h) is group invariant (under conjugation), and is called the signature of F. The area of D = F \ IHI is given by the Gauss-Bonnet formula IDI

(2.17)

= 2g-2+E 7-1

(1 -

\

+ h. 7

An interesting class of discrete groups with compact fundamental domains is made from division algebras over Q, called quaternion groups. For example, let us consider the group

r(n,p) _ { 1

fp

(cad

(cad V)_l E SL2(R) : a,b,c,d E Z

I

where p is a prime number which is - 1 (mod 4) and n is a positive integer which is not a square modulo p, i.e. (zi) = -1. We shall show that, aside from the identity, F(n, p) has only hyperbolic motions. To this end suppose ry c F(n, p) has ItryJ < 2. Then lal < 1, so a = 0 or ±1. Since a2

- b2n - (c2 - d2n)p = 1,

we must have a = ±1 because otherwise b2n - -1 (mod p), which violates

the hypothesis that n is not a quadratic residue mod p. For a = ±1 the determinant equation simplifies to b2n = (den - c2)p,

and this is possible only for b = c = d = 0, for the same reason that n is not a square mod p. Therefore we have ry = ±1, proving that I'(n, p) is a hyperbolic group. By J. Nielsen's theorem F(n, p) is discrete, and by Proposition 2.3 it has a fundamental polygon without cusps.

2. Automorphic Forms in General

34

2.4. Congruence groups Let q be a positive integer. Then (2.18)

1 ) (mod q) }

F(q) = -y E SL2(Z) : 'Y

is called the principal congruence group of level q. It is a normal subgroup of the modular group I'(1) = SL2(Z) of index

[F(1) : F(q)] = q3 J(1 - p-2).

(2.19)

pIq

To prove (2.19), consider the exact sequence

1- F(q) - SL2(Z)

(2.20)

SL2(Z/qZ) --> 1.

The only question is the surjectivity of the residue class map SL2(Z) -> SL2(Z/q7G), which is established by the Chinese Remainder Theorem. By (2.20) we have

[F(1) : r(q)] = ISL2(Z/qZ)I

For any y = (') E SL2 (7G/qZ) we have (c, d, q) = 1, and for a given lower row (c, d) we can find exactly q solutions (a, b) of the congruence ad - be - 1 (mod q). Hence we compute that jSL2(Z/qZ)j= q I {(c, d) (mod q) : (c, d, q) = 1}I

L\

= q> u(r)

(q)2

r

=q 3 f[(1 _

p-2).

pjq

rjq

Notice that -1

F(q) if q > 2. Moreover, if q > 2 then P(q) has no -qv) elliptic elements, and the parabolic elements are ( 1+qt 4u 1-qt with u, v, t integers such that uv = t2. All parabolic elements have trace 2. Since r(q) is a subgroup of SL2(Z), the cusps for P(q) are rational numbers and oc. If one writes cusps as rational points in lowest terms, say a = with (a, c) = 1

and a' _

with (a', c') = 1, then they are equivalent, i.e. a' = ya for e F(q), if and only if

f [a] c

[ac]

(mod q).

Hence the number of inequivalent cusps for r(q) is given by

h ={(c, d) (mod q) : (c, d, q) = 1}I

2.4. Congruence groups

35

which yields

h = q2 fJ(1 - p-2).

(2.21)

plq

Any subgroup of the modular group which contains P(q) is called a congruence subgroup of level q. Here are two examples; (2.22)

Po(q) = S 'Y E SL2(Z) : 'Y =

(* *)

(2.23)

IF, (q) =

(1

1' E SL2(Z) : y =

1

(mod q) }

,

I (mod q) } .

The first one, called the Hecke congruence group (it is not a normal subgroup of SL2 (Z)), is most often encountered in arithmetic. Therefore we are going to examine I'o (q) in great detail.

Proposition 2.5. A set of representatives for 17o(q) \ Po(i) is given by (2.24)

(*

v ) E I'o(1)

with vI q and u (mod q/v).

Hence the index is (2.25)

vq = [FO(1) : I'o(q)] = gfl(1

+p-1).

piq

Proof. We have

(y S) (c d)

(u v)'

= (rya+Sc yb+6d) = say. Hence (d, q) is invariant under multiplication in Po(1) by Po(q) on the left side. In fact one can bring any (a d) E Po (1) by multiplication with a suitable (a E Po (q) to a matrix (u v) with v = yb + 6d = (d, q), i.e. v1q. All solutions to the linear equation y'b+8'd = v form a one-parameter

family y' = y + dt, 6' = 6 - bt with t ranging freely over integers t - 0 (mod q/v) (this congruence ensures that y' = 0 (mod q)). These solutions move the other entry u into u' = u + t, so one can change u freely modulo q/v. Having fixed u, v in the prescribed range, there is no room left for additional changes. This proves the first part of Proposition 2.5. Hence the formula for the index is derived as follows: q) : (u,v) = 1} [Po(1) : I'o(q)] = E {u(mod vjq

(Pa + pa-1 _ pa-2 p°Ilq

+Pa-2 _ pa-3 +...) = q n(1 + p-1). plq

2. Automorphic Forms in General

36

Proposition 2.6. A set of inequivalent cusps for ro(q) is given by the following fractions: u

with vIq, (u, v) = 1, u (mod (v, q/v)).

(2.26)

Hence the number of inequivalent cusps is

h = E cp((v,w))

(2.27)

vw=q

Proof. All cusps are equivalent to rational numbers since ro(q) C ro(1). We transform the representatives T =

of of type (2.24) into (1 -1) T-1 =

*) and evaluate these at oo, getting the rational points v with /vjq and V (u, v) = 1. Suppose two such points are equivalent, say u' v'

a

Q

u

ry

b

v

for (ry a) E ro(q). Hence v' = ryu + bv, vIv', v'I v, v' = v, b - 1(mod q/v) and u' = au +,iv - au - bu u mod (v, q/v), as claimed. Our selection of inequivalent cusps of type (2.26) for r = ro(q) will be convenient in future computations. Recall that all cusps for IF = ro (q) are equivalent to oo under the modular group action. We choose Ta = ( uV

(2.28)

ESL2(Z),

so Taoo=a.

Then Ta 1ryaTa C rte, because any element of o-alraaa fixes oo. Since both groups are cyclic, we must have Ta 1'faTa =

{

f (1

1 bTn,

)

:bcZ}

for some positive integer ma. To interpret ma, observe that the fundamental domain for r = ro(q) can be filled by vq copies of the fundamental domain

for ro(1); the number of such copies which touch the cusp a must be the same for equivalent cusps and is exactly equal to ma. To compute ma, consider the relation

I'a = Tar.TaI n ro(q)

(u _ = {

(

M)

1 -muv -mv2

mug

1 + muv) : my

E ro(q) 2

-=

(

0 mod

q) }

2.5. Double coset decomposition

37

Here m ranges over all integers divisible by q/(q, v2), and this shows that q

M =

(2.29)

.

(q, v2)

a

Now the index formula (2.25) can be written in another way:

vq > ma = E

(2.30)

(q, 2)

P((v, q/v))

vq

The integer ma is called the width of the cusp a. The scaling matrix Qa reduces the width to 1, so it is given by Qa = Tap,,, where

( ma

(2.31)

Pa

2.5. Double coset decomposition Throughout r is a Fuchsian group of the first kind which contains parabolic motions. Select a complete set of inequivalent cusps for I', say a, b, ..., and the corresponding scaling matrices u,,, ub,... in SL2(IR). Recall that Qa la = Qb 1b = ... = oo and Ca 1Faaa = Ub 1rbab = = B, where

B={fl

i ):nEZ}

1

is the group of translations by integers. For any pair of cusps a, b (not necessarily distinct), we shall partition the set ca 1Fab into double cosets with respect to B. First let us examine the subset

'.rub:woo =o0} which consists of the upper-triangular matrices in a 'Fab. Suppose Q,,,, is not empty, say w = ca 1'ycb (for some y in F) belongs to Q. Evaluating -y at b, we get yb = aawvy 1b = vaoc = a, showing that the cusps a, b are equivalent; hence a = b, y E ]Pa, w E B. Therefore 1l = B if a = b, and it is the empty set otherwise. Any other element of Qa -']Pub is represented by a matrix w = (a d) with c > 0. The relation

li) C1

(a+cm

(1 1n

(a d)

)

=

c

d+')

2. Automorphic Forms in General

38

shows that the double coset Qd/a = B d) B determines c uniquely and d modulo integral multiples of c. Moreover, given c, d with w = (c d) E Qa 1FUb, the double coset 1d/c does not depend on the upper row of w. Indeed, if w' = I c d

E Ua

y = Ua

1

Ua 1,

'ya=a, yEra,w'w-1EB,w'= (11''w,a'=a+ cm for sgmemEZ. We conclude the above analysis with the following:

Proposition 2.7. For any pair of cusps a, b for r the set Qa lrub is partitioned into disjoint double cosets

Ua1rUb=babBUU U B(* d)B

(2.32)

c>O d (mod c)

where babB = B if a = b or else it is the empty set, and the union is taken over c > 0 and d (mod c) such that (* d) E o'a 'rub . In applications we need to control the number of elements in the cosets

B(*")B. Put (2.33)

C(a,b)={c>0:

(*

I E oa'rub

c

J

Let c(a, b) denote the smallest element of C(a, b), and put c(a) = c(a, a). That c(a) exists is seen from the construction of the standard polygon for the group o-a 1rua; c(a)-1 is the radius of the largest isometric circle. Hence c(a, b) also exists; in fact one can show that (2.34)

c(a, b)2 >, max{c(a), c(b)} = cab, say.

Since the standard polygon contains the semi-strip {z : 0 < x < 1, y > c(a)-1} whose area is c(a), it follows that c(a) is bounded by the area of the fundamental domain, c(a) < ID1.

(2.35)

Proposition 2.8. For any X > 0 we have c-1 {ci(mod c) : (c d) E o-a1FUb }

(2.36)

<, Cab X.

o
Hence for any c c C(a, b) we have

{ d (mod c) : (c d) E ora 'rub

.

cab c22

2.5. Double coset decomposition

39

Proof. By symmetry we can assume without loss of generality that c(a) > c(b). If w = (* d) and w' = d) with 0 < c, c' < X are both in 0a-1F0.b,

then w" = w'w-1 = (C * ) with c" = c'd - cd' is in a 1raa. If c' = 0 then the cusps a, b are equivalent, so equal, w" c' = c and d' = d. If

c0then Ic"I >c(a) and

- dc d c'

(2.37)

c(a)

>

c(a) .

cX Summing this inequality over 0 < c < X and 0 < d < c, where d'/c' is chosen to be the successive point to d/c, we get (2.36). cc'

Applying (2.36) for X = c(a, b), we infer the inequality (2.34) from the trivial bound

{d(mod c): (c d) EOra1rubJ > 1,

if c E C(a, b).

Proposition 2.9. Let a be a cusp for r, z E IHI and Y > 0. We have (2.38)

I{yEFa\F:IM aQlyz>Y}I <1+

10

c(a)Y

Proof. Conjugating the group, we can assume that a = oo, o'a = 1 and Fa = B. Then the strip P = {z : 0 < x < 1, y > 0} is a fundamental domain of Fa. Let D be the standard polygon of IF, so D consists of points in P of deformation less than 1. For the proof we may assume that z E D, so icz+dl > 1 for any -y = (* d) E r with c > 0. Since Im ryz = ylcz+d1-2 > Y, this implies y > Y, c < (yY) and lcx+dl < (y/ Y) 12. By the last inequality and the spacing property (2.37) are estimate the number of pairs {c, d} with 21

C
1+

)

(Yy'

2

)

<

10C y1 c(a)

(Y) 2

Adding these bounds for C = 2-'(yY)21 with n > 1, we get 10/c(a)Y. This is an estimate for the number of relevant y's not in Fa. Finally adding 1 to )account for Fa, we obtain (2.38). As an example consider the Hecke congruence group Fo(q) (note that

-1 E Fo(q)) and the cusps at oo, 0. The scaling matrices are a = (1 1) We have a001Fo(q)aoo = Qo'ro(q)ao = ro(q), and the set a.1Fo(q)ao = v0 1Fo(q)o-. consists of matrices of type

and ao = (2.39)

(-').

(c / ale ) y Vq-

6 /

with a, ,Q, -y, b F_ Z, abq - 0y = 1.

Hence c(oo) = c(0) = q and c(oo, 0) = c(0, oc) = ,Fq.

2. Automorphic Forms in General

40

Remark. Propositions 2.7, 2.8 and 2.9 are formulated for a Fuchsian group of motions rather than for a discrete group of matrices. Remember to take into account the factor -1 = (-1 _1) when translating the correspondence between the linear fractional transformations and their matrix representations (if -1 belongs to the group of matrices).

2.6. Multiplier systems For a complex number z 0 we choose its argument in (-7r, 7r]. We denote the principal branch of the logarithm by log z, so it is real for positive z and

logz=logIzI +iargz

ifzEC*.

Then we define the power zs for any s E C by z8 = exp(s log z).

Let A, B E SL2(R). By the chain rule jAB (Z) =jA(Bz)jB(z) it follows that the expression (2.40)

27rw(A, B) = -arg jAB(z) + arg jA(Bz) + arg jB(z)

does not depend on z c H. More precisely, w(A, B) takes only three values -1, 0, 1, because Iw(A, B) I is an integer 5 2. By examining numerous cases one can establish the following properties (cf. [Pet] or [Ran]):

(2.41)

(2.42)

w(AB, C) + w(A, B) = w(A, BC) + w(B, C) w(A, B) = w(B, A)

if A, B commute

(2.43)

w(DA, B) = w(A, BD) = w(A, B)

(2.44)

w(AD, B) = w(A, DB)

(2.45)

w(A-1DA, B) + w(A, A-'DAB) = w(A, B)

2.6. Multiplier systems

41

(2.46)

w(A, D) = w(D, B) = 0

(2.47)

w(ADA-1, A) = w(A, A-'DA) = 0

where A, B, C are arbitrary and D = (1 i) Here (2.45) is just a special case of (2.41) with A, B, C replaced by A, A-1DA, B respectively. Moreover .

we have

w(AD, A-') = w(A, DA-1) = w(A, A-') = 0 except for A = (o d ) with d < 0, in which case w (A, A`1) = 1. For any real number k we define the factor system of weight k by setting w(A, B) = e(kw(A, B)).

(2.48)

Note that w(A, B) depends on k (mod 1) and w(A, B) = 1 if k is an integer. We have

w(A,B)7AB(z)k =7A(Bz)k7B(z)k.

(2.49)

For any

tions f : H , (2.50)

SL2(]R) we define the "slash" operator JA acting on funcby

fIA(z) = iA(z)-k f

(Az).

This satisfies the rule of composition (2.51)

fIAB = w(A, B)(fIA)18

Let IF c SL2 (]R) be a discrete subgroup. A multiplier system of weight k for r is a function 19 : F -> C such that (2.52)

I,9(7)I = 1

(2.53)

'9(7172) = w('Yl,72)'d(71)?)(72).

Since w(-1, -1) = e(k), the above properties imply that if -1 belongs to IF, then i9(-1) = ±e(-k/2). Throughout we shall require that (2.54)

i9(-1) = e(-k/2)

if - 1 E IF,

which is called the consistency condition (the other choice yields the zero automorphic form only).

2. Automorphic Forms in General

42

Observe that if 19 is a multiplier system of weight k, then it is also a multiplier system of any weight k' = k (mod 2). The complex conjugate 19 is a multiplier system of weight -k. If 191,192 are multipliers of weight k1, k2 respectively for a group I', then 191192 is a multiplier system of weight k1 + k2 for F. If 19 is a multiplier system for IF and a E SL2(R), then',(2.55)

19a(-Y) = i9(a'Ya-1)w(Q'YQ-1, a)w(a,'Y)

is a multiplier system for the conjugate group a-1FQ. Note that _9 UT = (19 U)T

(2.56)

19a=19

(2.57)

ifaEF.

A multiplier system of weight k E 7G is just a unitary character of IF which satisfies the consistency condition 19(-1) = (-1)k. For example, given a Dirichlet character X to modulus q, we set (2.58)

19('y)

= X(d)

if ry =

(a

d) E I'0 (q).

This is a multiplier system for I'o (q) of any integral weight k such that (2.59)

k =_ 1(1 -

X(-1)) (mod 2).

2.7. Automorphic forms Throughout I' is a discrete subgroup of SL2(Is). It is not at all obvious that a multiplier system for r of given weight k exists. This problem is equivalent to the existence of a non-zero meromorphic function f : Ill1 -> C satisfying the transformation rule (2.60)

fv = 19('Y)f

for any yEF.

Notice that (2.60) is obvious for -y = -1, due to the consistency condition (2.54).

If I' has no parabolic elements, then a multiplier system of weight k exists for P if and only if k = 47rn/ JD J[ml, ... , me]

where n E Z, IDI is the volume of the fundamental domain, m1,. are orders of elliptic generators of I, (see Proposition 2.4) and [ml, .

.. , mp . . , me]

2.7. Automorphic forms

43

denotes the least common multiple (equal to 1 if £ = 0). In particular if r is a hyperbolic group of genus g > 2, then this condition asserts that 4(g -1)k is an integer. The holomorphic functions satisfying (2.60) are called automorphic forms for r of weight k with respect to the multiplier system V. We shall construct automorphic forms for groups having parabolic elements in due course (Poincare series). In this book we shall require an automorphic form to be

holomorphic not only in H but-lso at every cusp. The latter condition needs some explanation.

Suppose a is a cusp for F. Let Qa E SL2 (Il) be a scaling matrix, so aaoO = a and oa 11Faaa is the group of integral translations generated by Q = (1 11, together with -/3 if -1 E F. Then l iJ (2.61)

rya=Qa

(1

1) -1 1)O'

generates the stability group ra, together with -rya if -1 CI`. By (2.60) we obtain flaa

('3z)

=

jaa

(/3z) kf (cal3z) = jaa (/3z)-kf (ryaaaz)

= .7va (Qz)-ki-Ya

=

(O'az)

jaa (Qz)-k7_ya (O'az)k.7aa (z)k19(7a)fj. (z)

Here the product of j-functions is equal to

Qa) by (2.49), and each of the w-factors is 1 by (2.46) and (2.47) respectively (insert rya = Qa/3o'a 1) . Therefore we have (2.62)

fiaa(/3z) = 9(rya)f1aa(Z)

We set (2.63)

19(rya) = e(rca)

with 0 < rca < 1.

By (2.62) it follows that e(-rcaz)fjaa(z) is periodic of period 1, so we can write e(-rcaz)flaa(z) = g(e(z))

where g(q) is holomorphic in the punctured plane C*. We say that f is holomorphic at the cusp a if g(q) is holomorphic at q = 0. In this case the power series expansion of g(q) at q = 0 yields W

(2.64)

fiaa (z) = e(rcaz) E fa(n)e(nz) n=o

2. Automorphic Forms in General

44

where fa(n) are complex numbers called the Fourier coefficients of f at the cusp a. The above series, called the Fourier expansion of f at a, converges absolutely and uniformly in half-planes Im z > E. Denote by Mk (F,19) the linear space of automorphic forms for r with multiplier 19 of weight k. If 19 is trivial on the subgroup Fa, i.e. (2.65)

19(-ya) = 1,

so Ica = 0, then a is said to be a singular cusp for the multiplier system 19. An automorphic form f E Mk(F,19) is said to be a cusp form if

f.(0) = 0

(2.66)

for any singular cusp.

Thus, by Fourier expansion, a cusp form decays exponentially at every cusp (whether singular or not). Hence a cusp form f (z) has exponential decay as z tends to any point on the boundary i9H = R U {oo}. We denote the linear subspace of cusp forms by Sk (I',19) The space Mk(I",19) has finite dimension, but in general one does not know its exact value. If 19 is the trivial multiplier and k is even integer > 2, then we have .

where (g; m1, , mj; h) is the signature of F. Hence by the Gauss-Bonnet formula (2.17) we get the bound . . .

dimMk(I') < (47r)-1IDlk+1. In full generality one can prove that (2.68)

dimMk(F,19) < c(k + 1)(IDI + 1)

where c is an absolute constant.

2.8. The eta-function and the theta-function We shall conclude our general considerations by giving two examples of automorphic forms of weight 1/2 to present two interesting multiplier systems. The first one is constructed from the discriminant function (2.69)

0(z) = (27r)12e(z)fl(1 - e(nz))24, 1

2.8. The eta-function and the theta-function

45

which is a cusp form of weight 12 with respect to the trivial multiplier on the modular group. Since 0(z) does not vanish in IH[, one can define for every k > 0 a holomorphic function f (z) = (27r)-k0(z)k/,2,

which is a cusp form of weight k, but, of course, its multiplier system is no longer trivial. For k = we obtain the eta-function 2

i(z) = e (_z 2) fl(1 - e(nz)).

(2.70)

oo

Dedekind determined the multiplier system for q(z). Precisely, we have 7)(-(z) = 19(y)jy(z)27)(z)

(2.71)

if y c SL2(7G)

where 19(-y) = e(1/4)19(y) for any -y, 19(y) = e(b/24) if y = (1 b) and a + d - 3c

t9(y) = e

24c

-

1

2

s(d c)

)

a

if

c

bl ,c>0.

d

Here s(d, c) denotes the Dedekind sum

s(d' c) _ O<_n
c. (dc )

where,O(x) = x - [x] - 2. The Dedekind sums can be computed by a simple algorithm based on the following reciprocity law: s(d, c) + s(c, d) = 12

(++_3)

for c, d > 0, (c, d) = 1. Moreover, the eta-multiplier can be expressed in terms of the Legendre-Jacobi symbol. For example, we have 19(y)

= (Idl )

if y =

(a d)

E F(24), c > 0.

The Fourier expansion of 77(z) is a lacunary series e(n2z/24)

77 (z) =

n=±1 (mod 12)

-

E n-f5 (mod 12)

e(n2z/24).

2. Automorphic Forms in General

46

The inverse

rj(z)-1 has the Fourier expansion given by 00

00

e(z/24)77(z)-1 = fl(1 -

e(nz))-1

= > p(n)e(nz) 0

1

where p(0) = 1 and p(n) is the number of unrestricted partitions of n > 0 into sums of positive integers. The Fourier coefficients of are quite i7(z)-1 has weight -z but it is not holomorphic at oo). large (note that G.H. Hardy and S. Ramanujan invented the circle method, by means of which they established the asymptotic formula 77(z)-1

p(n) - (4,43-n) - 1 exp (r 2n/3)

as

n - oo.

A close relative of the Dedekind eta-function is the classical theta-function defined by its Fourier series:

8(z) = E e(n2z).

(2.72)

-00

It is an automorphic form (not a cusp form) for the group ro(4) of weight k = 1/2 with respect to a multiplier system slightly different from that of the eta-function. Precisely, we have (see Theorem 10.10) (2.73)

8('Yz)

=

Ed

(C) d,7.y(z)ao(z)

if -y = ( c db) E ro(4)

where (2.74)

Ed =

51

ifd =- 1(mod 4)

i ifd - -1(mod 4) and (a) denotes the extended quadratic residue symbol; namely, it is the Jacobi symbol if 0 < d - 1 (mod 2) extended to all d - 1 (mod 2) by (2.75)

d (0

(2.76)

d)

if c # 0

d

Icl 1

-{0

ifd=fl otherwise.

The theta-function has the Jacobi product representation 00

(2.77)

0(z) = fl(1 - e(nz))(1 + e((n + 1/2)z))2 1

which shows that 0(z) does not vanish in H.

Remark. The theta multiplier system coincides with the eta multiplier system on the subgroup r(24).

Chapter 3

The Eisenstein and the Poincare Series 3.1. General Poincare series A very important class of automorphic forms is constructed by the method of averaging. Suppose a is a cusp for r which is singular with respect to a multiplier system e9 of weight k. Let p : H -' C be a holomorphic function which is periodic of period 1. Define 7r: r x H - C by (3.1)

ir('y, z) = 9(-y)'W (aa 1,'Y) joa 1ry(z)-kp (Qa 17z)

Actually ir(-y, z) depends only on the coset ray. To prove this, consider 7r (-y', z), where 'y' = iy with rl E ra, so 17 = va/3va 1, where Q is an integral translation. We have p (ca ,7,z) = p (Qka "yz) = p (ca 1`yz)

ja.

(z)-k

11,(z)-k

j".

l.r(Z)-k

t9('y') = i9(t17) = w(17,'Y)'9(7) by (2.53) and

w(71,1')w (cn 1,1'x) = w (aa , 7)

by (2.45). Collecting these results, we arrive at n(y, z) = ir(ry, z). This property allows us to write without ambiguity the infinite series

pn(z) = E irl'y, z) 7Ero\r 47

3. The Eisenstein and the Poincare Series

48

provided it converges absolutely. For example, if p(z) is bounded the series (3.2) is majorized by 17(z),_k

Ijao

k

(Im oalryz)'

= yk

,

-yEro.\r

'YEr,\r

and this converges absolutely if k > 2 by virtue of Proposition 2.9. For any singular cusp b we deduce the following:

Palab(z) = jab(z)-kPa(abz)

(Z) -k E

=

7r ('Y,abz)

7Era\r iab

(Z) -k

1

,

?f (aaryab 1 , abz)

7EB\aatrab =lab

(z)-k

(aaryab 1) w (aa 1, aa'Yab 1) i7a61(abz)-kp(7z)

7EB\ap lrab

Since ja(z)'kj7a-1(az)-k = w(rya-1,a)j7(z)-k, this gives

Pa1,b (z) = E

t9ab(7)j7(z)-kp('Yz)

7EB\aa 1rab

where (3.4)

l9ab('y) = t9 (aa1'97b 1) w (aa

aaryab

w (ryab 1,ab)

Using the relation (2.41), one can derive a handful of expressions for i9ab(ry). For example, we have (3.5)

i9ab('y)W(aa,'Y) =19 (aaryab') to (aaryab n, 0'b)

Hence, in particular, for a = b the series (3.3) becomes (3.6)

Paiaa(z) _ E

'j'('Y)j7'(z)-kp('Y'z)

' Er, 00 \r, where t9' is the multiplier system for the conjugate group I" = aal'aa given by (3.7)

19'(ry')w(aa, 7) = '9('y)w(ry, aa)

if ry' = as lryaa.

The function Pa(z) defined by (3.2) is called the Poincare series associated with the cusp a and the generating function p (a is required to be singular with respect to the multiplier system, and p is periodic such that the series (3.2) converges absolutely).

Proposition 3.1. The Poincar6 series Pa(z) is an autotnorphic form, P0Ir(z) = t9(T)Pa(z) if T E F. (3.8)

3.2. Fourier expansion of Poincare series

49

Proof. By conjugating the group we may assume that a = oo and aoo = 1, in which case the series (3.2) looks simpler, namely Poo(z) =

(3.9)

9(-Y)jry(z)-kP('Yz)-

ryEra,\r

Hence for r E IF

Poo(rz) _

F, '9('Y)jy('rz)-kP(-YTZ)

.yEr.\r

= C`

j7r_i (rz)-kp(yz).

ryEroo\r

Here we have

(yr-i) = w('y, T-1)d(7)d(T-1) = j.yr-1(Tz)-k = w(7,r-1) jry(z)-k jr-i

w('y,r-1)w(,r, r-i) 9(r),y(7),

(rz)-k = w('y,T-'),w(,r,r-1) jr(Z)k j l(z)-k,

and by these expressions we arrive at (3.8). Clearly Pa(z) is holomorphic in H. To prove the holomorphy at cusps we need to expand Pa(z) in Fourier series.

3.2. Fourier expansion of Poincare series Let a, 6 be singular cusps for a multiplier system t9 on r. We seek the Fourier expansion of Pa(z) at the cusp b, i.e. for the series (3.3). Applying the double coset decomposition (2.32), we split the series into Po1g (Z) = aabP(Z) +

l9ab(7)I-r(Z) 1; 7ES\oo 'rab/B

where the first term comes from the contribution of 7 = 1 (which exists only if a = b), and for any 'y = ( :)Ean'rcbwithc>Owehave

I.y(z) = > jyr(Z)-'P(7rz) rEB nEZ nEZ

c z+n +d )

P

)

a

1

c

c(c(z + n) + d)

By Poisson's summation we get

I.y(z) = E J uEZ

(c(z + v) + d) -kp I 00

a

- c(e(z + v) + d)) e(-nv)dv.

3. The Eisenstein and the Poincare Series

50

In the sequel we specialize the generating function to p(z) = e(mz)

where m is a non-negative integer. For this function we can compute the Fourier integral quite explicitly. First by a linear change of variable we obtain / ma C nd)\

1,(z) E e I nz + nEZ

where

Jc(m., n) =

roo+iy

J

/ -m

(ev)-ke I

\\

oo+iy

(TV-

- nv I dv.

Notice that this integral does not depend on y by Cauchy's theorem. If n < 0, then, moving the horizontal line of integration upwards, we see that the integral vanishes,

3c(m, n) = 0

if n <, 0.

If n > 0 but m = 0, then we have (see [G-R], 8.315.1) (3.10)

nk-1

27r

J.(0, n) = (ic)

T(k)

For n > 0 and m > 0 we have (see [G-R], 8.412.2) (3.11)

Jc(m, n) =

21r

i

(n) a

Jk-l

(4)

where J.y(x) is the Bessel function of order v, defined by 00

(3.12)

(-1)t :C V I.2f J (x) = F- Q!r(e+ 1 + v) (2) e=o

Exercise. Derive (3.11) from (3.10) using power series expansion for e(z).

Collecting the above computations, we obtain the desired Fourier expansion for the Poincare series generated by the function p(z) = e(mz), namely

P0!, (z) = Sahe(mz) + > e(nz) n=1

Spb(m, n; c)J,(m, n) n>n

3.2. Fourier expansion of Poincare series

51

where J (m, n) are given by (3.10)-(3.11), and Sab(m, n; c) is the Kloosterman sum defined by (3.13)

''ab(l)e

Sab(m, n; c) _

/ ma + nd (

\

c

'Y=( d)EB\°O'rab/B

Recall that i9ab(-y) is given in terms of the multiplier system by (3.5). Since there are no negative terms in the Fourier expansion at any cusp, it proves that the Poincare series is an automorphic form in our strict sense. For m = 0 we denote Pa(z) by (3.14)

Ea(z)

(7)w

_

a

1,'1) 700

y(z)-k

'yEr,\r

which is called the Eisenstein series of weight k. This has the Fourier expansion 00

(3.15)

jab (z)-kEa(O bz) = bab + > rlab(n)e(nz) n=1

with k

(3.16)

%b (n)

(2z)

k

1

T(k) 0 c ksab(0, ni c).

For m > 0 we denote Pa(z) by

(3.17)

Pam(z) _ E ('Y)'w (Qa 1, ry) joo ,Y(z)-ke (mca 1ryz)

7Er,\r and call Pa,,,(z) the m-th Poincare series of weight k. This has the Fourier expansion 00

(3.18)

jab(z)-kPam(Qbz) =

ab(m,9b)e(nz) n=1

with (3.19)

pab (m n) r l

(-)

k-I

/41r

k b abbmn + 27ri-E c-1Sab(m, ni c)Jk-1 l\

c>0

Since there is no constant term in the Fourier expansion (3.18), we obtain

Proposition 3.2. For m > 1 the Poincare series

is a cusp form.

3. The Eisenstein and the Poincare Series

52

3.3. The Hilbert space of cusp forms Let f, g be automorphic functions with respect to a multiplier system '0 of weight k for a group r. Thus fh7 = &(y)f,

g17 = 'd(7)g

for all y E r. Hence the expression (a measure on IIll) (3.20)

(f, g) (Z) = ykf (z)9(z)dµz

is r-invariant. Therefore we can define the inner product (due to H. Petersson) (3.21)

(f, g) =

J\a(f, g) (Z)

provided the integral converges absolutely. For f = g we set

r\

IIf112 = (f, f) = f Ykif(z)I2dµz. A cusp form f has exponential decay at cusps, so IIf II < oo. The linear space of cusp forms Sk(r, i9) equipped with the Petersson inner product is a finite dimensional Hilbert space.

The Poincare series Pa,,,(z) associated with a singular cusp a and a positive integer m belongs to Sk(r, z9). We shall compute the projection of

any f E Mk(r, i9) onto Pa,,, by the unfolding technique. Without loss of generality we can assume that a = oo and as = 1 (take the conjugate group a; lraa if necessary). In this case we obtain

(f,

_k

f \MYkf (z)

f

e(myz)dpz

(IM yz)k f (yz)e(myz)dµz \8i 7Er.\r 0o

I

= f f yk f (z)e(-mz)dµz. 0

o

Inserting the Fourier expansion of f at cusp a = oo, say

f (z) _

=o

f (n)e(nz),

3.3. The Hilbert space of cusp forms

53

we get a contribution from the m-th term only: 00

(f, Pr) = f (m) J

yk-2 exp(-41rmy)dy

= r(k - 1)

0

(41rm)k-1

f

In general, for a singular cusp this argument yields

Theorem 3.3. Let Pp,,, be the m-th Poincare series attached to a singular cusp a for a multiplier system 0 of weight k > 2. Let f be an automorphic form with respect to the same multiplier system. Then (3.22)

(f, 1'asn) =

r(k -1)1 (4Trrrn) '-

fa(m)

where fp(m) denotes the m-th Fourier coefficient off at the cusp a, i.e. one has the expansion

fp(n)e(nz).

(z) _ n=0

Combining (3.22) with (3.19), we obtain Corollary 3.4. Let a, b be singular cusps for a multiplier system i9 of weight k > 2. For m, n positive integers we have (3.23)

(1 am, Pbn)

(4uj)k_1

c

Sab(m, n, c)Jk-1

c>O

In particular, (3.24)

=

((4irm){1+2iri>c'Saa(mm;c)Jk.l k - 1)

rli1 c

}

c>0

Corollary 3.5. Let a be a singular cusp for a multiplier system 17 of weight k > 2 on r. The space of cusp forms S1. (r, i9) is spanned by the Poincare series Pp,,, for m = 1, 2, 3, ... .

Proof. The linear space spanned by Pa,,, is closed in Sk(F, i9) because the whole space has finite dimension, and a function orthogonal to this subspace must be zero because all its Fourier coefficients vanish by virtue of (3.22). This proves Corollary 3.5. Since Sk(I', z9) is spanned by with in = 1,2,3,... and on the other hand s, (r, d) has finite dimension, it. follows that the Poiucar6 series

3. The Eisenstein and the Poincar6 Series

54

are linearly dependent. There are many open problems about Poincarb series such as:

Problem 1. Find all the linear relations between Pam(z). Problem 2. Find a basis of Sk(r, t9) consisting of the Poincar6 series. Problem S. Which Poincar6 series do not vanish identically? For the modular group it is known that the first Poincar6 series Pm(z) with m < dim Sk (r) span the space Sk(r). Recall that dim Sk(r) = k/12 +

0(1) by Theorem 1.4. R. Rankin has proved that Pm(z) does not vanish identically if m < c(e)k2-f. C.J. Mozzochi extended this result for the group ro(q), showing that P,,,(z) # 0 if m < c(e)(kq)2-. These results depend on Weil's bound for Kloosterman sums.

Exercise. Prove in general that Pam(z) does not vanish identically for any m < ckIDI, where IDS is the volume of the fundamental domain. Hint: estimate (3.24) using the trivial bound for Kloosterman sums (4.2), and apply (2.35). Choose an orthonormal basis of Sk(r,19), say .P, and expand Pam into this basis. By (3.22) we get (3.25)

P..(z) =

-

r(k - 1) (41rm)k - 1

f a(m)f (z)

fEF

where fa(m) denotes the m-th Fourier coefficient of f at the cusp a. Comparing the n-th Fourier coefficients at the cusp b on both sides of (3.25), we obtain by (3.19) Theorem 3.6. Let a, b be singular cusps for a multiplier system 19 of weight k > 2 on r. Let Jr be an orthonormal basis of Sk(r,19). Then for any positive integers m, n we have (3.26)

I'(k 1) (47r mn)k_1

fE

fa(m)fb(n)

=bmnaab+21ri-kEc 1Sah(m,n;c)Jk_1 c>O

(47 r

mn' Vr-

.

e

Remark. For k = 2 the Poincar6 series (3.17) does not converge absolutely; nevertheless, the formula (3.26) remains valid (the series of Kloosterman sums and Bessel functions converges absolutlely).

3.3. The Hilbert space of cusp forms

55

Example 1. Let IF = SL2 (Z) and k be an even integer, k > 2. The Kloosterman sum for the trivial multiplier system and for the cusps at oo becomes the one which was originally introduced by Kloosterman [Klol, namely

ma + nd

S(m, n; c) _

(3.27)

eC

c

ad=1 (mo d c)

For in = 0 this degenerates to the Ramanujan sum 's(0' n; c)

d(mod c) e

\dc

/

E . ()a

6I(c,n)

where the star restricts the summation to the classes prime to the modulus. Hence for k > 2 sl-k = ak--i(n)/n k- '((k) E C kS(0, n; c) _ ((k) c>0

61n

so by (3.16) the n-th Fourier coefficient of the Eisenstein series of weight k is

77(n) =

(27ri)k

((k)r(k) Qk-1(n)-

This agrees with the previous result (1.49). Now suppose in > 0. Then the Poincar6 series

P,n(z) _

7-y(z)-ke(m7z)

ryer.\r

can be written more explicitly as

PMW = 2

(cz + d) -ke I and (c,d)=1

- c(cz + d) )

The Fourier expansion (3.18) becomes 00

Pm(z) = E p(m, n)e(nz) n=1

where

(!)

p(m, n) = in

{6rnn +

27ri--k

E

c-1S(in,

47r cmnl

n; c)Jk-I

,

c>o

and S(m, n; c) is the original Kloosterman sum (3.27). By the Selberg relation (4.10) for S(m, n; c) and by a similar relation (3.33) for b117 we deduce

that (3.28)

p(m, n) =

dk-1 p (1, mnd-2)

in1-k

.

dl(m,n)

This formula belongs to the theory of Hecke operators (see Chapter 6).

3. The Eisenstein and the Poincarc Series

56

Example 2. Let r = SL2(Z) and k = 12. The space of cusp forms S12(r) is one-dimensional, spanned by

O(z) = (27r) 12 E r(n)e(nz). 1

Therefore any Poincar6 series P7 (z) is a multiple of 0(z), say ,,,,(z) c(m)A(z). To compute the constant c(m) we appeal to (3.22), getting c(m)IIOII2 = (0, Pm) = 27rF(11)(2m)-"r(m) Hence (3.29)

P,,,.(z) = 27rP (11 ) r(m) A (Z)

(2m)" II0II2

.

Applying (3.26), for f (z) = 0(z)/IIAII we obtain (3.30)

r(m)r(n) = v(mn) z 6m,n + 27r> c-1S(m, n; c)J11

(4)

r>0

where v is an absolute constant. One can express this constant by the value at s = 1 of the symmetric square L-function associated with f ; namely, we get v = IIIII2/47r13r(11) =

by (13.62). In particular for n > 1 we have (3.31)

r(n) = 27rvn 21 > c-'S(1, n; c)Jll

(47r c

c>o

Ramanujan Conjecture (proved by P. Deligne in 1974). Ir(n)I <, n 2 d(n)

(3.32)

Applying Selberg's formula (4.10) to the Kloosterman suns in (3.30) and the simple identity (3.33)

S1,nind-2

dj(m,n)

we obtain the relation (3.34)

r(m)r(n) _

d11r (mnd-2) dI(n,,n)

.

Chapter 4

Kloosterman Sums

Kloosterman sums are intimately related to Fourier coefficients of automor-

phic forms. They also play a very important role in the modern analytic number theory. Most recent results and applications of Kloosterman sums lie beyond the scope of this book. We shall only prove a few elementary estimates which, nevertheless, yield quite good bounds for the Fourier coefficients of automorphic forms if r' is a congruence group.

4.1. General Kloosterman sums For a general group and multiplier system the Kloosterman sum (3.13) is quite intricate; we cannot even describe explicitly the set of moduli (2.33). We can only estimate trivially (4.1)

SO(m,n,,c)

cab c2

for any c E C (a, b) (see Proposition 2.8). This is a rather poor bound; however we can do slightly better on average (apply (2.36)): c 1ISab(m,n;c)I

(4.2)

cab X.

O
Hence the series (4.3)

Z(s) _

c-23Sab(71l, tn; c) c>O

converges absolutely in Re s > 1. In the spectral theory of automorphic forms one shows that Z(s) has meromorphic continuation to the whole complex s-plane. This is a very deep result (which is due to A. Selberg, [Se1]) and technically difficult to establish in full generality (cf. (1w2)). 57

4. Ifloosterman Sums

58

4.2. Kloosterman sums for congruence groups Kloosterman sums associated with a congruence group and the trivial multiplier system can be expressed in terms of the classical Kloosterman sum

ma + nd ad=_ 1(mod c)

c

Consider the Hecke congruence group r = ro(q) and the cusps 00, 0 (recall the last paragraph of Section 2.5). The sets of moduli are C(oo, oo) = C(0, 0) = {c = eq : e E N}

C(oo, 0) = C(0, oo) = {c = £ ,/ : e E N, (e, q) = 1} ,

and the Kloosterman sums are given by (see (2.39)) (4.5)

S.. (m, n; c) = Soo(m, n; c) = S(m, n; eq)

(4.6)

So. (m, n; c) = Sao(m, n; c) = S(mq, n; e)

where q denotes the multiplicative inverse of q modulo e, i.e. (4.7)

qq = 1 (mod e).

This notation for the multiplicative inverse will be used throughout the book. Be aware that q depends on the modulus e. Exercise. Let r = ro(q) with q = rs, (r, s) = 1. Consider two cusps, oo and 1 /r with the respective scaling matrices a. _ (1 t) and at /,. Show that the set of moduli are

C(1/r,1/r) = {c = 1q: e E N} C(oo, 1/r) = {c=ervrs :QE N, (e, s) = 1}

and the Kloosterman sums are given by S11,.,11,.(m, n; c) = e ((m - n) s)

S(m, n; eq)

Soo,1/,.(m, n; c) = e (mr-) S(ms, n; er). s

4.3. The classical Kloosterman sums

59

4.3. The classical Kloosterman sums In this section we shall evaluate the Kloosterman sum (4.4) for special moduli. First we record the following elementary properties:

S(m, n; c) = S(n, m; c),

(4.8)

(4.9)

S(am, n; c) = S(m, an; c)

(4.10)

S(m, n; c) =

if (a, c) = 1,

dS(mnd-2,1; cd-1) dj(c,m,n)

The third property is due to A. Selberg [Se2]. In particular this gives (4.11)

S(m, n; c) = S(mn,1; c)

if (c, m, n) = 1.

Moreover we have (4.12)

S(m, n; c) = S(qm, 4n; r)S(rm, fn; q)

if c = qr with (q, r) = 1, where q, r are multiplicative inverses of q, r to moduli r, q respectively. The last property (twisted multiplicativity) allows one to reduce the problem of evaluating S(m, n; c) for any c to that for prime power moduli.

Lemma 4.1. If c = pea with a >, 1 and (c, 2n) = 1, then (4.13)

(2c )

S(n, n; c) = c2 (e

+e

2n)

)

Proof. Put d = a(l + bpa), where a ranges mod pea, a is prime to p, and b ranges freely mod pa. Then every primitive class mod plc is covered pa times. We have d - a(l - bpa) (mod p2c), so

S(n n, p2a)=p a

E e (na \ ++ d+ n (a - d)b)/I

I:.

a (mod p 2a )

E*

a (mod p--a)

L

(mod

p)

elna+d

`. P2a l

a=_a (mod pa )

The solutions to the last congruence are a = ±1 + tpa with t ranging freely modulo pa, and a = ±1 - tpa. Hence we deduce (4.13).

4. Kloostermazi Sums

60

Lemma 4.2. If c = p2a+1 with a >, 1 and (c, 2n)) = 1, then (4.14)

S(n, n; c) = 2

( c } c2 Re e,e (?)

where (n) is the Legendre-Jacobi symbol and Ec = 1, i according to whether

c - 1 or -1 (mod 4). Proof. Putting d = a (1 + bpa+i), we obtain as before a+a

e n p2a+1)

S(n, n; pea) _

.

a (mod pea+I )

a=a (mod pa)

The solutions are a = ±1 + tpa with t ranging freely mod

pa+i, and a =

±1 - tpa ± t2p2a. Hence

e(n

S(n, n; pea+') = Me

2+ t2 2a ) pea

t (mod p°+i )

= 2Re pae

2n

9(n, p)

p2a-t t

where 9(n, p) is the Gauss sum 2

(4.15)

g(n,p) _ t (mod p)

e

(_) = EP (p) p2.

This formula will be established later in Lemma 4.8. Hence (4.14) is proved. We put both formulas (4.13) and (4.14) (which are due to H. Salie, 1936) in a unified form

Proposition 4.3. If c = pa with J3 > 2 and (c, 2n) = 1, then (4.16)

S(n, n; c) = 2 (c) cz Re Eye

2cn

One can generalize this result slightly for sums S(m, n; c) with c = plj, Q

2, such that (c, 2mn) = 1. We have S(m, n; c) = 0 unless m

(mod c), in which case (4.9) and (4.16) yield S(m, n; c) = S(Cn, Cn; c) = 2 ( In any case we conclude

- c)

c^ Re e,e

(261). c

C271

4.4. Power-moments of Kloosterman sums

61

Corollary 4.4. If c = p3 with Q > 2 and (c, 2mn) = 1, then IS(m, n; c) I <, 2c!2.

(4.17)

The same arguments work for c = 213, giving similar results.

The Kloosterman sum S(m, n; p) to a prime modulus resists any explicit evaluation in the above fashion. In 1948 A. Weil proved the Riemann hypothesis for curves over finite fields, from which he deduced that S(m, n; p) = ap + Op

(4.18)

where ap, Op are algebraic numbers such that ap = Op and apfp = p. Hence

Iapl=I1pl=pa and (4.19)

I S(m, n;p)I 5 2p2

Combining all cases, one can derive by the multiplicativity property (4.12)

that

Theorem 4.5. For any c > 1, m, n we have (4.20)

I S(m, n; c)I

(m, n, c) Z c2,r (c)

where r(c) denotes the divisor function.

4.4. Power-moments of Kloosterman sums Today the Riemann hypothesis for curves, hence the Weil bound for Kloosterman sums, can be established by elementary means (due to S.A. Stepanov, W. Schmidt and E. Bouibieri), yet the arguments are quite involved. It. is much easier to estimate S(m, n; p) on average with respect to na, n. In this section we estimate a few of the power-moments defined by E,.

(4.21)

VA- (p) _

Sk(a,1;p)

a (mod p)

By (4.9) we obtain

(p - 1)Vk(p) = EY: Sk(a, b;p) abAO (niod p)

J:E

SA(a,b;p)

- (p- 1)A

2(-1)'`(p- 1)

a,b (mod p)

= r'k(p)2r' - (p -

1)

4. Kloosterman Sums

62

where vk.(p) is the number of solutions to the system

(modp) 0

(modp).

For k = 1 we have no solutions, so vl (p) = 0 and V1(p) = 1.

(4.22)

For k = 2 the solutions are x1 = -x2 - 0 (mod p), so v2(p) = p - 1 and V2(p) =p2 - p - 1.

(4.23)

For k = 3 the solutions are x2 = xly, x3 = -X1(1 +V) where xl $ 0 (mod p) and y2 - y + 1 0 (mod p). The number of solutions in y is 1 + (=3), so

V3(p) = (p - 1) (1 + (=3)). Hence p (4.24)

-3) p2 + 2p + 1.

V3 (P) _ (

\ p

For k = 4 we write

V4 (P) _ EE 7I(u>v)2 u,v (mod p)

where rj(u, v) is the number of solutions to the system

x+y-u (modp), xJ1+y-1v (modp). If u=0then v-0and r7(0,0)=p-1. Ifu 0then v$0,x+y=u, xy - uv-1, so x, y are solutions to the quadratic congruence z2

- tiz + uv-1 = 0 (mod p).

ty-1, so the number of solutions is The cliscriminant. is A = v2 - 42(A) y(u,v)=1++(1-4/uv)

p

p Hence

EY: uvOO (mod p)

?(U, V)2

?(U, 1)1

= (p - 1)

u O (mod p)

(i+()) =(p-1) E ml (mod P) = (p - 1) 4L ::-l + 1

-4

(p - 1) (2p - 5),

4.4. Power-moments of Kloosterman sums

and v4(p) = (p (4.25)

-

63

1)2 + (p - 1) (2p - 5) = 3(p - 1) (p - 2), giving

V4(p) = 2p3 - 3p2 - p - 1.

By (4.25) we infer by dropping all but one term in (4.21) that (4.26)

IS(m,n;p)I <2pi,

if p f 2mn. This result and the above methods of proof are due to H.D. Kloosterman (from his thesis of 1926; see [K1o]). By (4.23) and (4.25) we get V4(p)V2(p)-1 > 2p- 2, whence (4.27)

IS(a,1;p)I > (2p - 2)21

for some a (mod p), showing that the constant 2 in Weil's bound (4.19) cannot be replaced by any number < V-2. Therefore some of the roots ap are closer to the real axis than to the imaginary one. Incidentally, one can show that S(m, n; p) does not vanish, for S(m, n; p) _- -1

(mod 7r)

where 7r = 1 - (p is the prime factor of p in the cyclotomic field Q((p). In 1931 H. Salie and H. Davenport (independently) estimated the sixth power-moment (still elementarily) (4.28)

V6 (p) << p4

from which they deduced that (4.29)

S(a, l; p) < p:

.

Higher moments of Kloosterman sums resisted any attack until 1979 when N. Katz skillfully employed P. Deligne's profound theory of exponential sums over varieties over finite fields. It turns out that more natural than the power-moments are the mean-values of Tchebyshev polynomials Uk.(x) at normalized Kloosterman sums. In view of Weil's bound we have (4.30)

p ^i S(a, l; p) = 2 cos Op(a)

where 0 < Op(a) <, 7r is called the Kloosterman sum angle. The Tchebyshev polynomials are defined recursively by Uo(x) = 1,

U1(x) = x

4. Kloosterman Sums

64

xUk(x) = Uk-1(x1 + Uk-F1(x)

One shows that

(-1)t(k - E)ixk-2t

Uk(x) _

8I(k-2811

Another way of defining the polynomials Uk(x) is by sin(k + 1)0 Uk(2 cos 0) =

sin 0

The generating series of Uk(x) is 00 y2)-1

Uk(x)yk = (1 - xy + k=0

Theorem 4.6 (N. Katz). For k

1 and p > 2 we have

I(k+1)pa.

1 E* Uk(2cosOp(a))I

(4.31)

a (rnod p)

The key feature of the Katz estimate lies in the fact that the Tchebyshev polynomials Uk(x) form an orthonormal basis in G2([O,ir],µ) with respect to the Sato-Tate measure µ = 21r-1 (sin O)2d0.

(4.32)

Thus Theorem 4.6 implies that the Kloosterman angles Op(a) are asymptotically equidistributed for the Sato-Tate measure as a ranges over the primitive classes (mod p) and p - oo. In other words, for any continuous function f on [0, 7r] we have linl

op

f (O (a)) _

1

a

? ff(O)(sinO)2dO.

(mod p)

A much harder question is, how is the Kloosterman sum angle distributed with respect to the modulus? Conjecture. For a fixed a 0 0 the Kluusterrnan angles Op(a) are asymptotically equidistributed for the Sato- Tate measure as p ranges over primes,

i.e.

limn

00

E f (Op(a)) _

()p<x x

_

f (0)(sin0)2d0. fo

This conjecture seems to be out of reach by present methods. We do not even know if (the case of f (0) = 2 cos O) (4.33)

line

1 E S(a,1; p) = 0 7r (X) p
VP

4.5. Sums of Kloosterman sums

65

4.5. Sums of Kloosterman sums In this section we establish estimates for sums of IS(n, n; c) I over special moduli by completely elementary means. These estimates are strong enough to serve as a substitute for Weil's bound (4.20) in various important applications (see our elementary proof of (5.19)). We begin by showing that (4.34)

r a (mod r)

IS(an, an; r)12 <, 4(n, r)rr(r).

Indeed, opening the Kloosterman sum, squaring it out and then summing over a (mod r), we see that the left side is equal to the number of solutions to the congruence n(x +.t) _ n(y + y) (mod r). This is equivalent to n(x - y)(xy - 1) _ 0 (mod r) in (xy,r) = 1, whence by a local consideration one can easily deduce that the number of solutions is S 4(n, r)rr(r), as claimed. Next we infer by (4.34) that A(Q, R) _ EY: I S(4n, 4n; r) 12 q
r
(Qr + 1 I

IS(an,an;r)12 a (mod r)

S E 4(Q + r)(n, r)rr(r) 5 4r3(n)R2(Q + R) log 4R. r
Hence by (4.12) and Cauchy's inequality

13(Q,R) EE IS(n,n;gr)I q
4r3(n)QR(Q + R) log 4QR. Finally we remove the condition (q, r) = 1, getting A(Q, R) 2 A(R, Q)

C(Q, R) = 1: 1: I S(n, n; qr)I q
> > 13(Ql a, R/b) 5 3r.3(n)QR(Q + R) (log 4QR)2. a
Hence by partial summation we deduce

Lemma 4.7. For Q, R > 1 we have (4.35)

EE q
(qr)-1 {S(n, n; qr)I < r3(n)(Q + R)(log4QR)3.

4. Kloosterman Sums

66

Remark. By Weil's estimate (4.20) one could derive (4.35) with Q + R replaced by (QR) 2 , which has the same order of magnitude if Q = R.

4.6. The Salie sums They are obtained by twisting the classical Kloosterman sums with the Jacobi-Legendre symbol:

(d) e (m2 +nd)

T(rn,n;c) _

(4.36)

d (mod c)

c

c

J

Salie sums occur in the context of automorphic forms with respect to the theta multiplier system (see (2.73)) by means of the quadratic reciprocity law. These sums can be evaluated explicitly in terms of the Gauss sum (4.37)

g(n,c) _

e t (mod c)

We begin by evaluating the Gauss sum.

Lemma 4.8. If (c, 2n) = 1 then g(n, c) =

(4.38)

where

ec _

(4.39)

11 i

ifc ifc

1 (mod 4)

-1 (mod 4).

Proof. Put c = qr2, where q is squarefree, and change the variable t (mod c) in (4.37) to t = x + qry with x (mod qr) and y (mod r). We get t2 = x2 + 2qrxy (mod c), whence

g(n,c) _

j e (nx) --L t\) C2n.xy

x (mod qr)

y (mod r)

r

The innermost sum is equal to r if x - 0 (mod r), and it vanishes otherwise. Therefore (4.40)

g(n, c) = rg(n, q)

For q a square-free number we have

#{x (mod q)::r.'-=y (mod

q)}(1 +\p// rlr

/

(d).

Ir

4.6. The Salie sums

67

Hence

9(n,q)=E E (d)e(q ) dlq y (mod q) The inner sum vanishes save for d = q, giving (4.41)

(nq)

9(n,q)=

e(q) =

q

y (mod q)

g(1, q).

ve Combining (4.40) and (4.41), we der(nC) (4.42)

9(n, c) =

9(1, c).

Finally we get by (1.10) (4.43)

9(1,c) = e,c2

and complete the proof.

Now we evaluate the Salie sums. Suppose (c, 2n) = 1. Let us consider the function

F(x)_

E

d (mod c)

(d)e(md+cndx2 ) J

defined in x (mod c). The Fourier transform of F(x) is P(y)

=xE F(x)e (-f!) (mod c) _

d d

(

r

(c)

e

and

a

(ndx2_ C

x (mod c )

)

xE /

Complete the square; the innermost sum becomes e (-Qndy2) g(nd, c). If we substitute 9(nd, c) = (c) g(71, c)

the Legendre-Jacobi symbol disappears in the Fourier transform F(y), so that it becomes the Ramanujan sum F(y) = g(n,c)

e

(md -

I

C

d (mod c)

= g(n, c)

E_

e

((4mn

d (mud c)

= g(n., c)

E

dV2 )

dI(4,nn-i 2,c)

- y2)., dc

dtt(c/d).

4. Kloosterman Sums

68

Hence by the Fourier inversion

F(x) _ 1 y (mod c)

F(y)c ` c

9(c+c)>dµ(c) r l

l

r

e\xc/

y (mod c)

d1c

y2-4mn (mod d)

If (x, c) = 1 this simplifies to F(x) = g(n, c)

e

F%Z-mn (mod c)

Taking x = 1, we obtain the following.

Lemma 4.9. If (c, 2n) = 1 we have (4.44)

E

T (m, n; c) = (c) g(c)

y2ctnn (mod c)

e (v).

Corollary 4.10. If (c, 2n) = 1 we have (4.45)

IT(m,n;c)l <, car(c).

In the case of m = n we can write explicitly all the solutions to y2 = 712 (mod c), namely y = (ad - bb)n, where a, b range over integers with ab = c and (a, b) = 1. Thus (4.44) becomes

Corollary 4.11. If (c, 2n) = 1 we have (4.46)

T (n, n; c) = (c) g(c)

c (2n 1 ttb=c,(tyb)=l

G

-

J

I

111 /

Remark. This result was first established by H. Salie (Sal), but only for c a prime power. Originally we [Iwl] extended the Salie formula to the above form by an appeal to the quadratic reciprocity law; however, the present derivation (due to P. Sarnak [Sar]) is more direct. It is important that our formula does not depend on prime factorization of the modulus. The very specific shape of (4.46) (a kind of Dirichlet convolution) will be crucial in its application to estimate the Fourier coefficients of cusp forms of half integral weight (see Section 5.3).

Chapter 5

Bounds for the Fourier Coefficients of Cusp Forms For various constructions to be carried out in later chapters, such as the Lfunctions associated with a cusp form, we need to control the growth of the Fourier coefficients. In this chapter we shall establish the required estimates. Throughout we assume that r is a discrete group for which oo is a cusp of width 1, i.e. the scaling matrix is v00 = 1 (the more general case of an arbitrary cusp can be obtained by conjugating the group). We also assume that i9 is a multiplier system of weight k > 0 for r which is singular at the cusp oo.

5.1. General estimates Let f be a cusp form for r with respect to the multiplier system 19. Thus f has the Fourier expansion

f (z) =

L a(n)e(nz).

Our problem is to estimate the coefficients a(n) = a f(n). Quite strong results are deduced rather easily from the observation that (5.2)

F(z) = y4lf(z)I

is a bounded function on the whole upper half-plane H. This is clear because F(z) is F-periodic and has exponential decay at every cusp. Therefore, we 69

5. Bounds for the Fourier Coefficients of Cusp Forms

70

have f (z) G< y-k/2

(5.3)

if z E H

where the implied constant depends on f. Conversely, if F(z) is bounded in H then for any a E SL2(R) we have I fro (z) I = y-12F(az) << y- 3 . Hence fla,(z) vanishes as y - oo, proving that f is a cusp form. Therefore we have established the following criterion.

Criterion for Cusp Forms. I. Suppose f is an automorphic form of weight k > 0 for a group r. Then f is a cusp form for r if and only if (IM z)1/2I If (z) I is bounded in the upper-half plane.

By the Parseval formula and (5.3) we obtain 1

la(n)

I2e-4any = f If 0

n

(z) I2dx <<

y-k,

whence

E la(n)I2 << y-ke4nNy n
for any y > 0. Choosing y = N--1, we obtain

Theorem 5.1. For any N >, 1 we have (5.4)

E la(n)I2 K Nk n
where the implied constant depends on f.

Remark. The upper bound (5.4) is best possible, for one can prove (by the Rankin-Selberg method, see Section 13) the more precise asymptotic formula (5.5)

E Ia(n)I2 - cNk n_
as N -> oo, where c is a positive constant which depends on f. The upper bound (5.4) shows that the Fourier coefficients of cusp forms satisfy (5.6)

a(n) << n2

on average. For any individual coefficient it yields a(n) << n 2.

Using Cauchy's inequality, we deduce by (5.4) the following estimate:

71

5. 1. General estimates

Corollary 5.2. For any N >, 1 we have (5.8) n<'N

This bound can be improved substantially if we drop the absolute value to allow cancellation between the terms. Indeed we established the following:

Theorem 5.3. For any real a and N 3 1 we have (5.9)

a(n)e(an) << N L21 log 2N n<,N

where the implied constant depends only on f (not on a). Proof. The Fourier coefficients are given by

a(n) =

'I

J0

f (z)e(-nz)dx.

Hence the sum of coefficients twisted by the additive character e(an) is equal to

a(n)e(an) = J f (z + a)SN(z)dx 1

n<,N

0

where

SN(z) = E e(-nz) = e(-Nz)

1

K e27rNyl1 - e(z)l-1.

O
Since f (z + a) << y-k/2 and 1

11 - e(z)l-ldx << log(2 + y-1),

Jo

we obtain

E a(n)e(an) << y-k/2e2iNy log(2 + y-1) n_
where y is arbitrary positive number. For V = N-1 this yields (5.9).

The improvement in the estimate (5.9) is due to the variation in the sign of a(n), since (5.8) is essentially best possible. The logarithmic factor in (5.9) cannot be reduced to a small constant, since it would contradict (5.5) by Parseval's identity. Theorem 5.3 shows that the Fourier coefficients of a cusp form are asymptotically orthogonal to additive characters. The fact that the bound (5.9) does not depend on a makes it possible to derive (by means of additive characters) the same result for a sums of the Fourier coefficients restricted to any arithmetic progression.

5. Bounds for the Fourier Coefficients of Cusp Forms

72

Corollary 5.4. For any q 3 1 and a (mod q) we have a(n) << N 2 log 2N

(5.10) n_< N

n-a (mod q)

where the implied constant depends only on f.

The argument used in the proof of Theorem 5.3 can be modified to estimate a bilinear form in the Fourier coefficients of the following type:

Theorem. For any complex numbers a,,,,13n we have

EE

(5.11)

m) << NZ IIaII IIf3II

m
where hall, 111311 denote the £2-norms of a = (a,n) and /3 = (13n) respectively.

Proof. The restriction m < n can be ignored because a(e) = 0 if t 5 0. We estimate the resulting bilinear form by applying the Cauchy-Schwartz inequality and the Parseval identity as follows: 1

in

fo

n

E,Qne(-nz) dx

f (z) E arne(rnz)

a,n13na(n - m) =

m

<< y 2 a >

n

(Ia,2)

t

2

22

Iaml2e-4amg

in

N4

t IQn12eaany

n

2

(l/3l2 l 2

=

N4IIall11,11

n

by choosing y = N.

5.2. Estimates by Kloosterman sums In the general case the bound (5.7) is essentially the best possible; definitely (5.6) cannot hold for all n. However, if one has a non-trivial estimate for Kloosterman sums, then one can improve the individual bound (5.7) by an appeal to the Fourier expansion of Poincare series ro

(Y)?y(z)-tie(nvyz) _

Pm(z) _ 'yEr-,\r

p(m, n)e(nz).

=1

5.2. Estimates by KIoosterrnan sums

73

We assume that k > 2, so any cusp form is a linear combination of P,,,(z) with m 3 1; therefore, it suffices to estimate p(m, n). By (3.25) and the Cauchy-Schwartz inequality we obtain (5.12)

Ip(m, n) I2 < (m

p(m, m)p(n, n)

Hence it suffices to estimate the diagonal coefficients (see (3.19)) (5. 13)

p(n, n) = 1 + 2 rik E cS(n, n; c)Jk_1

(!) C

c>O

Recall that (see (3.25)) p(n,n) =

(5.14)

r(k - 1)

(47rn)k-1

11(n) 12 ! E.F

where 2 = (f) is an orthonormal basis of Sk(r, z9). Suppose the Kloosterman sums S(n, n; c) are bounded by c2°-1 on average; more precisely, suppose we have E c-2a IS(n, n; c) I << ne

(5.15)

c>O

< or < 1 and any e > 0 with the implied constant depending on or, e and the group r. We have for some (5.16)

J" (X) < lain{x",x z }

x6,

if - 2 < 6 < v.

By (5.15) and (5.16) with v = Ic - 1 and 6 = 2a - 1 we derive (5.17)

p(n,n) < n

Hence by (5.14) we get

Proposition 5.5. Suppose the Kloostersaun sums associated with a multiplier 79 for a group IF satisfy (5.15). Then the Fourier coefficients of any cusp form f E Sk (r, 79) of weight k > 2 are bounded by (5.1$)

a(n) <
If r is the Henke congruence group and 19 is the trivial multiplier system, we have the Weil hound (4.20), which gives (5.15) with any a > In this case (5.18) becomes .

(5.19)

iE a(71) «71.2-i.

5. Bounds for the Fourier Coefficients of Cusp Forms

74

We shall prove (5.19) by elementary means without appealing to the Weil bound for individual Kloosterman sums. Our argument makes use of (4.35) instead of (4.20), but not directly. Suppose f is a cusp form for ro(N) with respect to the trivial multiplier system of weight k > 2. Then f is also a cusp form for any ro(q) with q = 0 (mod N). Let IIfIIq denote the norm of f with respect to the group ro(q), so we have (5.20)

IIfIIq = [ro(N) : ro(q)]IIf1IN

where the index is determined by (2.25). If a(n) is the n-th Fourier coefficient of f (see (5.1)), then (5.21)

f (n) = a(n)/II f II q

is the n-th Fourier coefficient of the normalized form fq = f /IIf IIq. Take an orthonormal basis F of the space SkjI'o(q)) which contains fq. By (5.13), (5.14) and (5.21) we get

c-1S(n,n;c)Jx_u

1+2iri_A

(5.22)

(47rn)k)IIIfII2'

(.±)

c-0 (mod q)

by dropping all but fq from F. Next we sum this inequality over q = 0 (mod N) with Q < q <, 2Q. Since for fixed N the sum 7n f(Q) _

(5.23)

IIf IIq 2

Q
q=_0 (mod N)

is bounded by positive constants (see (5.20) and (2.25)), we obtain 00

n1-Aa(n)2 << Q +

>(qr)-'IS(n,n;gr)Jk_1 Q
4I. (-)

By (5.16) for v = k - 1 > 1 and (4.35) we get (5.24)

nl _k

a(n)2 < Q + (Q + R) min QR, (Q)

(nQR)c

for some R > 0. The worst bound occurs in (5.24) for R = nQ-l, namely (5.25)

nl-ka(n)2 < (Q + nQ-1)n

Here the best choice is Q = n°, giving (5.19).

5.3. Coefficients of cusp forms with theta multiplier

75

5.3. Coefficients of cusp forms with theta multiplier In the study of representations by quadratic forms in an odd number of variables (see Chapter 11) we shall encounter cusp forms with respect to a multiplier system derived from the standard theta function (2.72). By the theta multiplier of weight k = 1/2 + 2 where t E Z we mean (5.26)

(cll2k (Ed

if r =

kill

a c

b

d

E re(4).

The Fourier coefficients of cusp forms in Sk(ro(N), i9) satisfy the bound

a(n) <
(5.27)

by virtue of the Salie estimate (4.45) and Proposition 5.5. This result is essentially best possible; namely the exponent 2 - a cannot be reduced if n = m2. However if n is squarefree, one expects that hi-t a(n) « n Z +E

(5.28)

but this (an analog of Ramanujan's conjecture) has not yet been proved. We shall do slightly better than (5.27) by exploiting a cancellation in sums of Salie sums.

Theorem 5.6. Let k

z, 41N and f E Sk(ro(N),V). Then for n square-

free the n-th Fourier coefficient of f satisfies the bound (5.29)

a(n)<
-az3-r(n) logn

where the implied constant depends on f. This improvement on the standard bound (5.27) is very slight; nevertheless it is vital for applications, in particular to establish the equidistribution of integral points on the sphere (see Section 11.6). One can assume that 8IN by considering the subgroup Fo(2N) if necessary. We begin by modifying the arguments used in the proof of Proposition 5.5. For any p we have the inclusion Sk(ro(N),19) C Sk(F1)(pN),19), from which we deduce the following basic inequality (by positivity as in the proof of (5.22)):

F(k -

1)(41rn)1-kIa(n)l2lIf11-2 [Fo(N)

I+

:

Fo(pN))`1

c-1

27ri.-k

c_0 (moil pN)

S(n, n; 64-1

(!!)

5. Bounds for the Fourier Coefficients of Cusp Forms

76

where S(n, n; c) is the Kloosterman sum,

S(n, n; c) = E* v9(r)e I nd + d

(5.30)

I

d (mod c)

Our goal is to create some kind of bilinear forms in the Kloosterman sums. To this end we average the above inequality over primes p t n in the interval

P < p < 2P, each such p being weighted by logp. We shall choose P later subject to N(logn)2 < P < ne . Since [to(N) : Po(pN)] = p + 1 and

E (p + 1) -' log p "log 2 P
the resulting left side is asymptotically pn' Ia(n)12, where p is a positive constant which depends only on f. On the right side of our basic inequality we obtain a sum of Kloosterman sums to moduli c - 0 (mod N) weighted by

loge.

w(c) = P
Note that w(c) < log c if c > 0 and that the constant term 1 (the diagonal term) on the right side of our basic inequality yields w(0) P. Therefore we have

nl-loa(n) I2 << P + I SI

(5.31)

where S is the weighted sum of Kloosterman sums,

w(c)c_.1S(n,n;c)Jk-t

S

(n)

c-u (mod N)

c

Recall that the individual Kloosterman sum satisfies

S(n,n;c) << (c,n)2c2r(c), and the Bessel function is bounded by

Hence the terms with c < C contribute to S at most =2I

(5.32)

Sb << n

J>, n)-21,r(c) loge << Cn. .i (7-(n) log n)2, c<,C

5.3. Coefficients of cusp forms with theta multiplier

77

and the terms with c 3 D contribute to S at most (5.33)

SO «n2 >(c, n)2T(c)c`2 loge << D-ln2(T(n) log

n)2.

c>_D

We shall choose C and D quite close to n, with C < n < D, so the above bounds will be admissible. Now we are left with the central terms

S* =

w(c)c-1S(n, n; c)Jk

C4C 1rn 1

C
JJ

We set c = qr, where q denotes the largest factor of c coprime with nN, so r has its all prime factors in nN and is divisible by N. Therefore w(c) = w(q), and S* splits into

S* =

(5.34)

E

r-' T,

rl(nN)°O r=_0 (mod N)

where 47rn

Tr= E w(q)9lS(nn;qr)J ,

q1'

C
First we estimate Tr trivially as follows:

qrnqr {()

3

1

T,- <<(n,r)Zr2T(r)T(q)q z' min 9

logq

« T (')') (11, 1 )'l 'it 1 (lug 7)2.

This will he used only for r sufficiently large, say r > R. Hence the contribution to S= of terms with r > R is bounded by (5.35)

71-1T,- « R =1 n2 (log n)2

r(1') (n, I-) ^ r-'

r>R

78

It remains to estimate T, for each r < R. The Kloosterman sum in Tr factors into S(n, n; c) = S(nq, nq; r)T (nr, nr; q)

(5.36)

where qq - 1 (mod r), rr = 1 (mod q) and T(nr, nr; q)

(5.37)

x (mod q)

(T)

e q(nr(x

+ x))

is the Salie sum. Observe that the Kloosterman sum S(nq, nq; r) depends only on q (mod r). Splitting into residue classes, we obtain

Tr S E_ 1S(n3, ns; r)T,.sj

(5.38)

s (mod r)

where

w(q)9-1T(nr, nr; q)Jk-1

Trs

C
4(irn

qr

) /J

q=s (mod r),(q,n)=1

The Salie sum was evaluated in Corollary 4.11; namely, we have

T(nr, nr; q) = eq I

q

I

\ /

, aEq a (2nr

bQ

(a,b)=1

In the sequel we omit writing the condition (a, b) = 1 whenever it is redundant vis-a-vis the meaning of a (mod b) and b (mod a). By the reciprocity formula

x+y y

x

xy

(mod 1)

we write (

`E e (2n? q

\

//

= 2Re ae (In a + 2n a
a-

b)

Inserting this expression into !rs we get L

2r

Trs=ES(s )(n)

:;

Re

w(ab)lable(2n C
b

+2n

)j \ _--

5.3. Coefficients of cusp forms with theta multiplier

79

where

j(x) = s e(-x)JA._1(27rx). Next we remove j(2n/abr) by partial summation in b. Using the estimates j(x) «x2, j(x) << 1 and j(x) << 1 we get

EI

Trs << C-1(nr)1

`"'(ab) (b) a (2n(1 + rr) a)

I

a< A a< b < B (a,nr)=1 abas (mod r)

for A = (D/r) 2 and some B which depends on a such that C < arB < D. Since w(ab) = w(a) +w(b), the innermost sum splits into Va +w(a)VQ, where

E w(b)(n)el2n(1+rf)a ),

Va=

a
and Va' is given by the same sum without the weight w(b). Thus

E (jVaj+w(a)jVQj)

T,. << C-l(nr)2

a
We shall estimate each Va separately, so the dependence of B in a is irrelevant (the case of Va is similar, actually slightly simpler). We have Va = P
(loge) (p.)

a/p
()

e1

2n(a+r))

.

\\\

apl_s (mod r)

If a < P we split the summation over a into residue classes (mod ar) and apply the Polya-Vinogradov estimate n 1

E t
(

«n= logn

f=a (mod ar)

(note that the character (e) is non-trivial on any arithmetic progression to modulus ar because n is squarefree and larger than ar). We obtain

Va «aPn2 logn.

(5.39)

If P < a < A we regard Va as a bilinear form in 2 and p. By Cauchy's inequality

/

f
Pi ape=-$ (mod r)

p)(n)a(2n(1+rr)L!)l2 P \

5. Bounds for the Fourier Coefficients of Cusp Forms

80

where L = BP-1, P1 = max(P, a/e) and P2 = min(2P, B/e). Squaring out and changing the order of summation, we get

E

Vat << L(log P)2

e

1

P
(2n(1 + rr)(p - p')we

I.

a/p'
p=-p' (mod r) apt=-s (mod r)

The terms with p = p' contribute O (L 2p log p). If p p' the innermost sum over a is an incomplete Kloosterman sum to modulus a. After completing the sum by Fourier technique we can apply Weil's bound (4.20) to deduce

that

E <<(p-p',a)a-1LlogL+(p-p',a)la21 T(a)loga. t Summing over p # p', we obtain a2 << (L2P + a-1 L2P2

LP2) (,r (a) log n)2.

+ a2 Since a > P the middle term can be ignored; hence

Va << (BPI +a4B2P2)7-(a)logn.

(5.40)

The same estimate holds true for Va' by similar arguments. Summing over a (use (5.39) if a < P and (5.40) if P < a < A), we infer

that

T,,, «C-1(nr)2 (n2p3+DP 21 +DSP2) (logn)2. Inserting this into (5.38) and then summing over s (mod r), by the trivial estimate IS(ns,ns;r)I < r we get

Tr<
<

RLC-1 (DP z

+ DRP2) n2 (7-(n) logn)2.

r
From (5.41), (5.35), (5.33) and (5.32) we deduce that

S<[Cn-21 +D-1nz +R z1n2 +R2C-1 (DP We choose C = (5.42)

1151

112

14

nil], D = n lit , P = n.1 T 11

S < 711

+DBPz)nz](T(n)logn)2. 2

and R = nil, , getting

(-T (71) log n)2.

Finally, inserting (5.42) into (5.31), we deduce (5.29).

5.4. Linear forms in Fourier coefficients of cusp forms

81

Remark. We could easily get a slightly sharper estimate if we did not waste various factors in r. However, any further improvement seems to be

insignificant for applications. Our choice of the parameters C, D, P, R seems to be very delicate, leaving the impression that we were lucky that at the end of the proof we have room to accommodate these parameters. However, this could be predicted quite easily before computations if one assumes there is a better estimate for sums of Salie sums (over the moduli) than that which follows from the estimates for individual Salies sums. Our improvement is obtained from cancellation due to the variation in the sign of Salies sums, which we were able to exploit by creating bilinear forms.

5.4. Linear forms in Fourier coefficients of cusp forms Choose an orthonormal basis of cusp forms in Sk(F, i9), say F. Any f E F has the Fourier expansion (we assume that oo is a singular cusp of width 1) 00

f(z)

=

f(n)e(nz).

n=1

The orthogonality of distinct forms in F has profound impact on the Fourier coefficients. One of these can be seen in estimates for general linear forms (5.43)

,Cf(a) = > an'/'J(n) n
where (5.44)

iJ, f(n) =

(I'(k - 1)/(47rn)k

1) z

f (n)

are normalized coefficients. Given a sequence of complex numbers a = (a,,), one can show that there is a considerable cancellation between the terms of £f(a) for almost all f E.P. We shall give precise estimates of L f(a) for the Hecke group F = Fo(q) in terms of the level.

Theorem 5.7. Let F be an orthonormal basis in S(Fo(q)) with k > 2. Then for any complex numbers a,, we have (5.45)

E 1G f(a)12 = {1 + O(N/q)} 1IaIh2 JE.F

where the implied constant is absolute.

5. Bounds for the Fourier Coefficients of Cusp Forms

82

Proof. By (3.26) the left-hand side of (5.45), say L(q), is equal to IIaII2 +

47r mn

c-1 1:1: a .,,,a,,S(m, n; c)Jt;_1

27ri-k

c= o (mo d

m,n, N

q)

Opening the Kloosterman sum, we derive by Cauchy's inequality

I EE

C dm

n; c)I <, E I E ame

11 E ane

d (mod c) m<,M

m,n<_N

dn(l JI

n<,N

ane

C

(dn)I2 C

d (mod c) n<-N

l

= C EL aman <,(c + N)IIa1I2. m=n (mod c)

In order to attach the factor Jk_ 1 (4 7r VCmn) we expand the Bessel function into power series: 00 Jk-1(x) _ E (-1 e!r(k + e)

)e(x/2)k-1+ze

and apply the above inequality for each term separately, getting

I E: amanS(m,n;c)Jk_1 \C47r mnl) c

m,n,
where

00

Ik-1x) = e=o

Ik-1

I

(47rNl (c+N)IIa1I2

\

c

J

(x/2)k-1+2e

e!r(k + e)

Next we estimate (1 + c-1N)Ii,-1 (mod q)

C47rN)

\

c

C1 +

N1

qJ

IR_1

(47rN) q

JJ

C(k

Hence (5.46)

L(q) = (1+0(N/q)}IIaII2

with IB(x)I S 21r((k - 1)Ik_1(47rx)(1 + x). This proves (5.45) if q >, N. For q < N the estimate (5.46) is rather poor because Ik_ 1(x) has exponential growth. To improve the result we apply the inequality (5.47)

L(q) < (p+ 1) L(pq),

5.5. Spectral analysis of the diagonal symbol

83

which follows from the embedding Sk(ro(q)) C Sk(ro(pq)) and the estimate for the index [ro(q) : I'o(pq)] 5 p + 1. By (5.47) and (5.46) we obtain L(q) S (p + 1){1 +9(N/pq)}11x112.

Here we choose p with N < pq < 2N, getting (5.45). To get some perspective on what Theorem 5.7 offers, first one needs to know the size of the normalized Fourier coefficients (5.44) and how many forms are in the basis T. Applying (5.45) for the sequence consisting of one term an, we get (5.48)

E I'+Gf(n)I2 = 1 + O(n/q) fE.F

Hence I f(n)I has the size (5.49)

ITI-112 on average provided n << q, where

IFl = dim Sk(ro(q)) x kq H (1 + p Pi q

\\

.

/

In view of these results Theorem 5.7 asserts that the linear forms L f (a) are bounded by (5.50)

Ilall = i lanl2 n<,N

no matter what the coefficients an are, provided only that N is not too large; namely N « q. For example, for the von Mangoldt function an = A(n) this assertion is as strong as the Riemann hypothesis for the Hecke L-function (5.51)

Lf (s)

c*

Of (n)n-'

but, of course, only for almost all cusp forms f.

5.5. Spectral analysis of the diagonal symbol In this section we consider the space Sk(Fo(q)) of cusp forms with trivial character. The effect of orthogonality on estimates for general linear forms in the Fourier coefficients of cusp forms, which we have already seen in Theorem 5.7, can be strengthened if one employs a larger spectrum. There are several possibilities. One can perform extra averaging over the level q, over the weight k, or both. Since the q-aspect. was the issue in the previous

5. Bounds for the Fourier Coefficients of Cusp Forms

84

section, we focus here on the k-aspect. Note that either k is even or else the space S.(ro(q)) is zero. For notational convenience we put e = k - 1.

(5.52)

We begin by recalling Theorem 3.6 for cusps a = b = oo. The left side of (3.26) is (5.53)

De(m, n)

(m)'bf (n) f C- -Fl.

where .Fk, is an orthonormal basis in Sk(ro(q)) and ib f(n) are the normalized Fourier coefficients given by (5.44) (we do not display the dependence on q because the level is fixed throughout this section). As we know, De(m, n) is

basis independent. Indeed, (3.26) asserts that (5.54)

De(m, n) = 6(m, n) + ie+i Jt(m, n)

where 6(m, n) is the diagonal symbol (5.55)

6(m, n)

1,

j` 0,

if m=n otherwise

(4).

and Je(m, n) is given by the series of Kloosterman sums and Bessel functions (5.56)

c 'S(m, n; c)Je

Jt(m, n) = 27r c0 (mod q)

One may think of (5.54) as a kind of spectral decomposition of S(m, n), since Jt(m,, n) can be shown to be quite small uniformly in m, n « e. This range of uniformity is satisfactory for a few applications; however, it is not sufficient for some advanced problems. Therefore we shall refine (5.54) by averaging over e. To control the range of e we choose a test function g(y) which is smooth and compactly supported on R4-, and we count (5.54) with weight g(e). It is of interest to average separately over (5.57)

k=0

(5.58)

k = 2 (mod 4),

(mod 4),

f - --1 (mod 4) and e - 1 (mod 4) respectively. This distinction is particularly important in the context of the Hecke L-function (5.51) because it has the functional equation with sign ik = 1, -1 respectively (see Theorem 7.2). For a given a = ±1 we consider i.e.

(5.59)

D(na, n) =

n), fair (mod 4)

5.5. Spectral analysis of the diagonal symbol

(5.60)

1:

G(m, n) =

85

g(2)Je(m, n).

P=_a (mod 4)

F om (5.54) we obtain (5.61)

>2

D(m, n) = S(m, n)

g(e) + iai"1G(m, n).

t =-a (mod 4)

Our goal is to estimate G(m, n). Inserting (5.56) into (5.60) and changing the order of summation, we arrange G(m, n) as (5.62)

G(m, n) = 2-7r

>2

c-1S(m, n; c)G

147r

mryi I

c=_0 (mod q)

where (5.63)

G(x) = > g(f)Je(x) e=_a (mod 4)

Hence we are led to the problem of computing the series (5.63) of Bessel functions of integral order, which is called the Neumann series. Various expansions into series of Bessel functions are thoroughly developed in the context of the spectral theory of Kloosterman sums (see the comments at the end of Chapter 9 of [1w2J). A complete characterization of the functions G(x) which can be represented by absolutely and uniformly converging Neumann series was given by G. H. Hardy and E. C. Titchmarsh (HTJ. Our task is simpler, since we start from the coefficient function g(y). We shall not compute G(x) exactly, but rather give a strong approximation, which is friendly in applications. We begin by applying the integral representation (see [GR]) 1/2

Mx) =- ill /2 e(et)e--"'ill This can be regarded as the Fourier transform of the periodic: function exp(-ix sin 21rt); therefore, by Poisson's formula, g(C)JJ(x) =

00

J

9(t)e

where g(t) is the Fourier transform

g(t) _

[()etY)1i.

5. Bounds for the Fourier Coefficients of Cusp Forms

86

The above formula can be automatically generalized for series of Bessel functions twisted by an additive character

g(Q)e(cxt)Jt(x) = J

g(t)eixsin2n(a-t)dt

by change of variables. Now we use the additive characters to detect the congruence e = a (mod 4), getting 4G(x) =

J

g(t)c(t)dt 00

where

c(t) = e-iasin2nt + i--a eixcos21rt + i-2a eixsin2nt + 2i1-a -2i sin(x sin 27rt) + sin(x cos 21rt)

because a is odd. Since we think of the original function g(y) as being supported in a large segment, the Fourier transform g(t) concentrates on a short segment. Therefore, there will be no loss if we simplify c(t) by applying the approximations sin 27rt = 27rt + O(ItI3) cos 27rt = 1 - 27r2t2 + 0(t4). Hence

c(t) = -2i sin(27rtx) +

2i1-a

sin(x

27r2t2x) + O(xjtl3).

The error term O(xItl3) contributes to G(x) at most O(xc3(g)), where (5.64)

cl,(.9) =

By the involution formula 9(y) contributes to G(x) exactly

f

oo ao

g(-y) we see that the term -2isin(27rtx)

9(x) - 9(-x) = 9(x)

(5.65)

because g is supported on R+. To compute the contribution of the term 2i1-a sin(x - 21r2t'x) we need the formula XI

(5.66)

J

a g(t)e(t2x)dt = (1 + i.)

Jn

g(2 :ry)e(-y)dy.

5.5. Spectral analysis of the diagonal symbol

87

This can be established by using the rule f §f = f g f with f (t) = e(t2x), for which we have (see (10.11) )

f(y) =

(p2) 2Ve

Inserting this into the integral and changing the variable y to 2 xy, one obtains (5.66). Put (5.67)

h(x) = 2

J

g(t) sin(x - 27r2t2x)dt. 00

By (5.66) we derive that 0C

(5.68)

h(x) = 1

g(

2xy) sin(x + y -- 4 )(iry)-1'2dy.

0

Finally adding (5.64), (5.65) and (5.68), we obtain

Lemma 5.8. Suppose g is a smooth, real function compactly supported on R+ and a = ±1. Then (5.69)

4 E g(e)Jt(x) = g(x) + ii-ah(x) + O(xc3(g)) tea (mod 4)

where It is given by (5.68), and the implied constant is absolute. One can refine (5.69) by using more terms in the power series for sin 27rt and cos 27rt. For example, the error term O(xc3(g)) can be replaced by Tgn. (x) + O(xc.4(g) + x2cc, (g))

From now on we assume that the test function g satisfies the following conditions: (5.70)

(5.71)

supp g C j K, 2K] g(i) << K

for all j > 0, with the implied constant depending only on j. Then we have (5.72)

g(t) < Ii (1 + jtj )--.

This yields (5.73)

Ig(t)t"Idt < K--".

5. Bounds for the Fourier Coefficients of Cusp Forms

88

) « (xK-2)', whence we deduce by

Moreover, we have (O1/8yi)g( partial integration that h(x) <<

(5.74)

(xK_2)j

for any j > 0. By virtue of these estimates Lemma 5.8 yields (5.75)

4G(x) = g(x) + O(xK-3 + (xK-2)j)

for any j >, 0, where the implied constant depends only on j. Now we return to D(m, n) and G(m, n). Inserting (5.75) into (5.62) and applying the trivial bound for Kloosterman sums, we derive the approximation (5.76)

4G(m, n) = 9(m, n) + O

)lo2mn) qh + (=mn)j) K 2K2

where (5.77)

g(m,n) = 27r

E C'S(m,n;c)g (47r c

c=0 (mod q)

mnl /J

Next, by Poisson's formula we have (5.78)

9(e) = 9(0) +

4

O(K-?).

e ma (,nod 4)

Combining (5.61), (5.76) and (5.78), we obtain

Theorem 5.9. Assuming the conditions (5.70) and (5.71), we have (5.79) If

4D(m, n) = g(0)S(mn., n) +ia'''lg(rn, n) +O

mn

+

' `m.n ` ' (d

log 2mr,.

where D(m, n) is the spectral sum defined by (.5-58), (5.59) and 9(m, n) is the sum of Kloosterman sums defined by (5.77). This approximation holds for any j >, 0, with the implied constant depending only on j.

Note that, g(m, n) is void unless V7-nn > qK, in which case we have

the trivial estimate g(m, n) < i/qK. A slightly better estimate follows from Weil's bound for Kloosterman sums. However a more interesting result can be established on average by applying the large sieve inequality; namely (5.80)

EE

N<m. ,,<2N

(1t + NK ')IloII111311.

5.5. Spectral analysis of the diagonal symbol

89

On the other hand the diagonal terms contribute

g(0) E an Nn << KIIaIIIIQII N
Note that the bound (5.80) does not beat the contribution of the diagonal terms, although it comes tantalizingly close. There are problems (such as evaluation of power moments of central values of Hecke L-functions) which require the coefficients a,n, 3n to be special. In such cases one can do considerably better to get bounds for g(m, n) which beat the contribution of the diagonal. Unfortunately these subtle and important cases lie beyond the scope of this book. So far we have employed separately each half of the spectrum corresponding to (5.57) and (5.58). However, in many problems one can afford to employ both halves together, which amounts to summing up (5.79) for a = +1. Then the sum of Kloosterman sums g(m, n) cancels out, and we are left essentially with the diagonal term (5.81)

2 E g(l) /i(m, n) = g(0)5(m, n) + D 9 +

(=mn)j) log 2mn q

.

1 odd

At this point it is interesting to observe how the synthesis of two spectra of distinct parity dramatically improves the approximation of the diagonal symbol in terms of Fourier coefficients of cusp forms. In Section 11.4 we shall express the symbol b(m, n) in terms of additive characters by using the circle method of Kloosterman. We apply (5.81) to extend Theorem 5.7 (a similar result was established first in [1w4)).

Theorem 5.10. Let y be a function satisfying (5.70) and (5.71). Let £1(a) denote the linear form (5.4y) with complex coefficients a for 1 n < N. Thcn. (5.82)

g(k - 1)

2

k even

l

f(a)12 =

fE.F,

where g(0) is the integral mean value of g(y) and 1t(K, N) satisfies

rl(I ,N) < {N/qK3+(N/gIi2)j}Nlog2N for any j >, U, with the implied constant depending only on j.

Chapter 6

Hecke Operators 6.1. Introduction It was observed long ago that the Fourier coefficients of basic classical modular forms have remarkable arithmetical properties. A fascinating example is the multiplicativity of the Ramanujan T-function,

T(m)T(n) =

d11T(mnd-2). dj(m,n)

We have derived this formula by an appeal to the Fourier expansion of Poincare series using a similar property of Kloosterman sums (4.10). However, in quite a different way, this formula was first established by E. Hecke (1936) by means of certain self-adjoint operators. The theory of Hecke operators explains numerous other identities. More important, Hecke's original ideas proved to be vital for developments of modern fields such as Galois representations. In a general setting the Hecke operators are averaging operators over a suitable finite collection of double cosets with respect to a group: therefore a great deal of the Hecke theory belongs to linear algebra. But when one considers the spectral analysis of these operators the problems become more delicate, and complete results are known only for arithmetic groups. In this chapter we shall present the theory of Hecke operators in the context of the congruence group I'o(q) and the Dirichlet character (mod q). For economy of exposition we replace the double coset constructions with specific representatives. Throughout we assume that k is a fixed integer. For any A E GL.; (R) the slash operator is defined on functions f : H -+ C by (6.1)

ff.t(:) = (detA)R/-JA(z)-k f(Az). 91

6. Hecce Operators

92

This is the same operator as in (2.50) if A E SL2(R); it is invariant under multiplication by positive scalars. For any real number a # 0 we have lA (1a

flaA

a/

fill

Since k is a integer, the slash operation is associative:

flits = (flA) IB Obviously the slash operator maps a holomorphic function into a holomorphic function.

6.2. Hecke operators T Throughout r = SL2(Z) is the modular group. For a positive integer n we consider the set (6.2)

d):

a, b,c,dEZ, ad--be=?t}.

In particular, G1 = r. The modular group acts on G from both sides, so that (6.3)

G = ran =

Lemma 6.1. The collection l

d): ad=n, 0<,b
(6.4)

fwms a complete set of right coset representatives of G modulo F. i.e. we have the disjoint partition

an = U Pp. PEA,,

Proof. Let p = ( ) E G,,.

First one can find integers -y, b such that.

ya 4- 6c = 0 and (-y,6) = I (take y = c/(a,c) and h = -a/(a, c)). Hence there exists r = (7 ) E r which gives rp Changing the sign of r if necessary, we get rp = (n I,) with ad = n, d > 0. Finally, multiplying

6.2. Hecke operators T

93

on the left by ( 1 1) with a suitable u E Z, we bring b to 0 < b < d. As p ranges over 0 the cosets rp are disjoint, for (a

(y 6)

d) =\a' d'/

implies -y = 0, a = S = 1 and 3 = 0. This completes the proof of Lemma 6.1.

Note that A,, is a finite set with the number of elements o, (71) = > d.

,ti There exists a one-to-one correspondence between 0,, x r and r x 0,,, i.e. for any p E 0,,, T E r there exist unique r' E r, p' E 0 such that PT =T'p.

(6.7)

For a fixed T E F, as p ranges over the whole set 0,,, p' does also. This intertwining relation is the key for resolving problems of automorphy of many functions and operators. We shall need (6.7) in a somewhat more explicit form. Writing (6.7) as

(a'

a

(6.8) a

d)

'y

6)

y'

d'/

b')

we get by multiplication (6.9)

Since

as + yb

,d

1°

bd

V+

= I we deduce that

a' = (aa+yb,yd) a' = (aa + 'yb)/(aa + yb, -yd)

y' = yd/(aa + yb, yd) d _ n./(aa + yb, yd)

because a'd' = n. Furthermore, notice that a'6S' = I (mod Iy'I) by the determinant equation, so 6S' is given modulo Iy'I. Then, equating the right lower entries in (6.9), we find b' modulo d'. If we require 0 <, b' < c!' then such b` is determined uniquely, and so is 6'. Note that our argument. for

6. Hecke Operators

94

the uniqueness proves also the existence of the intertwining relation (6.7)

(except for the case -y = 0, which can be verified directly; see (6.16)).

Let X : Z -+ C be any function. This yields a function on GL2(Z) by setting (6.10)

X(p) = X(a)

if p

For a given x and a positive integer k we define the operator Tn on functions

f : H -+ C by the formula (6.11)

Tf = nz-1

X(P)fip

Using (6.4) we shall write T formally as (6.12)

T. = n E X(a)ak > { a nd=n

b

d

0_
This expression means that for any function f : H --, C (6.13)

f

(Tnf)(z) = n Y X(a)ak ad=n

(t)

0<,b
Remember that T depends on X (well, also on k); we shall display this fact by writing T,i whenever it is necessary to avoid confusion.

6.3. The Hecke operators on periodic functions From now on we assume that (6.14)

ti(1) = I and k(

IN

_ (

1)r.

We begin to examine the Hecke operators T on functions f which are automorphic with respect to the group (6.15)

r, =

1

:uEz},

i.e. satisfying fir = X(7-) f for any r E rte,. This is a fancy way of saying that f is periodic of period 1, i.e. f (z + 1) = f (z). Theorem 6.2. The operator T,, maps periodic functions to periodic functions, i.e. T :.M,:(r;>:,, X) -# M1 (r.., .N:).

6.3. The Hecke operators on periodic functions

95

Proof. The correspondence (6.7) holds between 0 x ra and r. x 0,,; more explicitly, we have the identity (6.16)

Ca d)(1 1)=(1 1)\a d/

where b' = b + au - dv. Hence, if pr = r'p' with p, p' E On and T, T' E ro, then flp. = fir'p' = X(r')flp, and X(A)X(r') = X(p')X(r); therefore (Tnf),T = J-1 E X(A)fp,pE A nk

=

2

X(r)X(A )fop' = X(T)Tnf

1

p'EA This completes the proof. Of course, one can see Theorem 6.2 more directly; nevertheless the above argument prepares us for more subtle considerations. Next we examine how the operator T. acts on the coefficients of a Fourier series

f (z) =

(6.17)

a(m)e(mz). m=o

Proposition 6.3. Suppose f is given by the series (6.17) which converges absolutely in H. Then Tn f is given by the series (T,L f)(z) _ E a (mn)e(mz)

(6.18)

rn=0

whose coefficients are (6.19)

a,,(m) = > X(d)dk-la(mnd-2). dj(n,,n)

Proof. Applying (6.12) we get 1

n n,=0

a(m)

:i(a)ci all=n

.

c b (nod (1)

00

_

E

z)

t=0 ad=,r OG

E E x(a)a'-'-'a(de) n,=0

e(mz)

ud

which yields (6.18) with coefficients given by (6.19).

nz+b (711 c1

6. Hecke Operators

96

Corollary 6.4. For n >, I we have an(0) = ak_I(n,X)a(0)

where Q3(n,X) =

(6.20)

X(d)d8. din

Corollary 6.5. For m, n > 1 we have an(m) = a,n(n).

(6.21)

From now on we assume that X is completely multiplicative, i.e.

X(ala2) = X(a,)X(a2)

if al, a2 E Z,

and we show that the Tn are also multiplicative in the following style.

Theorem 6.6. For any m, n > 1 we have

T.T. =

(6.22)

X(d)dk-1T,nnd-2. dI(m,n)

Proof. By (6.12) we get mnT,7-n _

E

b2

dbi

X(ala2)(ala2)A

ald1=,n

b1 (rood (11)

a2d2=n

Cal

1) \aL d2)

,

b2 (m,od

and after multiplying we reduce the resulting matrix by c5 =

(al,d2) to

obtain mnT,.,,T,r = (al,d2)=1

(aia2

(a, a2)'

Y(b)bk

ajd1=m/h

b, (mod d1)

a2r12=,r./h

G2 (mod d2)

b

dld2

where b = al b2 + bldg. Given a,, a2, d, , d2 as above, the upper-right entry b covers every class modulo dl d2 exactly b times when bl,b range over all classes nlodulo d1, bd2 respectively. Indeed b determines b2 - bal (mod d2) and bl = (b - (.,b-2)d2 l (mod di). Therefore

.

91b927;mTr, _

aI a.2

(<5)bk+l

(11111 =14's

(rr1 .d2)=1

n.d,=m/,'I

b (mod(1td_1

b

d, d-,

6.3. The Ilecke operators on periodic functions

97

Put a = aja2 and d = d1d2i so ad = mn6-2. Conversely, given a factorization ad = mnb"-2, there exist unique factorizations a = aja2,d = d1d2 with (al, d2) = 1, a1d1 = m/b and a2d2 = ?t/b; indeed al = m/(m, bd) and d2 = 6d/(m, bd). Hence we can write

a ad=mnd-2

dl(m,n)

d)

G (,t,od d)

which yields the formula (6.22).

Corollary 6.7. The Hecke operators commute: T,nTn = TnT,n .

(6.23)

One can invert the multiplicative relation (6.22) into (6.24)

,L(d)x(d)dk-IT..IdTnid

T,,,,, =

dl(m,n)

where p(d) is the Mobius function. Hence in particular Tmn = IM T,,

if (m, n) = 1.

Taking in = p" and n = p, we get by (6.24) the following second order recurrence formula:

71,Tp, - x(p)p"--T,,,-i.

(6.25)

This property can be captured at once in the formula for the formal power series (6.26)

X(p)pk-1X,)

T1,"X- _ (1 - T1,X +

JJ

For X = p-' this becomes (6.27)

(1 - I1,p

1

s

+

X(p)pl:-i-2sl

If X = 1 then the recurrence formula (6.25) is similar to that satisfied by the Tchebyshev polynomials U,,(X); in this case we obtain (6.28)

'T,,, =

U,,(p1;..1T,).

Let T denote the algebra over C generated by all T,, (called the Hock o algebra). By the above properties we conclude that T is a counuutaa(ivr' algebra generated by the operators 7 ,tn primes p.

6. Hecke Operators

98

6.4. The Hecke operators for the modular group Next we consider the Hecke operators T on the linear space Mk(r) of modular forms of weight k. Since r = SL2(Z) contains -1, we assume that k is an even integer; otherwise the space Mk(r) is zero. Under the restriction of Tie to Mk (r) one can exchange every g in (6.11) with anything in the coset rp, so one can write without ambiguity

T,tf =

(6.29)

n-1

flp

pEr\G

because flip = flp if 'r E r and f E .Mi. (r). Theorem 6.8. The Hecke operator Tn maps a modular form to a modular form and a cusp form to a cusp form: Tn .Mk(1') M k(r), T,t : s, (r) -, Sk,(r).

Proof. By the correspondence pT = r'p' we get (Tnf)l,r =

n2-1

flpT

=

flp,

n2-1

PEA

f

p'E0,.

since fipr = flr'p' = flp,. A cusp form f (z) has the property that it decays exponentially as z tends to any point on the boundary of H. However this property is preserved by the slash operators, so it is carried over to Tn f . Since we already proved that T, ,f is a modular form, it must be a cusp form.

Let us recall that the space of cusp forms SR(r) with k > 2 is spanned by the Poincare series jT(z)

(6.30)

k

e(nir z)

TEr.;;.\r

for m -- 1, 2, 3, .... Therefore T,, Pnl is a linear combination of various Poincard series (not unique). We shall find a short combination in the following.

Theorem 6.9. For k > 2, m >, 0 and n 3 1 we have. (6.31)

1 Pm = E dl(t,n)

uI-=

6.4. The Heclce operators for the modular group

99

Proof. First notice that (6.32)

nR'-1

(TTP,)(z) =

E j9(z)-ke(mgz). gErx\G,,

Let f and 9 be any sets of right coset representatives of r,,,\F and r\G respectively. Then naturally fG is a set of right coset representatives of r \Gn, but also !9H is a set of right coset representatives of I',,,,\G,,. Hence

nk-1 >: E jpr(z)-ke(mfrz)

TTPni(z) =

pEc rE?{

ad=n

nk-1

> jr (z)-ke \(m arz + b

d-k

nk-1 >

=

E ad=n dim

d

b (mod d)

dl--k E jr(z)`ke ( am Tz

\d

rE?t

which yields (6.31).

For m = 0 we get the Eisenstein series jr(z)-k =

Ek. (z) =

rEr,,\r

(cz +

d)-k

(c.d)=1

In this case Theorem 6.9 implies that Ek is an eigenfunction of all the Hecke operators Tn with eigenvalue ak-1(n), i.e.

T,Ek = ak-l(n)Ek

(6.33)

Corollary 6.10. For m, n >,I we have

mk-1 T Pvn = n 1;-1 T,,, p".

(6.34)

A similar symmetry occurs for the inner product of T,,f against P,,,. Indeed, if f has the Fourier expansion (6.17), then T, ,f has the expansion (6.18), and by Theorem 2.12 and Corollary 6.5

h(k Hence

1)-'(41rm)k-I (T,,f,

Pm) = (6 (m) _ (t,,,(11).

6. Hecke Operators

100

Proposition 6.11. For in, n > 1 and f E Mk(r) we have p"). rn1-1(Tnf, p,,.) = (6.35) Let S be a finite dimensional Hilbert space over C. Any linear operator T : S -, S has the adjoint operator T* : S S defined by

(Tf,g) = (f,T*g) T is said to be normal if it commutes with T'. If T = T* then T is said to be self-adjoint.

Theorem 6.12. The Hecke operators T acting on the space of cusp forms for the modular group are self-adjoint, i.e. (f,Tng) for all f,g E Sk.(I').

(6.36)

Proof. Since Si.(I') is spanned by the Poincare series (6.30), it suffices to verify (6.36) for the Poincare series. By Corollary 6.10 and Proposition 6.11 we get Pt) = (T.PP, P,n) = (P.,,T.P,) because the Fourier coefficients of Poincare series, and so the inner products, are real. Now we can appeal to the following result from linear algebra.

Proposition 6.13. Let A C

be a commuting family of normal ma-

trices. There exists a unitary matrix U E M,,(C) such that U-'AU is diagonal for every A E A. This result can be stated in terms of linear operators.

Proposition 6.14. Let S be a finite dimensional Hilbert space over C and let T be a commuting family of normal operators T : S -, S. Then there e:c:isis an u,

basis .P of S which consists of common eigenfunctions

of all the operators in T.

Proof. Let

be an orthonornial basis of S. Then

Tft = E A7.i(T)fj with some \jj(T) E C. In this way T is represented by a matrix A(T) _ (Aij(T)) which commutes with its adjoint A(T)" = A(T)', i.e. A(T) is a normal matrix. These matrices commute; therefore by a suitable linear transformation with a unitary matrix the system T can be diagonalized. Applying Proposition 6.1-I to the flecke algebra, we get.

6.5. The Hecke operators with a character

101

Theorem 6.15 (Hecke). In the space Sk(F) of cusp forms for the modular group there exists an orthonormal basis .F which consists of eigenfunctions of all the Hecke operators T,,.

Let f E Sk(r) be a Hecke eigenform,

T, ,f = .(n) f for it = 1, 2, 3, .. .

(6.37)

Suppose f has the Fourier expansion 00

f (z) _

a(m)e(mz).

Combining (6.37) with (6.19), we obtain dk_..ra(mnd--l).

A(n)a(m) = E

(6.38)

dI (nt,n)

For m = 1 this gives a(n) = A(n)a(1).

(6.39)

Hence a(1) # 0, as otherwise f would vanish identically. Therefore the Fourier coefficients of a Hecke cusp form are proportional to the eigenvalues of the Hecke operators. As an example, consider the space S12(f), which is one-dimensionally spanned by 0(z); therefore 0(z) is automatically a simultaneous eigenfunction of all the Hecke operators, namely

r(n)0(z)

(6.40)

where r(n) is the Ranianujan function. Hence one obtains another proof vi the formula (3.34).

6.5. The Hecke operators with a character The significance of Hecke theory is most evident in the context of the group

f ( q) =

n

EP:-y-0(moclq)

and a Dirichlet character N, (niod q) (here h is the full modular group). NotsV that X(p) vanishes for p = (`; ;) E GL.,(Z) with (a, q) > 1, so the set of right

6. Hecke Operators

102

coset representatives An in the definition of Tn (see (6.11)) can be reduced to (6.41)

On=11 a

d

I EOn:(a,q)=1}.

Observe that the correspondence between On x r and r x 0,a given by (6.7)

projects to that between on x ro(q) and r'o(q) x On. Indeed, if r'p' = pr with (p,r) E On x r'o(q), we deduce by (6.9) that

a'a' = as (mod q)

(6.42)

so (a', q) 1, i.e. p' E An, and -y' = 0 (mod q), i.c. r' C r'o(q). Moreover (6.42) shows that (6.43)

X(p)X(r) = X(r')X(P) if pi = 'r A'

for (p,r)EAn' xro(q). As before we require that the character x (mod q) satisfies the consistency condition x(-1) = (-1)k; otherwise the space Mk(r'o(q), x) of automorphic forms of weight k for the group ro(q) with respect to the multiplier x is zero.

Theorem 6.16. The Hecke operator Tn = Tn maps automorphic forms to automorphic forms and cusp forms to cusp forms: Tn : Mk(ro(q), X) - Mk (IFO (q), X)

T Sk(ro(q), X) - Sk(ro(q), X) Proof. We proceed as in the proof of Theorem 6.8 by using (6.43), namely

x(P)fipr = x(p)fir,p, = x(p)x(r')fiP' = X(r)x(p )fid, whence (Tn f )1r = x(r)Tn f for any r E ro(q). Moreover if f is a cusp form then so is Tn f , because any slash operation preserves exponential decay of a function at any boundary point.

Recall that we have already established the multiplicativity (6.22) of the operators Tn, actually in a bigger space of periodic functions, hence in Mk(ro(q), x). This shows that the Hecke algebra T is commutative, generated by Tp for all primes p. We have p

1

(6.44)

Tp = p

+ X(p)pk-l

b (mod p)

1

r

/

6.5. The Hecice operators with a character

103

If p divides the level, this expression reduces to (6.45)

Tp = 1

1

E

if plq,

p

P 6 (mod p)

P,

and (6.22) yields Tp- = (Tp)"

(6.46)

if p1q.

The situation becomes somewhat complicated with the action of T,z = Tn on the Poincar6 series (6.47)

E

Pm(z) =

rEroo\ro(q)

One needs these for all m to span the space of cusp forms, but only the TnPm with (mn, q) = 1 possess the exact symmetry which lies behind the self-adjointness. More precisely, TnPm is a combination of the Poincare series of the following type.

Theorem 6.17. Suppose k > 2 and X(-1) = (-1)k. For all m >, 0 and n 3 1 we have (6.48)

TnPm =

X

rnl

l\nd

1k

1 Pmnd-2

ld

The proof of Theorem 6.17 is similar to that of Theorem 6.9 (one needs obvious modifications of the argument to accommodate the character X). For m = 0 we see that the Eisenstein series X(T)jr(z)-k

Ek(z) = E rEroo\ro(q)

is an eigenfunction of all Tn with eigenvalue

ak-1(n,X) =

X(d)dk-1 dIn

If nlq°O then vanishes unless nom, in which case TnPm = (n, q) = 1, we can write (6.49)

TnPm = X(n)nk-1 E di(rn,n)

Hence

X(d)d1-'Pmnd

-.

/n. If

6. Hecke Operators

104

Corollary 6.18. If m, n > 1 and (mn, q) = 1, then (6.50)

X(m)mk-1TnPm = X(n)nk-1TmPn

Our next aim is to find the adjoint of T. One could argue as in the proof of Theorem 6.12 using the symmetric relations (6.21), (6.50) and Theorem 2.12 to establish that (6.51)

(Tnf, g) = X(n)(f,Tng)

if (n, q) = 1,

but this method works only for special f, g E Sk(ro(q), X) which are in the linear subspace spanned by the Poincare series P,,, with (m, q) = 1. Since

not every cusp form can be represented as a linear combination of such Poincare series, we cannot conclude that Tn = X(n)Tn. To establish (6.51) completely we make use of the original idea of Hecke. To this end one needs the concept of the absolute inner product. We begin by recalling the expression (3.20), (f,9)(z) = ykf(z)9(z)dµz.

Let t9 be a multiplier system of weight k for a group r (here I' is arbitrary). If f , g E Sk (r, t9), then (f , g) (z) is a r-invariant measure, and the inner product is just the volume of a fundamental domain for this measure,

(f, 9) = Notice that for any a E GLZ (R) we have (6.52)

(fig, 9i) (z) = (f, g) (az).

One can also use any subgroup r' C r of finite index and scale down by the index to get the same value: (6.53)

(f,9) = [r

1

w(f,9)(z)

For computational convenience we wish to have r' sufficiently small so that the multiplier t9: r --+ C becomes trivial on r, i.e. f, g E Sk(r'). This is certainly possible if r is a congruence group and t9 is a character (take the kernel of t9). Furthermore, for any given a E GL2 (Q) one can choose

r' small enough, but of finite index, such that the a-slash operator maps Sk(r, 9) into Sk(r'), i.e. (6.54)

fI, r. Sk(r') if f E Sk(r,19).

For example, if a E G,, and r = ro(q), then r' = r(nq) is the group to take.

6.5. The Henke operators with a character

105

Proposition 6.19. Let r be a congruence group and a E GL2 (Q). Then the a-slash operator is an isometry of Sk(r, o9) with respect to the absolute inner product, i.e. (6.55)

(fl0, 910) _ (f, 9)

for all f, 9 E Sk(r,'o)

Proof. Let r' be a sufficiently small subgroup of r of finite index such that (6.54) holds true, and let D' denote the fundamental domain of r. Then

(60,910) = [r : I'll -1 fDhhla)(x)

_ [r: r')-' ID'

g)(az)

= [r : r')-1 f D,(f,9)(z) where OD' is a fundamental domain for the conjugate group r" = ar'Q-1.

Choose r' small enough that r" C r. Then [r: r"J = [r: ar'a-lj = [r: r] by computing volumes. Hence we get (6.55). One can write (6.55) in the following way: (fro, 9) = (f, 910-1)

saying that the a-1-slash operator is adjoint to the a-slash operator with respect to the absolute inner product. Notice that gl0-1 = gI0,

where a'a = det a.

Now we are ready to prove (6.51) in all of Sk(ro(q), X).

Theorem 6.20. If (n, q) = 1 and f, g E Sk(ro(q), x), then (6.56)

(Tnf,9) = X(n)(f,T,19)

In other words, the adjoint of Tn is Tn = k(n)T,; thus T is normal while (n)1/2Tn is self-adjoint.

Proof. Computing via the absolute inner product, we get (T,,.f,9) =

rte-' (X(P)fIp,9) pEOn

6. Hecke Operators

106

Here (fip,g) = (f,glp,), where pp' = n. If p = (a d), then p' _ (d X(p) = X(a) = X(n)X(d) = X(n)X(p') Therefore

Q)

and

(X(p)fjp) = X(n) (f, X(P )91p,)

For any r, -r' E Fo(q) we deduce by (6.55) and the automorphy of f, g that (f, X(P )9ip') = (f, X(P ")9Ip")

where p" = r'p'r-r. Assuming (a, d)1b, we can make p" = p. To this end we choose integers a, b such that

as+6d+bq0 (mod n) a6

1 (mod q).

Then we put

a' _ (aa + bq)d-1,

6' _ (bd + bq)a-1.

Note that a', b' are integers with a'6' = ab + bq(aa + bd + bq)n; 1 (mod q). Therefore

r- \q

r,-

ab

q

are in ro(q). We check that r'p' = rp, so p" = p. This proves that (x(p)fip, 9) = x(n)(f, X(p')9ip')

for p = (a d) with ad = n and (a, d) fib. Suppose that n is squarefree. Then the condition (a, d),b is automatically satisfied for any p E 0,,. Thus, summing over p, we get (6.56). If n is not squarefree, then the above construction needs some modifications. However for the proof of Theorem 6.20 we do not need extra work because the algebra of Hecke operators Tn is commutative and generated by the operators Tp, for which (6.56) is already established.

Applying (6.56) to an eigenfunction f = g # 0, we infer that (6.57)

x(n).1(n)

if Tnf = \(n)f, (n, q) = 1.

By Theorem 6.20, Corollary 6.7 and Proposition 6.14 we obtain the following fundamental result, due to Hecke.

Theorem 6.21. In the space of cusp forms SA(I'o(q), x) there exists an orthonormal basis .F which consists of eigenfunctions of all the Hecke operators Tn with (n, q) = 1.

6.6. An overview of newforms

107

6.6. An overview of newforms Let f E Sk(ro(q), X) be a Hecke eigencuspform, i.e. a cuspform which is a common eigenfunction of all Tn with (n, q) = 1, say

Tn f = \(n) f if (n, q) = 1.

(6.58)

The condition (n, q) = 1 is a hassle in practice. Suppose f has the Fourier expansion

f (z) = E00 a(m)e(mz).

(6.59)

M=1

Combining (6.58) with (6.19), we obtain

A(n)a(m) _ E X(d)dk-1a(mnd-2) dl(m,n)

but only if (n, q) = 1. For m = 1 this gives (6.60)

a(n) = A(n)a(1)

if (n, q) = 1.

Unfortunately we cannot deduce from (6.60) that a(1) 0 0, because the condition (n, q) = 1 does not allow us to control all the coefficients in (6.59). In fact there exist Hecke eigencuspforms f 0 with a(1) = 0. They come from lower levels if x is not a primitive character. Let q* denote the conductor of X. Take any positive integers q', d such that q*lq' and q'dlq. Let X' (mod q') be the character induced by X. Then

f (z) E Sk(ro(4 ), X) = f (dz) E Sk(ro(q), X). This follows immediately from the identity (6.61)

d 1)('Y 6)-(7/d

1) 6)(d

Now if f (z) has the Fourier series (6.59), then

f (dz) = E a(m)e(mdz) = E +n

a(m/d)e(mz);

n &=_O (mod d)

therefore f (dz) has first coefficient zero if d # 1. We denote by SR(ro(q), X), the linear subspace of Sk(ro(q), x) spanned by all forms of type f (dz) where djq and (6.62)

f E Sk(ro(4'), X')

6. Hecke Operators

108

for some q' < q such that q'dlq. Here X' (mod q') is the character which is induced by X (mod q). Then we define Sk(ro(q), x) the linear subspace of Sk(ro(q), X) orthogonal to SS(ro(q), X) with respect to the inner product. Thus we have the orthogonal decomposition (6.63)

Sk(ro(q), x) = Ss(ro(q), x) ® Sk(ro(q), x).

Proposition 6.22. The subspaces Sk, So are stable under the Hecke operators T with (n, q) = 1.

Proof (sketch). That Tn : Sk - Sk can be verified directly on individual forms f (dz). Moreover, T preserves the orthogonality to Sk, because its adjoint is Tn = X(n)T,,. Hence T : Sk -' S. By Proposition 6.22 it follows that each of the two subspaces Sk, Sk has orthonormal bases consisting of eigenfunctions of the Hecke operators Tn with (n, q) = 1. The Hecke eigencuspforms in the space Sk(ro(q), X) are called newforms.

A cusp form f E sk (ro(q), X) which is an eigenfunction of Tn only for (n, q) = 1 is not quite adequate for the study of various L-functions attached

to f. For this reason we wish f to be a common eigenfunction of all the Hecke operators. An easy way of getting this property would be to show that the joint eigenspaces for all of the Tn with (n, q) = 1 are one-dimensional, i.e. for each sequence {A(n)}(n,q)=1 there is at most one cuspform f 96 0 (up to a scalar) which is an eigenfunction of Tn with prescribed numbers A(n)

for (n, q) = 1 as eigenvalues. Then any other operator T which commutes with all Tn for all (n, q) = 1 preserves each a(n)-eigenspace, so T f must be a multiple of f, i.e. f is automatically an eigenfunction of T. One can show that the multiplicity-one principle holds true in the space SS(ro(q), X) Consequently, if f is a newform then (6.58) extends to all n. By (6.60) we -

deduce

Proposition 6.23. Suppose f is a newform. Then (6.58) holds for all n. The first coefficient in the Fourier expansion (6.59) does not vanish, so one can normalize f by setting a(1) = 1. In this case

a(n) = \(n) for all n.

6.7. Hecke eigencuspforms for a primitive character We shall give details for the assertions made in the last section in the case of a primitive character X (mod q). Thus in this case SR is zero and SO is the whole space Sk(ro(q), X).

6.7. Hecke eigeucuspforms for a primitive character

109

For any d the operator Ad defined by Ad

1

d

( b (niod d)

b/d 1

acts on the Fourier series f (z) = >2 a(m)e(mz)

(6.64)

in

by selecting the terms with m - 0 (mod d), i.e. we have

E

(Adf)(z) =

(6.65)

a(m)e(mz).

,n_0 (mod d)

Hence the sifting operator Sq defined by (6.66)

µ(d)Ad

Sq = dIq

acts on the Fourier series (6.64) by selecting the terms with (m, q) = 1, i.e. we have

(Sq f) (z) = E a(m)e(mz).

(6.67)

(m,q)=1

Remark. One can show that S. : Sk(ro(q), X) -, Sk(ro(gg1), x), where q1 is the product of prime factors of q. Lemma 6.24. If f E Sk(ro(q), X) has the Fourier coefficients a(n) = 0 for all n with (n, q) = 1, then f = 0. In other words, the sifting operator Sq has zero kernel.

Proof. We begin with the equation (CZ)-'f

(ca

cz) _ X(d)f (

d

+ c)

for any T = (a * ) E I'0 (q), which is just the automorphy relation fi. = X(T) f written at the point (z - d)/c. Expanding both sides into Fourier series, we obtain a kind of Poisson's summation formula,

(CZ)`k E m=1

a(m)e

cam C

-

- rnl = X(d)>a(n)e (-dn + nz 00

cz J

n=1

c

C

6. Hence Operators

110

for any 0 < c - 0 (mod q) and ad - 1 (mod c). Hence, summing over a (mod c with (a, c) = 1 for c = q, we get 00

(qz)-""E a(m)S(m, 0; q)e

\-

=

m=1

n=1

a(n)Sx(-n, 0; q)e l

z

\q J

where S(m, 0; q) is the Ramanujan sum and Sx(-n, 0; q) = X(-n)T(X) is the Gauss sum. Since every term on the right-hand side vanishes by our hypothesis, we conclude that every term on the left-hand side also vanishes (the uniqueness of Fourier series expansion), i.e. a(m)S(m, 0; q) = 0 for every m.. If q is squarefree, then the Ramanujan sum never vanishes, namely we have S(m, 0; q) = µ(q/(m, q))W((m, q));

therefore a(m) = 0 for every m, proving Lemma 6.24 in this case. If q is not squarefree then the assertion of Lemma 6.24 remains true, but our proof needs substantial modifications. To this end we generalize the above relation between the Ramanujan and Gauss sums. First we multiply the Poisson formula by a function V;(a), and only then we sum over the primitive classes. We obtain (qz)_A

E a(m)S(m)e

(-ml

m=1

where

S(m) =

*

a (mod q) e

Sx(n)

=

qz J

E SS(n)e (nz"\ q

n=1

(am q) V (a)

d (mod q) X(d)e

(

q')0(a)

and ad - 1 (mod q). For a given m we want S(m) # 0, whereas SS(n) = 0 for all n with (n, q) # 1. We can permit i 1(a) to depend on m. To construct such 0 we factor the modulus q = q, q2 so that q1 has primes in m and qq is coprime with m. We put

qo= II p. pl(m,q)

We use only the residue classes a (mod q) of type a=bg141+ I 1+cq1 I g242 qo/

6.7. Hecke eigencuspforms for a primitive character

111

where b runs over the classes modulo q2 with (b, q2) = 1 and c runs over the classes modulo qo with (1 + cql/qo, ql) = 1. For O(a) we choose a character of conductor q2 restricted to the above classes. Then we have

S(m) = E i'(b)e (bmL)

e

b (mod q2)

(m

c (mod qo)

(mi!) qo [I 1 1- p

11+

41 \

qo

0

as desired. In order to compute S((n) we factor X (mod q) into the primitive characters Xl (mod qj) and X2 (mod q2). Since

d = bglgl + (1 + c41)

g2g22,

go

where the ' is the mapping to the multiplicative inverse modulo ql, we obtain

X1(1+c9o)e(-nql (i+C)').

TX2(b)ebn I

Sx(n) b(mod g2)

c(mod go)

Assuming that 1/.'X2 is primitive, the first sum is equal to z/.X2(-41n)T(zbx2), so it vanishes if (n, 92) 1. Since Xl is primitive, the second sum vanishes if (n, qj) # 1. Combining both facts, we conclude that SX(n) = 0 if (n, q) 96 1. We still have a problem if X2 is the only primitive character to modulus

q2. This is the case for 92 = 3 and q2 = 4. If q2 = 3 we take b the principal character, and the above argument still works because (b)e (bm 3

)

= 2cos

(3 mql)

0.

b (mod 3)

If q2 = 4 then X2(b) = X4(b) We apply the above arguments for the modulus 2q in place of q. We take z/i(b) = X8(b) = (2). Since X8 and X8X4

are both primitive, the previous result is valid. This completes the proof of Lemma 6.24 for any primitive character X (mod q).

Remark. The above arguments are sensitive to the primitivity of the character X, as they should be. Now suppose that f E S,.(ro(q), X) is an eigenfunction of all the Hecke

operators T with (n, q) = 1, say (6.58) holds, and that f has the Fourier expansion (6.59). Then we have (6.60). Hence by Lemma 6.24 it follows that the first coefficient of f is a(1) 0. Therefore we can normalize f by setting a(1) = 1. Moreover, Lemma 6.24 implies the multiplicity-one principle. These observations yield the following conclusion.

6. Hecke Operators

112

Lemma 6.25. A Hecke eigencuspform f E Sk(ro(q), X) can be normalized to have its first Fourier coefficient a(1) = 1, and the normalized form is determined uniquely by the eigenvalues of T with (n, q) = 1. Since Tn commutes with T,,, for any m, we obtain

Tn(Tmf) = Tm(Tnf) = A(n)Tmf if (n, q) = 1. In other words, Tmf is a Hecke cusp form with the same eigenvalues as f for all (n, q) = 1. By the multiplicity-one principle (Lemma 6.25) it follows that f is a multiple of f . Therefore we get

Theorem 6.26. If f E 3k(ro(q), X) is an eigenfv.nction of every T with (n, q) = 1, then it is an eigenfunction of the whole algebra T = {T,,,; m E N}, i.e. Tn, j = ).(rn)f for all r E N.

Besides Hecke operators Tn there is yet another important operator acting on cusp forms for ro(q), namely the l ricke involution, which is defined by (6.68)

W f= few

where w = (q

-1

\ I

(cf. [AL)), i.e.

(Wf)(z) = q-k/2z`kf(-1/qz) Note that w2 = -q, so = (-1)k. Moreover w normalizes the group ro(q), i.e. wro(q) = ro(q)w or, explicitly, W2

WT = T W

where

/ l T=I bq -aI4I

if

\ T=(c d).

Hence it follows that for f E Sk(ro(q), X) and T E ro(q) (fiw)lr = f1wr = fj7-1w = X(T')ffw =

Therefore (6.69)

W : Sk(ro(q), X) -, Sk(ro(q), X)

Since w E GL2 (Z), the rricke involution is an isoinetry, i.e. we have the equality (W f, W g) = (f, g).

Here (,) on the left and right are the inner products in Sk(ro(q), k) and Sk(ro(q), X) respectively (one can see that this equality holds, because either side coincides with the corresponding absolute inner product).

6.7. Hecke eigeiicuspforms for a primitive character

113

Theorem 6.27. If (n, q) = 1 we have WT,x = X(n)TTW.

Proof. We have WTn =

n2-'

X(P)fl, PEOn

Tn W = n-f

X(P)flwp. PEAn

We also have wpw-;

if P=

_

d)

Hence for r = (a 6) we compute that P! = T-iwpw-1 _

6d +,3bq

---yd

-,f3a

as

Let (n, q) = 1 and suppose p E On. Choose a = d/(b, d) and 'y = -bq/(b, d). For this choice there are unique 0 < Q < a and 6 such that T

-

P-

d/(b, d) -bql (b, d)

*

n/(b,

E rfl(q)

d)

E On.

This establishes a one-to-one correspondence on the set On given by p 1-4 P, = T-1wpw-1. We have flwp = flrp'w = X(T)flp'w and

X(T)X(P) = X(dl(b,d))X(a) = X(n)X((b,d)) = X(n)x(P') Hence Theorem 6.27 follows.

In order to bring the character X back to X we shall compose W with the complex conjugation operator K defined by

(Kf)(z) = 1(-2).

(6.70)

Note that K acts on a Fourier series by simply conjugating its coefficients,

f (z) =

a(n)e(nz) ' (K f) (z) =

a.(n)e(nz).

6. Hecke Operators

114

Be aware that K is not linear over C, since

K) f = AK f if A E C. We have (6.71)

(6.72)

(6.73)

(6.74)

K2 = 1, WK = (-1)kKW K : Sk(ro(q), X) -' SL(ro(q),

(K f, Kg) _ (f, 9) KT,x =

for all n.

Set (6.75)

W = KW.

Note that W = KW. From the above properties of K and W we deduce that (6.76)

W2=1, Wrrf =j)Wf if

(6.77)

W : Sk(ro(q), X) -+ Sk(ro(q), X)

(6.78)

Tn.W = X(n)WTT if (n, q) = 1.

77 EC

Now suppose f E Sk(ro(q), X) is a non-zero eigenfunction of all the Hecke operators T,, with eigenvalues A(n) for (n, q) = 1 (we have called f a Hecke eigencuspform). Then by (6.58) and (6.57) we get

f = X(n)WTn f = X(n)WA(n)f = X(n) \(n)Wf = \(n)W f showing that W f is also an eigenfunction of all T,, with the same eigenvalues A(n) for (n, q) = 1. Therefore by the multiplicity-one principle (Lemma 6.25) we obtain

6.7. Hecke eigencuspforms for a primitive character

115

Theorem 6.28. A Hecke eigencuspform f E SR;(ro(q), x) is also an eigenfunction of the involution operator W, i.e. (6.79)

ITV f = ref.

By (6.76) we deduce that the eigenvalues of W (for non-zero functions) are complex numbers on the unit circle; indeed f = W 2 f = Wrl f = t7Wf = rin f , i.e. (6.80)

I771 = 1.

The. eigenvalue of the involution W from Theorem 6.28 will appear as a factor in the functional equation for the L-function attached to f. We shall compute this eigenvalue in terms of the Gauss sum (6.81)

r(X) _ a (mod q)

X(a)e

(q)

and the q-th Fourier coefficient of f. Suppose that f is normalized, i.e. its first Fourier coefficient is one. Then all the Fourier coefficients of f coincide with the eigenvalues of the Hecke operators T, i.e. 00

f (z) _

(6.82)

A(n)e(nz). n=1

Recall that for all m, n > 1 A(m)A(n) _

(6.83)

X(d)dk-1A(mnd-2).

In particular, for any m and djq we obtain the exact factorization A(dm) _ A(d)A(m). Hence

(Ad f) (z) = E

A(n)e(nz) = A(d) f (dz);

n_0 (mod d) in other words

(6.84)

Adf = A(d)d -k/2 f, (d 1)

if dlq.

Theorem 6.29. The eigenvalue of the involution W of a normalized Hecke eigencuspform f E Sk(ro(q), X) is given by (6.85)

47 = T(X)A(q)q_'

l'

6. Hence Operators

116

Proof. We begin by writing the identity

uiq) (q -1/q) (1 v/q) _ (u (uv 1)/q\

(1

.

We choose uv = 1 (mod q) to bring this matrix to ro(q), then we divide by (1 v/q) and we slash f with the resulting matrix to obtain 1

fl(1 u19)(9 -1/q) = X(v)fl(1 -1/9)

by the automorphy of f. Next we sum over the reduced residue classes modulo q, getting

g = E* fl(1 u/q) ( -1/q) _ i X(v)fl(1 -v/9\. u (mod q)

l

it (trod q)

q

1

1

/

If we apply the Fourier expansion (6.82), the right-hand side yields

g(z) = E X(v)f (z - 9) _ EX(n)

X(v)e

n=1

it (mod q)

it (mod q)

(_!!)) q

e(nz)

00

= T(X) E X(-n)A(n)e(nz). rt=1

Hence K9(z) = T(X)(S9f)(z)

because X(-1);r(X) = -r(X) and a(n) for (n, q) = 1 (see (6.57)). On the left-hand side we relax the condition (u, q) = 1 by the Mobius formula and apply (6.84) as follows:

E{ fl(1 u/q) =ad=q L /_t(a)

u (mod q)

1

L

u (mod d)

fl(1 u/d) 1

1: µ(a)dAdf =

E

ad=q

ad=q

1

Slashing with ((I -1/q), since (d 1) (9 -1/q) = (q -1) (1 1/a)I we obtain

9=

E* fl(1 u/q)(

u (mod q)

1

µ(a)X(d)d1-A/2

--1/q) = q

ad=q

1a) /

6.7. Ifecke eigencuspforins for a primitive character

117

Now suppose f is an eigenfunction of W, say (6.79) holds. Then K fj,, _ KWf = W f = f and K fl- (1 ) = ri f (1 _ rjak/2f(az); hence 1/a

R

1/a )

Kg(z) = rl E

(az).

ad=q

Comparing this with the former expression, we obtain the identity (6.86)

T(X)(Sgf)(z) =,q E µ(a)A(d)(a/d)k/2df(az).

ad=q

From the first Fourier coefficients of (6.86) one gets (6.87)

T(X)

=

Hence by virtue of (6.80) and (6.88)

IT(X)I = q2

we first deduce that (6.89)

JA(q)l = q42-',

and applying this to (6.87) we complete the proof of (6.85).

From the identity (6.86) one can also infer Theorem 6.30. For plq the eigenvatues A(p) of TT in the space Sk(Fo(q), X) satisfy (6.90)

IA(p)I =

pT.

Proof. Comparing the p-th Fourier coefficients in (6.86), one gets (p)p(q)g1-k/2

= A(q/p)(p2/q)k/2q/p

which yields (6.90) because \(p)A(q/p) = A(q).

6. Hecke Operators

118

6.8. Final remarks If X is not primitive, then the space of newforms So(I'o(q), X) is not necessarily the whole of Sk(ro(q),X); nevertheless the basic results remain true in this subspace. In particular, a newform is automatically an eigenfunction of all the Hecke operators Tn, n = 1, 2, 3, ... , as well as of the involution W (but the formula (6.85) still requires X to be primitive). As a matter of fact the converse is true, i.e. if f E SS(ro(q), x), f 0 0, is an eigenfunction of all

the Hecke operators T, n = 1, 2, 3, ... , and of W, then f is a newform. A newform has the first Fourier coefficient different from zero; hence, letting this coefficient be one (normalization), we have the Fourier expansion f(z) = j a(n)e(nz)

(6.91)

1

with coefficients equal to the cigenvalues of Hecke operators. By virtue of (6.83) the associated Dirichlet series 00

L f(s) = E,\(n)n-'

(6.92)

I

(which converges absolutely for Re s > k21 by Corollary 4.2) has the Euler product (6.93)

Lf(s) = 11(1 - A(p)p s P

Winnie Li [Li] established a kind of converse assertion by showing that if f E Sk(ro(q), X) is an eigenfunction of W and the associated series L1(s) has the Euler product of type (6.93), then f is a normalized newform. Suppose X is a real character. Then both operators K, W map the space Sk(ro(q), X) to itself. Moreover K commutes with all T,a for (n, q) = 1. Therefore we can make a Hecke basis r of Sk(ro(q), X) which consists of eigenfunctions of K, say K f = ±f since K2 = 1. Replacing f by if, we can change the eigenvalue -1 to 1, so we can always require that (6.94)

Kf = f.

This means f has real Fourier coefficients. For such f we have W f = WI, and if f is a newform then (6.95)

W f = 77f

with q = fik because W2 = (-1)k. Recall that n is also given by (6.85). However, even in the case of a real charecter X (mod q) for which the Gauss sum T(X) is computed in Lemma 4.8, the exact determination of the eigen-

value q = fik remains a problem since the Fourier coefficient A(q) of f is intractable (except for some forms f such as these which are associated with elliptic curves).

Chapter 7

Automorphic L-functions

7.1. Introduction A connection between automorphic forms and L-functions can be traced in the celebrated memoir of B. Riemann (1860) on the zeta-function 00

(7.1)

C(S)=En-8,

Re s > 1.

i Starting from the gamma function oo

r(s) = je_1'Y3_1d by termwise integration Riemann established the formula (7.3)

(27r)-sr(s)c(2s)

=

2

j(O(iy) -

where 0(z) is the standard theta function (2.72). Then by the Jacobi inversion (10.55) he arranged this formula in a symmetric fashion: (7.4)

i-zr (2) ((s) +

s(1

s) =

where

jt(v) (y2 +y'-$) y-'dy 00

2I

u(y) =

(0(,Y)

e-jrn2y

2 i

119

7. Automoiphic L-functions

120

Since u(y) << e-'y the above integral converges absolutely and uniformly on

vertical strips, giving analytic continuation of ((s) over the whole s-plane with only a simple pole at s = 1. Moreover (7.4) implies the functional equation (7.5)

1r

2r

`2l ((s) = .7r- Y r (1 2 s) S(1 - s)

(actually Riemann's original transformations were slightly different). Conversely, in 1921 H. Hamburger showed that any Dirichlet series 00 a(n)n_9

L(s) _

satisfying (7.5) and some regularity conditions is necessarily a multiple of

These ideas were greatly generalized by E. Hecke (1936). We shall present this profound theory in the context of automorphic forms for row) of integral weight k > 0 and a character X (mod q), not necessarily primitive.

7.2. The Hecke L-functions We begin with a cusp form f E Sk(ro(q), X). Consider simultaneously the Fourier expansions at cusps oo and 0, say f(z) _ 00 a(n)e(nz)

(7.6)

00

g(z) = Eb(n)e(nz)

(7.7)

1

where (7.8)

g

with w =

(q-1}

Recall that g E Sk(ro(q), X) by (6.69). To these Fourier series we associate the Hecke L-functions (7.9)

L1(s) _

00 1

a(n)n-'

7.2. The Hecke L-functions

121

00

b(n)n-'.

L9(s)

(7.10)

By virtue of Corollary 5.2 the above Dirichlet series converge absolutely in

the half-plane Re s >. In this half-plane we can show by termwise integration that (7.11)

(27r)-8r(s)Lf(s) = (21r)-,r(s)L.9(s)

(7.12)

=

f f

oo f(iy)y8-ldy

00 g(iy).ys-ldy.

0

Since f and g have exponential decay at 0 and oo, we have

Theorem 7.1. The L-functions L f(s) and L9(s) associated to the cusp forms f E Sk(ro(q), x) and g = flw E Sk(ro(q), X) have analytic continuation over the whole s-plane, and they are entire functions. Actually the so-called complete L -junctions

A f(s) =

(7.13)

12

s 1

r(s)L1(s)

I

r(s)L9(s)

\\

A9(s) =

(7.14)

are entire functions, bounded on vertical strips.

Both Af(s) and A9(s) are connected. Indeed, by (7.8), for z on the imaginary line we get

f(iy) = q-k/2(i/y)kg(z/qy) Hence

f00 0

f

(iy)y9-ldy

= ikgk12

f

9(iy)yk-s-ldy

0

which together with (7.11), (7.12) yields

Theorem 7.2. The complete L-functions satisfy the functional equation (7.15)

A f(s) = ikA9(k - s).

If f is an eigenfunction of the operator W = KW (see (6.75)), say (7.16)

Wf = nf,

7. Automorphic L-functions

122

then b(n) = ra(n), so Lg(s) = 77L f(s), and the functional equation (7.15) becomes VY

(7.17)

(2

k

s

VY

k-s

r(k - s)L f(k -s).

r(s)L1(s) = i

In particular, if X is a real character and f has real Fourier coefficients and is an eigenfunction of W, say Wf = fikf,

(7.18)

then (7.17) becomes

(7.19)

= f ()'r(k_8)Lf(k_s).

Of

Thus for k = 2 (this includes the I-lasse-Weil L-function; see the next chapter) the eigenvalue of W coincides with the sign of the functional equation.

The assumption that f, g are cusp forms is not absolutely necessary; however without this requirement A f(s), Ag(s) may have simple poles at s = 0 and s = k. Most of the arguments used above have little to do with automorphic forms and can be reversed by Mellin's transform to establish the following result.

Theorem 7.3 (Hecke). Suppose f and g are given by the Fourier series 00

(7.20)

f(z) = E ane(nz) 0

(7.21)

g(z) = 00 E bne(nz) 0

with coefficients an, bn bounded by O(nk) for n > 1, where a is a positive constant. Put 00

(7.22)

L f (s) _

(7.23)

L9(s) _

ann-s

00

bnn s

and (7.24)

Af(s) =

(7.25)

A9(s) =

()3r(s)Lf(s) ()Sr(s)Lg(s)

7.2. The Hecke L-functions

123

where q is a given positive number (not necessarily an integer). Put w = = (fz)-k f (-1/qz), where k is a given positive inte( q -1) and (fl..,) (z) ger. Then the following assertions are equivalent: (Al) The functions f, g are connected by

9 = Al.

(7.26)

(A2) Both Af(s) and A9 (s) have meromorphic continuation over the whole s-plane, (7.27)

A f(s) + aos-1 + boik(k -

(7.28)

A9(s) +

bos--1 + aoi-k(k

-

s)-1

s)-1

are entire and bounded on vertical strips, and they satisfy (7.29)

A f(s) = ikA9(k - s).

Proof. Assuming (A1), we establish the following integral representation: 00

A1(s) + aos-1 + boik(k - s)-1 = J

(.f (iv/vT) -

ao)y'-ldy

00

+ ik 1

bo)yk-s-ldy

1

Since f (z) - ao and g(z) - bo have exponential decay as y = Imz --> oo, the above integrals converge absolutely, and they are entire functions, bounded on vertical strips. A similar integral representation holds true for (7.28). By symmetry we infer the functional equation (7.29), but with the polar terms included. However the simple fractions of the polar terms alone satisfy (7.29), so we can subtract these. This completes the proof of the implication (A1) = (A2). The converse implication goes by routine application of the Mellin inversion, the Phragmen-Lindelof convexity principle and Stirling's estimate for the gamma function. From the functional equation (7.29) one

first deduces (A1) on the imaginary half-line z = iy with y > 0, and one extends this relation to the upper-half plane by analytic continuation. Note that the holomorphy of (7.27) and (7.28) implies that ao = -L f(0),

bo = -L9(0) and (s - k)L f(s) N bo(2iri/ f)k, (s - k)L9(s) ti ao(21ri/ f)k ass -->k.

7. Autolnorp.hic L-functions

124

7.3. Twisting automorphic forms and L-functions Automorphic forms behave nicely with respect to twists by primitive characters.

Theorem 7.4. Let f E Mk(ro(q), X), where X is a character of conductor q' with q* l q. Let ip be a primitive character of conductor r. If f has the Fourier series 00 f (z) a(n)e(nz), then the twisted series fp(z) _ E00bb(n)a(n)e(nz) 0

belongs to Mk(ro(N), X'iG2), where N is the least common multiple of q, q'r,

r2. Moreover if f is a cusp form, so is f,p.

Proof. Let r(V)) be the Gauss sum T(O)

/,

_

`d(u)e

(±).

u (mod r)

Since i/i is primitive we have IT(V))l = r1/2, and for any n T(`Y)Y'(n) _ u (mod r)

'd(u)e (ur )

Hence we can express fp in terms of f as (7.30)

T( )ff _ E G(u)fi(1 u (mod r)

u/r 1

Also, one has the identity due

T =

C1

Hence if ry = therefore

r

1) \c d) (a b

1_(a Cr

b-

uc

bcdu

r

E ro(N), then r E ro(N) and T

T(')).f

I7 = E

(a d) (mod q*);

(u)X(d)fl(1 d' /r ) \

u (mod r)

= X(d)I2(d) E ' u (mod r)

= X(d)ik2(d)T(' )fo.

r u'::du'

I

(u)fI(1 u/r) 1

7.3. Twisting automorphic forms and L-functions

125

This completes the proof of the automorphy of f V,. That f o is a cusp form

whenever f is can be seen by writing the Fourier expansion of fo explicitly at every cusp. However, we only need to check that f p does not grow too fast at the boundary of H, or for that matter f V, satisfies the criterion of Section 5.1. This is seen immediately from (7.30).

Theorem 7.5. Let f E Sk(ro(q), X), where X is a character to modulus q. Let rk be a primitive character of conductor r with (q, r) = 1. Then (7.31)

APIWN = w(Ogi*L

where wN = (N -1) for N = qr2, g = fl,,, for w = w4 = (9- 1) and (7.32)

w(1P)

=

X(r)'tP(q)T('O)2r-1

Proof. /Recall that g E Sk(ro(q), x) by virtue of (6.69). Check the identity (7.33)

1 l

(9 -1\ (-uq *) (1

u1 r) (N -11 _r

uir

We choose uvq = -1 (mod r) to bring (_rUq *) into ro(q), so this identity yields (7.34)

f, (1 u/r)WN = X(r)gj(1 u/r). 1

1

Moreover we have

T('P)*(n) u (mod r)

'd(u)e ( ru)

for any n, because 1/i is primitive. Multiplying (7.34) by (u) = Eli(-vq) and summing over the reduced residue classes to modulus r, we obtain

?P(-1)r.

This yields (7.31), since

Now let L f(s, ,O) denote the Hecke L-function associated with the twisted

form fj,, i.e. 00

(7.35)

L f(s, tG) _i(n)a(n)n-9,

and put (7.36)

A f(s,,O) =

(,,I-N

By Theorems 7.5 and 7.3 we deduce

s

r(s)L f(s, ).

7. Automorphic L-functions

126

Theorem 7.6. Suppose f E Sk(ro(q),x),

is a primitive character of conductor r with (q, r) = 1, N = qr2 and w(O) is given by (7.32). Then

A f(s,1/)) is an entire function, bounded in vertical strips, and it satisfies the functional equation Af(s,i)) = ikw(V))A.9 (k - s, i)

(7.37)

where 9= fl,,, forty=wq= I q-1 1. In the particular case of k = 2, X principal, 7li real, W f = 77f (this includes the Hasse-Weil L-functions), the functional equation (7.37) becomes (7.38)

2

(2

s

2-s

r(2 - s)Lf(2 -

where N = qr2 and (7.39)

since

n1L(-q) I7-(iP)I2

= r and T('+J,) = Zb(-1)T(ir) =C-1)T(O).

7.4. Converse theorems For q = 1 (the case of the modular group) Theorem 7.3 shows that an automorphic form is fully characterized by a single, suitable functional equation; precisely, we have

Theorem 7.7 (Hecke). Let k be a positive, even number. Suppose f is given by Fourier series 00

f (z) _ E ane(nz) 0

with the coefficients an « n° for all n >, 1. Then f E Mk(r0(1)) if and only if the function

Af(s) =

(27r)-sr(a)00E

ann-s

1

can be analytically continued over the whole s-plane, Af(s) + a,,s-1 + ikao(k - s)-1 is entire and bounded in vertical strips, and satisfies the functional equation (7.40)

A f(s) = ik A f(k - s).

7.4. Converse theorems

127

Proof. The modular group is generated by the translation Tz = z + 1 and the involution Sz = -1/z. If q > 1 one needs more functional equations to show the converse theorem; these are obtained by twisting with characters. There are many possibilities for choosing a sufficient collection of characters. We shall follow closely the original version established by A. Weil [Wei].

Theorem 7.8 (Weil). Let k be a positive, even number, and X a character to modulus q >, 1. Suppose f, g are given by the Fourier series (7.20), (7.21) with coefficients bounded by O(na) for all n 3 1, where a is a positive constant. Suppose A f (s), A9 (s) defined by (7.24 ), (7.25) satisfy (A2). Furthermore let 1Z be a set of prime numbers coprime with q which meets every

primitive residue class, i.e. for any c > 0 and any a with (a, c) = 1 there exists r C 1Z such that r - a (mod c). Suppose for any primitive character zJi of conductor r E 1Z the functions A f(s,1J)), A9(s,,O) defined by (7.36)

with N = qr2 are entire, bounded in vertical strips and satisfy the functional equation (7.37) with w(7b) given by (7.32). Then f E Mk(ro(q), X), g E Mk (ro(q), X) and g = fl,,,, where w = wq = (q- 1 ). Moreover f, 9 are cusp forms if L f(s) or L9(s) converges absolutely on some line Re s = a

with 0
fop=Eafla if /3=>act a

or

where a ranges over GLZ (1R). We begin by constructing special O's. For any v with (v, r) = 1 there exists u such that (7.41)

ry(v/r) =

r

uq

* ) E ro(q). -v

Put (7.42)

a(v/r) _

and (7.43)

O(v/r) = (X(r) -'Y(v/r))a(v/r)

Actually a(v/r) E GLl (Q) and /3(v/r) is in the group algebra C[GL; (Q)].

7. Atitomorphic L-functions

128

Lemma 7.9. Suppose r is a prime number with (r, q) = I. Suppose f and g = fl,,, satisfy (7.37) for any primitive character ip of conductor r with the constant w(O) given by (7.32). Then glp(v/r) does not depend on v for (v, r)

= 1.

Proof. First recall the formula (from the proof of Theorem 7.4)

r( )f+, = E VP(u)fIa(u/r).

(7.44)

u (,nod r)

Slashing with WN, we get by (7.31)

w('+G)TW9o = E '

(u)fla(u/r)wr4.

u (niod r)

Then applying (7.44) to g,y, we arrive at (7.45)

X(r)V)(-q) E is(u)91-(u/r) = E (u)fWu/r)wN u (mod r)

u (mod r)

Next for each (u, r) = 1 we take v with uvq = 1 (mod r); this yields ry(v/r) E I'o(q) (see (7.41)). Recall the identity (see (7.33))

a(u/r)WN = rwgry(v/r)a(v/r) Hence fla(u/r)WN =

and we can write (7.45) as follows: {v}gla(vlr)

= 0.

v (mod r)

Since this holds for every 0 other than the trivial character, it implies by orthogonality that 9jp(,,,/r) = 9100v2/r) if (VI V21 r) = 1 (to see this, multiply by O(v1) - ' ,(v2) and sum over -0 (mod r)). Lemma 7.10. Let m, n be primes with (mn, q) = 1. Suppose f and g = few satisfy (7.31) for any primitive character -0 of conductor r = m or n with the constant w(,O) given by (7.32). Then (7.46)

9iy = X('Y)9

it - n) . for all ry E I'p(q) of type ry = ( -uq

7.4. Converse theorems

129

Proof. Applying Lemma 7.9 for -y and 11 = ("` v } we get uqn 91(x(m)-y)a(vlm) = 9I(X(+n)-y,)a(-v/vn)5

whence (7.47)

9I(X(m)-y) - 9I(X(m)-7j)a(--2v/in)-

Similarly, applying Lemma 7.9 for -y,1 = (_' ,n) and

ry-1

=

(n t) , we

get

(7.48)

91(x(n)-y, ') =

9I(X(,n)--y-1)a(-2v/n).

Since X(mn) = 1, we have the relations

X(n) -'yl 1 = -X(n)(X(m) - Yl)7i 1 X(n) --y-1 = -X(n) (X(m) - 7)'Y-1 and inserting these into (7.48) we get (7.49)

91(X(m)-7i)y1 T 9I(X(m)-y)j-1a(-'2v/n).

Combining (7.49) with (7.47), we arrive at 9I(X(m)-y) = 9I(X(m)-7)7-la(-2v/n)7ia(-2v/m).

Putting h = gi(x(,n)-y) and tion asserts that

y-la(-2v/n)-yja(-2v/m), the above rela-

(7.50)

hIQ = h.

We have

Q_

1

-2v/m

2uq/n -3 + 4/mn

hence Itr/3I = 12-4/mnj < 2 and itr/31

'

0, 1, and so Q is an elliptic element

of SL2(Q). The eigenvalues of /3 are not roots of unity. Indeed, since they are in a quadratic field, they would have to be among the numbers ±1, ±i, ±(6, ±(3, giving Itr/31 = 0, 1, 2, which is not possible. Let zo be the fixed point of /3 in the tipper half-plane H. Put p = (z -- Nv)-1 (1

-zo) E GL2(C),

7. Autoulorphic L-functions

130

so p brings f3 to the Jordan form,

\

pop-1 = (

(7.51)

with n E C.

n_1 J

Moreover p maps IHl onto the \\unit ball B = {w E C : IwI < 1}. Therefore the function P(w) _ (hip-1 )(w) = jp-1 is holomorphic in B. By (7.50) and (7.51) we get (w)-kh(p-1w)

P(n2w) = n-"P(w) Looking at the Taylor expansion of P(w) at w = 0, we find that all the coefficients vanish because n is not a root of unity. Therefore P(w) - 0. Hence h(z) - 0, which means gay = X(m)g = X(n)g = X(ry)g, and this completes the proof of Lemma 7.10. Finally we are ready to prove Theorem 7.8. We need to show that (91-f

=X(7)9

if

'Y

-

dbl ac

E ro(q)

This is obvious if c = 0, so let c > 0, c =- 0 (mod q). Applying suitable integral translations on both sides of 'y, we can bring -y to a matrix of the type considered in Lemma 7.10. Indeed, by the prime number theorem for arithmetic progressions (due to Dirichlet) there exist m = a + ct E 7 and

n=d+csE7.,andwehave

d)

( rn

1t) -uq

n

\1

1sl with -u = c/q and -v = b + sm + to + cst. By Lemma 7.10 (note that the Cc

hypothesis g = fl,,, is satisfied since (A2) implies (A1) by virtue of the Hecke converse Theorem 7.3) we get 91, =X(n)9 = X(d)9 = X('y)9

since the integral translations do not alter the character nor the multiplier factor. This relation shows the automorphy of g with respect to ro(q) and the character X. Hence f is automorphic with respect to ro(q) and the character X by virtue of (6.69). Suppose L f(s) converges absolutely on the line Re s = a, the case of Lg(s) being the same by applying the involution w. Then we have 00

jamIe-2nmy << 1 + max

If (x)I

(n°e-2xny)

n=0

Hence f is a cusp form by virtue of the following

<< 1 + y-O.

7.4. Converse theorems

131

Criterion for Cusp Forms. H. Suppose f is an automorphic form of weight k > 0 for a group r such that (7.52)

ykf(z)-->0

as

y-->0

uniformly on H. Then f is a cusp form. Proof. The zero-th coefficient of f at a cusp a satisfies lao! S fo I fto(z)I dx =

f

o

1(y-1Im az) z l f (az)ldx

where a E SL2(R) is a scaling matrix. We can assume that a is not at oo (by choosing an equivalent cusp on R), so a is not an upper triangular matrix.

Thus as y -- oo we have y-1 : Im az - 0 so the above integral vanishes. Hence ao = 0, proving that f is a cusp form. Remark. If L f(s), L9(s) converge absolutely in Re s > k - 6, then A f(s), A9(s) are holomorphic in Re s > k - 6, so ao = bo = 0 by (7.27), (7.28); but the vanishing of the Fourier coefficients at cusps other than 0 or oo does not follow by the holomorphy of Af(s), A9(s). For q prime every cusp is equivalent to 0 or oo, so in this case the holomorphy condition alone permits one to deduce that f, g are cusp forms.

Chapter 8

Cusp Forms Associated with Elliptic Curves 8.1. The Hasse-Weil L-function An elliptic curve E is an algebraic curve (a projective algebraic variety of dimension 1) of genus 1 over a field K. If char(K) yc- 2, 3, then E is given by the Weierstrass equation (8.1)

y2=x3+Ax+B

with A, B E K. The discriminant of E is the discriminant of the cubic polynomial (8.2)

g(x) = x3 + Ax + B,

and it is equal to (8.3)

A = -16(4A3 + 27B2).

It turns out that (see Figure 9) E is non-singular

A

0,

Ehasnode taA=0,A0, Ehascusp s.A=0,A=0. 133

8. Cusp Forms Associated with Elliptic Curves

134

y2 = x3 + x

y2 = x3 - x

smooth

smooth

y2 = x3

y2 = x3 + x2

Cusp

node

A=-64

A=64

00

=0

Figure 9. Elliptic curves

If char(K) = 2 or 3, the Weierstrass equation and the description of singularities are slightly different. Suppose E is given by the Weierstrass equation (8.1) with A, B E Z and

A 36 0. For each prime p consider the reduced curve E/Fp over the field Fp of p elements. Let v(p) denote the number of points on E/Fp, i.e. the

8.1. The Hasse- Weil L-function

135

number of solutions to the congruence y2

= 9(s) (mod p).

We do not count the point at infinity. It turns out that v(p) is well approximated by p; more precisely, the difference (8.4)

\(P) = p - v(P)

satisfies

IA(P)I < 2 j. This estimate is due to H. Hasse and is essentially best possible. In order to understand how the X(p) vary with p, Hasse began and Weil continued to investigate the L-function for E defined by the following Euler product: \(P)P-9)-1 [j (1 (8.6) LE(S) = 11(1 - \(P)P-' + pl-2s)-1 (8.5)

-

PIo

PIo

Remarks. Although we assumed that E/Q is smooth (since A ,E 0), the reduced curve E/]FP is singular if pl0, and E is said to have bad reduction at such primes. One can show that for primes of bad reduction a(p) = 0, 1, -1

according to a type of singularity which occurs, namely a cusp, a node with rational slopes for the tangents, or a node with quadratic irrational slopes for the tangents. If p f i then the reduced curve E/1FP remains smooth, so E is said to have good reduction at p. In this case the local factor 1 -

\(P)P-8 + pl-2s

appears naturally in the so-called congruence zeta-function of E defined by 00

CE/Fn(s) = exp(E, m-1V(pm)P m4) M=1

where v(q) denotes the number of points on E over the finite field E. (it is not the same as the number of points modulo q). We have -2s). (E/Fp(s) = (1 - pl-s)-l(1 - )1(P)P-' +P1 The estimate of Hasse (8.5) can be interpreted as the Riemann hypothesis for CE/FP (s), which asserts that the roots are on the line Re s = 1/2. Write LE(s) as the Dirichlet series 00

LE(s)

Note that the Euler product (8.6) and the Dirichlet series (8.7) converge absolutely for Re s > 3/2.

8. Cusp Forms Associated with Elliptic Curves

136

Conjecture (Hasse). LE(s) has analytic continuation to an entire function, and it satisfies the functional equation (8.8)

(2)2-sr(2

(A)r(s)LE(s) 2= 77

- s)LE(2 - s)

where q is a positive integer composed of prime factors of A, the so-called conductor of E, and 71 = ±1 is called the root number.

Conjecture (Shimura-Taniyama). The Fourier series 00 f(z)

(8.9)

A(n)e(nz)

is a cusp form of weight 2 for ro(q) and the principal character; it is a newform with

T f = A(n) f Wf =77f.

(8.10) (8.11)

Recently these conjectures were proved by A. Wiles (at least if q is squarefree). We shall give a simple proof for special curves, the so-called congruent number curves.

8.2. Elliptic curves Er In this chapter we shall examine a family of elliptic curves Er given by the equation y2 = x3

(8.12)

- r2x

where r is a positive, squarefree integer. These curves were studied by J. `I unnell in connection with the ancient problem of the so-called "congruent

numbers." A positive rational number r is called a congruent number if it is the area of some right triangle with rational sides. Equivalently, there exists x such that all three number x - r, x, x + r are squares of rationals. This also means there are infinitely many rational points on E,.. Multiplying by suitable squares, we may require r to be a positive, squarefree integer. Tunnell used the curve Er to establish an effective method of checking if r is a congruent number. The smallest congruent numbers are r = 5,6,7. Note that the discriminant of E,. is A,. = 64r6, and if plA,. then vr(p) _ p, so Ar(p) = 0. In this case the Hasse-Weil L-function reduces to 11 (1 - a,.(p)P s +p1-2s)-1 = (8.13) L1=r(s) = ,(n)n-s

142,'

(, 2r)_I

8.2. Elliptic curves E,.

137

In general if a curve E is given by

y2 = 9(x) with g E Z[x], then the number of points of E/]Fp is equal to

V(p) _ E (mod p)

px

(i+(i\

where (p) is the quadratic residue symbol (the Legendre symbol). Hence gy(p)=-

(8.14)

(g(x)\.

p

x (mod p)

In particular, if p fi 2r, then

r(p) x_(mod- p) (x3 pr2x/ - (Pr) x (mod p) \x3p x/ = (Pr)

(P)

by changing x -> rx. Hence for all n (8.15)

Ar(n) = Xr(n)A1(n)

where Xr(n) is the Jacobi symbol, (8.16)

X,- (n) = (n)

This shows that the .-function for Er is obtained from that for El by twisting with the character Xr, (8.17)

Xr(P))'1(p)P-s

LE,.(s) = [J(1 -

(p)p'-2S)-1

+ X?

p#2

= 1: Xr(n)AI (n)n-s 2in

By virtue of the above connection it will be sufficient to prove the Masse

and the Shimura-Taniyama conjectures for El. In this case we simplify notation by omitting the subscript r = 1, so we write E = El, A = X1 and (8.18)

LE(s) =

rj(1 - A(p)p +pl-2s)-1 p712

= .

2{,i

A(n)n-'.

8. Cusp Forms Associated with Elliptic Curves

138

After establishing the conjectures for the curve E, one can extend the results for E, by an appeal to a general principle about twisting automorphic forms with characters (see Section 7.3, Theorem 7.4 and the formulas (7.32), (7.33)).

8.3. Computing A(p) The curve E given by the equation (8.19)

has many automorphisms; for example, if (y, x) is on E then so are the points

(-y, x) and (iy, -x). We do not dwell on explaining what really happens here, but only say that this observation is tacitly used in the course of computing \(p). The discriminant of E is A = 64. For p = 2 we have v(2) = 2, so A(2) = 0.

(8.20)

For p = -1 (mod 4), since (p1) = -1 and g(-x) = -g(x), we derive by (8.14) that \(p) = -.\(p), and so (8.21)

A(p) = 0

if p = -1 (mod 4).

In the remaining case p =- 1 (mod 4) we shall carry out computations by passing to another curve E' given by the equation

Y2 =X4+4.

(8.22)

There is a map from E - (0, 0) to E' given by

(y,x) -> (2x -

y2x-2, yx-1)

This has the inverse from E' to E - (0, 0) given by (Y,X) -+

(x(Y + X2), 2(Y+X2))

Therefore the number of points on E/Fp and E'/Fp are related by v(p) 1 = v'(p). The key advantage of dealing with E' is that E' has a diagonal equation. Let p - 1 (mod 4). The multiplicative group FP* is cyclic of order p-1 0 (mod 4), and so is the character group F. For any z E FP* we have

#{x E Fp : x4 = z} = E X(z). X4=1

8.3. Computing A(p)

139

Hence

v'(p) = 2 + > J(X) X4=1

where

E

J(X) =

X(Y2

- 4).

Y (mod p)

There are four characters of exponent 4, all given b y X = 1 , n, 72, rj is a fixed character of order 4. For X = 1 we get

7

7, where

J(1) = p - 2.

For X = 7?2 (it is the Legendre symbol) we get j(172)

_

E X((Y - 2)(Y + 2)) Y

=EX((Y-4)Y)=> x(YY4) Y Y:A0 _ 1: X(1 - 4Y) _ -1 + > X(Y), Y

YOO

whence

j(r12) = -1. For X = n3 we get j('73) =

,7(n) From the above evaluations

we infer that &J(P) = p - 1 + ,7(m) + .707),

whence

v(P) = p + J(r!) + Y(rl), and (8.23)

)(P) _ -,T(rl) - .707)

Now we proceed to compute J(r?) (the Jacobi sum). First we establish

that (8.24)

P'.

S. Cusp Forms Associated with Elliptic Curves

140

Indeed, by squaring, factoring y2 - 4 = (Y - 2)(Y + 2) and changing the variables several times we derive the following expressions: 2

_

77((x - 4)x)

IJ('7)12 =

77

X

x,y9E0,4

_ E 77(z) z

E

77

l

(

\ (y - 4)y/

(z_4) _ z

=p-2+>77(z) >

4(z - 1) Cz + y-4 /

y90,4

77(z+(z-1)v)

V00,-1

z5E1

= p - 2 - > 77(z) (17(z) + 1) 201

=p-> z

Z

Next we determine the argument of J(77). There are not many possibilities to choose from. Since 774 = 1 the terms of J(77) take values 0, ±1, ±i; therefore ,7(77) is a Gaussian integer, J(77) E 7G[i]. On the other hand, p - 1 (mod 4) factors in Z[i] into

p=7r7; where 7r is determined up to complex conjugation (7r is not distinguished from ir) and a unit c = ±1, ±i (by the unique factorization in the ring Z[i]). Combining the above facts, we deduce that J(77) = 7r

(8.25)

for some prime factor of p in Z[i]. To determine which factor (out of eight possibilities) is correct, we test the equation (8.25) modulo the ideal

a = ((1 + i)3) = 2(1 + i),

Na=8.

Since the character 77 takes values 0, ± 1, ±i, each of which except for 0 is congruent to 1 (mod (1 + i)), we infer that 7(17) =

'J(Y2

- 4) =

Y (nod p)

77(Y2

0<1'<'P2 -'

\

1+21

p2111

Hence for p - 1 (mod 4)

/

(8.26)

1+2E Y#2

p-2 (mod a).

J(77) = -1 (mod a).

- 4)

8.4. A Hecke Grossencllaracter

141

The above congruence together with (8.25) determines J(77) up to complex

conjugation (surely one cannot be more exact as long as rl is not distinguished from n).

We say that a Gaussian integer a is primary if a - 1 (mod a). Every odd a (i.e. coprime with a) is conjugate to exactly one primary integer. The only primary unit of 7L[i] is 1. The product of primary numbers is primary, and every primary number factors uniquely (up to permutation) as a product of primary numbers which are Gaussian primes. By (8.25) and (8.26) it follows that -,7(7) is a primary prime. Finally by (8.23), (8.25) and (8.26) we conclude that (8.27)

A(p) = it + Tr

if p - 1 (mod 4)

where 7r: = p and ;r is determined up to conjugation by the congruence (8.28)

it - 1 (mod a),

i.e. IT is a primary factor of p.

8.4. A Hecke Grossencharacter Consider the multiplicative group (7L[i]/a)* of residue classes in 7L[i] to mod-

ulus a and prime to a; it is a cyclic group of 4 elements represented by the units. For a odd we define p(a) to be the unit which makes p(a)a primary, i.e. p(a) = 1, i, i2, i3 is such that (8.29)

p(a)a - 1 (mod a) if (a, a) = 1.

If (a, a) 1 we set p(a) = 0. Thus p is a character on Z[i] to modulus a. Then we put (8.30)

X(a) = p(a)a.

The function X is one of many kinds of Grossencharacters which have been invented by E. Hecke. This can be regarded as a character on ideals r C 7L[i]. Every ideal is a principal ideal, say r = (a), with generator determined up to a unit. If (r, a) = 1 we can fix a by requiring a 1 (mod a), and we set

X(t) = a.

(8.31)

If (t, a) # 1 we put X(r) = 0. With the character X Hecke associated the L-function defined by the Euler product (see Chapter 12) (8.32)

L(s, x) = [J(1 - X(p)(NP)-`)V

8. Cusp Forms Associated with Elliptic Curves

142

Verifying the local factors in (8.18) with these in (8.32) by means of (8.20), (8.21), (8.27), we see that the Hasse-Weil and the Hecke L-functions coincide,

LE(s) = L(s, X).

(8.33)

Then comparing the coefficients in the Dirichlet series we get

a(n) = E X(r) =

(8.34)

Nr=n

4

E p(a)a IaI2=n

where t ranges over the ideals of 7L[i) and a ranges over the elements of 7L[i) (there are four units in 7L[i), whence the factor 14).

8.5. A theta series By (8.34) the Fourier series (8.9) associated with the curve (8.19) is expressed as a sum over Gaussian integers

AZ) = 4 E p(a)a e(zIaI).

(8.35)

aEZIi)

This expression constitutes a vital step in showing that f (z) is an automorphic form; it marks a transition from the multiplicative to the additive structure of 7L[il. At this point it is worth recalling that the sign of X(p) is captured by the congruence (8.28), which allows us to determine a(n) without inspecting the factorization of n into primes. Now, having f (z) expressed in additive terms, we can apply tools of harmonic analysis. Considering Z[i] as a lattice, we shall treat the series (8.35) by using Poisson summation; the arguments are standard but details are cumbersome because of powers of 2 in the conductor. Much more general theta functions will be considered in Chapter 10. We begin by recalling the formula (1.9), E e-7r(n+r)2v = v-1/2 n

En

e-7r?t2t?-1

e(nx).

Differentiating in x, we derive another formula:

E(n + x)e-'r(71+x)2v =

-iv-3/2

71

En

ne-7rn2v-1

e(nx).

Combining both formulas, we find that J :(m + x + i(n + y))e-7r[(m+y)2+(n+y)2}v 7)t

7T

_ -iv-2 E E(m + 771

71

in)e-'r(?n2+n2)t,-1

e(mx + ny).

8.6. The automnorphy of f

143

This formula is established for x, y, v real with v > 0; however it remains true for any complex numbers with Re v > 0 by analytic continuation. In the complex numbers notation we rewrite the above as follows (put ce =

m+in,)3 = x+iy,v = -2iz): aEZ[i]

2

E

(a + /3)e(zIa + $12) = 4z2

aEZ[i]

ae

4z+ Re a,(3)

.

This holds for any complex numbers ,Q, z with Im z > 0. Hence we infer our basic formula for the following theta-series: (8.36)

F(z; 0, y) = E

ae(ZjaI2);

aEZ[ij

a-p (mod y)

namely, for any 6, y E Z[i] with y 36 0 we have (8.37)

F(z; N y) = aEZ(ii]

ae (4 II7II2 + Re aR)

.

We shall exploit the transformation rule (8.37) to establish automorphic properties of the theta-series f (z) twisted by the character p(a) by splitting into residue classes to suitable moduli. Our target is to show that f E S2(r(,(32)).

(8.38)

8.6. The automorphy of f First we show that f is an eigenfunction of the involution of weight 2 and level 32 with eigenvalue 1, i.e.

fl,,, = f for w = More explicitly, we need to show that (8.39)

32z f

(32z)

z f (z).

Let y = 2(1 + i), so Iry12 = 8. By (8.37) we obtain

E p(Q)F(z; 0,,y)

f (z) =

/3 (mod y) i-Y

(322)2

aEZ(i]

ae ( -I al ,i G(a,y) 32z

8. Cusp Forms Associated with Elliptic Curves

144

where

P(Q)e (Re mo/

G(a,'Y) _

\

p (mod -y)

.

If

Since p is a primitive character to conductor a = (-y), it is clear that G(a, y) = 0 unless (a, y) = 1, in which case on changing Q to a,Q and observing that 1a12 = 1 (mod 4) we obtain G(a, -y) = P(a)G(1, -y) -

Finally we compute G(1, -y) term-by-termn as follows: 2

) + ile (Re i) + i2e (Re )

G(I, y) =e (R.e 'Y

+ i3e I Re

'Y

\

a3

)

'Y

=e4I-ie4)-el- 1+ie(-4)=2i(1-i)=iy. Gathering together the above results, we arrive at (8.39). Next we show that f is automorphic of weight 2, i.e.

fl. = f

if T E ro(32).

This is obvious for r = (1 i) because f is defined by a Fourier series. Suppose z = (a d) with 0 < c = 0 (mod 32) and ad = 1 (mod c). We need to show that

z-1f (-z

(8.40)

c-a) =zf (z+d)

We start from the right-hand side as follows:

zf(zcd)= 4

P(a)ae

(_IaI2)

«ezli)

r

E P(Q)el

\

/3 (mod 2)

IQI2JF(zc ;Q,2 //

(note that a - Q (mod c) implies Ja12 = 1,012 (mod c)). Applying (8.37), we obtain

_ 2cz

aEZ(il

ae (

2l CP(Q)e (!/3!2 / 13 (mod

2)

+ 2Re a/3)

.

8.6. The autoinorphy of f

145

Here we have d1/312 + 2Re a/3 - d113 + aa12 - a1«12 (mod c); hence

z f (z

(8.41)

d) 2cz

c

(_z1`c

E ag(a, c)e

aEZ,i]

-

ala121

where g(a, c) is a kind of Gauss sum given by

g(a, c) _

p(Q)e I d IQ + aa12 I

\

p (mod 2)

43 - aa)e (d 1012 J jI

(mo d 2 )

If (cx, 2) 0 1 then -a - a (mod a), so changing /3 to -/3 we infer that 9(a, c) = -g(a, c), i.e. g(a, c) = 0. Suppose (a, 2) = 1; then (1 + i) 1,6, and so

1 g(a, c) = 2 E p((1 + i)/3 - aa)e (2d IQ12 I

`

3 (mod 4)

/

If Re /3 - I (mod 2), then the terms for /3 and /3+ s cancel out. If Im /3 1 (mod 2), then the terms for /3 and /3 + s cancel out. Therefore we are left with /3 - 0 (mod 2), which give (8.42)

g(a,c) _ -2p(aa)

e

0 (mod g)

(lfi,2) c

To evaluate the last suns we appeal to the following formula of Dirichlet (see(1.10)): a

(8.43)

e

(N 1 = (l +i)Nz

n (mod N)

which is valid for any N > 0 with 41N. Hence

E e ( alN / = ((1 +i)NS)2 = 2iN. l

13 (mod N)

For (d, N) = 1 one can find 6 E Zli] such that d - ±1612 (mod N), whence p(d) = ±1 and

e M) = 13 (mod N)

/l

e /3 (mod N)

1131

2

±2iN = 2ip(d)N.

146

8. Cusp Forms Associated with Elliptic Curves

Applying this for N = c/8, we get by (8.42)

g(a, c) = C p(a) Inserting this into (8.41), we obtain (8.40). Finally we claim that f is an eigenfunction of all the Hecke operators. Since the coefficients of f are derived from the Euler product (8.18) which coincides with (6.93) (the character is principal), one checks that Tn f = a(n) f for all n by using Proposition 6.3. Thus f is a newform.

Chapter 9

Spherical Functions Spherical fimctions belong to the theory of representations on symmetric spaces associated with specific Lie groups. The classical theory goes back to Legendre, Laplace, Jacobi (late eighteenth century), and the modern development began with Cartan and Weyl (1930's); other notable contributors are Gelfand and Harish-Chandra. For our purposes we need classical spherical functions in the context of a positive definite quadratic form.

9.1. Positive definite quadratic forms Throughout we use the following notation and simple facts:

A=(ai?) a real r x r matrix, to = (aji) the transpose matrix, A I = det A 54 0

nonsingularity,

A-' = (a! .) the inverse matrix, 2.7

1A= A symmetry. Spectral Theorem. Any real, symmetric matrix A is orthogonally equivalent to a diagonal matrix, i.e. A = tUDU with'UU = I and D =

Here \l, ... , A, are the eigenvalues of A. 147

9. Spherical Functions

148

A matrix A E M(r) is positive definite if all its eigenvalues are positive. Equivalently

AA'

with lAjI>0 for all I

We shall write vectors x in R'' as column vectors so the transpose is the row vector tx = (x1, ... , x,.). Hence txx

= IxI2 =

x?.

With a matrix A = (aij) E M(r) we associate the quadratic form (9.1)

aijxixj.

A[x] = LxAx = ij

If A is positive definite, then so is its quadratic form, i.e. (9.2)

IxI2 < A[x] < IxI2

for all real x. A complex vector u ,E 0 is called isotropic for A if A[u] = 0.

(9.3)

For a positive definite matrix A = (aij) its inverse A-1 = (aij) is also positive definite. We attach to A the Laplace operator (9.4)

AA = > atj i

j

8z dxi8xj

Moreover we shall consider the inner product with respect to A defined by (9.5)

(f,9)A =

f

f(x)(x)dx A

for functions f, g on the ellipsoid (9.6)

KA={xEIR':A[x]<, 1}.

We shall regard lie? as a Riemannian manifold with the matrix (of curvature zero) derived from the differential form given by

ds2=1y?j

9.1. Positive definite quadratic forms

149

in the rectangular coordinates yl, ... , yr. The volume element is

dy = fl dyj, and the Laplace operator is given by 0 = div grad = E

To simplify computations we shall often make a suitable change of variables

x (a diffeomorphism of W). The Jacobian of this transformation is a nonsingular matrix y

3 = (8y:/8xi). The arc element is expressed in new coordinates by

ds2 = I: gijdx=dxj

where G = (g,i)

4j

the volume element is given by

dy = g1/2 fl dxj where g = [GI,

i and the Laplacian takes the following form: g-1/2

,

xd8-g1/2i =i axJ

Let A be a symmetric, positive definite matrix and U be an orthogonal matrix which diagonalizes A. Then say,

B=

r

U = (6ij),

gives

Al

A=tU

U=tBB. Ar

Changing variables y = Bx, we transform the quadratic form A[x] into A[x] =

t(B-ly)A(B-1y) = ty(tB)`[AB-1y

= tyy =

[y[2.

9. Spherical Functions

150

For this linear change of variables the Jacobian is 9 = B, and G tBB = A; hence IGl = JAI = 1B12,

ds2 = > aijdxidxj, t)

dy = IAI1/2 ll dxj,

A=Ea!. axiaxj

AA.

ij

The unit ball

K = {yElllr:lyl<11 is transformed to the ellipsoid ICA, and the inner product with respect to A on KA becomes (LOA = JAI-1/2 fK f (x)g(x)dy

The boundary OK of the unit ball K is the sphere S''-1

= {y E llt' : lyl = 1}.

The surface area of Sr-1 is equal to 2ir''/2I'(r/2)-1.

9.2. Space spherical functions A smooth function f on D C it is called a harmonic function if it satisfies the Laplace equation AA f = 0.

(9.7)

In dimension r = 1 the harmonic functions are linear functions. In higher dimension the basic examples are (for A = I) f (x) = log l xl

if r = 2

f(x)=lxl2^r ifr>3. These are the fundamental solutions for the Laplacian A (up to constant factors), and they are (must be) singular at the origin x = 0. However, if f is harmonic everywhere in Rr and is bounded by a polynomial, then f itself is a polynomial. This result is an analog of Liouville's principle for holomorphic functions. Clearly the homogeneous parts of f of every degree are also harmonic functions.

9.2. Space spherical functions

151

The harmonic functions which are homogeneous polynomials are called spherical functions or harmonic polynomials.

Theorem 9.1. Let f (x) be any homogeneous polynomial of degree v. Then the following assertions are equivalent: (i) f (x) is a spherical function, (ii) f (x) is orthogonal to any homogeneous polynomial of lower degree, (iii) f (x) = any constant, if v = 0, f (x) = any linear form, if v = 1, f (x) = any finite sum of forms (tuAx)" where u is an isotropic vector,

i.e. A[u] =0, if v,>2. Remark. At first glance the implication (ii)

(i) looks somewhat surprising. In a certain setting one could detect here a variational principle for the lowest eigenfunction of a self-adjoint operator in a Hilbert space. Our proof does not reveal this principle; it is more or less direct.

For the proof we make the change of variables y = Bx and v = Bu, which yields tuAx = tvy and tuAu = tvv. Therefore we can reduce the general case to the identity matrix. From now on A = I. Before starting the proof let us recall some facts from differential calculus

on W. Denote the differential forms

w? _ (-1)j-ldxl ...

j ... dx,.

dx=dxt...dxr=dx?w? for each 1,<j,
where 71 is an (r -- 1)-form on K. For 77 = f - w we get

di=d> where 6 denotes the differential operator (of Euler) (9.8)

Sf =

a xjdxi

Therefore for ri = f w Stokes' theorem gives (9.9)

f (bf + rf)dx =

fw.

9. Spherical ,Functions

152

Next, by Stokes' theorem for the form rl = (b f )w (or by the divergence theorem) one gets (9.10) fK

Af dx=fx(bf)w

where 0 is the Laplacian

Af

(9.11)

82f axe

If f is a homogeneous polynomial of degree v, then

bf = vf,

(9.12)

i.e. f is an eigenfunction of b with eigenvalue v (check this on monomials xi' ... xa, with al +... + a,. = v). Combining (9.9), (9.10) and (9.12), we conclude that (9.13)

fK

=v fx a

f=v(v+r)J

x

for any f which is a homogeneous polynomial in r variables of degree v. Finally let us record the following property of the Laplacian:

A(fg)=fAg+gOf+2

(9.14)

of ag ax; ax;

Now we are ready to prove Theorem 9.1 (for A = I). First we show that (iii) implies (i). This is obvious if v = 0, 1. Suppose v >, 2 and f (x) = (tux)" with tuu = Ju12 = 0. Then

Next we prove that (i) implies (ii) by induction on v. But first notice that if f is a spherical function, then so are its partial derivatives a f /axe because g commutes with 8/ax3 (all partial derivatives commute, and 0 has constant coefficients). Assuming that deg g < deg f and A f = 0, we proceed as follows:

f fgdx = cl JrI 0(7g)dx = cl f fLgdx t

Ic

= C2

li

0(1Og)dx = C2

f JA2gdx li

9.2. Space spherical functions

153

by (9.13), (9.14) and the induction hypothesis. We can continue this way until we get 0"g = 0, whence (f, g) = 0. It remains to show that (ii) implies (iii). Let us denote the space of all homogeneous polynomials of degree v by

P = {f E C[x] : f (ax) = a' f (x)}.

(9.15)

This has finite dimension over C; precisely, dim P" =

(9.16)

Cv+r - 1

r-1

The linear space spanned by all polynomials of type (tux)" with tuu = 0 is closed in P". Therefore the problem reduces to showing that if f E P" is orthogonal to all P. with µ < v and all polynomials of type

g(x) = (tux)" with tuu = 0

(9.17)

then f = 0. Since g and its partial derivatives satisfy (iii), they satisfy (i) and (ii). Using this information we infer the following:

22f 19 dx = cl J A(fg)dx = 2c, J ( E Nax ax j K

;

1C

i

dx

.i

by (9.13) and (9.14), since Og = 0 and g10 f , the last property being a consequence of the implication (i) times, we arrive at 2"c"

J

a"f

(ii) for g. Repeating this argument v

ax;, ... ax;,, ax;,

a1g .

ax;,,

dx = 2"c"vol (K)a

where o denotes the above sum of derivatives, which is a constant since deg f = degg = v. To compute a apply the formula (see (9.12))

of =2-., xi

of ax;

successively to the resulting partial derivatives until we get

V! f = E x;, ... ..i,, a"f /ax;, ... ax;,.. .ii...;t,

9. Spherical Functions

154

Moreover, we have

avg/ax ... 8xj = v!uj, ... uj,,. From these two formulas it is plain that or = f (u). On the other hand, a = 0 because f 1g. Therefore we have shown that

f (u) = 0

if E u = 0.

This means that f (x) vanishes on the algebraic variety V = {x E C' : p(x) _ 0} defined by the polynomial

p(x) = Ex" By Hilbert's Nullstellensatz the ideal of V contains a power of f. Except when r = 1, the polynomial p(x) is either irreducible (if r 3 3) or factors into two coprime irreducible polynomials (if r = 2), so f (x) is divisible by p(x), say f (x) = p(x)h(x). By (9.13) we infer that h(x)h(x)dx = cl 10K h(x)(x)w

c10K f(x)h(x)w

Jx

= c2

J f(x)h(x)dx = 0

K

because p(x) = 1 on 6K and deg h < deg f. Hence h = 0 and f = 0. For r = 1 we have f (x) = axv, and if v > 2 then p(x) = x2 divides f (x), so the former argument applies. For r = 1 and v = 0, 1 there is nothing to prove. This completes the proof of Theorem 9.1.

Corollary 9.2. The space lv of harmonic polynomials of degree v > 2 has complex dimension

/---1 v+r

dimW, = t

(9.18)

r-1

_

v+r-3

r--1

Proof. By Theorem 9.1

EP,,: f1Pµ for all it
But P,,-LP, if lc product), so

7-v = {f E Pv : f 1Pv-2, Pv-4, ... }. We also have

(f, 9) =

J!i

f (x)9(x)dx = (J ti +IL+rdt) o`

JO

f (x)g(x)dx

9.3. The spherical functions reconsidered

155

where v = deg f, µ = deg g and r is the dimension. Hence

_

v+µ+r+3

v+p+r-I-1(f'Pg)

Rv = {f E P, : f 1Pv-2}. Hence by (9.16) we get (9.18).

9.3. The spherical functions reconsidered A homogeneous polynomial f (x) of degree v is determined by its values on the unit ball. Namely, we have

f(x)=IxIvfCHI This formula establishes a one-to-one correspondence between the space spherical functions and the surface spherical functions. The surface spherical functions are eigenfunctions of the Laplacian on the sphere

Si- ={xER":IxI=1} with eigenvalues depending on the degree v and the dimension n (we changed

notation for the dimension from r to n to avoid confusion with the radius coordinate). We shall now briefly present the surface spherical functions from the point of view of representations of the special orthogonal group. The space X = R" is usually viewed as a homogeneous space of the additive group G = R" acting on itself by translations. A differential operator is G-invariant when it has constant coefficients, and the joint eigenfunctions of all such operators are multiples of exponential functions. Therefore harmonic analysis on the flat space R' (curvature zero) reduces to the study of classical Fourier integrals. One can be more demanding by taking the whole group of orientation-

preserving isometrics on ll8", say M(n), which is generated by the translations and rotations. Then R" acted on by the whole group M(n) is the euclidean space of dimension n. The group of rotations 0(n), which is the isotropy group of the origin, is represented by real orthogonal matrices

O(n)={U:tUU=I}.

9. Spherical Functions

156

Hence the euclidean space R" can be viewed as the space of cosets

R" = M(n)/O(n) which makes II8" a homogeneous space. Now an invariant differential operator is not only translation invariant but also rotation invariant. This additional requirement implies that such an operator is a polynomial in the Laplacian

a2

92

8x2

+ ... + ax2 n

i

with constant coefficients. In other words, the algebra of invariant differential operators is generated by A. The basic eigenfunctions of A are e('xy) = e(xiyl + ... + x:tyn), and harmonic analysis on II(" reduces again to the classical Fourier inversion. With the above point of view it is not surprising that the Fourier transform in l[8" commutes with rotations. In particular the Fourier transform of a

radial function, i.e. a function which depends on the distance

r(x) _

(:xi

2

is also a radial function (see (1.12)). Now we consider the sphere S' 1 C R' for n > 2. The special orthogonal group

SO(n) = {U E 0(n) : detU = 1} acts transitively on Sn-1, and the isotropy group of each point can be identified with SO(n -1). To be specific, fix a point, say s = (0,... , 0, 1) E Sn-1,

for which the isotropy group is the group of rotations around the x,t-axis and is isomorphic to SO(n - 1). Think of U in SO(n - 1) as an element u (?) of SO(n). By means of this embedding the sphere can be regarded as the homogeneous space

S"-1 = SO(n)/SO(n -1). As a Riemannian manifold S"-1 has positive curvature. On convenient to work in the spherical polar coordinates x1 = rsinB"_1...sin 92 sin91 X2 = r sin 9,t_ 1 ... sin 02 cos 01 x,t_1 = rSin0,t_1 COS9n_2

xn = rcos9,t_1

S"-1 it is

9.3. The spherical functions reconsidered

157

where 0 < r < oo, 0 < 91 < 21r and 0 < 0.7 < ir if 1 < j < n. The rotation invariant measure on Sn-1 is given in terms of the angles 9j by dµ(9)

r (2)

fir

(sin 9j)jd9j. 1<,j
The Laplacian 0 on R" and the surface Laplacian 0' on Sn-1 are related by

-0 +r1 O'.

_

02

(9.21)

8r2+nr

Let P = C[x1, ... , xn] be the algebra of polynomials in n variables over the complex numbers. Clearly P is SO(n)-invariant, and it splits into the direct sum of finite dimensional invariant subspaces

-P

P

where P denotes the space of homogeneous polynomials of degree v. We have d1ln Pv

=

v+n

-1

n-1

because the monomials xi ... xn" with eel + ... + an = v make a basis for P. We endow P with a nondegenerate bilinear form (an Hermitian inner product) with respect to which the distinct monomials are orthogonal, Ai

a

r al!...an! if (a) =

(i3)

otherwise.

0

To be specific (though it will not be of much use), for any q, r E P we put.

(q,r) = (a(r)q)(0) where 8(r) is the differential operator a ar() r(ax1'

a

8x

Clearly the multiplication p -, rp is adjoint to the differentiation q --4 8(r)q, i.e.

(q,rp) = (a(r)q, p) (it is sufficient and easy to check this for r(x) = xj, NO = 09/axe;). In particular, multiplication by r(x) = xi +... + x2 is adjoint to the Laplace operator A = a2/8xi + ... + a2/az a, i.e. 71

(q,?'p) = (Oq,p).

9. Spherical Functions

158

Observe that rP C P,,+2 and AP,, C

The kernel of 0 consists of harmonic functions on Rn, i.e. polynomials annihilated by A. We have the direct sum

7l = ker to = ®7-l where H = H fl Pv is the space of harmonic polynomials of degree v. In general the kernel of a linear operator is equal to the orthogonal complement to the range of its adjoint, so we have

P,, = f'l,, ® r2P.,-2 Hence by induction we get the orthogonal decomposition

P,, =

EBr21"/21 i-l

Here each subspace r2kxv-2k is SO(n)-invariant because r2 and A commute with rotations.

We denote by £ the space of functions in H restricted to the sphere Sn-1, and call these the surface spherical functions (or simply spherical functions). By (9.21) it follows that a spherical function Y E £ is an eigenfunction of the surface Laplacian (9.22)

(A* + v(v + n - 2)) Y = 0,

and since the eigenvalues -v(v+n-2) are distinct for different v, the spaces £v are mutually orthogonal. By the Stone-Weierstrass theorem the space of all spherical functions (direct sum)

V

is dense in the space C(Sr-1) of all smooth functions on Sn-1 in the uniform norm by compactness. Hence we have the orthogonal decomposition L2(Srn-1) =

® £,,.

By (9.21) and (9.22) it follows that the whole spectrum of 0* in L2(S1-1) consists of numbers of the type

A=-v(v+n-2) and that the A-eigenspace is £,,. Since 0' is rotation invariant, each eigenspace £ is SO(n)-invariant. We shall show that the eigenspace representations are irreducible. Suppose £, = F 0 9 is an orthogonal decomposition

9.4. Harmonic analysis on the sphere

159

Figure 10. The sphere

into nonzero SO(n)-invariant subspaces. By the SO(n)-invariance of E there are two functions f E F, g E G which take value 1 at some point, say s = (0, ... , 0, 1), f (s) = g(s) = 1. We may assume that f, g are SO(n - 1)invariant by averaging over SO(n - 1). Such functions f,g depend only and the Laplace equation (9.22) for such functions on the angle 0 = becomes the ordinary, second order differential equation 2

Cd02 +(n-2)cot0d0

+v(v+n-2)) f =0.

This has a unique solution normalized by f (s) = 1/(the other linearly independent solution is ruled out since it is singular at 0 = 0). Therefore f = g, and this shows that the eigenspaces E,, are irreducible with respect to rotations.

9.4. Harmonic analysis on the sphere The case of three dimensions is particularly illuminating. We shall give an

account of this theory in quite explicit form. Our aim is to construct a complete, orthonormal system of spherical functions on S2. To this end it is natural to work in the spherical polar coordinates (see Figure 10) xl = r sin 0 cos cp,

0 < r < oo,

x2 = r sin 0 sin gyp,

0<0<7r,

x3=rcos0,

0
9. Spherical Functions

160

In the new coordinates we have ds2 = dx2 + dx22 + dxz3 dr2 + r2d92 + r2 sin2 9d 2 dµ = dxldx2dx3 = r2 sin 9drd8dcp,

If = r2(sin9)1 (Q(r2sin9i) +

(sing

) + (sin©)-1 'Q

)

.

On the unit sphere S2 these formulas give (take r = 1) ds2 = d92 + sin2 9dcp2,

d1 = sin 9dOdp,

and the Laplace-Beltrami operator on S2 is

A` _ (sin 9)-1 -(sin

g--)+(sin 9)`2

2

cP

2

2

a2 + cot 9 a9 + (sin o) -2

22 .

= The group of rotations SO(3) leaves ds, dit and 0' invariant. Applying 0 to a spherical function of degree v of the type f (x) = r"Y(9, cp),

we obtain the differential equation for Y = Y(9, cp), (9.23)

(sin9)-1

(sin1) + (sin

+ v(v + 1)Y = 0. cP

In other words, Y is an eigenfunction of A` with eigenvalue -v(v + 1). C is an eigenfunction of i , then its eigenvalue must be -v(v + 1) for some integer v >, 0, and the function f (x) = r"Y(9, cp) is a spherical function of degree v. Therefore the surface spherical functions Y(9, cp) of degree v are just the eigenfunctions of the surface Laplacian A* with eigenvalue A = -v(v + 1). The eigenfunction equation A*Y = AY can be solved by the method of separation of variables. We search for solutions of type

Conversely, one can show that if Y : S2

Y(9, cp) = P(cos 9)E(cp).

Inserting this into (9.23), we obtain two independent second order differential equations: d2E dcp2

(sin g)-1

9

E

(sinOP(cosO)) + (v(v + 1) - p(sin 9)-2)P(cos 9) = 0

9.4. Harmonic analysis on the sphere

161

where ate is an unknown parameter. Since E(w) must be periodic of period

2ir, the first equation has such solutions only for µ = m2, where m is an integer; they are linear combinations of E(W) = eti,np

(9.24)

Putting this value of µ into the second equation, and changing the variable 0 (recall that 0 < 0 < ir) to z = cos 0

(9.25)

(note that -1 < z < 1), we get. (9.26)

((1-z2)P')'+(v(v+1)-

1mz2)P=0.

First let us consider m = 0; then Y(0, cp) = Y(O) = P(cos 0) does not depend on cp (such functions are called zonal spherical functions). The equation (9.26) simplifies to ((1 - z2)P')' + v(v + 1)P = 0.

(9.27)

It is not difficult to verify that the Legendre polynomial defined by U

(2"v!)--1 dv (z2

P"(z) =

(9.28)

dz"

- 1)v

satisfies (9.27). These polynomials have the generating power series 00 (9.29)

E P"(z)r" = (1 - 2rz +

r2)-1/2,

v=o

has degree v. We have

The Legendre polynomial

(-1)k (2v - 2k)! -2k 2kk! (v - k)!(v - 2k)!z

P"(z)

=2"

()2(z_1)(z+1)"3. O;i

V

1 3 2 --2. P((z)=1, P1(z)=z, P2(z)=2z

9. Spherical Functions

162

Moreover, the polynomials satisfy a handful of recurrence relations; for example, for all v 3 1 we have (9.30)

(v + 1)P.,+1(z) - (2v +

0,

and they also satisfy differential recurrence relations, such as zv-1P,,(z)

(9.31)

= (z"PY-1(z))'

We already know that the P, (z) are orthogonal on the segment -1 <, z 5 1 because they come from spherical functions of different degree. However one can derive directly that (9.32)

r1 r 1

1

1

P,(z)P,(z)dz = (v +

Indeed, by (9.27) we have ((z2

(v - A) (V + p +

- 1)(P'P1 -

which gives (9.32) if v # µ by integration along -1 5 z 5 1. For v = p we use the recurrence relation (9.30) to get

v > 2, whence integrating and using orthogonality we get the recurrence

relation (2v + 1)

/ 1

I l

P,2(z)dz = (2v -

r

1) /

1

Pv_1(z)dz,

1

which proves (9.32) by induction (verify it directly for v = 1). By (9.32) the normalized functions (v+ form an orthonormal system in G2([-1,1]) which is complete by the Weierstrass theorem.

More precisely, if f E G2((-1, 1)) is piecewise smooth, then the spectral series 00

(9.33)

f (z) = E(v + 2)(f, P,.)P,.(z) V=o

converges to f (z) at every continuity point of f (z) and to 2 1f(z+0)+f(z-0)] if z is a discontinuity. Note that the orthonormal system is not uniformly bounded with respect to the spectrum. Indeed, at z = 1 we have 1, whence

(v+2)ZP(1)=(4-A)'

9.4. Harmonic analysis on the sphere

163

This property may have something to do with the fact that S2 is a Riemannian surface of positive curvature (equal to 1). The spectral expansion (9.33) can be written as the integral equation

f (z) = with kernel

f

K(z, w) f (w)dw

00

K(z, w) = >2(v + =o

Using the recurrence relation (9.31), one can compute the partial kernel K,, (z, w) which is obtained by restricting the sum to v < n. We obtain Kn(z, w) = 2 (Pn(z)P,,-1(w) - Pn-1(z)Pn(w)](z -

w)-1

By orthogonality f1

Kn(z, w)dw = 1. Hence we derive an integral formula for the tail of the spectral series (9.33), namely

(v+ 2) (f,., P,.)P,.(z)

f (z) = v
///

+2

(Pn(z)Pn-1 (w) - Pn-1(z)Pn(w)]F(z, w)dw 1-1

where (z) - f (w)

z-w This expression is very useful for establishing effective spectral approximations.

As an example of spectral series (9.33) we expand the characteristic function of a segment

f(z)=

if -1
1I

We get

f (") = 1+ a+ E[PY+1(a) - P v>O

i

9. Spherical Functions

164

Now we return to the equation (9.26) for any integer in. We consider in >, 0 only because the equation is the same for -m. One can check that the so-called associated Legendre functions defined by (9.34)

z2),n/2d- P ,,(Z)

P" '(Z) = (1 -

satisfy (9.26). Note that P,m(z) - 0 if m > v, because degP,, = v. For 0 5 m 5 v the associated Legendre functions are linearly independent. There is a simple integral representation ( 9 . 35)

P (cos 9 ) _

(Tri

i v)! i r

f

(cos 6 + i sin B cos

We conclude the above investigatiuus with the following (cf. [CHI, [Bocj and [Ter]). Theorem 9.3. The spectrum of the spherical Laplacian 0' consists of numbers of the type

A = -v(v + 1),

(9.36)

v = 0,1, 2,... .

The A-eigenspace has dimension 2v + 1, and the functions y- (O, W) = C,,,"', P,""`! (cos 9)eimw

(9.37)

with Im[ 5 v, where C",t=(2v+1(v-m)!)

(9.38)

(v + m)!

4Tr

form a complete, orthonormal system of the A-eigenspace. All the functions (9.37) with Iml 5 v form a complete, orthonormal system in ,C2(S2, dµ) with respect to the measure dµ = sin 9dOdcp (a rotationally invariant measure). Every smooth function g : S2 -> C has the spectral expansion

9(e, cp) _ EE

(9.39)

cP)

V>'ImI

where (9.40)

9;;'

n

27

91'v"dA --

fo

fo

9(e,

The series converges absolutely and uniformly.

W)Ym (9,

cp) sin 9d©dcp.

Chapter 10

Theta Functions 10.1. Introduction Given a symmetric, positive definite matrix A and a spherical function P with respect to A, we define the theta function (10.1)

P(m)e (A[mJz).

O(z)

Clearly 6(z) is holomorphic in the upper half-plane H because the series converges absolutely (its terms decay exponentially, because A[m] >> Jm12 and P(m) is a polynomial). One can generalize the theta function by introducing another complex variable, say v E C, as follows: (10.2)

P(m)e (A1m]z+B(xm)v)

O(z, v) _ fnEZ'

where x E C' and B(x,y) = txAy is the bilinear form associated with A; O(z, v) is called a Jacobi function. However we shall achieve our main goals by means of the special congruent theta functions (10.3)

O(z; h) =

P(nz)e

())

vn=_h (mod N)

for h E Z", where N is a positive integer. Note that O(z; 0) = N"O(z), where v = deg P. Throughout we assume that A is an integral matrix, i.e. (10.4)

A=(aij)with a;j =a1,EZ, 1<, i,j<, r, 165

10. Theta llinctions

166

and the same holds true for the matrix A' = NA-1, but we do not require N to be the least positive integer with this property. Clearly N = IAI satisfies this condition; however a smaller number suffices quite often. Note that IAj]A'! = N'; hence we always have

N' M 0 (mod JAI)

(10.5)

so every prime factor of JAI is in N. Observe by symmetry that the quadratic form associated with A has even coefficients save for the diagonal, (10.6)

A[x] = txAx = 2 E aijxixj + > aiix?. i<.j

i

We shall say that A[x] is even if all its coefficients are even, i.e. if (10.7)

diag A - 0 (mod 2).

However this assertion will not be needed for many results. Our objective is to show that A(z; h) and 6(z) are automorphic forms of weight

k=v+2

(10.8)

with respect to a congruence group of a certain level and for a multiplier system which depends on A and N (but not on P at all!). The theta functions are a vast source of automorphic forms. There are immediate applications to the theory of representations by quadratic forms and less immediate ones to analytic studies of L-functions for number fields. We shall present some of these topics in the subsequent chapters.

When the quadratic form A[x] has an odd number of variables, the multiplier system for the associated theta function is quite sophisticated because the weight k is not an integer. Needless to say, in this case one must be careful with multiplying the square roots of various complex numbers.

Let us recall that z9 is defined for any z E C` and S E C in Section 2.6. Observe the following rules:

zn+v =

(z1/2)s = zs/2, Z-' = (1/z)s unless z < 0, (zlz2)m=z2 z2 , ifmE7L,

zuzv,

(cz)s = cszs, ws(z/w)s = zs

if c > 0, if z,w E H.

10.2. An inversion formula The automorphy of various theta functions will be derived from the following basic formula (a generalized Jacobi inversion).

10.2. An inversion formula

167

Proposition 10.1. Let A be a symmetric, positive definite matrix (not necessarily integral) and let P be a spherical function with respect to A of degree v. For z E H and z E Cr we have (10.9)

(i)k

EP(m+x)e (2A[m + x]z } =

Z

EP (m)e l \\

IAI

In particular (for x = 0)

-A- 1

m + rrtxJ

) ()k

(10.10)

9(z; A, P) =

0(-z-1; A-1, P')

i-"IAI-1/2

Z

where P*(x) = P(A-1x) is a spherical function for the matrix A'1. The formula (10.9) constitutes a far reaching generalization of (1.9), and its proof is based on the Poisson summation f (m)e(tmx).

f (m + x) = m

m

We also use the familiar Fourier integral (for u E R and z E 1111) (10.11)

J e(2y2z-yu)dy= (z)112P(2z2)

Proof. We apply Poisson's formula to the function f (x) = e

(A[xJz).

To compute the Fourier transform

J(v) =

J

f (x)e (-txv) dx

we change the variable x to y = Bx by the linear map B which diagonalizes

A as in Section 9.1. We have tBB = A, A[x] = tyy, v = tBU, txv = tyu, dx = IBI-ldy, IB12 = j (v)

IAI.

Hence

= IBI-1 f e (1 tyyz - tyu dy , =IBI-1TT f

=IBI

1(z)2e(_tuu)

10. Theta. Functions

168

by (10.11). Now we return to the original variables. Using the transformation tuu = t(tB-iv)tB-lv = = ttvA-lv =A-'Jv], we get tvP-ltB-iv

f (v) = JAI- "2

(Dr!2 e

(i1).

Hence Poisson's formula becomes

(10.12) >e (A1m+x]z)

= CAI

1/2 (zl

r/2

>2e in

in

C-A2z1[m ]

+tmx 1\

This is a special case of (10.9) for P = 1. We shall derive (10.9) for any P by differentiating (10.12) with respect to x by means of a suitable operator. Without loss of generality we may assume (see Theorem 9.1) that P(x) = ('cAx)", where c is an isotropic vector if v >' 2 and c is arbitrary if v = 0, 1. Given this vector, one defines the linear differential operator

L=>cjO/ax;. We have LA[x] = 2tcAx and L2A[x] = 2A[c] = 0 if v > 2. Hence applying L" to (10.12) we obtain (divide throughout by (21riz)" and observe that z-"(Z/z)r/2 = 2-"(i/z)k)

>P(m+x)e C 1 A[m + x]z

(t

(!)

to

in

)"e

+ 2z

Here we insert (tcwn)" = P(A-lm), completing the proof of (10.9).

Remark. If JAI = 1, then 9(z; A-', P*) = 9(z; A, P) by changing m to Am in the summation. Therefore in this case (10.10) becomes (10.13)

9(-1/z) = i-"(-iz)k9(z).

10.3. The congruent theta functions Let A, P, h, N be as in the first section. Note that (10.14)

9(z; -h) = (-1)"9(z; h).

Moreover, observe that 9(z; h) depends only on h (mod N). We shall consider the theta functions 9(z; h) exclusively for classes in f = {h (mod N) : Ah - 0 (mod N)}. Note that grad A[x] = 2Ax - 0 (mod 2N) if x E R.

Lemma 10.2. The set 1-1 is a finite abelian group (additive) of order (10.15)

mWi = CAI.

.10.3. The congruent theta functions

169

Proof. Think of A as a linear map A : RI _4 R'' which transforms the unit box 5 = [0, 1]' to the fundamental parallelogram P = A!3 of volume IAI. Since A is integral, P has integral vertices, so the number of integral vectors in P is equal to the volume; IP n Z'* I = IPI = IAI. Also the matrix N`IA

maps the box NB = [0, N]' to P, and the integral vectors h in NB with Ah - 0 (mod N) correspond to the integral vectors in P. This completes the proof of (10.15). That 7-l is a group is obvious.

For a diagonal matrix A = diag [a,,... , a,] the group f is the direct sum of Z/ajZ,1 j S r. Since the bilinear form B(x, y) = txAy is nondegenerate and 1 is isomorphic to its dual, it follows that every character on fl is given by (10.16)

O(i(h, l) = e(N-2B(h,1)),

h E f-i,

for some I E 1-l. Note the symmetry 0/i (h, l) = zy(l, h). We have the orthogonality formula

-

(10.17)

if h = 0 (mod N)

( IAI

otherwise.

0

ILE%

Proposition 10.3. Let h E W. We have (10.18)

6(z + 2; h) = e

(1) 6(z; h),

and if diag A is even, then (10.19)

6(z + 1; h) = e

( 2N2 ) O(z; h).

Proof. Since A is symmetric we have (10.20)

A[x + y] = A[x] 4- 21 xAy + A[y].

Hence, writing m = h + nN, we get A[m] = A[h] + 2N`nAh + N2A[n] = A[h]

(mod N2).

Moreover if diag A is even, then the above congruence holds true to the modulus 2N2 because A[n] is even (sec (10.6)). These congruencies lead to (10.18) and (10.19) immediately.

Next we shall establish a transformation formula for 6(z; Ii) with respect to the involution z --> -1/z.

10. Theta Functions

170

Proposition 10.4. For any It E N we have

9(-1/z; h) =

(10.21)

E?P(h,

l)6(z; 1).

IE1(

Proof. We apply (10.9) for x = hN-1. The left side is just N-"9(z; h). On the right-hand side we change m to n = NA-1m. The condition m E Zr is equivalent to n E Zr and An - 0 (mod N). Moreover, under this change

we have P(A-lm) = N-"P(n), A-1[m] = N-2A[n] and tmx = N-2tnAh. Therefore (10.9) becomes (multiply both sides by N" and change z -- -1/z)

9(-1/z; h) = it l AI.1/2(-iz)k

P(n)e

(I!!iz

A)

An-o (mod N)

We shall generalize (10.21) for some transformations T = (:) in SL2(Z) which depend on A and N. We assume that d# 0, because otherwise T = ± (1 -1) and we are done. Changing r to -T, if required, we may also assume that (10.22)

d > 0.

Next let us assume that one of the following two conditions holds:

b - c - 0 (mod 2) diag A - 0 (mod 2).

(10.23) (10.24)

We begin with the transformation

(a

-1) = (d d) (1

-ac).

Since d-yz = b - (dz - c)-1, we have

P(m)e

9(yz; h) = m-h (mod N)

A[m] 2N2

b

1

(d - d(dz --c) )

Note that e(bA[m]/2dN2) depends only on m (mod dN) (see (10.20), the condition h E 71 and (10.23)-(10.24)). Therefore we can split the summation into residue classes to modulus dN, getting IA[g]

9(yz; h)

P(m)e

N2

g (mod dN) .9=h (mod N)

m=_g (mod N)

dA[m]

-1

2(dN)2 dz - c

10.3. The congruent theta functions

171

Here the innermost sum is the theta function associated with the matrix dA and the residue class g (mod dN) evaluated at the point -(dz - c)-1. Since dAg - 0 (mod dN), Proposition 10.4 is applicable to this sum, giving di )2

('IAg) P(m) J ml (mod dN)

e

I (mod dN)

e (j(dz - c)l

AI==-O (mod N)

Note that I dAI = dr IAi and cA[m] _- cA[l] (mod 2dN2). Hence we deduce

that e(tix,

h) = (i(c - dz))2 2d

I AI

p(h,1) I (mod dN)

m)e (A[ m=I (mod dN)

\ .2N2]

z1 ///

AI-O (mod N)

where

E

e ((bA[g] + 2tlAg - cA[l])/2dN2) .

9 (mod dN) g=h (mod N)

We shift g to g+cl so the new variable ranges over classes modulo dN which

are = h - cl (mod N), and in the exponential we get (bA[g] + 2adtlAg + acdA[g])/2dN2

by applying ad - be = 1. In the middle term 2adtlAg we can replace g by its class h - cl (mod N), getting cp(h,1) = e

(10.25)

=e

(2atlAh_acAEl)) acA[1]

2N2

cp(h - cl,0)

(ah,1)
This expression shows that cp(h, l) depends only on I (mod N), and we get

()(7z; h) = i-'d- /2IAI-1/2(i(c - dz)) E W(h, h')e(z; h'). h'EN

Now we come to rz by changing z to -1/z (note that ry(-1/z) = rz). On the right-hand side we apply Proposition 10.4 (again!) to each e(-1/z; h') and obtain (10.26)

e(rz; h) = i-2'd

-''12IAI-1(cz

+ d)' E (D(h,1)e(z;1) tEN

10. Theta. Functions

172

where cp(h, h')V)(h',1).

41(h, 1) =

(10.27)

11,01

The automorphy factor in (10.26) came from the equality

(!(cz+d))k (z) k _ (cz+d)k. Check this multiplication rule separately for c > 0, c = 0, c < 0. To proceed further we need one of the following Lwo condiLions: (10.28)

c

(10.29)

c

0 (mod 2N) 0 (mod N) and diag NA-1_ 0 (mod 2).

Observe that the hypothesis diag NA-1 - 0 (mod 2) implies A[l] __ 0 (mod 2N) for any l with Al - 0 (mod N). Indeed we have m = N-IAl E Z'

and NA-1[m] - 0 (mod 2), whence A[l] = NIA-1[m) - 0 (mod 2N). Therefore under either of the above conditions the formula (10.25) simplifies to

cp(h, l) = /i(ah, l)cp(h, 0).

Hence by the orthogonality of characters (see (10.17)) we infer that (D(h,1) = tp(h, 0) E O(ah, h')?,l'(h', l) = cp(h, 0) IAA h'EN

if l - -ah (mod N), and it vanishes otherwise. Hence there is only one terns on the right side of (10.26), and taking (10.14) into account we arrive

at (10.30)

e(Tz; h) = cp(h, 0)d^''12(cz + d)k®(z; ah).

It remains to compute cp(h, 0)

-

E g (mo d dN )

e ( 2A[9, )

2dN2 l

g,h (niod N)

Since ad = 1 + be - 1 (mod N), we can write g = adh + xN with x ranging freely modulo d. We get cp(h, 0) = e(a2bdA[h]/2N2)G, where (10.31)

G = E e(bA[x]/2d) x (mod d)

10.3. The congruent theta functions

173

is a Gauss sum associated with the quadratic form In the exponential factor of cp(h, 0) we can replace ad by 1, because either ad =_ 1 2A[x].

(mod 2N), Alit] = 0 (mod N) in case of (10.28), or ad = 1 (mod N), Alit] - 0 (mod 2N) in case of (10.29). Inserting this, we get

cp(h,0)=e(a2 2] ) G.

(10.32)

Before computing G we make a convenient transformation. We assume

that

d - I (mod 2).

(10.33)

Then (d, 2cIAI) - 1, and changing x to 2cx cnodulo d we get (apply be -= -I (mod d)) .(-2rA[Tl /d),

G=

(10.34)

x (rnod d)

Lemma 10.5. Let d be a positive integer with (d, 2clAI) = I. Then G -- (LA

(10.35)

dl

(Z-d

(2c)

y

d

Proof. The one-dimensional case was considered in Lemma 4.8. We have z

e (d) _ \dl eddl/z

(10.36) x (rnod d)

if d > 0 and (d, 2c) = 1, where

_1)1/2 (10.37)

e

`t = {

d= i

I

1 (mod 4)

if d if d

-1 (mod 4).

We shall exploit this result to evaluate G by a local diagonalization. First recall that a nonsingular symmetric matrix over a field of characteristic # 2 is equivalent to a diagonal matrix. Hence, applying a p-adic approximation

argument to A, one shows that there exist an integral matrix V and an integral diagonal matrix M = diag (ml, ... , m,.) such that

tVAV - M (mod d) with (d, I V I) = 1. Changing x to V y modulo d, we get A[x] - M[y] _ nt j yj (mod d) and

G=

ry,

e(-2cnijyY /d) (rood d)

=

lnnl ... m,.l

\

J 1\

,

(2c) /I

by (10.36). Moreover we have nil ...nt,. = IMI ItVAVI = IAIIV'I2 (mod d), which leads to (10.35). Now we combine the conditions and collect our results into the following:

10. Theta Functions

174

Proposition 10.6. Let r = (c b) E SL2(Z) with d -= 1 (mod 2). Suppose that any one of the following three conditions holds true:

0 (mod 2N), b - 0 (mod 2). 0 (mod 2N), diag A = 0 (mod 2). 0 (mod N), diag A diag NA-1 - 0 (mod 2). Then for any It E N we have

(i) c (ii) c (iii) c

(10.38)

(,r z; h) = e

(abA[h])'0(r)

(cz + d)'A(z; ah)

where

(10.39)

'0(r)=

(A) (Ed

d(v))".

Remark. Observe that for c = 0 the result was already established in Proposition 10.3. The original assumption d > 0 was relaxed in Proposition

10.6, since one can change r to -T. This change does not alter the righthand side of (10.38) if one observes the convention (2.75), which extends the Legendre-Jacobi symbol (d) for 0 < d =- 1 (mod 2) to all d =- I (mod 2) by the rules (10.40)

\d)

Fcl

( d)

and p (10.41)

(d)

= {

if c

1

ifd=fl

0

otherwise.

0

Corollary 10.7. Let h E W. The congruent theta function 9(z; h) is an automorphic form for the principal congruence group I(4N) of weight k _ v + r/2 and the multiplier (10.42)

i. e. for r = (')

'0(rr) = Mr' (1 1) (mod 4N)

we have

d)'®(z; h). e(rz; h) _'0(r)(cz + Moreover, if the spherical function P is not constant, i.e. v > 0, then 6(z; h) is a cusp form.

(10.43)

Proof. The multiplier (10.42) is a simplified expression (10.39), since d - 1

(mod 4N) and (A1) = 1 by (10.5). Then (10.43) is a simplification of (10.38). That ®(z; h) is holomorphic and vanishes at cusps if v > 0 can be seen from Propositions 10.3 and 10.4 (as in the proof of Theorem 10.8).

10.4. The autonlorphy of theta functions

175

10.4. The automorphy of theta functions For h = 0 we obtain O(z; 0) = N"O(z). Observe that v must be even; otherwise the theta function O(z) vanishes identically. The transformation formula (10.38) simplifies to (10.44)

O(rz) = 19(r)(cz + d)kO(z)

with 19(T) given by (10.39) for any T satisfying any one of the alternative conditions of Proposition 10.6. In particular, from the second group of these conditions we infer

Theorem 10.8. Let N be such that NA-1 is integral, and let diag A be even. Then O(z) is an automorphic form for ro(2N) of weight k = v + r/2 and multiplier 19 given by (10.39), i.e. (10.44) holds true for any T E

r0(2N). If v > 0, then O(z) is a cusp form. Proof. We only need to show that for every cusp a of ro(2N) the function is holomorphic at infinity and vanishes as Im z -+ OIQa (z) = jpa 0o if v > 0. It is simpler to show these properties for all the congruence theta functions O(z; h) with h E. W. The point is that the whole linear space generated by these functions is preserved under the translation T = (1 1)

\1

by Proposition 10.3 and under the involution S = (1 -1) by Proposition 10.4. Since T, S generate the modular group SL2(Z) and every cusp of ro(2N) is equivalent to oo with respect to SL2(Z), it suffices to show the holomorphy of each O(z; h) at oo. However this fact is obvious by the very definition of O(z; h). Moreover O(z; h) decays exponentially at oo if v > 0, as does O(z) at every cusp.

If r is even, we can do slightly better with respect to the level. Since e2 = ( 1), the multiplier system (10.39) reduces to the fixed real character (D\)

(10.45)

19(r) = XD(T) =

where D =

1),,12

JAI.

The weight is an integer k = v+r/2 -r//2 (mod 2). Observe the consistency condition 19(-1) = (-1)k. We infer

Theorem 10.9. Let A be a symmetric, positive definite integral matrix of even rank r. Let N be a positive integer such that NA-1 is also an integral matrix. Suppose both A and NA-1 have even diagonal entries. Let P be a spherical function with respect to A of even degree v. Then the associated theta function O(z) is a modular form for ro(N) of weight k = v + r/2 and a multiplier z9(T) given by the character (10.45). Moreover O(z) is a cusp

form ifv>0.

10. Theta Functions

176

Proof. The automorphy of O(z) with respect to the group I'o(N) is already established by virtue of Proposition 10.6 (the third alternative) except for

transformations r = { a ) E i'o (N) with d even. If such a Tr exists, then 2 { c, which implies 2 [ N and 2 t A. In this case one can show that

D = (-1)r/2 JAI - 1

(10.46)

(mod 4)

(exploit that r and diag A are even). Hence by the quadratic reciprocity law XD(d) is a character to modulus D' = the squarefree kernel of D, so XD(d) is periodic in d (mod N) for D*IN by (10.15). Now applying the automorphy of O(z) for T,

-(a d) (1 1)-(c c-4-d)

at z' = z - 1, we verify the automorphy for T at z (we have rz = r'z' and XD(c + d) = XD(d)). Finally, the holomorphy and vanishing at cusps are verified as in the proof of Theorem 10.8.

One should ask if the conditions of Theorem 10.9 can be satisfied with

N = 1, so that the theta function O(z) is a modular form for SL2(Z). In that case we must have i-"(-i)k = 1 by virtue of (10.13), which implies that A has rank r - 0 (mod 8). That such matrices with diag A = diag A-' = 0 (mod 2) exist was shown by H. Minkowski. As an example for r = 8 we have the Jacobi type matrix A (i.e. a tridiagonal matrix) with

diag A = 12,2,4,4,20,12,4,2) diag A = diag A = 11, 1, 3,5,3, 1, 1].

10.5. The standard theta function The simplest example of a theta function is given by the series 8(z) = 1: e(n2z)

(10.47)

nEz

which was first studied thoroughly by Jacobi. Besides automorphic transformations, Jacobi established the infinite product representation 8(z) _ fl(1 - e(nz))(1 -- e((n + 1/2)z))2. I

10.5. The standard theta function

177

Therefore 0(z) does not vanish in H. There is a close connection between 0(z) and the Dedekind e

(24)

e(nz)),

namely 0(2z - 1) = 772(x)/77(2x).

Both are automorphic forms of weight 1/2, but they have slightly different multiplier systems (they coincide on the group I'(24)). In Section 2.8 we have described the multiplier systems in terms of the Dedekind sums and the Jacobi symbol. Now we can deduce the automorphy transformation (2.73) from (10.38). Actually the method works for theta functions twisted by Dirichlet characters. Denote (10.49)

0x(z) =

>00 X(n)n"c(nlz) -00

where v = 0, 1 (there are no spherical functions in one variable of degree v > 1) and X is a character to modulus N such that (10.50)

otherwise 0.,(z) vanishes identically.

Theorem 10.10. For any -r = (a d) E I'o(4N2) we have (10.51)

0,((rz) = X(d)Ed (a) (cz + d)"+1/20, (z).

Moreover, if X is a primitive character of conductor N, then (10.52)

0,(-1/2Nz) =

i-"E(X)(-iz)v11/20, (z/2N)

where E(X) is the normalized Gauss sung (10.53)

E(X) =

N_112

X(h)e

(N)

h (niod N)

Remark. Observe that the involution (1 -1) switches X to,i. If

is real,

then f(z) = 0.(2Nz) is an eigenfunction of the slash operator I (i "'1 ).

10. Theta. Functions

178

Proof. We split into residue classes n = It (mod N) to obtain

0.(z) = E X(h)e(2Nz;h)

(10.54)

h (mod N)

where 6(z; h) is the congruent theta function associated with the matrix

A = N and the spherical function P(x) = x". For r = (a d) we have 2Nr(z) = r'(2Nz), where r' _

(ca

2Nb).

If r E Fo(4N2), then r' satisfies the first alternative of Proposition 10.6, so (10.38) is applicable, giving 2N

e(2Nrz; h) - ©(r'(2Nz); h) - cd ( ) (cz + d)"+1/2e(2Nz; ah). Multiply this relation by X(h) and sum over It (mod N), getting (10.51) after changing h to A. To prove (10.53) we write (10.54) for -1/2Nz and apply (10.21) for each

h (mod N), getting

9Y(-1/2Nz) = i-"N-1/2(-iz)k

X(h)e ($) 6(z; l). h,1 (mod N)

Here the sum over It (mod N) is a Gauss sum which for primitive characters is equal to X(l)e(X)N1/2. Inserting this, we arrive at (10.52) by an appeal to (10.54) again. For the trivial character we have 9x(z) = 9(z), and Theorem 10.10 yields 2.73. Moreover, (10.52) becomes (10.55)

9(-1/2z) = (-iz)1/29(x/2)

which agrees with (1.9) for x = 0 and 29 = -iz. For the real character x(n) _ (n), which is even and primitive of conductor N = 12, we have the following formula, due to Hecke: (10.56)

28. (z)

"0

-n

e(n2z) = i(24z).

In this case v = 0, E(X) = 1 and (10.52) together with (10.56) yield (10.57)

1)(-1/z) = (-iz)1/271(z).

Chapter 11

Representations by Quadratic Forms 11.1. Introduction Let Q(x) be a positive definite quadratic form in r >, 2 variables. In Siegel's notation Q(x) = 2A(x], where A = (ai?) is a symmetric, positive definite matrix of rank r. Hence

Q(x) =

ai,jxixj +

aiixi

i<.j

Suppose A has integral entries which are even on the diagonal, i.e.

aii - 0 (mod 2). Then Q(x) has integral coefficients. A central problem in the arithmetic theory of quadratic forms is solving the equation (11.1)

Q(m) = n

with m E Z' for a given positive integer n. In two variables this was studied by Fermat, Lagrange and Gauss. The theory of binary quadratic forms is special; it corresponds to the arithmetic of ideals in an imaginary quadratic number field. General cases began to be considered systematically in the late nineteenth century, and the theory matured in the hands of Minkowski, Hasse, Siegel, to mention a few original contributors. Minkowski created the geometry of numbers to answer the question of equivalence of quadratic forms. His methods of reduction of forms yield practical algorithms for the 179

11. i epresentatious by Quadratic Foals

180

determination of all solutions. Hasse's work is oriented towards the algebraic aspects. In particular, his local-global principle produces intrinsic conditions for the existence of solutions to (11.1). Siegel developed analytic methods (first brought to the theory by Dirichlet) to find, among many other things, a formula for the number of representations. He is also credited with the extensive use and development of automorphic forms in this connection, such

as theta functions (cf. [Sie]). Other analytic tools, in particular the circle method of Hardy-Ramanujan, considerably developed by Kloosterman, turned out to be alternatives to the theta function methods of Siegel. After the full number of representations is well estimated one next asks a deeper question about the equidistribution of representations. Most recent ideas for addressing such problems came from ergodic theory (due to Linnik) and from exponential sums methods. In this chapter we shall present in detail a few result-, of analytic type. There are numerous surveys of the subject written by A. Malyshev, who was a champion in the classical theory of quadratic forms; cf. [Mal].

11.2. Siegel's mass formula Here we give a brief review of fundamental concepts and results in the theory of positive definite, integral quadratic forms. Occasionally we abuse notation by referring to the matrix representing a form rather than to the form itself.

The determinant of A is also called the determinant of Q(x) = 2A[x]; det Q = det A = JAS > 0. Not every positive integer is a determinant, since

JAI _ (-1),/2 (mod 4) if r is even and IAA - 0 (mod 2) if r is odd. We define the discriminant D = D(Q) E Z by D = (-1)r/2 JAI =_ 0, 1 (mod 4)

if r is even

D = 2(-1) 2 IAI

if r is odd.

Two quadratic forms Q1, Q2 (it is assumed that all forms under consideration have the same rank r >, 2) are equivalent if one can be obtained from the other by an invertible linear change of variables, i.e. if (11.2)

Al ='UA2U for some U E MM(Z).

Since (11.2) also holds for A1, A2 interchanged (with possibly different U in M,.(Z)), it is clear that both A1, A2 have the same determinant, and so JUG = ±1; in other words, U is a unimodular transformation. The equivalent forms form a class. The determinant is a class invariant. In general, for a given determinant the number of classes can be infinite, except for binary forms.

11.2. Siegel's mass formula

181

A quadratic form is equivalent to itself in a number of ways. Denote by (11.3)

O(A) = {U E Mr(Z) : LUAU = Al

the group of automorphs of A (this is also called the isotropy group of A with respect to the action of Mr(Z) on A given above). The group O(A) is always finite and is the same for equivalent forms; denote its order by

o(A) = #O(A).

(11.4)

For binary forms, o(A) depends only on the determinant IAl, but if r > 2 then o(A) varies from class to class. Obviously the representation number (11.5)

r(n,Q)_#{mEZ' :Q(m)=n}

depends only on the equivalence class of Q. Finding an exact and relatively simple formula for r(n, Q) is one objective; this problem has been a driving force throughout the history of quadratic forms. Only in very special cases does a satisfactory formula exist. For instance, if Q(x) = xi + x2 (determinant 4) we have

r(n, Q) = 4 > X(d)

(11.6)

din

where

(d) -

(-4)

(d) =

1

if d = 1 (mod 4)

-1

if d

-1 (mod 4) is the character of the field Q(i) (this was known in a different setting by Fermat). Dirichlet generalized (11.6) to primitive quadratic forms of any determinant (11.7)

Q(x) = axi + bxl x2 + cx2,

(a, b, c) = 1.

Put D = - det Q = b2 - 4ac < 0 (the discriminant of Q), and (11.8)

XD(d) = (s).

Then for n > 0, (n., D) = 1 the character sum (11.9)

r(n.; D) = WD E \D(d) din

11. Representations by Quadratic Forms

182

gives the number of all representations of n by representatives of forms of all classes of discriminant D. Here WD stands for the number of automorphs:

WD=

6 ifD=-3 4 ifD=-4 2 ifD<-4.

The Dirichlet formula solves the problem for discriminants D for which the class number h(D) is one. However there are only finitely many D with h(D) = 1, as can be seen from another formula of Dirichlet: h(D) = WD Since

DI._e I

2

L(1, XD)-

< L(1, XD) « ID1e (the lower bound is ineffective, due to

Siegel), it follows that (11.12)

IDI2-f << h(D) << IDI2- e.

Actually the Dirichlet formula (11.9) requires fewer classes than h(D), since certain forms do not represent n at all. If n > 0, (n, D) = 1, is represented by a primitive form (11.7) (assume without loss of generality that (a, 2D) = 1), then n must comply with various congruence conditions, for example Xp(n) = Xp(a)

for every pID, p odd.

Therefore the admissible classes must have the same values of Xp(a) for all pID, p # 2. For p = 2 one needs slightly different characters, namely By means of these characters Gauss was able X4 (n) = (-) n or x8 (n) _ (2). 'n

to classify equivalent classes into genera. For forms of arbitrary rank r >, 2 the Gauss theory of genera was beautifully generalized by Hasse. In modern fashion a genus of forms is created by p-adic equivalence or rational equivalence. We have several equivalent definitions. Two forms Qt, Q2 are in the same genus if they are equivalent over all p-adic rings Zp and over the real numbers (the latter assertion is redundant since both Ql, Q2 are positive definite). Actually it suffices to establish the equivalence over Zp for all p dividing 2IAdIIA2I Another characterization asserts that Qt, Q2 are in the same genus if they are rationally equivalent by matrices U having denominators coprime with any prescribed integer q; actually it suffices for q = 2IA1IIA2I. A third, quite appealing characterization goes as follows: two forms Qt, Q2 of the same determinants IAl I = IA21 are in the same germs if they are equivalent over Z to forms congruent modulo 81A1IIA2I.

11.2. Siegel's mass formula

183

The determinant is a genus invariant. The number of classes in a genus of a fixed determinant is finite. This number can be larger than one. As an example take Q1(xl,x2) = 5xi + 11x2 and Q2(xl,x2) = xi +55x2; both forms are in the same genus but not in the same class (the second form does not represent n = 5). Except when r = 2, 3, the genus is no longer characterized by the residue classes of a number n being represented. We return for a moment to the primitive binary quadratic forms. Suppose D is a fundamental discriminant. The equivalence classes form a finite abelian group (the composition of classes corresponds to the multiplication of ideals in the ring of integers of Q(./D)): its identity element is the class of the form

X - 4x2

if D = 0 (mod 4),

X1+X1X2-D41x2 if D =- 1 (mod 4). The genus containing the identity class is called the principal genus. It was discovered by Gauss that the principal genus consists of squares of classes. The number of genera is equal to 2t-1, where t is the number of distinct prime factors of D. Hence the number g(D) of classes in every genus satisfies (11.13)

2"(D)-lg(D) = h(D).

Since 2W(D) << DIE, the estimate (11.12) for the class number h(D) remains valid for g(D). Therefore if IDI is large, every genus contains several classes, and we have strict inequality for the representation numbers

r(n, Q) < r(n, gen Q) = r(n; D). For certain n's one may establish estimates in the opposite direction. Indeed, using the structure of class group as described by Gauss, one may show that if for any class there exists a divisor of n represented by a form of that class, then n is represented by any class of forms in its genus (we tacitly assume it is represented by some form of discriminant D). In fact for such n's we have

r(n, Q) >> r(n, gen Q). Moreover, as the number of divisors of it in each class grows, then

r'(n, Q) ^' g(D)-lr(n, gen Q) For such "good" n's the Dirichlet formula (11.9) at least solves the problem in the asymptotic sense for r(n, Q).

11. Representations by Qtmdratic Forms

184

C.L. Siegel found a wonderful generalization of the Dirichlet formulas (11.9), (11.11) for forms of any rank r >, 2. Since the number of automorphs varies over classes, even within a given genus, it is necessary to weight the representation numbers r(n, Q) with appropriate coefficients w(Q) = o(Q)-'m(gen Q)-1

m(gen Q) = E o(Qj)-' Q, Egen Q

is called the genus mass. Observe that the total mass is

E w(Q;) = 1.

(11.16)

Q:Egen Q

Accordingly the averaged number of representations of n by forms of a given genus is defined as the weighted sum (11.17)

r(n, gen Q) _ E w(Q:)r(n, Q:) QiEgen Q

Siegel established the following formula: (11.18)

r(n, gen Q) = e,. fJ bp(n, Q) 6,. (n, Q) p

where e,. = 1/2 or 1, depending on whether to r = 2 or r > 2, and bp(n, Q) represents the density of solutions Q(x) = n in 7Lp. To define bp(n, Q) precisely, consider Q(x) as a map from 7Lp to 7Lp and take the limit

bp(n, Q) = lim

U-+n

vol Q-' (U) vol U

where U ranges over p-adic neighborhoods of n and the volumes are taken with respect to the Haar measures on 7L7, and Z. respectively. One can show

that the above ratio stabilizes when U falls into a sufficiently small ball centered at n. In other words, we have (11.19)

bp(n, Q) =

p^'1'-1)#{m

E zr : Q(m) __ n

(mod pn)}

for sufficiently large a. More precisely, (11.19) holds for any a > 2 ordp8lAI.

For p = oo the real density bo,, (n, Q) is given by integrals with respect to Lebesgue measure on R' and R, and it is (11.20)

6,,-(n, Q) = (2ir)1r (2) -1

n2-1IAI-1/2

11.3. Representations by Eiseustein series and cusp forms

1.85

(recognize here the volume of the unit ball). Clearly the local densities bp(n, Q) are genus invariants. The product (11.18) converges absolutely if r > 4. For r = 2,3 the convergence is a subtle matter. We shall arrive at the Siegel product (11.18) for r > 4 from another direction when analyzing the singular series, which arises naturally in the circle method. But we do not prove the exact formula (11.18) in this way, only sharp approximations, however for each individual r(n,Q2)(see Theorem 11.2). The p-adic local densities bp(n, Q) will be expressed by Gauss sums.

11.3. Representations by Eisenstein series and cusp forms The representation numbers r(n,Q) are the Fmirier eoefficients of the theta function {11.21)

e(Q(m)z) _ >2 r(n, Q)e(nz).

®(z, Q) _

n=O

mEZr

On the other hand, we know by Theorem 10.8 that e(z, Q) is an automorphic

form for ro(2N) of weight k = r/2 and multiplier (11.22)

(LA) (-Fd

(2c))"

(a

b

E ro(2N).

Here N is a positive integer such that NA-1 is an integral matrix and might not be the minimal level. The space Mk(ro(2N), t9) of all such automorphic forms has finite dimension, and if we can write ®(z, Q) as a linear combination of some basic forms in Mk(ro(2N), t9), then by comparing the Fourier coefficients we obtain an exact formula for r(n, Q). However the question whether the resulting expression for r(n, Q) is appealing depends on our familiarity with these basic automorphic forms which make the theta function. In the next paragraph we expand 6(z, Q) into basic forms. From now on suppose r > 4, so k = r/2 > 2 and the space M,;(ro(2N), t9) is spanned by the Eisenstein series (one for each singular cusp with respect. to t9) and cusp forms. Hence (11.23)

e(z, Q) = E(z, Q) + F(z, Q)

where (11.24)

E(z, Q) _

(pn(Q)E.(z) n Singular

11. Representations by Quadratic Forms

186

is the unique linear combination of the Eisenstein series and F(z, Q) is a cusp form. To determine the coefficients cpa(Q), recall the Fourier expansions (3.15): 00

(11.25)

En1cb (z) = bab + > 77ab (n)e(nz)

where bb is the diagonal symbol. Hence V. (Q) is just the value of E(z, Q) at the cusp a; more precisely, (11.26)

coa(Q) _ li

nelna(z,Q)

The function E(z, Q) is called the Eisenstein series associated with the qua-

dratic forn Q as well as with the theta function e(z, Q). How to find the values cpa(Q) in practice remains an open question. For the standard theta function e(z) associated with the quadratic form (11.27)

Q(x) = xi + ... + X2

one uses the group ro(4), which has three inequivalent cusps at a = oo, 0, 1/2, and one finds that cp °(Q) = 1, WO(Q) = i-k and cpl/2(Q) = 0, so that (11.28)

E(z, Q) = E00(z) + i-kEo(z).

If Q(x) is the sum of s 3 2 squares, we denote rs(n) = r(n, Q). This form has received much attention, and a lot is known about the representation numbers rs(n). Here are samples of results, gathered in two groups depending on the parity of s: If s = 2, the Dirichlet formula (11.9) yields r2(n) = 4 E X4 (n) din

where X4(d) = 1, -1, 0, depending on whether d - 1, -1 and 0 or 2 modulo 4. Hence r2(n) # 0 if and only if every prime p - -1 (mod 4) appears in n with even exponent. If s = 4, it was already known to Lagrange that every positive integer can be represented as the sum of four squares. Jacobi gave the formula r4(n) = 8(2 + (-1)n) > d. dIn,2ld

11.3. Representations by Eisenstein series and cusp forms

187

If s = 6, we have

r6(n) = 16X4 (d) d2 -4E X4(d)d2 din

din

If s = 8, we have r8(n) =

161:(-1)n-dd3

din

In the above cases rs(n) is exactly the Fourier coefficient of the corresponding Eisenstein series, but if s > 10 the cuspidal part is non-trivial. For s = 10 we have

rio(n) =

64

4

(d)d4 +

EdinX4 (d) d+ EdinX4

E xa

5

Nz=n

where z = a + bi runs over all Gaussian integers with norm n. For s = 12 we have

r12(n) = 8(_1)n-1 E(-1)d+n/dd5+4 din

r x4

Nlzz=n

where z = a + bi + cj + dk runs over all quaternions with norm n. The above formulas are due to Glaisher (1907). For an odd number of squares the formulas are more subtle. It was already known to Gauss that a positive integer n is representable as a sum

of three squares if and only if n # 4k(81 - 1). Moreover Gauss expressed the number r3 (n) of primitive representations (i.e. without common divisor different from 1) in terms of the class number h(-n) of primitive positive binary quadratic forms of the determinant n, namely (11.29)

r (n) = 3

( 12h(-n) if n = 1, 2 (mod 4) 8h(-n) if n = 3 (mod 8).

1

In the other cases r3(n) = 0. Note that by Siegel's theorem (11.30)

n^-E

« h(-n) < 7112"';.

There are known explicit formulas for rs(n) for a few odd s >, 5. For example, it was shown recently by G.A. Lomadze that for nine squares

ra(n) = p9(n) +

7

(X1 xi+X +:c3=3n

X31 a

3

11. Representations by Quadratic Forms

188

where p9(n) is the Fourier coefficient of the corresponding Eisenstein series

(he computed p9(n) explicitly, but the expression is rather long to write here; see [Lom]).

For a general quadratic form Q the problem of finding the associated Eisenstein series E(z, Q) is not yet solved adequately. However, whenever E(z, Q) is known its Fourier coefficients, say p(n, Q), are relatively simple and usually give a very good approximation to r(n, Q), especially if the number of variables is large. An interesting case is for the quadratic forms of level one, i.e. Q(.s) _ 'A[x] with diag A-' =_ 0 (mod 2).

diag A

(11.31)

For such a form N = 1, ;A; 1, A is equivaleiA Lu A"-'-, and the number of variables is r - 0 (mod 8); see the comments and an example after the proof of Theorem 10.9. By Theorem 10.9 the corresponding theta function is a

modular form for the modular group r0(1) = SL2(Z) of weight k = = 0 (mod 4). Therefore

e(z, Q) _ E0"r(n, Q)e(nz) = Ek(z) + F(z, Q) 0

where Ek(z) is the unique Eisenstein series whose Fourier expansion is (see (1.49)) ti

Ek(x) = 1 +

00 Y:ak-l(n)e(nz),

(( )r(k)

and

F(z, Q) _ >2 00 a(n, Q)e(nz) is a cusp form. Comparing the Fourier coefficients, we obtain k

(11.32)

r(n,Q) =

S(k)r(k)ak-t(n) +a(n,Q).

Here a(n, Q) is regarded as an error term; we have the bound (11.33)

a(n, Q) < nkz l r(n)

by Ramanujan's conjecture (Deligne's theorem). Note that the elementary bound O(nk/2) is sufficient. For k = z= 4,8 there are no cusp forms with respect to the modular group; therefore a(n, Q) = 0, and (11.32) becomes (11.34) (11.35)

r(n,Q) = 240a3 (n) if r = 8 r(n, Q) = 480x7 (n.) if r = 16.

11.3. Representations by Eisenstein series and cusp forms

189

As an example of a quadratic form of level one we have 8

Q(x) _

2

(Exi) 1

8

+ E x' - 2x1x2 -- 2x2x8. 1

Siegel has found two distinct quadratic forms in r = 24 variables of level

N = 1, say Q1 96 Q2. Thus 6(z, Q1) - 6(z, Q2) = c0(z), where c # 0 is a constant; hence the Ramanujan function can be expressed by representation numbers for the two quadratic forms, cT(n) = r(n.Q1) -- r(n, Q2).

We return to general Q. Although the Eisenstein part E(z, Q) is not satisfactorily understood, it is a relatively minor problem compared to that of the cuspidal part F(z, Q). Considerable research has been conducted on classes of quadratic forms for which the cuspidal part of the corresponding theta function can be determined explicitly by a special kind of cusp form. In particular one considers the following categories of forms: - Q is of Eisenstein type if the theta series coincides with the Eisenstein series; i.e. if the cuspidal part vanishes,

F(z, Q) - 0. - Q is of Liouville type if the cuspidal part is a theta function for a quadratic form in two variables twisted by a spherical function, i.e.

F(z, Q) = 9(z, B, P) where rank B = 2 and P is a spherical function associated with B of degree necessarily k - 1. We also allow linear combinations of such theta functions. - Q is of Mordell type if

F(z, Q) = 9(z, B, P) where rank B < r and necessarily z rank B + deg P = k. We also allow linear combinations of such theta functions.

Q is of Weil type if F(z, Q) is a cusp form associated with an elliptic curve over the rationals of conductor dividing the level. We also allow linear combinations of such cusp forms. Since the cusp form associated with an elliptic curve has weight k = 2, these quadratic forms are in r = 4 variables. For example, the quadratic form

Q(x) = xi2 + 4x22 + 4x.3 + 16x2

11. Representations by Quadratic Forms

190

is of Weil type. For p = 1 (mod 4) we have (11.36)

r(p, Q) = p + 1 -

-

/u3

- u\

p u (mod p)

In order to generalize the notion of Weil type quadratic forms to any even number of variables r = 2k we must create in some way a cusp form of weight k out of a cusp form of weight 2 associated with an elliptic curve E/Q. Suppose E/Q has complex multiplication, and the L-function for E/Q is given by (up to local factors of bad reduction) 00

(11.37)

LE(s) = fl(1 - app-s)-1(1 - app

E AE(n)n--'.

P

1

Define. 00

[[(1-ak-lp-s)-1(1-s(p)an-lp-s)-1 `

(11.38) L(k) (s) _ p

E,\Fk)( 1

where e(p) = 1 except for k odd and A(p) = ap + ap = 0, in which case e(p) = -1 (note that e(p) is the character of the imaginary quadratic field which gives the ring of endomorphisms of E). For bad primes (those which divide the discriminant) the local factors can be chosen so that LEk) (s) is an entire function whose Mellin transform is a cusp form of some certain level and character, 00

(11.39)

fLk)(z) = j>(k)(n)e(nz) E sk(ro(q),x).

In view of this result L. A. Kogan says that (see (Mall) - Q is of Weil-Eichler type if F(z, Q) is a linear combination of cusp forms of type (11.39) of level q dividing N. For example, the quadratic form

Q(x) = xl +x? +x3+x22+4x2 +4x2 is of Weil-Eichler type. For p = 1 (mod 4) we have (11.40)

(u3_u))

r(p,Q)=4(p-1)2+4

2

u (mod p)

The circle method was invented by Hardy and R.amanujan (1918) for the purpose of estimating the partition function p(n) (see Section 2.8). Shortly

11.4. The circle method after Kloostermani

191

afterwards Kloosterman (1926) made an essential refinement to estimate the number r(n, Q) of representations of n by a quadratic form Q(x) in four variables. He considered only diagonal forms (11.41)

Q(x) = a,x1 + a2x2 + a3x3 2 + a4x2,

but, of course, the method works for any positive definite quadric (see [Klo)). We begin by establishing a Fourier-Kloosterman type decomposition of

the distribution b(n.) =

(11.42)

ifn=0 ifn#0.

1

{ 0

Proposition 11.1. Let C be a real number > 1. We have 1/cd

/

el

b(n)=2R.e

(11.43)

c<,C
\

fo

n-xdx. C

Remark. For n = 0 our formula (11.43) gives the cute identity (try to prove this directly)

EE 2(cd)-1

(11.44)

= 1.

c<,C
(c,d)=1

For the proof of (11.43) we may assume that C is a positive integer. Let f : R -' C be a periodic function of period 1. We shall evaluate the mean-value

AM =

f 0

1

f (x)dx

by dissecting the interval with Farey's points of order C, a

a, 0
Let

< SC <

be adjacent points; they are determined by the conditions C < c + c' '< C,

ac

C < c + c' <, C,

ac"

1 (mod c) -1 (mod c).

Between the adjacent points there are mediants °°(a sim<°< pleminded rule for addition of rationals); they define the Farey segments of different length 71_°x,

M

ra) _ a'+a a+a" _

a

\c/ - (c'+a' c+c" - (c

1

a

1

c(c+c')' c + c+c")j

11. Representations by Quadratic Forms

192

The Farey segments make a disjoint partition of the unit interval

UU M

C

1

+T C+

o
n)c

(a,c)=1

Hence by the periodicity of f we get

l'(f)

EY, 0a
f ( + x) dx c

o

(a,c)=1

:i

r1/cd

c
\

//

if

J

/

i C

d+xJdx.

(c,d)=1

Suppose that f has the symmetry f (-x) = Y (x); then we have (11.45)

z(f) = 2ReEE

c

c
Changing the order of summation and integration, we write (11.46)

57

µ(f) = 2Re

c
r1/CC

J0

Kf(x,c)dx

where K f (x, c) is a kind of incomplete Kloosterman sum associated with f, (11.47)

Kf(x,c) _

f (x - d) C < d
.

Applying (11.45) for f (x) = e(nx), we obtain (11.43). Now let Q be a positive definite, integral quadratic form in r >, 4 variables and let n be a positive integer. In order to pick up the Fourier coefficient r(n,Q) of the theta function ®(z) = 9(z, Q) (see (11.21)) we apply (11.46) to f (x) = 9(z)e(-nz), where z = x+iy with any y > 0 to be chosen later. We obtain (11.48)

r(n, Q) = 2Re.

rC

f

1/cC

T(e, n; x)e(-n.z)dr

11.4. The circle method after Kloosterman

193

where

e(n-JO c/

T(c,n;x)=

(11.49)

C
C

\\\

(c,d)=1,cdx<1

Next we split the theta function into congruence theta functions for residue classes h (mod c):

O (z - dlJ

= F e (_Q(h))

e(Q(nz)z). »i=h (mod c)

h (mod c)

Now we transform each congruence theta function by the Jacobi inversion formula (10.12) with x = h/c, getting r

e(Q(m)z) = IAI-1/2c

/k

`

\z/

m=1t (mod c)

QC2x)

mEZr

- the J

where Q' (x) = A-1 Ixj is the adjoint quadratic form. Combining these two 2 equations, we obtain

0 (z -

IAI1/2c(i)k

=

G

r E

(

(Q + m)J e(-Q* (m)/c2z)

where G is the Gauss sum defined by (11.50)

GI

(Q + m)/) _

e

(-

h (mod c)

thm)\

(Q(h) +

)

C

Inserting this into (11.49), we obtain .

(11.51)

/ T,n(c,n;x)e(-Q*(n1)IC2z)

T(c,n;x) = IAI-1/2c-r ()'

TILE

Zr

where (11.52)

T,,,.(c, n; x) =

e

C
(n) C

-(Q + m))

.

C

Observe that n.; x) is constant for x < 1/c(c+C); we write T,,, (c, n). This is a complete sum in r + 1 variables to modulus c, namely (11.53)

T,,,(c,n) _

*

d (mod c)

/

e (n

d-1

\c

/d

G \\(-(Q+711)1 c

11. Representations by Quadratic Fornis

196

11.5. The singular series The infinite series (11.58) converges absolutely by virtue of (11.54); nevertheless in the literature it is called a singular series. We shall express a(n, Q) in terms of the local densities bp(n, Q). To this end notice that the sum over reduced classes d (mod c) in (11.59) is just the Ramanujan sum

e(d) _

(11.60) d (mod c)

(f) q gl(b,c)

q

Hence

li (9) q#{h (mod c) : Q(h) =_ n (mod q)},

qc(n, Q) qlc

c-' gc(n, Q) = > µ (q-) ql-r#{m (mod q) : Q(m)

n (mod q)}

9lc

and

E c rgc(n, Q) = sl-r#{m (mod s) : Q(m) = n (mod s)}. cps

If every prime exponent of s is larger than that of 641A12, then the result does not depend on these exponents; it becomes the product of local densities (11.20):

c rgc(n,Q) = flsp(n,Q)

(11.61)

pas

cps

This shows that the complete product of local densities converges absolutely to the singular series

a(n,Q) = f op(n,Q)

(11.62)

P

Let p(n, Q) denote the main terns for r(n, Q) obtained by the circle method (the singular series with infinite valuation included), i.e. (11.63)

p(n, Q) =

(21r)''r(k)-ins'-1

JAI - 1120,(n, Q)

(see Theorem 11.2). By (11.62), (11.20) and Siegel's formula (11.18) we find

that (11.64)

p(n, Q) = r(n, gen Q).

11.5. The singular series

197

Moreover Siegel proved that the p(n, Q) are the Fourier coefficients of the Eisenstein series E(z, Q) for Q (see (11.23)); therefore (11.65)

E(z, Q) _

p(n, Q)e(nz) 0 00

_ E r(n, gen Q)e(nz) 0

1:

w(Q2)e(z, Qs)

QiE gen Q

by (11.17). Hence we observe that E(z, Q) is genus invariant. Inserting (11.23) for each E) (z, Q;) in (11.65), we find that the Eisenstein series cancel because the total mass is 1 (see (11.16)), and we conclude that the weighted mean-value of the cusp forms F(z, Qt) over the genus classes vanishes:

E w(Qi)F(z,Q;) = 0.

(11.66)

QiE gen Q

Next for primes p { 2IAI we shall compute bp(n, Q) explicitly using the formula (10.35) for Gauss sums, which translates to (11.67)

C \CQ) =

(!J)

)r

(()

in the notation of (11.50), provided (c, 2IAI) = 1. Hence

gc(n,Q) _

:s G (cQ) a (-nd)

d (mod c)

r (11.68)

_ (AI) C

d (rood c)

c

e

(_). c

Further computations depend on the parity of r. First we consider r even; thus k = z is an integer > 2 and the discriminant of the quadratic form Q(x) = A [x] satisfies (11.69)

D = (-1)r121A1 =-O,1

Put

D (11.70)

9

(mod 4).

11. Representations by Quadratic Forms

194

The main contribution to the final formula (see Theorem 11.2) comes from m = 0. In this case To(c, n) is also denoted by gc(n, Q) whenever it is important to display the dependence on the quadratic form Q (cf. (11.59)). In any case we have the following estimate: Tm(c,n;x) K (c,n)ZCY+£

(11.54)

where e > 0 and the implied constant depends only on E and the quadratic form Q. For the proof of (11.54) one first completes the sum Tm(c, n; x) using a standard Fourier technique. Then one factors the complete sum into sums to moduli c1, c2 with clc2 = c, (cl, 21A1) = 1 and c2 having all prime factors in 21A1. In the sum to modulus cl one can transform the relevant Gauss sum (by completing the square) to one of the standard Gauss sums evaluated in Lemma 10.5. After inserting this evaluation one. is left with a single summation in d1 (mod cl ). Such a sum is either a Kloosterman sum if r is even or a Salie sum if r is odd. In both cases we have the same estimate (cl, n)1/2c1/2z(cl), due to Weil and Salie, respectively (see Chapter 4). For the other sum to modulus c2 one can proceed similarly, but the details are cumbersome. However one can be satisfied with an almost trivial bound for this sum, since we allow the implied constant to depend on the quadratic form Q. Multiplying the resulting estimates for complete sums to moduli cl and c2, then summing trivially over the frequencies which occurred in the process of completing the sum (11.52), we arrive at (11.54). Exercise. Following the above guideline, give a complete proof of (11.54). We shall use (11.54) to estimate T(c, n; x) except for To(c, n; x) = To(c, n)

in the range 0 < x < 1/c(c+C). Isolating this term, we deduce from (11.51)

that k

T(c,n;x) =

IAl-1/2c-r (!.)

To(c,n)

+ O((c, n)1/2C1/2+e(clzl)-k E' exp(-27ryQ*(m)/c21z12) mEZ*

where E' means that m = 0 is excluded from the summation if 0 < x < 1/c(c + C). We take C = n1/2,

(11.55)

y=7%1.

This choice implies clzly-1/2 (C-1 + Cy)y-1/2 = 2 and clzly-1/2 1/2Cy1/2 = 1/2 if 1/c(c+C) < x < 1/cC. Since Q*(m) 3 Im12/2A where A is the largest eigenvalue of A, we obtain trivially E / exp(-21ryQ*(m)/c21 z12) << (clzly-1/2)k mEZ*

11.4. The circle method after Kloosterman

195

(actually this estimate holds true with any positive exponent in place of k). Hence (11.56)

T(c, n; x) _ A

-1/2c-

(!) k To(c, n) + O((c,

n)1/2c1/2+enk/2).

Inserting this in (11.48), we get r(n, Q) = IAA-1/2 E c-TTo(c, n) C

ff 1/CC / i 1 k e(-nz)dx +

0(nk-;

+e).

I/cc

Here the integral is equal to (see (3.10))

00 (i) J-

k

e(-nz)dx =

(21r)kr(k)-ink-1

up to the error term O((cC)1-1). Summing over c < C, these error terms contribute no more than the error term already present (apply (11.54) for To (c, n)); therefore we have (27r)kr(k)-1IAI-1/2nk-1

r(n, Q) =

` c rTo(c, n) + O(nz-a+e c
Finally, extending the summation to all of c (estimate the tail of the series by using (11.54)), we obtain

Theorem 11.2. If r > 4, then for any n > 0 we have (11.57)

(27r)AF(k)-1nk-1 JAI-1/2a(n, Q)

r(n, Q) =

+ O(n2 a+e)

where u(n, Q) is given by the absolutely convergent series (11.58)

a(n, Q) _

c-rgc(n, Q)

and

(11.59)

gc(n, Q) = E* E e

(Q(h) - n))

d (mod c) h (mod c)

The implied constant depends on a and the quadratic form Q.

11. Representations by Quadratic Forms

198

We have EC = (ii), and the sum over reduced residue classes d (mod c) is

the Ramanujan sum (see (11.59)). Therefore, if (c, 2D) = 1,

9c(n, Q) = XD(C)C k E p (9

q.

gl(c,n)

Hence, if (s, 2D) = 1, EC-rgr(n,Q) =

>2XD(C)C-k

gl(c,n)

cls

cls

µ (q) q

XD(q)gl-k 7T(1

gI(n,s)

- XD(p)p k).

PI 19

In particular, if every prime exponent of s is larger than that of n, we have (11.71)

> c-rgc(n, Q) = fl(1 - XD(p)T-k) E cis

PIs

XD(q)gl-k.

qIn

Taking s = p` with v = 1 + ordp(n), we deduce by (11.61) and (11.71) that (11.72) Sp(n,Q) _ (1 - XD(p)P k)(1

-

XD(p)pl-k)-1(1

- XD(PY)p"

(1-k))

for any p f 2D. In particular this shows that (11.73)

Sp(n, Q) 0 0 if p f 2D.

Another consequence of (11.61) together with (11.71) is that (11.74)

o(n, Q) = fi 6,(n, Q)L(k, X4D)-1 E

X4D(d)dl-k

din

p12D

where L(s, X4D) denotes the Dirichlet L-function. Observe that if (11.75)

Sp(n,Q)

0

for all p dividing 21A1,

then Sp(n, Q) is bounded from below and above by positive constants depending only on Q. The non-vanishing conditions (11.75) are equivalent to the existence of m E zr such that (11.76)

Q(m) = n (mod 271AJ3).

Assuming this congruence is solvable, we infer that a(n, Q) x 11 (1 + Pin

XD(p)p1-k)

11.6. Eq uidistribu tion of integral points on ellipsoids

199

where the implied constants are absolute. This is bounded by constants if k > 2. For k = 2, i.e. r = 4, the product may vary slightly between (log log 3n) -1 and log log 3n. If r is odd, the exact computations of the local densities bp(n, Q) are more

involved. However, since r > 5, it is quite easy to estimate the series or(n, Q) by absolute constants. Indeed, by (11.68) we have trivially 1gc(n, Q)l < c2+1 if (c, 21AJ) = 1; hence

c -'I' = (1 - 2-3'2)((3/2) < 2,

E c-''gc(n, Q) < (c,21A1)=1

21c

and the lower bound is 2 - (1 - 2-3/2)((3/2) > 0. Therefore a(n,Q) x 1; provided (11.76) has integral solutions. By Theorem 11.2 we get (whatever the parity of r is)

Corollary 11.3. If r > 4, then any n sufficiently large for which the congruence (11.76) is solvable is represented by the form Q. The number of representations satisfies

r(n, Q) x nk-1,

if r > 5,

r(n,Q)xnfl(1+XD(p)p 1), if r = 4. pin

11.6. Equidistribution of integral points on ellipsoids Let Q be a positive definite quadratic form with integer coefficients. If the number of variables r is greater than three, then any large integer n satisfying a finite number of necessary congruences is represented by Q in many ways (see Corollary 11.3). Thus it makes sense to ask how these representations

are distributed on the surface of the ellipsoid Q(x) = n. Let F be a nice domain (e.g. convex with smooth boundary) on the unit ellipsoid

6= {xER'':Q(x)=1} and put (11.77)

rF(n, Q) = {m E Z' : Q(m) = n,

Vnm

E F}.

One expects that, as n tends to infinity through the integers satisfying the relevant local conditions (the solvability of (11.76)), rF(n, Q) - BFI r(n, Q)

11. Representations by Quadratic Forms

200

where BFI denotes the surface area of F. In other words, we say the integral points on Q(x) = n are equidistributed with respect to the invariant measure on E. Applying a suitable approximation, this property can be stated as (11.78)

r f(n, Q) -

(

f (x)dx ) r(n, Q) E

where (11.79)

rf(n,Q) _

f+

)

for any smooth function f : E -. C. Actually one can restrict the test functions to the spherical functions (with respect to the matrix A) from a complete orthonormal system on 6 (see Chapter 9). If f is a constant, then (11.78) becomes an identity. If f is a harmonic polynomial of degree v > 0, then n lir f (n, Q) is just the Fourier coefficient of a cusp form (see Theorem 10.8) 00

(11.80)

e1(z) _ E n2 r f(n, Q)e(nz).

This has weight k = v + 2, and a multiplier z9 on the group I'o(2N) (z9 becomes trivial on a congruence subgroup). Proposition 5.5 yields (11.81)

rf(n,Q) << r41+e

Hence we obtain

Proposition 11.4. Let r > 4. If f is a linear combination of spherical functions, then (11.82)

rf(n,Q) _

(ff(x)dx)r(nQ)+o(n5).

Here r(n, Q) can be replaced by the complete singular series p(n, Q) (see (11.63), (11.62), (11.57)).

If r >, 4, then the error term in (11.82) has lower order of magnitude compared with r(n, Q) (see Corollary 11.3), so (11.82) implies (11.78), proving

11.6. Equidistribution of integral points on ellipsoids

201

Theorem 11.5. If r > 4, then the vectors m E Z" with Q(m) = n are asymptotically equidistributed as n - oo over integers satisfying the congruence (11.76).

For ternary forms (r = 3) Proposition 11.4 is still valid, but the result is (just) too weak to imply the equidistribution. First we need estimates for the total number of representations Q(m) = n. The circle method does not work directly as for Corollary 11.3; however there are other ways to estimate r(n, Q). If n is squarefree with (n, 21AI) = 1, one can show that (11.83)

ni/2-E << r(n, Q) << n1/2+e

provided n is represented by Q modulo 271A13, i.e. (11.76) is solvable. The upper bound is obvious, and the lower bound can be esta.hlishedl by means of the theory of binary quadratic forms discussed in Section 11.2 using some

ideas of Linnik from ergodic theory. Since the details are formidable, we leave (11.83) as a hypothesis. Next we need a bound for the number r f(n, Q) of representations weighted by harmonic polynomials f $ constant. In view

of (11.83) the standard estimate (11.81) for r = 3 is just insufficient, but Theorem 5.6 is applicable (for a cusp form a f(z) of weight k = v + 2 % 2)+ giving (11.84)

r f(n, Q) << n2 222+e.

By (11.83) and (11.84), applying the same principles of the equidistribution theory as for r >, 4, we obtain

Theorem 11.6. Let Q be a ternary, positive definite quadratic form with integral coefficients. Then the vectors m E 7L3 on the ellipsoid Q(m) = n are asymptotically equidistributed as n -' oo over squarefree numbers satisfying (11.83).

This result is due to W. Duke [Dul] (actually he makes slightly weaker assumptions about n).

Chapter 12

Automorphic Forms Associated with Number Fields

In this chapter we shall illustrate with numerous examples the unproven principle that the L-functions which are encountered in algebraic number theory are derived from automorphic forms of various kinds. To accomplish this we make use of the Hecke-Weil converse theorems, which transfer the problem to knowledge of a large collection of adequate functional equations. Another way of establishing the automorphy is by expressing the suspected

form in terms of theta functions. The latter method is constructive, but it is less general.

12.1. Automorphic forms attached to Dirichlet L-functions We begin with the Riemann zeta-function 00

En-s = fl(1

((S) = 1

-

p-a)-1

p

This alone does not correspond to an automorphic form, because it has the wrong gamma factor in its functional equation: (12.1)

7r

21P (2) ((S) = 7f-'2r (1

2

s) S(1- s). 203

12. Autotnoiphic Forms Associated with Number Fields

204

Moreover, the Euler product for S(s) has local factors of degree 1 at each prime while the automorphic L-functions have local factors of degree 2 at almost all places. These observations suggest that one should consider the product of two zeta-functions. Put L(s) = t;(s)((s - k + 1)

(12.2)

where k is an even integer > 2. Applying (12.1) for S(s) and ((s - k + 1), we find that the complete L-function A(s) = (21r)-9r(s)L(s)

(12.3)

satisfies

A(s) = ikA(k - s).

(12.4)

Here to check that the gamma factors agree one needs the recurrence formula

r(s + 1) = sr(s)

(12.5)

and the duplication formula

r G) r

(12.6)

7r1/221-91'(s).

(s 2

By Hecke's Theorem 7.7 there exists a modular form of weight k for the modular group whose Fourier coefficients are the Dirichlet coefficients of L(s). Actually it is easy to find that correspondence. Namely, we have 00

L(s) = E Ork-1(n)n-s; 1

therefore L(s) is the L-function associated with the Eisenstein series (see (1.49)) x

00

Ek(z) = 1 + ((k)r(k)

ak.-t (n)e(nz).

Next we generalize the above construction to the Dirichlet L-functions 00

L(s, X) _

X(n)n-9 = fl(1-

X(p)p-s)

n

For a primitive character X (mod q) with X(-1) = (-1)", u = 0, 1, we have the functional equation g

(12.7)

(! 2rl s 2ulL(s>X)=e l7r/

1-s 2

r(1-2+u\L(1-s,

12.1. Automorphic forms attached to Dirichlet L-functions

205

where eY = i-uT(X)q-'/2 and r(X) is the Gauss sum. Let X1 (mod ql) and X2 (mod q2) be primitive characters of conductors ql,q2 respectively, not both trivial, i.e. q = qjq2 > 1. Then X = X1X2 is a character to modulus q, not necessarily primitive. Let k be a positive integer such that X(-1) _ (-1)k. We consider the product of L-functions 00 ann-s

L(s, X1)L(s - k + 1,X2) _ >2 where

1.

o, = an(X1,X2)

X1(a)X2(d)dk

ad=n

Put 00

(12.8)

f(z) = fx,,xz(z) =>ane(Tz) 0

where ao = 0 unless k = 1 and q2 = 1, in which case ao is defined by ao =

1

L(0, X)

2

Theorem 12.1. The function (12.8) is an automorphic forma of weight k for the group r0(q) and character X (mod q), i.e. (12.9)

f E M .(ro(q), X)-

Proof. Consider also the function 00

g(z) _ E bne(nz) 0

with b,, = bn(X1, X2) = wan(X2, X1), where (12.10)

w = (-1)kT(X1)T(X2)q-112(g2lg1)2

Put

2 ) r(s) E s

of (s) =

00

J A9(S) =

-) r(s)

AY

00

> 1

an-s

12. Automorphic Forms Associated with Number Fields

206

By (12.5), (12.6) and (12.7) one derives the functional equation (7.29). Furthermore one verifies the holomorphy of (7.27) and (7.28) (if ao or bo does not vanish one uses ((0) = -1/2). Therefore the assertion (A2) in Theorem 7.3 is completely satisfied, and consequently (A1) holds, i.e. 9 = fl,,,

with w = (q

-1 )

.

Next let 0 (mod r) be a primitive character of conductor prime to q. Consider

Lf(s,

00

t(n)ann

.4

= L(s,'+GX1)L(s - k + 1,'WX2)

and

L9(s, 4h) _ 00E O'(n)bnn-s = wL(s,''X2)L(s - k + 1, 0X1

Put Af(s, ali) =

2

r(s)L f(s, V))

2r B

A9(s, G) =

(s) L9(s, VG)

where N = qr2. Using (12.5), (12.6), (12.7), one derives the functional equation (7.37) with w(O) given by (7.32). To get exactly w(v)) one needs X(r)'1I1(q)7_(X1)T(X2)7_(1p)2. There(12.10) and the formula r(V)Xj)-r(?kX2) = fore the pair of functions f, g satisfies all the hypotheses of Theorem 7.8, and consequently we have (12.9). Note that (12.8) is an eigenfunction of all the Hecke operators; more precisely, Tn f = an f if n > 1; however f is not a cusp form, even if ao = 0. In fact, given X (mod q) with X(-1) = (-1)k, one can show that all the functions ff1iX2(dz) with Xl (mod ql), X2 (mod q2) (both characters are primitive such that X1X2 = X) and dqlq2lq span the whole subspace ek(ro(q), x), which is the orthogonal complement to Sk(ro(q), X) in MA(ro(q), x). We call ek (ro(q), x) the space of Eisenstein series, because if k > 2 it is indeed spanned by the Eisenstein series associated with cusps of ro(q) (the cases k = 1, 2 are somewhat delicate).

12.2. Hecke L-functions with Grossencharacters The next generation of Dirichlet L-functions emerges from characters on ideals in a number field. Let K/Q be a finite extension of rationals of degree

12.2. Hecke L-functions with Grossencharacters

207

n = [K : Q]. There are exactly n isomorphisms of K into the field of complex numbers

a:K->K"CC x-+X". Among these there are r1 real embeddings K° C R and 2r2 complex embeddings, pairwise complex conjugate, where rl + 2r2 = n. We say that K is totally real if r1 = n or totally imaginary if r1 = 0. Let 0 C K denote the ring of integers, so 0 is a free Z-module of rank n:

where {w ...... ,,nn} is an integral basis. This gives us the discriminant D = det(Tk w=w,) E Z

where Tr : K Q denotes the trace. Furthermore we introduce the different D C 0; its inverse is given by the fractional ideal

a-1={xEK:TlxOCZ}. The prime ideals p C 0 which divide the different also divide the discriminant. Prime divisors of D, and only they, are ramified in the extension K/Q. We have sign D = (-1)r2.

The group of units, say U C 0, is the direct product of a finite cyclic group

E C K of roots of unity and a free abelian group of rank r = Ti + r2 - 1, i.e. every unit can be written uniquely as the product

with C E E and n1, ... , n,. E 7G, where {E1 i ... , e,-} is a fixed system of free generators, called fundamental units of K. Having a system of fundamental units, we define the regulator of K by R = j det(log Ie"

where aj ranges over r = Ti + r2 - 1 isomorphisms no two of which are complex conjugate (R does not depend on the choice of fundamental units nor the choice of the isomorphisms, and it does not matter which a is missing

in the determinant for R).

12. Automorphic Forms Associated with Number Fields

208

Let I be the group of fractional ideals # 0,

I = { at : al, 02 C O, ala2 # 0} a2

and P C I the subgroup of principal ideals

P={(a)=aO:aEK*}. Then R = I/P is the class group, and its order h = [I : P] < oo is called the class number of K. For any integral ideal m C 0 we consider the groups

Im = {a E I : (a,m) = 1}

Pm={(a)EP:a-1 (mod m)). Here (a,m) = 1 means a = al/a2, al,a2 C 0, (ala2im) = 1; and a - 1 (mod m) means a = al/a2, al, a2 E 0, (ala2, m) = 1, al - a2 (mod m), i.e. al - a2 E m. Clearly Pm C Im, and hm = [Im : Pm] is finite. The factor group Im/Pm is called the ray class group. In what follows a ranges over a fixed system of isomorphisms having exactly one isomorphism from each pair of complex conjugates. For any a from the system we take numbers u0, v, subject to the following restrictions: uo = 0, 1 uo E Z

if a is real, if a is complex,

v, E R

such that E vQ = 0.

Let S' = {z E C : Izl = 1}. Define the homomorphism G. : K* - S' by the product [J(aC/Ia°I)°'Ia°I'vv.

.(a) = Cr

The condition E vQ = 0 is introduced for normalization; it implies that &," is trivial on the multiplicative group of positive rational numbers. However 1;. may not be trivial on the group U of units of K, so oo is not a function of principal ideals (a) E P. Let m be an integral ideal, small enough, such that the group (12.11)

Um={riEU:77-1 (modm)}

is in the kernel of Cm. Then &,o can be regarded as a function on Pm, and we call m a modulus for Coo.

12.2. Hecke L-functions with Grossencharacters

209

A group homomorphisin : I,,, -p S' is said to be a character to modulus m if it coincides with ,,, on P,,,, i.e. (12.12)

(a) = &(a) if a = (a) with a = 1

(mod m).

Of course if C is a character to modulus m, then it is also a character to any modulus n c m. Moreover, given C (mod m), there may exist a character C* (mod m*) with m c m* such that C*(a) = C(a) on I,,,. The largest ideal m* with this property (which is the greatest common factor of all such ideals) is called the conductor of C. If m* = m, then is called primitive. For any character C (mod m) there exists a unique primitive character C* (mod m*)

such that e*(a) _ C(a) if (a,m) = 1. Any Hecke character extends to C : I - C by setting

Im -> S'

(a) = 0 if (a, m) # 1.

(12.13)

These so-called Grossencharacters were introduced by E. Hecke (1917-18) (cf. [Hel]). If all uo and vQ are zero then is a class character; that is, is induced by a character on the ray class group Im/Pm. The L-function associated with a Hecke character (mod m) is defined by (12.14) o96aco

where a ranges over non-zero integral ideals and N : I -> Q* is the norm. In the case of integral ideals Na is just the number of residue classes modulo a:

No = #(O/a),

(12.15)

and it extends to fractional ideals by multiplicativity. The series (12.14) converges absolutely in Re s > 1, and it has the Euler product over prime ideals [J(1 -

(12.16)

4(P)(NP)-s)-1

P

For the trivial character we get the Dedekind zeta-function (12.17)

E(Na)-s = 11(1

(x(s) = a

-

(Np)-s)-'

P

By a method similar to that applied by Riemazm to CQ(s) = ((s) one proves the following theorem.

210

12. Automorphic Forms Associated with Number Fields

Theorem 12.2. The Junction (K (s) can be holomorphically continued over the whole s-plane except for a simple pole at s = 1 with residue (12.18)

ress=1 (K (s) = 2'''

(27r)'ZIDI-1/2hRw-1

where D is the discriminant, R is the regulator (we take R = 1 if rl+r2 = 1) and w is the number of roots of unity in K. Moreover the complete zetafunction (12.19)

AK(s) =

2-r27r -n/2IDI1/2r

(2)r` r(s)r2(K(S)

satisfies the functional equation (12.20)

AK(s) = AK(1 - s),

and s(s - 1)AK(s) is bounded in vertical strips.

Hecke generalized this result for L(s,1;) with a primitive character (mod m). To this end one needs Gauss sums. Put for a E K*.

!;m(a) =

Note that ,,, (a) is periodic of period m, i.e. £. (a) = lm(b)

if a, b E 0, a - b (mod m).

Choose an integral ideal r prime to m such that rdm is principal, say tom = (b) with b E 0, and define the Gauss sum by (12.21)

(Tb)

ins(a)e(TY). aEr/rm

One can show that W(Z;) does not depend on the choice of r and b. For primitive (12.22)

IW

I2 = Nm.

If 1 (mod ml) and 1;2 (mod m2) are primitive with (ml,m2) = 1, then (12.23)

W(41e2) = 41(m2)42(m1)W(fl)W(42)-

12.3. Automorphic forms associated with quadratic fields

Theorem 12.3 (Hecke). Let

211

(mod m) be primitive, nontrivial. Put

(12.24) (12

A(s,C) = (2r'(21r)-"IDINm)2 jir

(luol+no(s+iva)) )

where no = 1 if a is real and nQ = 2 if a is complex. The function A(s,t;) is entire and bounded in vertical strips, and it satisfies the functional equation (12.25)

A(s, t;) = w(C)A(1 - s, t;)

where (12.26)

w(l;) = i-uW(6)(Nm)-1/2,

u=

uo

In the simplest case K = Q the Hecke L-functions coincide with the classical L-functions for Dirichlet characters. A character X (mod q) extends to fractional ideals of Q by

b with a, b E N.

X(a) = X(a)X(b) if a =

This makes X a Hecke character of modulus q with u = 0, 1 chosen so that X(-1) = (-1)u. Any Hecke character of Q is obtained in the above way. Of course, the functional equation (12.25) in this case is just (12.7).

12.3. Automorphic forms associated with quadratic fields Our primary objective is to construct cusp forms from Hecke characters (mod m) of a number field K. First notice that if L(s, ) corresponds to the L-function of a cusp form, say f E Sk(ro(N), X), then (12.27)

Lf(s) = L (s

-- h 2

1,t;)

because the center points for the relevant functional equations (12.25) and (7.37) must agree. Moreover K must be a quadratic field, because the number of gamma factors in the functional equations is two. Next, examining the parameters in the arguments of the gamma factors, we infer that Z;00 must have all its imaginary exponents ivv, vanish, and in the case of a real field we also must have u = u1 + u2 = 1, i.e. either ul = 0, u2 = 1 or ul = 1, u2 = 0 (because a different parity is required by the duplication formula (12.6) to make one gamma function out of two). Therefore, if K = Q(V' ) is imaginary (so D < 0), we must have (12.28)

(a) =

a

(Tall

z ,

a E K',

212

12. Automorphic Forms Associated with Number Fields

for some integer u, and if K = Q(-/D) is real (so D > 0) we must have either too(a) =

(12.29)

IaI,

a E K',

or

a E K',

!;te(a) = la;I,

(12.30)

where a' denotes the conjugate of a over Q.

Before stating the main results we recall a few basic facts about the quadratic field K = Q('/). We assume that D is the discriminant of K. The field character is given by the Kronecker symbol XD(n) =

D) n

Note that XD(-l) = sign D = 1, -1 if K is real or imaginary, respectively. Moreover the values of XD at primes characterize the factorization into prime

ideals in K, namely p = p2

p = pp' p=p

if XD(p) = 0 (ramified prime), if XD(P) = 1 (split prime), if XD(P) = -1(inert prime).

The norm of an ideal is Na = aa', where a' denotes the conjugate (a' = a if K is imaginary). If ' (mod r) is a Dirichlet character on Z, it gives rise to a character on the fractional ideals of K by composing with the norm

(0 o N)(a) = O(Na),

a E I.

If /i is primitive of conductor r with (r, D) = 1, then 0 o N is a primitive Hecke character. We need to know the Gauss sums for XD and 0 o N.

Lemma 12.4. Let K = Q(v), where D is the discriminant. We have (12.31)

( D1/2 r(XD) = S l ilDI1/2

ifD > 0 if D < 0.

If i (mod r) is primitive with (r, D) = 1, then (12.32)

W(-+p o N) = XD(r)'tlb(I DI)r(P)2

12.3. Automocphic forms associated with quadratic fields

213

Proof (sketch). One can give a direct argument, but it is simpler to derive these results from the functional equations stated earlier. To check (12.31) we appeal to a factorization of the Dedekind (12.33)

(K (S) = ((s)L(s, XD).

Hence, comparing the functional equations (12.20), (12.1) and (12.7), we deduce (12.31). To check (12.32) we appeal to the factorization

o N) = L(s, ,)L(s,'OXD).

L(s,

(12.34)

Hence, comparing the functional equations (12.25) and (12.7), we deduce (12.32).

Now we are ready to present the main results. For clarity we state the imaginary field and the real field cases separately.

Theorem 12.5. Let K = Q(fD) be an imaginary quadratic field with discriminant D < 0 and ( (mod m) a Hecke character such that (12.35)

((a)) =

(

if a = 1 (mod m m)

jai)

where u is a non-negative integer. Then. (12.36)

f (z) _

((a) (Na) 2 e(zNa) E MR(ro(N), X)

where k = it + 1, N = IDINm and X (mod N) is the Dirichiet character given by (12.37)

X(n) = XD(n)6((n))

if n E Z.

Moreover, f is a cusp form if u > 0.

Remark. We assume that u > 0 by conjugating ( if necessary.

Proof (sketch). Note that ( (mod m) is not required to be primitive; nevertheless we only sketch the arguments for ( primitive. Consider another function (12.38)

g(z) = C

Z(a)(Na) Z e(zNa)

214

where C =

L (s - 2,

12. Automorphic Forms Associated with Number Fields i-2u-'W(1)(Nm)-1/2. By definition of f and g we have L f(s) =

and Lg(s) = CL (s - 2, '). To these we attach the factor

(vW-/21r)9r(s) to make the complete L-functions

Af(s) =

Sr(s)L ( -- 2

21r

3

Ag(s) =

2 r(s)CL

(s-

2,

and verify by (12.26) that A f (s) = i' Ag (k - s) (recall that u = k -1). Hence by the converse Theorem 7.3 it follows that

9=flwiv)

WN=(N

1).

Next let 0 (mod p) be a primitive Dirichlet character of conductor p t N. Then the twisted L-functions are given by L f (s, i/') = L (s - 2, ,0 o N) and Lg (s, ?P) = CL (s - 2 , £ . o N) . We set Af(s,-0) =

(P)8r(s)L(soN) S

A9(s, i) =

p22

r(s)CL (s - 2 ,

tG o N)

and verify by (12.25) that Af(s, 0) = ikw(,O)A9(k - s, with the appropriate root number )2p-1 w(O) = X(p)V,(N)T(' (to get exactly this number, use (12.37), (12.32), (12.23) and (12.22)). Hence

by the converse Theorem 7.8 we obtain (12.36). If it > 0, then L1(s) = L (s - 2, C) converges absolutely in Re s > 2 + 1, where 2 + 1 = 1 < k, so f is a cusp form.

Remark. If 6 (mod m) is primitive, then f given by the Fourier series (12.36) and g = fl,,, given by the series (12.38) yield L-functions with adequate Euler products (i.e. of type (6.93)); therefore f is a newform, namely an eigenfunction of all the Hecke operators T,),( with eigenvalues an = n a E 6(a), Na=n

and f is also an eigenfunction of the operator W = KW with eigenvalue C= i2k-'W()(Nm)-1/2.

12.4. Class group L-functions reconsidered

215

Theorem 12.6. Let K = Q(v' ) be a real quadratic field with discriminant D > 0 and l; (mod m) a Hecke character such that (12.39)

i;((a)) = lal

if a =- 1

(mod m)

-

ifa - 1

(mod m)

or (12. 40)

((a)) =

Via'

where a' denotes the conjugate over Q. Then (12.41)

f (z) _

l;(a)e(zNa) E S1(T'o(N), X)

where N = DNm and the character X (mod N) is defined as in (12.37).

Proof (hint). This follows from the converse Theorem 7.8 by arguments similar to those used in the proof of Theorem 12.5. Note that a Hecke character on a real quadratic field always yields a cusp form of weight k = 1. If (mod m) is primitive, then f given by the Fourier series (12.41) is a newform with Hecke eigenvalues (12.42)

a._ E (a) Na. n

12.4. Class group L-functions reconsidered Our arguments used for Theorem 12.5 and Theorem 12.6 were sketchy and the proofs were not really complete, since we appealed to Hecke's general results about analytic continuation and functional equations for L-functions

attached to Grossencharacters. Our purpose was merely to illustrate an application of the Hecke-Weil converse theorems. Now we take a direct approach to show that the automorphic forms associated with L-functions of a quadratic field correspond to theta functions. For simplicity we confine

the demonstration to characters of conductor m = 0, so such a character is primitive. Actually we only consider the class group characters. For an imaginary field this means u = 0 in (12.35). However if K is real and every unit of K has norm 1, then neither (12.39) nor (12.40) defines a function on principal ideals. In this case in order to speak of a class group character one needs a more subtle definition of ideal classes.

12. Automorphic Forms Associated with Number Fields

216

Let K = Q(') be any quadratic field of discriminant D, positive or negative. Two ideals a, b E I are said to be equivalent in the narrow sense if

b=(a)awith c K, Na>0.

This is a new concept only when K is real and all its units have norm 1. Here we may replace Na > 0 by the condition a >- 0, which means a > 0 and a' > 0 in case of a real field and a 0 in case of an imaginary field. We put

P+={(a):aEK,a>-0}.

Then R+ = I/P+ is the class group of narrow classes and h+ = [I : P+] is the narrow class number, h+ 2h if K is real and Ne = 1 otherwise where E is the fundamental unit. By class character we mean a group homomorphism X S'. With X we associate the L-function Sl h

(12.43)

Ltt(s, X) = E X(a)(No)-8 CL

where a ranges over non-zero integral ideals. We split this series into narrow classes, getting

LK(s, X) = E X(A)Cx(s, A) AEN

where (K (s, A) is the zeta-function of the class A, (12.44)

A) = E(No)-s. aEA

It turns out that every class zeta-function (K (s, A) has meromorphic continuation over the s-plane and satisfies a functional equation of the same type. Hence the whole function LK(s, X) inherits these properties. From now on we give further details only for an imaginary quadratic

field K = Q(v/D) of discriminant D < 0. Let w = #U be the number of units of K. For every class A E f we introduce the theta function (12.45)

eA(z) = w-1 + E e(zNa), aEA

and for any character X E fl we put (12.46)

fx(z) = E X(A)eA(z) AEf

= w-1hb(X) + EX(a)e(zNa) a

12.4. Class group L-functions reconsidered

217

where 5(X) = 1 if X is trivial and 6(X) = 0 otherwise. Every class A contains an integral primitive ideal (i.e. not divisible by a rational integer > 1). Every primitive ideal can be written as (12.47)

a= [ai b +2 with a> 0, b2 - 4ac = D, (a, b, c) = 1. j

The above notation means a is a free Z-module,

a - aZ + Note that b+

as = a,

Z.

2

E 0 and a = Na: indeed, we obtain a,

Z

b-*"D 2 = a 2,a

b+b-fD ,a 2

2

c

= [a2, ab, ac) = a[a, b, cJ = a0.

With the generators of a we associate the quadratic form

con(x) = axi +bxlx2 +cx2 =

2A[x]

where A = C 2y c ) . This establishes a one-to-one correspondence between

the ideal classes A E 1-1 and the equivalence classes of primitive binary quadratic forms cpn of discriminant D. We choose fD = iv/I'-Dl so that za

b+ -fD =

2a

E H.

Then the inverse ideal a-1 is a free Z-module generated by 1 and a-1

= [1, zaJ = Z ±

b-VD2a

Z.

Now, given a class A which contains a, we can write

®A(z) = w-1 + E e(zNb). b-n

Here the equivalence b - a means b = (a)a with a E a-1, a # 0, i.e. a = m + n.2a with m, n E Z, not both zero. As m, n range over the integers

12. Automoiphic Forms Associated with Number Fields

218

every ideal b - a is covered exactly w times. Moreover we have Nb = I al22a = am2 + bmn + cn2, whence (12.48)

9A(z) =

w-1

e(coA(m,n)z) (m,n)EZ2

b This is indeed the theta function associated with the matrix A = r 2a b 2c)-

Note that -DA-1 = ( ? b 2a) Therefore by Theorem 10.9 eA E .M1(ro(IDI),XD)

(12.49)

(observe the consistency condition XD(-1) = -1). Furthermore, by the Jacobi inversion formula (10.10) we get (12.50)

9.4(z) =

ID1-1/22z-19,A-I(-1/IDIz)

where A-1 denotes the inverse class to A. Averaging over the ideal classes A E 1, each one weighted by X(A), we extend the above automorphy relations to fx. These show that

fx E M,(ro(IDI),XD)

(12.51)

f

(12.52)

xl(IDI

respectively (note that fx

1)

= -ifx

= fz)

Put (12.53)

AK(s,X) =

(viTi)5r(s)L(sx),

and observe that the functions f = fx and g = f XI (

1

= -i fx have

IDr

the Fourier series (7.20) and (7.21) with constant terms ao = w-1hb(X) and bo = -iw-lhb(X) respectively (to compute the constant terms, use the asymptotics for s -4 1 from the end of Section 7.2). Therefore (A2) of Theorem 7.3 asserts that the function (12.54)

AK(S, X) + w-lhb(X)(s(1 - s))-'

is entire and bounded in vertical strips, and AIj(s, X) satisfies the functional equation (12.55)

AK (s, X) = AI{(1 - s, X).

12.5. L-functions for genus characters

219

Actually we have the following integral representation: s))-1

Atc(s, X) + w-1hb(X)(s(1 -

=

f

00

(ys-1 + y-s)(E X(a) exp(-27ryNa/

I DI ))dy

a

from which the above assertions follow at once. As a by-product we derive from (12.53), (12.54) and (12.33) the Dirichlet class number formula (examine the residue at s = 1): (12.56)

h=

w

2,DI L(1, XD)-

If K = Q(/D) is real (D > 0), the above analysis fails since the group of units is infinite while the quadratic forms corresponding to ideal classes are indefinite. The latter topics lie beyond the scope of this book, so we don't proceed beyond what has been stated about LK (s, X) in Theorem 12.2 (if X is trivial) and Theorem 12.3 (if X is non-trivial).

12.5. L-functions for genus characters Next we shall examine LK(s, X) closely for special class group characters (for

either imaginary or real quadratic fields), the genus characters. The genus theory was created by Gauss in the context of binary quadratic forms. Here we recall its content in terms of ideals. A discriminant of a quadratic field is said to be a prime discriminant if it has only one prime factor, so it must be one of the following type: (12.57)

-4, ±8, ±p = 1 (mod 4).

The product of coprime discriminants is again a discriminant. Every discriminant can be written uniquely as a product of prime discriminants, say D = P1 ... Pt. Hence D can be arranged as a product of two discriminants D = D1D2

(12.58)

in 2t-1 distinct ways if we allow interchanging D1 with D2 (here and here-

after, t denotes the number of distinct prime factors of D). For any such decomposition we define a character XD,,D2 on ideals by setting (12.59)

XDi,D2(p) =

YD,(Np) X D2 (Np)

if p f D1 if p $ D2

220

12. Automorphic Forms Associated with Number Fields

(recall that Xd(n) = (4) is the Kronecker symbol). This is well defined on prime ideals because (12.60)

XD(Na) = 1

if (a, D) = 1,

and we extend XD1,D2 to all fractional ideals by multiplicativity. Therefore

XD,,D2 : I -> {±l} so XDl,D2 has order two, except for the trivial character which corresponds to the trivial factorizations D = D 1 = 1 D. Every such XD,,D2 is called the genus character of discriminant D; these are different for distinct factorizations (12.58), so we have exactly 2`1 genus characters. The genus characters are the narrow class group characters, XD1,D2 E H+, i.e. (12.61)

XD1,D2(a) = 1

if a = (a),a >- 0.

Theorem 12.7 (Kronecker). The L -Junction of K = Q(// ) associated with the genus character XD1,D2 factors into the Dirichlet L -functions, (12.62)

LK(s, XD1,D2)

= L(s, XD1)L(s, XD2)

Proof (hint). For the trivial character (D1 = 1 or D2 = 1) this is the factorization (12.33) for the Dedekind zeta-function. In general one can verify the Kronecker factorization (12.62) easily by examining the local factors

in the corresponding Euler products and using the law of factorization of primes in K expressed in terms of values XD(p) = 0, ±1. One shows that every real character of x+ is a genus character, and they all form a group of order 2t-1 which is isomorphic to (Z/2Z)t-1

Now let us describe the dual side of genus characters. We say that two non-zero ideals a, b E I are in the same genus if (12.63)

X(a) = X(b)

for all genus characters.

The ideals with x(a) = 1 for all genus characters form the principal genus (a subgroup of I). The same definitions apply to the narrow classes. We denote the subgroup of principal genus classes by (12.64)

9 = {A EH' : X(A) = I for all genus characters}.

The factor group F = ?-l+/9 is called the genus group. By duality F is isomorphic to

(Z/2Z)t-1

12.5. L-functions for genus characters

221

It was proved by Gauss that the principal genus consists of squares of classes

9 = {A2 : A E fl+}.

(12.65)

Hence two non-zero ideals a, b belong to the same genus if and only if (12.66)

a = bc2

for some c E I.

One can also show that a, b are in the same genus if and only if

Na = NbN-y for some -y r= K.

(12.67)

A class A E x+ is said to be ambiguous if A = A-1, i.e. A2 = 1. The group of ambiguous classes (12.68)

D= JA Efl :A2=1}

is isomorphic to the genus group .7 = 1-1+/9 (since V is the kernel of the homomorphism A H A2). Ambiguous classes are represented by ideal divisors of V/D; indeed if (12.69)

fD = 0102, (01,02) = 1,

then Cl(D1) = C102) and C1(D1)C1(D2) = 1.

From now on we consider a real quadratic field K = Q(v) of discriminant D. Much of the material presented below can be found in [Asa], (He2] and [Miy]. Though D > 0, it does not mean that (12.58) has positive factors. For any positive MID, consider matrices of type (12.70)

1

WAI _ VrM-

(a b) c

dJ

where a, b, c, d E Z, MIa, Mid, Dic, ad - be = M. From the determinant it follows that (M, DIM) = 1, i.e. D = MN with (M, N) = 1. In particular, for M = 1 we obtain all elements of the group ro(D). Each WA! normalizes I'o(D), i.e. Wnfro(D)W,II = ro(D), and W.421 E ro(D). The group of all such matrices, (12.71)

r*(D) = {WAJ : MN = D, (M, N) = 1},

is the normalizes of ro(D), and the factor group

4)(D) = r*(D)/ro(D)

222

12. Automorphic Forms Associated with Number Fields

is isomorphic to (7G/2Z)t. The elements of '(D), that is the classes of matrices W& f modulo ro(D), correspond to the ordered factorizations D = D1D2 into discriminants (take M = ID1I). We may regard fi(D) as a group

acting on M/ro(D). By the action of -t(D) any cusp for ro(D) can be brought to any other cusp; for example, WMOO = a/c is equivalent to the cusp a = 1/N. acts on automorphic forms (by the slash operator) The group

4P(D) : Mk(ro(D), XD) - Mk(ro(D), XD) (D(D) : Sk(ro(D), XD) - S1.(ro(D), XD) decomposing each of these spaces into 2t eigenspaces of type

Jtiik) = { f E Mk : f (z) = Y a(n)e(nz) with a(n) = 0 if Xp(n) = -bp for all pID} Ilere Xp(n) _ (n) is the Kronecker symbol associated with the prime divisor p of D and P = ±p, ±8,4 is the corresponding prime discriminant. To each such prime we assign a symbol by which takes two values by == ±1. Out of this we form the vector symbol (b) = (... , 5,,,...) with plD. Denote

P(b) _ f by = ±1. pI D

Any (b) with P(b) = 1 corresponds to the genus of classes in K = Q(v' which contains an ideal a such that (12.72)

)

Xp(Na) = by for all pID.

Therefore it is natural to single out the spaces

MA = ®Mk),

Mk = ® M.

P(6)=1

Y(b)=-1

We have a direct sum decomposition Mk = M+ ® M. The space M+ is particularly interesting; it consists of forms whose Fourier coefficients are supported on integers n > 0 with XD(n) = 0, 1, that is,

Mk = f E Mk : f (z) _

a(n)e(nz) YD(n)#'1

The above decomposition of MR into 21-1 linear spaces Mk) (one for each genus of K) can be examined further in terms of Eisenstein series. It turns

12.5. L-functions for genus characters

223

out that the co-dimension of MjjI fl Sk(ro(D), XD) in M(') is 1, and that MA a)

contains one Eisenstein series. More precisely it is a specific linear com-

bination of standard Eisenstein series which depends uniquely on the ideal a having the genus invariants (12.72). For example if k = 2, we have the following 2' standard Eisenstein series in M2(ro(D),XD), each one corresponding to an ordered factorization D = D1D2 into discriminants, namely EDI,D2(Z)

1

00 >dXD,(d)XD2(n/d) e(nz) = 2 L(-1,XD)eD,,D2 +E

n=1

dIn

where eD,,D2 = 0 except for ED,1 = 1. Then, out of these one can construct an Eisenstein series in .M2bl for any 6 with P(6) = 1 as followsE(b) (z)

=

(ri 6p

ED1,D2 (Z)

D1 D2=D

_

L(-1, XD)+E f n=1

ntn2=n

n1XD,(n1)XD2(n2Na)

e(nz)

D, D2=D

where a is an ideal satisfying the genus conditions (12.72) and Na is the norm.

Another benefit from the action of the normalizer r(D) on the space Mk(ro(D), XD) is that one can explicitly link the Fourier coefficients of an automorphic form at any desired cusp to these at the cusp oo. More precisely, suppose

a(n) e(nz) E Sk(ro(D), XD)

f (z)

is a newform, normalized by a(1) = 1. Given a decomposition D = D1D2 into discriminants, define twisted coefficients b(n) by the following requirements:

b(n) = XD1(n)a(n) if (n, D1) = 1 b(n) = XD2(n)a(n) if (n, D2) = 1 b(mn) = b(m)b(n) if (m, n) = 1.

Since a(n) = XD(n)a(n) if (n, D) = 1 (see (6.57)), one shows that the above requirements are consistent and define b(n) uniquely. The resulting twisted series

00

g(z)

b(n)e(nz) E Sk(ro(D), XD) 1

224

12. Automorphic Forms Associated with Number Fields

is also a newform, and it is related to f by (12.73)

WAI f = ANIg

for M = ID, I, where aas can be expressed by Gauss sums. For example, if M = p is prime then k12.

(12.74)

P = T(XP)a(p)p

For M = D this agrees with Theorem 6.29 (the proofs are similar).

12.6. Automorphic forms of weight one A lot of fascinating connections between number fields and automorphic forms have been revealed by Langlands' interpretation of Artin's L-functions. We shall give a glimpse of some results (without proof) in the context of two dimensional Galois representations vis-a-vis modular forms of weight one. References for these and more results can be found in the survey article by J-P. Serre [Ser]. We begin by recalling well known facts from arithmetic in number fields.

Suppose L/K is a Galois extension of degree n = (L : K] and the Galois group G = Gal(L/K). Let OK C OL be the rings of integers in K and L respectively, so n = (Of, : OK]. The central problem is to determine the way in which a prime ideal p C OK factors into prime ideals 3 C OL. We have

p = Mi ...F'9)e where E31 i

... , X39 C OL are all distinct prime ideals, mutually conjugate,

say to 3, each one of degree

f = (OL/T : OK /p],

efg = n.

The absolute norms are related by NT = (Np)f. Consider the decomposition group

G93={aEG:aT =T} and the inertia subgroup

Iq={aEG:ax=x (inodg3)forallxEOL}. To conjugate ideals there correspond conjugate groups. The decomposition group GT acts on the residue class field OL/g3 leaving OK/P fixed; hence we have a natural map

Gp -a Gal((OL/q3)/(OK/p))

J2.6. Automoiphic forms of weight one

225

whose kernel is the inertia group IT, so f = (Gq3 : Iqj]. Suppose p is unramified, i.e. e = 1 (there are only finitely many ramified

p in the extension L/K, namely the prime factors of the discriminant), so IT = 1, (GpJ = f and fg = n. For such T the Frobenius generator of the Galois group of the residue class fields extension can be lifted uniquely to the Frobenius automorphism in Gp; we denote such automorphism by aT = (3, L/K); it is characterized by the property 2°T _ xNp

(mod T) for all x E OL.

The Frobenius ap depends on T rather than p, but it is in the conjugacy class ap = {aaTO -1 : a c G},

which we also denote by ap = (p, L/K). In particular, ap is a single element if G is abelian. Very important problems concern the prime ideals of K which split completely in L, i.e. such that

p = Ti... T., a=f = 1. These are characterized as having the trivial Frobenius automorphism, ap =

ap = 1. According to Bauer's theorem the set Sp(L/K) of primes of K which split completely in L determines the Galois extension L/K uniquely. Therefore it becomes an attractive proposition to describe the set Sp(L/K)

in some kind of arithmetical terms. For K = Q by Cebotarev's theorem Sp(L/Q) has density 1/n. For abelian extensions L/Q the class field theory asserts that a prime number (ideal) splits completely if and only if it belongs to certain arithmetic progressions modulo the discriminant. For example, if

L = Q(v), then (DP )

Sp(L/Q) = {p :

1}.

If the extension L/Q is not abelian, the law for splitting primes is no longer governed by arithmetic progressions. However a great number of non-abelian extensions have been described by arithmetic properties of Fourier coefficients of certain automorphic forms. Much of this research is conducted in connection with Artin's L-functions.

Let L/K be a Galois extension, G = Gal(L/K) and p : G a representation. For each p C K unramified in L/K, define the local Lfunction by Lp(s,p) = det(I -

p(ap)(Np)-s)-1

226

12. Automoiphic Forms Associated with Number Fields

where op is the Frobenius conjugacy class (note that Lp(s, p) depends on p but not on primes in L which lie above p, because the characteristic polynomial depends only on the conjugacy class of p). If a(1), ... , ap"''1 are the eigenvalues of p(op), then m

Lp(s, p) = fI(1 - a,, (Np)`9)+1. i=1

For the ramified primes (a finite number of them) the local L-functions can

be defined in a similar fashion. Then the Artin L-function is formed by multiplying all the local L-functions,

L(s,p) = JJ Lp(s, p). p

Example. The one-dimensional representations correspond to the Dirich-

let characters. To this end take K = Q C Q(() = L, where S is the N-th root of unity. This is a cyclic extension of degree n = cp(N); its Galois group is isomorphic to the group of primitive residue classes mod N:

(Z/NZ)* - Gal (L/K)

a Dirichlet character X (mod N) one associates the representation

p : Gal (L/Q) -F GL1(G) = G* oa F-+ x(a).

For p } N the Frobenius automorphism op E G is the one which maps ( to (P, and we have p(op) = X(p). In this case the Artin L-function attached to p coincides with the Dirichlet L-function for X; L(s, p) = L(s, X). From now on L/Q is Galois with G = Gal(L/Q). There exists a continuous representation p : Gal(Q/Q) - GL,n(G) whose kernel is Gal(Q/L); this yields an injective homomorphism p : G -+ GL.,,F.(G). Therefore the set of splitting primes is given by

Sp(L/Q) = {p : p(up) = I}, and the local L-functions at such primes are Lp(s, p) = (1-p-') -'. Writing the Artin L-function as a Dirichlet series 00

L(s, p) = 1

ann _s

12.6. Automorphic forms of weight one

227

we see that p splits completely in L/Q if ap = m (i.e. ap takes the maximum value). Analytic properties of Artin L-functions are largely intractable. Artin conjectured that L(s, p) are entire functions except for some explicitly given representations (L(s, p) is expected to be entire if p is irreducible of dimen-

sion > 1) and that L(s, p) satisfies a suitable functional equation which connects values at s and 1 - s. Langlands has deeper vision; briefly the celebrated "Langlands program" predicts that L(s, p) coincides with the Lfunction attached to some automorphic representation on GL,,,, where m is the degree of p. If this is true, then the Artin conjecture would follow in a straightforward manner. What we know at present is that L(s, p) is meromorphic on the whole s-plane (due to R. Brauer), since L(s, p) can be represented as a quotient of products of zeta-functions corresponding to Hecke's characters in various fields. But the holomorphy question in general is not resolved. Yet, Artin succeeded in proving his conjecture for one-dimensional representations of abelian extensions as part of the "reciprocity law". From this perspective Langlands' program offers an attractive direction towards non-abelian class field theory. From now on we shall discuss various cases of two-dimensional Galois representations. Such representations (according to the Langlands program) should correspond to classical automorphic forms of weight one. Indeed, this correspondence has already been established in the direction from automorphic forms to number fields by P. Deligne and J-P. Serre.

Theorem (Deligne-Serre, 1974). Let f be a normalized cusp newform on lo(N) of weight k = 1 and character X (mod N) with X(-1) = -1. Then there exist a Galois extension L/Q and an irreducible two-dimensional

representation p : Gal(L/Q) - GL2(G) such that L f(s) = L(s, p). R.emark. In the above result the level N is the conductor of p, and the character X (mod N) corresponds (via class field theory) to the one-dimensional representation e = det p. This representation is odd, i.e. E(c) = -1, where c is the complex conjugation automorphism (think of p : Gal(Q/Q) --> GL2(G)

and e : Gal(4/Q) - G`). As an immediate consequence of the Deligne-Serre theorem one gets the Ramanujan conjecture for the Fourier coefficients of cusp forms in the space Si(ro(N),X). Namely, one obtains the estimate Ia(p)I S 2 for prime coefficients of 00

f (z) _

a(n)e(nz).

228

12. Automorphic Forms Associated with Number Fields

We assume that f (z) is normalized so that a(1) = 1. If

g(z) _

b(n)e(nz)

is a normalized newform in S1(ro(N), X) other than f , then Serre showed

that density {p : a(p) = b(p)} < 7/8,

and 7/8 is attained. Much is known about the Artin conjecture for two dimensional representations. Suppose

p : G - GL2(C) - PGL2(G) = GL2(C)/(C)* is continuous, i.e. its image is finite. On the other hand all finite subgroups

of PGL2(C) were described by Klein; they are: C cyclic of order n; D,, dihedral of order 2n, n > 2; A4 tetrahedral; S4 octahedral; A5 icosahedral. The cyclic groups appear for reducible representations; the corresponding Artin L-function is given as the product of two L-functions for Dirichlet characters and the corresponding automorphic form is an Eisenstein series. The remaining subgroups are in the image of irreducible representations. Let us recall that a finite group D is called dihedral if it has a subgroup

H of index 2 and an element x ¢ H such that x2 = 1 and xyx-1 = Y-1 for all y E H. In particular, D,, can be viewed as the group of rotations and reflections of the plane which preserve a regular polygon with n vertices.

Thus D is generated by a rotation r of angle 27r/n and a reflection s, and the only relations between these generators are r" = 1, s2 = 1, srs = r-1. The Artin L-function for a dihedral representation coincides with the Hecke L-function for some Grossencharacter; hence the Artin and Langlands conjectures are both true. For tetrahedral representations the Artin conjecture was established by Langlands, and then his ideas were adopted by Tunnell for all octahedral representations. However the icosahedral representations still resist any treatment, except for a few examples established by J. P. Buhler. Finally we make a few comments about the dimension of S1(r0 (N), X).

Serre has established a surprising fact: a newform f E S1(ro(N),x) has almost all Fourier coefficients zero (in the sense of density). Hence it seems there are not as many cusp forms of weight one as for weight > 2. Serre has examined the space S1(I'o(p), ,yp) for prime level and character )(1,(n) _ ().

Since XN(-1) = -1, we must have p = 3 (mod 4). Applying the Selberg

12.6. Automorphic forms of weight one

229

trace formula, one can showed that (private communication by P. Sarnak) (12.75)

dimS1(ro(p), Xp) <<

l gp

This is already an interesting estimate, because the fundamental domain ro(p)/III has volume 1(p + 1), which is larger than the bound (12.75) by the factor log p. There is a reason for such a good bound, namely the fact that the forms y1/2 f (z) lie at the bottom of the continuous spectrum of the Laplace operator 2

2

8x2 + 8y2) - Zy 8x

y2

Presumably the estimate (12.75) can be improved further, but not simply by manipulating the choice of test functions in the trace formula. A serious weakness of the trace formula in this respect is that Si (Po(p), Xp) is a subspace for the lowest eigenvalue AI = 1/4 which is not isolated, so any reasonable test function picks up a large chunk of eigenforms which are not holomorphic forms. This said, in order to get an accurate estimate for dim Si (170(p), Xp) one

must exploit the very arithmetical structure of holomorphic cusp forms of weight one. Serre established the formula (12.76)

dimS1 (to (p), Xp) =

h

-

1

2

+ 2s + 4a

), s is the number of quartic,

where h is the class number of K = Q(

normal fields with discriminant -p (any such a field has the Galois group S4) and a is the number of quintic, non-real fields with discriminant p2 whose normal closure has Galois group A5. He also conjectured that (12.77)

2s+4a<
whence (12.78)

dims1(ro(p), Xp) = h

2

i + OW).

Here the h21 linearly independent cusp forms in Si(Po(p), Xp) are actually theta functions constructed from the class group characters of K, namely

X(a)e(zNa)

f.<(z) _ O54aCO,c

for any X E ?l, X # 1 (see (12.46)). Note that X are complex characters (by

the genus theory, because K has prime discriminant), and f, = fc. There

230

12. Automorphic Forms Associated with Number Fields

are no other linear relations between the fx. Recall that the class number is bounded by p1/2-e << h «p"2log p.

While Serre's conjecture seems to be out of reach, a striking breakthrough was recently made by W. Duke [Du2), who succeeded in showing that (12.79)

dimSi (I'o(p), x) <<

p12-L' (log P)4.

His arguments combine arithmetical and analytic ideas beautifully.

Chapter 13

Convolution L-functions

13.1. Introduction We have already experienced how fruitful the idea of twisting automorphic

forms by Dirichlet characters is. One can regard the Dirichlet characters X (mod q) as a kind of automorphic forms on GLI, and furthermore one can regard the Hecke Grossencharacters (mod m) for a number field K of degree n as automorphic forms on GLn. Hence a question arises: what does one obtain by twisting two genuine automorphic forms? A quick guess is that one should get an automorphic form on some higher rank group. We cannot address these advanced topics in this book; however, we shall be able to make convincing observations in the context of relevant L-functions. To illustrate the general picture we consider two L-series L l (s) =

L9(s) =

I:b(n)n-s

1

which converge absolutely in some right half-plane. With these we form the convolution L-series by multiplying the coefficients,

L(f ®g, s) _

a(n)b(n)n-'. 231

13. Convolution L-functions

232

For example, if f, g are Eisenstein series whose L-functions are 00

Lf(s) = E aQ(n)n-s = ((s)((s - a) 1

00 Eab(n)n-s

L9(s) =

= ((s)((s - b),

1

then we get 00

L(f (& g, s) = > aa(n)ab(n)n-s = ((s)((s - a)((s - b)((s - a - b)((2s - a - b)-1.

This formula was proved by Ramanujan. Rankin and Selberg considered the convolution L-functions attached to arbitrary automorphic forms f, g. It is easy to show that if L1(s) and La(s) have Euler products of type Lf(s) = Ti(1 - app-s)--1(1

-

QPP-s)-1

P

La(s) = [J(1 - 7pp)'(1 - 6PP-s P

then L(f 0 g, s) also has an Euler product of type L(f O g, s) = fJ(1 - aPNP YP5Pp 2s)

(13.1)

P apypp8)-1(1

X

-

aP6PP~e)-1(1

-Qprypp 8)-'(1 - (3P6PP-8)-1.

P

Our main goal is to establish the analytic continuation of L(f 0 g, s) and some kind of functional equation.

13.2. Rankin-Selberg integrals We begin with two automorphic forms f E Mk(r,19), g E M1(r,21) for a group r and multiplier systems 29, n of weights k,1, respectively. We assume that one of these forms, say f, is a cusp form (to avoid problems of convergence of certain integrals). Let a be a cusp for r which is singular with respect to both multipliers. Then f and g have Fourier expansions at a of type (see (2.64)) f

fa(n)e(nz),

(z) _ 1

00

go(n)e(nz).

910, (z) _ 0

233

13.2. R.anlcin-Selberg integrals

For n >, 1 we normalize the coefficients by setting I-k fa(n) = (41rn)

1. (n)

ga(n) = (47rn) 2tt ga(n), and we define the convolution L-function by

L0(f 0

(13.2)

,

s) _

fa(n)9a(n)n-s

00

Note that AO0, glO0 are autoinorphic forms for the conjugate group aQ Tor. and the multiplier systems 19'a, 77'd given by (2.55). Although f. (n), ga(n) depend on the choice of the scaling matrix a0', the product fa(n)ga(n) does not, and La(f ®g, s) depends only on the equivalence class of cusps. We have

f

1

fiO0(z)giOp(z)dx

0" = E.fa(n)9a(n)e-

Avrny

1

Hence using the gamma integral 00

r(s) = f e

yldy

we get

/

(47r)-9r l s +

k+1 2

- 1) La(f (9g, s) =

ff °O

0

ysli(z)d;iz

0

where

h(z) = Y'2

f1Qa(z)g1O0(z)

By changing the variable z - o a z we can write the above integral as 1

fH

Jsh(z)dpz = fa\H(Im 0'Q 1z)h(a- 1z)dµz

f

(Im a.1ryz)sh(aa 1'yz)dµz. \H 7Ero\r

We have flan (a0' 1`yz) = 700 (an

,yz)-'.f

(7z)

=700(0'. 7z)-,77(z)"9('y).f(z) = 19(ry)LLk(0'a, 0'a 17)7ad I.y(z)''f(z)

13. Convolution L-functions

234

by (2.50), (2.60) and (2.49). A similar formula holds for g. Hence by (2.7) we get h(aa 17z) = X.('y, z)y s f (z)g(z)

where y(z)I)k-t

(13.3)

Xa(7,z) _

17)

Summing over 7, we create the Eisenstein series

EQ(z,s) = E XQ(7,.z)(Im as 17z)s, 7Er,\r

(13.4)

and we obtain

Proposition 13.1. If Re s is sufficiently large, then (13.5)

(4-7r)-9r (s + k

l - 1 1 LQ(f ®9, s)

r\ 2y 2 f (z)9(z)EQ(z, s)dµz. The result simplifies a bit if k = 1. Namely, we get (13.6)

(47r)-'F(s + k - 1)Ln(f (9 g, s) = J \H yk f (z)g(z)EE(z, s,19f)dµz

where (13.7)

E t9(7)i ey)(Im as "Yz)s.

EE(z, s,

At the cusp a = oo with a. = 1 for any k, l the Eisenstein series becomes (13.8)

E-(z, S) =

E t9(7)f(7)(9ry(z)/I.iry(z)I )t'-l(Im 7z)8 -yEr.\r

For t9 = 1 and 17 = Ed (d) (the multiplier of the standard theta function (2.72) of weight 1 = 1/2) on the group ro(4) we have (13.9)

=

2

ed (C) (c,d)=1 c=_0 (mod 4)

cz + d

(lcz+dI)

k

s

y

(Icz

I

dI2)

13.3. Selberg's theory of Eisenstein series

235

The integral representation (13.5) for the Rankin-Selberg L-function is a close analogue of (7.4) for the Rieniann zeta-function. In a similar fashion (13.5) will yield analytic properties of L. (f ®g, s) from those of Eg(z, s). Since f is a cusp form and g has at most polynomial growth, the associate measure (13.10)

µ19(z) = yk f (z)g(z)dµz

has exponential decay at the cusps. Hence the integral (13.5) converges absolutely and uniformly with respect to s on compacta on which the Eisenstein series E0(z, s) can be analytically continued.

13.3. Selberg's theory of Eisenstein series A general theory of Eisenstein series on GL2 was completely developed by A. Selberg in the 1950's and subsequently generalized to higher rank groups by R. Langlands in the 1960's. The real analytic Eisenstein series are indispensable for the spectral theory of automorphic forms; they are building

blocks for the subspace of continuous spectrum. At the root of this lies the problem of analytic continuation over the whole complex s-plane. For a congruence group the desired properties of Eisenstein series follow straightforwardly from explicit Fourier expansions (we shall rely on them), so the real difficulties are encountered for other groups. Although we don't need

to be general, it is worthwhile to establish some properties (such as the functional equations) in Selberg's original way. We restrict the presentation to Eisenstein series of weight zero (such as (13.7)), i.e. (13.11)

X('r)(Im a l-yz)s

E0(z, s, X) =

-rEr,\r

where X is a multiplier system of weight zero (X : r -* C* is a group homomorphism) singular at cusp a. The series (13.11) converges absolutely in Re s > 1 by virtue of Proposition 2.9. The Eisenstein series is in the space A(r, x) of functions f : H -> C which satisfy the automorphy equation (13.12)

f (yz) = X(ry) f (z)

if 'y E r.

Though Ea(z, s, X) is not holomorphic in z E 1111, it is an eigenfunction of the

Laplace operator

r

+02

L = y2 r ,9x2 2

13. Convolution L-functions

236

namely (13.13)

(A + s(1 -- s))E0(z, s, x) = 0.

This is clear because A commutes with the action of any r E GL2(R) on H, the power function (Im z)3 satisfies (13.13), and the Eisenstein series is built from (Im rz)s. We denote by A, (17, x) the space of functions f : 1H[ ---> C satisfying (13.12) and (A + s(1 - s))f = 0.

(13.14)

Note that s - s(1 - s) is a double cover of C since s and 1 - s yield the sauce eigenvalue A = s(1 - s), except for s = 1/4 which covers A = 1/4 once. The Laplace equation (13.14) replaces the Cauchy-Riemann equations

for holomorphic functions. Of course, we require f to be at least twice differentiable; actually, since i is an elliptic operator, its eigenfunctions are real-analytic. By arguments similar to those employed for the holomorphic Eisenstein series (3.14) we derive the Fourier expansion of Ea(z, s, x) at cusp b (singular) of the following type: X)yl_s

Ea(obz, s, X) = 5abY8 +'Pab(S,

+

(13.15)

'Pab(n, s, X)W8(nz), n,EO

where (13.16)

lPab (S, x) _ lr1/2 r(

E r(s) /2)c>o

C-2s S

b(0, 0; c),

(13.17) WO (n, s, x) =

lnls-1

r(s)

>

c-2s Sab(0,

n; c),

c>0

Sab(0, n; c) are (Kloosterman-Gauss-Ramanujan sums) given by (3.13) and (13.18)

W8(z) = 21y11/2Ks-1/2(21

lyl)e2,Ti3

is the Whittaker function on z E C\IR (an analogue of the exponential function e(z)). The Bessel function 2

f

0

has the integral representation

13.3. Selberg's theory of Eisenstein series

237

which shows that Kv(y) is entire and even in v and decays exponentially in y; more precisely, one can prove the approximate formula

K(y) _ (i;)

e-1{1

+ O(lvl2y-1)}

uniformly for Re v > 1/2, y > 0. Hence WS(z) - e(z)

as y -+ oo.

As an example, take the modular group r = SL2(Z). There is only one cusp a = b = oo (up to equivalence), and the multiplier is trivial. We have S(0, 0; c) = cp(c), whence (13.19)

W(S) _ 7r

1/2r(s - 1/2) ((2s - 1) r(s)

((2s)

where ((s) is the Riemann zeta-function. By the functional equation (12.1) we can write cp(s) symmetrically: (13.20)

cp(s) = 0(1 - s)0(s) -1

where 0(s) = 7r_.Sr(s)((2s).

(13.21)

The other Fourier coefficients of E(z, s) are given by (13.22)

co(n, s) =

0(s)-1Inl

1/2

E (a/d)s-1/2

ad=InI

In particular, for s = k - 1/2, where k is a positive integer, (13.23)

n1/2cp(n, k -

1/2) = 7rkr(k - 1)-1((2k - 1)ak_1(n).

By these formulas the Fourier expansion (13.15) shows that E(z, s) has meromorphic continuation over the complex s-plane, and it satisfies the functional equation (13.24)

0(s)E(z, s) = 0(1 - s)E(z,1 - s).

In the half-plane Re s >, 1/2 the Eisenstein series E(z, s) is holomorphic except for a simple pole at s = 1 with (13.25)

3

ress=iE(z,s) = 3.

13. Convolution L-functions

238

For a general group r the Fourier coefficients c'ab(s, X), (Pab(n, s, X) are hard to compute, but once the meromorphic continuation is taken for granted one can derive the functional equation in an elegant fashion. The so-called scattering matrix (13.26)

4'(s, X) = (Wab(s, X))

(its rank equals to the number of inequivalent, singular cusps) plays a central role. In this connection it is convenient to arrange all the Eisenstein series into the column-vector (13.27)

E(z, s, X) = t(... , E0(z, s, x),...).

Selberg established (among many other things) the following theorem.

Theorem 13.2. The scattering matrix and the Eisenstein series have meromorphic continuation over the whole s-plane. In the half-plane Re s > 1/2 the scattering matrix has at most a finite number of simple poles in the segment 1/2 < s < 1. The poles of Ea(z, s, X) in Re s >, 1/2 are all simple, and they are at the poles of Soaa(s, X), so they are also in the segment 1/2 < s < 1. The point s = 1 is a pole of E0(z, s, X) only when X is trivial, in which case the residue is a constant function of z; more precisely, (13.28)

ress_iEa(z, s) = 1/vol (r\H).

The scattering matrix satisfies the functional equation (13.29)

and is unitary on the critical line, i.e. (13.30)

4(s, X)tfi(s, X) = I if Re s = 1/2.

The vector Eisenstein series satisfies the functional equation (13.31)

E(z, S, X) = `1'(s, X)E(z,1 - s, X).

Proof of the functional equations. We have proved in previous sections the meromorphic continuation of the Eisenstein series for only congruence groups. However, assuming the first part of Theorem 13.2, we can establish the functional equations in full generality. There is no way to employ Poisson's summation as it could be, although not easily, in the case of a congruence group. Therefore, it is surprising that the functional equations are

13.3. Selberg's theory of Eisenstein series

239

a routine consequence of the apparently remote fact that A is a symmetric and non-negative operator in the Hilbert space

L2 (r, x) = { f E A(r, x), (f, f) < oo} where (f, g) denotes the inner product

(f, g) = f \H f (z)9(z)dpz. Indeed, by Green's formula one shows that

y2(f+whence

(-Af,g) = J

(A f, g) = (f, Ag) and (-A f, f) > 0 provided f, g, Af, Ag are all in. L2 (r\Illl). By these properties it follows that the eigenvalues of A in

L2(r\Ill<) are real, non-negative numbers. Based on this observation, we draw the following characterization of the Eisenstein series.

Lemma 13.3. Any f E A3(r, x) with Re s > 1, which at every cusp is bounded by a power function, is necessarily a linear combination of Eisenstein series. Proof. Since the Fourier coefficients of f (Qaz) are bounded by a polynomial while the Whittaker function decays exponentially, we get from the Fourier expansion

f (a z) = coy' + 0(1) as y - oo. Subtracting the Eisenstein series, we kill the leading terms toys and get a function

g(z) = f (z) -

c0E,(z, s, x)

bounded on H. Since g E A, (17, x) n L 2 (r, x), we must have A = s(1-s) > 0, which contradicts Re s > 1 unless g = 0, i.e. (13.32)

f (z) = E caEa(z, s, X) a

Now we apply Lemma 13.3 for f (z) = Eb (z,1 - s, x) with Re s > 1, getting Eb(z,1 - s, x) _

P6a(1 - s, x)E.(z, s, x)

by (13.32). This is just the functional equation (13.21) with s replaced by 1 - s. Here the condition Re s < 0 can be relaxed by analytic continuation. Finally, (13.29) follows by repeated application of (13.31).

13. Convolution I,-f actions

240

13.4. Statement of general results Since the analytically continued Eisenstein series Ea(z, s, X) for s not a pole is bounded by a power function of z at every cusp, we infer by the integral representation (13.6) and Theorem 13.2 the following important theorem.

Theorem 13.4. Let f E Sk(r,19), g E Mk(r, rl) and let

L(f ®9, s) _ t(... , La(f ®9, s), ... ) be the vector column of the convolution L-functions attached to the singular cusps a for r with respect to the multiplier system X = ft Then ,C(f ®g, s) has meromorphic continuation to the whole s-plane, and the vector function

A(f ®g, s) = (47r)-6I'(s+k- 1)L(f ®g, s) satisfies the following functional equation: (13.33)

A(f (9 g, s) = 4,(s, z9i)A(f ®g,1 - s)

where 4 (s, X) is the scattering matrix associated with the Eisenstein series on A, (IF, X).

One should be cautious when using Theorem 13.4 in general, because the scattering matrix is a meromorphic function of order up to 2. We don't have satisfactory estimates on -(D(s, X) except for congruence groups, in which case -t(s, X) can be expressed in terms of classical Dirichlet L-functions, and

consequently t(s, X) has order 1. By the integral representation (13.6) one sees that La(f (9 g, s) has at most simple poles at the poles of Ea(z, S, X). In the half-plane Re s 3 1/2 there can be at most a finite number of poles in the segment 1/2 < s 5 1; they are the poles of W.,, (s, X). There is no pole at s = 1 unless r) = 19 and f, g are not orthogonal. The residue is equal to (13.34)

ress_1 La(f ®g, s) =

47r

(f, g) L.

r(k) vol (L \>ii)

where

(f,g)k = J

\A[

y'f

(z)9(z)dpz.

13.5. The scattering matrix for ro(N) We shall give formulas as explicit as possible for the scattering matrix 4i(s, X) = X)) of the Eisenstein series in A3(ro(N), X), where X (mod N) is an even Dirichlet character. Recall that for a pair [a, b] of

13.5. The scattering matrix for ro(N)

241

singular cusps with respect to X the entry ,pnb(s, X) is given by the Dirichlet series (13.16) with coefficients (13.35)

Van C D a`);

Sab(0, 0; c) = (

(;

d)EB\aa-'rab/B

here and hereafter, r = ro(N). To compute this sum we need an explicit description of the set aa -, rab. Since cppb(s, X) depends only on the equivalence classes of cusps, we may stick to specific representatives. Recall that all cusps for ro(N) are represented by rational numbers

v with VIN, (u, v) = 1

(13.36)

and two such cusps are equivalent if they have equal denominator v and the numerators are congruent modulo (v, (see Proposition 2.6). However not v)(it is a property shared by the whole every cusp is singular with respect to X equivalence class).

Lemma 13.5. Suppose X (mod N) has conductor N* IN. Then the singular cusps for X of type (13.36) are exactly those with denominators vI N such

that (v, v) I N If X (mod N) is the trivial character, then every cusp is singular. If X (mod N) is primitive or N is squarefree, then the singular cusps are represented by

r = 1 with vw = N, (v,w) = 1,

(13.37)

v

so
of N. For such a cusp we take /a matrix (13.38)

Tp =

with a - 6

(v

0 (mod w) and a - b

E SL2(Z)

1 (mod v). Then for a pair of such

cusps [a, a'] _ [1/v, 1/v'] we have Tn 1 rTn,

1(c

E SL2(Z) : (v,v')Ic,

(v,'w) d}

d) The cusp a = 1/v has width rn = to (see (2.29)), and a = TQ ( is its scaling matrix. Similarly for a'; therefore L

_

{(a/i?7i c w'io

b/

'1

dw/w' J

(w, v')a, (w, w')Ib (v, v') I c,

(v, w') Id

13. Convolution L-functions

242

Recall that N = vw = v'w'. Writing w = (w, v')(w, w') and w'= (w', w)(w', v), we show that the set oo troal consists of matrices of type P

_ A ww'/(w, w') B(w, w')/ ww'

-(

D ww'/(w, w') )

C(v, v') ww'

where A, B, C, D are integers such that det p = 1, i.e. satisfying (w, V') (v, w) AD - (w, w') (v, v') BC = 1.

Now suppose X (mod N) is the trivial character. Then Saa, (0, 0; c) = cp((w, w') (V, V') C)

for c = (v, v') ww'C > 0, and by (3.16) we get 1 r(S - 1/2)

Paa' (s)=r 2

r( s)

(ww/)-9(v, v/)-29

T\(w,w )(vfv )C)

(C,(w,v')(v,w'))=1

This series can be computed easily by factoring over primes. The following result was first established by D. Hejhal (by a slightly different computation; see [Hej]).

Proposition 13.6. Suppose X (mod N) is the trivial character and N is squarefree. Then for any pair of cusps [a, a'] = [1/v, 1/v'] with vw = v'w' _ N we have Paa' (s) = (P(S)paa' (s)

where W(s) is the scattering matrix for the modular group (see (13.19)(13.21)) and Paa'(s) = co((w,w)(v,v'))

fl (p2s PI(w,v')(v,w')

pIINN

D. Hejhal used the above results to arrange the scattering matrix for ro(N) as the tensor product

'(s) = w(s) ®PI N NP(s) where

Np(s) =

1

p-1

ps-p1-s

p2s-1 p9-pl-s

p-1

).

This can be also written as NP(s) = .Mp(1 - s)Mp(s)-I, where

Mp(s) =

1

1S ) Cps

13.6. Functional equations for the convolution L-functions Hence

243

2-2s-1 det 4)(s) = cp(s)h fl pp2s - 1 PI N

where h = 2' is the rank of 4b(s), i.e. the number of inequivalent cusps. These formulas hold for N squarefree. If X (mod N) is primitive, all singular cusps are still of type (13.37) even if N is not necessarily squarefree. In this situation the former computations work but are somewhat longer, so we only formulate the results; they are taken from unpublished notes of N. Pitt (Rutgers, 1991).

Proposition 13.7. Suppose X (mod N) is a primitive even character and a, b singular cusps of type (13.37). Then cpb(s, X) vanishes unless a = 1//v, b - 1/u, with vw = N, (v, w) = 1, in which case (13.39)

Xv(w)X,u(v)N-sir1/2r(s - 1/2) L(2s - 1, XwXv)

Wab(s, X) =

r(s)

L(2s, XwXv)

where X = XwXv is the unique factorization into characters to moduli w, v respectively.

The above result shows that if we arrange the singular cusps in a suitable order, then the scattering matrix is skew diagonal. One can write (13.39) in an invariant form by applying the functional equation for the Dirichlet L-function (take (12.7) for 2s -1 in place of s and XwXv in place of X) and factoring the Gauss sum T(XwXv) = Xw(v)Xv(W)T(Xw)T(Xv)

One obtains T(Xw)T(Xv)N1-3sir2s-1

,Pab(s, X) =

In particular, for the pair of cusp a = 7 (13.40)

W000 (s, X) =

r(1 - s)L(2 - 2s, XUXw) r(s)L(2s, xvXw)

oo and h = i

T(X)Nl-3s7r2s-1

0 we get

r(1 - s)L(2 - 2s, X) r(s)L(2s, X)

13.6. Functional equations for the convolution L-functions We begin by considering the situation in which X = t9rl is a primitive character. Since (D(s, X) is skew diagonal, the vector functional equation (13.31) for the Eisenstein series reduces to the individual equations E. (z, s, x) = Wab(s, x) Et, (z,1 - s, x)

13. Convolution L-functions

244

where b is the complementary cusp to a, i.e. ON = 1. Accordingly the functional equation (13.33) for the Rankin-Selberg L-functions reduces to (47r)--sr(s

+ k - 1)La(f ®9, s) = (Pab(s, X)(47r)s-'r(k - s)Lb(f ®g,1 - s)

provided X = i9rl is primitive. In particular, for cusps a - oo, b - 0 we obtain

Theorem 13.8. Let r = ro(N), and let i9, r7 be two characters to modulus N such that i9(-1) = r7(-1) = (-1)k where k, N are positive integers. Suppose that X = t9r7 is primitive of conductor N. Let f c Sk(r,19) and g E .Mk(r, r7). Put

A. (f ® s) = (21r)-2sN 2'r(s)r(s + k - 1)L(2s, X)L,,,(f (9 9, s) Ao(f ®

s) = (27r)-2sN 2'r(s)r(s + k - 1)L(2s, X)Lo(f ®g, s)

where L,,(f ®g, s) and Lo(f ®g, s) are the Rankin-Selberg convolution Lfunctions attached to cusps oo and 0 respectively (see (13.2), and note that oo, 0 are always singular cusps). Then we have the functional equation (13.41)

A.(f ®g, s) = e,Ao(f ®g,1 - s)

where e), = T(X)N-112 is the normalized Gauss sum. To achieve further simplifications we assume that all 19, 77, i977 are prim-

itive of conductor N and that f, g are newforms. Therefore f , g are eigenfunctions of the involution operator W by Theorem 6.28. More precisely, by Theorem 6.29 we have (13.42)

Wf=

T(z9)N-1/2aN(f) f

21-A

where aN(f) = = (4ir)k-1 f,.(N) is the N-th Fourier coefficient of f at cusp oo (up to displayed normalization). Hence we can express any Fourier coefficient at cusp 0 by the corresponding one at cusp oo; namely, we have (13.43)

fo(n) =

T('9)N-1/2aN(f)f.(n)

A similar relation holds true between the coefficients of g. Combining both relations, we deduce that (13.44)

Lo(f 09,s) =

®g, s).

Inserting (13.44) into (13.41), we get a functional equation which connects the Ranldn-Selberg L-functions for the same cusp at oo (in this case we drop the subscript oo), namely (13.45)

A(f 0 g, s) =eA(f 0 g,1 --s)

13.6. Functional equations for the convolution L-functions

245

where

e=T(t9n)T('t9)T(1))N-3/2aN(f)aN(g).

(13.46)

Another interesting situation is for two newforms of distinct level, say f E Sk(ro(N1), X1) and g E Sk(Fo(N2), X2), where X1, X2 are primitive char-

acters to conductors N1, N2 respectively such that X1(-1) = X2(-1) = (-1)k. In addition suppose that (Ni, N2) = 1, so X = XIX2 is primitive to conductor N = N1N2 and the formula (13.41) applies. However. (13.43) reads

1/2aN,

fo(n) = T(Xi)N1 (f)f.(n) Consequently (13.45) holds true but with a different root number, namely s = T(XIX2)T(X1)T(X2)N-laN1 (f )aN2 (g). By the multiplication formula for Gauss sums the root number simplifies to (13.47)

E = X1(-N2)X2(Nl)aN,(f)aN2(g)

The third situation we wish to analyze is for one cusp form f = g of weight k = 0 (mod 2) with respect to the modular group IF = SL2(Z). In this case we write a(n)nti21e(nz)

(13.48)

f(z)

L(f ®f, s) _

(13.49)

Ia(n)I2n-s

so the integral representation (13.6) becomes (47r)1-k-

r(s + k - 1)L(f ®f, s) = f\H ykI f(z)I2E(z, s)dpz.

The Eisenstein series has the Fourier expansion

9(s)E(z,s) = 9(s)ys + 0(1 -

s)yl-9

+ 4y1/2 (> (a/d)9-112)Ky-1/2(2irInIy) cos(27rnx) n._1 ad=,

where 0(s) = 7r-3I(s)((2s) (see (13.15)-(13.22)). Hence the complete Lfunction (13.50)

A(f x 1, s) = (27r)-2sI (s)I (s + k - 1)((2s)L(f 0 7, s)

13. Convolution L-functions

246

is holomorphic in s E C except for a simple pole at s = 1, and it satisfies the functional equation (13.51)

A(f x f,s)=A(f x f,1-s).

The residue of L(f ®f , s) at s = 1 is (13.52)

res9=1L(f ®f , s) = 3 (r() llf 112.

The convolution L-function L(f 0I, s) is useful for estimating the Fourier coefficients of f. Applying standard complex integration methods, Rankin and Selberg derived the asymptotic formula (13.53)

E la(n) 12 = CX + O(X3/5) n-<X

where c is the constant given by (13.52). Hence it follows that a(n) << n3/10 Compare this bound with (5.19).

13.7. Metaplectic Eisenstein series Next we shall be interested in the convolution L-functions for a cusp form f of weight k on the modular group (its Fourier expansion is (13.48)) against the standard theta function

0(z) =

e(n2z) E ro(4).

Such convolution yields the following zeta-function: 00

Z f(s) _

a(n--)n-9, 1

which has the integral representation (apply Proposition 13.1 for l = 1/2 and + a in place of s) ; (13.54)

(47r)"-Sr' (s+ 2

1)

Zf(s) =

J

`H11(z)E(z, 2 + 4)ctµx

where h(z) = 2y^+3 f(z)B(z) and E(z,s) is the Eisenstein series for the group r = ro(4) at cusp oo given by (13.9). This is called the metaplectic Eisenstein series. The general theory of Selberg applies here, giving analytic

continuation of E(z, s) to the whole s-plane. In the half-plane Re s >, 1/2

13.7. Nletaplectic Eisenstein series

247

there is only a simple pole at s = 3/4 (whose residue is some kind of a theta function); therefore, by the integral representation (13.54) it follows that Z1(s) is holomorphic in Re s > 1/2 (including the line Re s = 1/2) except

possibly for a simple pole at s = 1. Actually we shall see later that s = 1 is a removable singularity (see the next section concerning the symmetric square L-function).

One can also deduce the above analytic properties of the metaplectic Eisenstein series directly from the Fourier expansion. We shall only indicate a few vital steps to observe the pole. First write E (z, s) = y s

where , =

-2s l E 'Ed Cdl + EC 41c>0 (d,c)=1 c

z+ d/c K z + d/c

y

s

Iz + d/c12

-

1. Then split the summation into residue classes d (mod c), 2 class apply Poisson's formula: and for each

E

z + m/c

(z+n/c)K(

JZ+m/cl2)

m_d (mod c)

=

s

y

(n

o00

(IZ+Yt12)S

cd) f o (z+t)K

n n

e(-nt)dt.

For n = 0 the integral is equal to /3(s)y1-8, where = / 7 /t+i)K (t2 + 1)-sdt = X41-sI'(2s - 1)/I'(s - r)I'(s + n), s a() J (\t - Z'

and for n # 0 it is equal to Vs(nz)lnls-1, where ()"

V$(z) _

L

z-t

(izt,2Ye(t)dt..

Note that V3(z) is holomorphic in Re s > 1/2, and it decays exponentially for z = x + iy with jyj --+ oo. More precisely, V3(z) << e-lay as y -> 00 (to see this bound, move the contour of integration up to Im t = 21ry). We have V3(z) = e(x)V8(iy) and Vs(iy) = i2'7rsr(s + K.)-1Wn,s-1/2(47ry)

if y > 0, where W, ,,(y) is the Whittaker function. Next, summing over the residue classes d (mod c), we obtain Gauss suns 9. (C)

> Ed (dc) e

d (mod c)

(nd c

13. Convolution L-functions

248

and summing these over c we form the Dirichlet series gn(cc-2s

ln(S) _ 41c, c>0

Collecting the above results, we arrive at the Fourier expansion E(z, s) = y8 + (13.55) + E Inj8-lln(s)Vs(nz) v(s)y'-8

ni40

where v(s) = Q(s)lo(s). Further explicit computations are technically complicated. According to the general theory the poles of E(z, s) in Re s > 1/2 are inherited from those of v(s). Here it is easy to compute lo(s); we have go(c) = i2 cp(c) if c = b2, or else it vanishes, and hence

E

+ i

10(s) = 1

,p(62)b-4s

=

(1 + i)((4s - 2)

1)((4s

21b, b>0

Therefore v(s) = /3(s)lo(s) is holomorphic in Re s >, 1/2, except for a simple

pole at s = 3/4 with residue (1 + i)/'r(I - s)r(4 + k). The other Gauss sums gn(c) can be computed in terms of the Jacobi symbol Xn(c) = (n) if (c, n) = 1, and consequently ln(s) can be expressed in terms of the Dirichlet L-function L(2s - 1/2, Xn), among other things. For n squarefree Xn is the trivial character, so the Fourier coefficient ln(s) has

a pole at s = 3/4.

13.8. Symmetric power L-functions The idea of Rankin-Selberg convolution provided inspiration for building many other L-functions. We shall close this book by discussing the simplest case of symmetric power L-functions. Throughout, r is the modular group, k is an even positive integer and f is a cusp form for r of weight k which is an eigenfunction of all the Hecke operators T, ,f = A(n)f,

n = 1, 2, ... .

We normalize f so that it has the Fourier expansion 00

f(z)

The Fourier coefficients

a(n)nkzle(nz).

a(n)nik-,

satisfy the multiplication rule

a(?n)a(n) = E a(mn/d2). dI(m,n)

13.8. Symmetric power L-functions

249

The 1-Iecke L-function associated with f has the Euler product 00 a(n)n-s

Lf(s) = E

=

fl(1 - a(p)p`

+

p

2s)-1

n

I

whereas the complete A-function (obtained by including a local factor at

p=00) A f (s) = iT-81 1 s+

k

2 1

J Lf (s)

is an entire function, and it satisfies the functional equation A f(s) = ikA f(1 - s). All these properties have been established in Section 7.2. We factor the Hecke polynomials of f into (13.56)

1 - a(p)p-' + p -2s = (1 - app-e)(1 - Qpp 8)

where ap + ,Op = a(p) and apf3p = I. Recall that the Ramanujan conjecture (proved by Deligne) asserts that lapj = INpl = 1, i.e. ap = f3p. By (13.56) we get am+1 - /4m+1

(13.57)

a(p") = n

p

=

ap - Qp

ap -7Qp. o
Next we reconsider the Rankin-Selberg convolution L-function no

L(f (9 f, s) = > a(n)2n-8 By (13.57) one derives that L(f 0 f, s) has the Euler product of degree four

L(f ® f, s) = fJ(1 + p-s)(1 - an p-8)-' (1 - p-8)-1 (1 - l3Np 4)-1 p

Here the first factor is somewhat different from the other three. Multiplying by C(2s), we obtain a more natural Euler product (still of degree four):

L(f x f, s) = C(2s)L(f ® f, s)

= fJ(1 p

-

app-s)-1(1

- p ")-2(1

-

/app-.v)-I .

13. Convolution L-functions

250

The corresponding A-function is defined by

A(f x f, s) = (27r)-2sr(s)r(s + k - 1)L(f x f, s). We have shown in Section 13.6 that A(f x f, s) is holomorphic in the whole splane except for a simple pole at s = 1; hence also L(f x f, s) is holomorphic

in the s-plane except for a simple pole at s = 1 with positive residue given by (13.52). Observe that the original Selberg-Rankin convolution L(f of, s) may have infinitely many poles on the line Re s = 1/4 at the zeros of ((2s). We also have the functional equation

A(f x f, s) = A(f x f,1- s). Closely related to L(f of, s) and L(f x f, s) are the following series: 00

(13.58)

a(n2)n-s

Z1(s) _

and (13.59)

L (2)(S) = S(2s) Z f (s)

00

a(2) (n)n-' t

where a(2)(n)

= E a(m2). m12 =n

By a(n)2 = 1: a(n2/d2)

On

it follows that (13.60)

((s) Zf (s) = L(f ® f, s)

and (13.61)

((s)L f)(s) = L(f x f, s).

These equations are quite interesting. First they yield meromorphic continuation of Zf(s) and L( )(s) to the whole s-plane. In the half-plane Re s >, 1/2 we already established (in the previous section) that Zf(s) is holomorphic except for a simple pole at s = 1. By virtue of this fact the equations (13.60)

and (13.61) imply that L(f 0 f, s) and L(f x f, s) are divisible by ((s) (i.e. each one has the zeros of ((s)) in Re s > 1/2 at any rate. Furthermore the

13.8. Symmetric power L-functions

251

simple pole of L(f 0 f, s) at s = 1 cancels out with that of ((s), so Z f(s) is holomorphic at s = 1 and does not vanish at this point. Precisely, we have (see (13.52)) (r(k)ti

11f112 > 0.

Zf(1) = 3

The same observation refers to Lf )(s), and we have k

Lf) (1)= 2([' k)

Ilfl12>0.

The fact that Zf(s) is regular at s = 1 implies, by virtue of (13.54), that the residue of the Eisenstein series E(z, s) at s = 3/4 is orthogonal to u2+d.f(z)6(z) for every cusp form f(z). Comparing the Euler products for S(s) and L(f x f, s), we deduce from (13.61) an Euler product for Lf ) (s), namely

Lj 1(s) = (1

(13.62)

2p-.,) `1(1

- ap3pp`S)`1(1

-(app-S)-1

P

This Euler product is called the symmetric square L-function. Combining the functional equations for c(s) and L(f x f, s), we deduce from (13.61) a functional equation for Lf 1(s). Before stating this equar tion, let us modify the relevant gamma factors by applying the duplication formula

r(s) = 7r-1/22s-lr

\2/ r (S 2 1)

This yields

r(s + k - 1) =

1122s-F "' -2. (S + 2

-1) r (s k) 2

.

Accordingly we associate with Lf 1(s) the complete A-function 2

2

JJ

2

and deduce the following functional equation:

Ar(s)=Af)(1-- S). (2)(S) Hence we conclude that Al is entire (holomorphic in s E C), and so is L()(s). These properties of the symmetric square L-function were first.

established by G. Shimura in 1973 [Sh1J.

13. Convolution L-functions

252

By obvious analogy to the preceding investigations one may get an idea how to construct L-functions of higher degree (associated with a newfor111 f). A somewhat simplified construction would be the series 00

Zf

n) (s)

_

a(n°n)n-s

1

or the Euler product

P

7,( 1)(.4) = L; (s) and ?J2)(s) Zf(s). However, the right thing to consider is

for any m > 0. Note that Zf )(s) (13.63)

L(-) (s)

_

(]

-ape-sQpp-s)- 1

P 0!5j<»e

This product converges absolutely in Re s > 1; it is called the m-th symmet-

ric power L-function of f. In particular we have L f)(s) = ((s), Lt )(s) _ L f(s), and L(2) (s) as before. Note that either Zf)(s) or Pf'n)(s) approximate L(T) (s) reasonably well in the half-plane Re s > 1/2, in the sense that the ratio is given by an Euler product which converges absolutely in Re s > 1/2 (to see this, compare (13.63) with (13.57)). However, such a similarity is false in deeper ranges, for one can show that if m is large the series

Z( ) (s) does not admit analytic continuation over the line Re s = 0 (the abscissa of meromorphy) while L("n) (s) is expected to satisfy the following properties

Conjecture. The symmetric power L-functions are entire. In fact, the complete function defined by

Af )(s) = 7r-(m+1)Z I

C3 + rj l L(n)(s)

ojItn

/J

where rj are suitable constants, is entire, and it satisfies the functional equation of type

Aje)(s)=CAf)(1-s). This conjecture is part of a larger program of Langlands which is rooted deeply in various subjects of contemporary mathematics. Even before its completion this profound program of Langlands has had a significant impact

13.8. Symmetric power L-functions

253

on the direction of modern number theory (see the comments on Artin Lfunctions in Section 12.6).

The meromorphic continuation of L!) (s) over the whole s-plane and the predicted functional equations have been established for m <, 4. Almost nothing is known about the symmetric power L-functions of higher degree. There are fascinating applications of the available results, in particular those for the symmetric square L-function (cf. [DI] and [Iw3]).

Bibliography

[AL]

A. Atkin and J. Lehner, Hecke operators on Fo(m), Math. Ann. 185 (1970), 134160.

[Asa] [Boc]

T. Asai, On the Fourier coefficients of automorphic forms at various cusps and some applications to Rankin's convolution, J. Math. Soc. Japan 28 (1976), 48-61. S. Bochner, Lectures on Fourier Integrals, Princeton University Press, Princeton,

NJ, 1959. (CH) R. Courant and D. Hilbert, Methods of Mathematical Physics, Vol. 1, Wiley-Interscience, New York, 1961. [Dav) H. Davenport, Multiplicative Number Theory, Markham, Chicago, 1967.

W. Duke and H. Iwaniec, Estimates for coefficients of L-functions II, Proceedings of the Amalfi Conference on Analytic Number Theory in 1989, Universita di Salerno, 1992, pp. 71-82. [Dul] W. Duke, Hyperbolic distribution problems and half-integral weight Maass forms, Invent. Math. 92 (1988), 73-90. [Du2] W. Duke, The dimension of the space of cusp forms of weight one, Internat. Math. Res. Notices 2 (1995), 99-109. [GR] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products, Academic Press, London, 1965. [Hell E. Hecke, Uber eine neue Art von Zetafunktionen, Math. Zeit. 6 (1920), 11-51. (He2) E. Hecke, Analytische Funktionen and algebraische Zahlen, Abhandl. Math. Sem. Hamburg, vol. 3, 1924, pp. 213-236 (see his Werke, p.381). [Hej] D. A. Hejhal, The Selberg Trace Formula for PSL(2,R), Lecture Notes in Math., [DI)

(HT]

(Iw1)

vol. 1001, Springer, 1983. G. H. Hardy and E. C. Titchmarsh, Solutions of some integral equations considered by Bateman, Kapteyn, Littlewood, and Milne, Proc. London Math. Soc. 23 (1924), 1-26. H. Iwaniec, Fourier coefficients of modular forms of half integral weight, Invent. Math. 87 (1987), 385 -401.

255

Bibliography

256

11w2]

(lw3]

H. Iwaniec, Introduction to the Spectral Theory of Automorphic Forms, Bibl. Rev. Mat. Iber., Madrid, 1995. H. Iwaniec, The lowest eigenvalue for congruence groups, in Topics in Geometry (memorial volume of Joseph D'Atri) (S. Gindikin, ed.), Birkhi user, 1996, pp. 203212.

[1w4]

H. Iwaniec, Mean values for Fourier coefficients of cusp forms and sums of Kloosterman sums, Journme Arithmetiques (1980); London Math. Soc. Lecture Note Ser., vol. 56, Cambridge University Press, Cambridge, 1982, pp. 306-321.

H. D. Kloosterman, On the representation of numbers in the form ax2 + bye + cz2 + dt2, Acta Math. 49 (1926), 407-464. [Lan] S. Lang, Introduction to Modular Forms, Springer-Verlag, 1976. W. Li, Newforms and functional equations, Math. Ann. 212 (1975), 285-315. [Li] [Lom] G. A. Lomadze, On the number of representations of positive integers as the sum of nine squares, Acta Arith. 68 (1994). 245-253. (R.ussian) [Mall A. V. Malyshev, On formulas for the number of representations of integers by positive quadratic forms, in Current Problems of Analytic Number Theory, Nauka i Tech., Minsk, 1974, pp. 119-136. (Russian) (Klo]

T. Miyake, Modular Forms, Springer-Verlag, 1989. H. Petersson, Zur analytischen Theorie der Crenzk eisgruppen. I-V, Math. Ann. 115 (1938), 23-67; 175-204; 518-572; 670--709; Math. Z. 44 (1938), 127-155. [Ran] R. A. Rankin, Modular Forms and Functions, Cambridge University Press, Cambridge, 1977. H. Salie, Uber die Kloostermanschen Summen S(u, v; q), Math. Z. 34 (1931), 91[Sall [Miy]

[Pet]

109. (Sar]

P. Sarnak, Some Applications of Modular Forms, Cambridge Univ. Press, Cam-

[Sell

bridge, 1990. A. Selberg, On the estimation of Fourier coefficients of modular forms, Proc. Symp.

[Se2]

(Shl]

Pure Math., vol. 8, Amer. Math. Soc., Providence, RI, 1965, pp. 1-15. A. Selberg, Collected Papers, Vol. I, Springer, 1989. G. Shimura, On modular forms of half integral weight, Ann. Math. 97 (1973), 440-481.

[Sh2] [Sie]

(Ser]

(Ter) [Wei]

G. Shimura, Introduction to the Arithmetic Theory of Automorphic Functions, Princeton Univ. Press, Princeton, NJ, 1971. C. L. Siegel, Lectures on Quadratic Forms, Tata Institute, Bombay, 1967. J-P. Serre, Modular forms of weight one and Cairns representations, in Algebraic Number Fields (A. Frohlich, ed.), Academic Press, 1977, pp. 193-268. A. Terras, Harmonic Analysis on Symmetric Spaces and Applications, Vol. I, Springer-Verlag, New York, 1985. A. Weil, Uber die Bestimmung Dirichletscher Reihen durch Funktionalgleichungen, Math. Ann. 168 (1967), 149-156.

Index absolute inner product, 6.5 adjoins operator, 6.4 ambiguous class, 12.5 angle of a Kloosterman sum, 4.4 Artin conjecture. 12.6 Artin L-function, 12.6 automorphic form, 2.7 automorphs of a quadratic form, 11.2 Bernoulli numbers. 1.3 B-group of integral translations, 2.5 chart, 2.3

circle method of Kloosterman, 11.4 class group, 12.2 class group L-function, 12.4 complete L-function, 7.2 complex conjugation operator K, 6.7 congruence group, 2.4 congruent number, 8.2 congruent theta function, 10.1 conjugacy class, 2.2 consistency condition, 2.6 converse theorem, 7.4 convolution L-functions, 13.1 criterion for cusp forms, 7.4 cusp, 2.3 cusp form, 2.7

cusp forms for the modular group, 1.6 cuspidal zone, 2.3 decomposition group, 12.6 Dedekind sum, 2.8 Dedekind zeta-function, 12.2 deformation factor, 2.1 determinant of a quadratic form, 11.2 diagonal symbol, 5.5 different of a field, 12.2 differential operator of Euler, 9.2 Dirichlet class number formula, 11.2 discontinuous group, 2.3 discrete group, 2.3 discriminant function, 1.2 discriminant of a field, 12.2 discriminant of a quadratic form, 11.2 discriminant of an elliptic curve, 8.1 double cosets, 2.5

cigenvalue rl of W, 6.7 Eisenstein series E(z,Q), 11.3 Eisenstein series for the modular group, 1.3 Eisenstein series of weight zero, 13.2 Eisenstein series with a multiplier, 3.2

ellipsoid KA, 9.1 elliptic curve, 8.1 elliptic function, 1.2 elliptic. hvperboic. parabolic motions. 2.2 equidistribution, 11.6 equivalence class of forms, 11.2 equivalent quadratic forms, 11.2 eta-function, 2.8 Euler product for llecke L-function, 6.8 even quadratic form A[xl, 10.1

factor system of weight k, 2.6 courier coefficient at a cusp, 2.7 courier expansion at a cusp, 2.7 courier transform, 1.1 fractional part, 1.1 Flicke involution W, 6.7 Flobenius aut.omorphism, 12.6 Frobenius conjugacy class, 12.6 Fuchsian group, 2.3 flichsian group of the first kind, 2.3 functional equation for Hecke L-function, 7.2 fundamental domain, 2.3 fundamental units, 12.2 gamma function, 7.1 Gauss sum, 4.3 Gauss sun

12.2

generating function of Poincare series, 3.1 genus characters, 12.5 genus group, 12.5 genus mass, 11.2 genus of quadratic forius, 11.2 geodesics, 2.1 Crossencharacter, 12.2

group of fractional ideals, 12.2

harmonic function, 9.2 harmonic polynomial, 9.2 Hasse-Weil L-function, 8.1 Hecke congruence group, 2.4 Hecke eigencuspform, 6.6 257

Index

258

Hecke eigenform, 6.4 Hecke 1-function, 7.2 Hecke operators, 6.2 Hilbert space of cusp forms, 3.3

homogenuous space R', 9.3 hyperbolic isometrics, 2.1 hyperbolic measure, 2.1 hyperbolic plane, 2.1

inertia group, 12.6 inner product, 3.3 integral basis, 12.2 integral representation for L-function, 7.2 isometric circle, 2.1

isotropic vector, 9.1

Jacobi function 8(z, v), 10.1 Jacobi inversion formula, 10.2 Jacobi sum, 8.3

Jacobi symbol extended, 2.8 Jacobian of a diffeomorphism, 9.1 j-function, 2.1 j-invariant function, 1.3 Kloostorman sum with a multiplier, 3.2 Kronecker symbol, 12.3

Laplace operator A, 13.2 Laplace operator AA, 9.1 lattice, 1.2 Legendre function P,m(z), 9.4 Legendre polynomial 9.4

level of a congruence group, 2.4 L-function with Grossencharacter, 12.2 linear fractional transformations, 1.3 local densities 69(n,Q),6o (n, Q), 11.2 metaplectic Eisenstein series, 13.7

modular function of weight k, 1.3 modular group, 1.3

multiplicity-one principle, 6.6 multiplier system of weight k, 2.6 Neumann series, 5.5 newform, 6.6 normal operator, 6.4 normal polygon, 2.3 number field, 12.2

principal genus, 11.2, 12.5

principal ideals, 12.2

quadratic form of level one, 11.3 quadratic form of Liouville type, 11.3 quadratic form of Mordell type, 11.3 quadratic form of Weil type, 11.3 quadratic form of Weil-Eichler type, 11.3 quaternion group, 2.3 Ramamijan sum, 6.7 Ramamujan r-function, 1.4 ramified primes, 12.2 Rankin-Selberg integrals, 13.1 ray class group, 12.2 real and complex embeddings, 12.2 regulator, 12.2 representation number r(n,Q), 11.2 residue class field, 12.6 Riemann surface, 2.3 Riemann zeta-function, 7.1 ring of integers, 12.2 root number, 8.1

Sali6 sum, 4.6 Sato-Tale measure, 4.4 scaling matrix, 2.3 scattering matrix, 13.2 Shimura-Taniyama conjecture, 8.1 Siegel mass formula, 11.2 sifting operator Sv, 6.7 signature, 2.3 singular cusp, 2.7

singular series, 11.5 slash operator, 2.6

space of modular forms of weight k, 1.6 special orthogonal group SO(n), 9.3 sphere 9.3 spherical function, 9.2 spherical function Y(O, sp), 9.4 spherical polar coordinates, 9.3 stabilty group, 2.3 standard fundamental polygon for the modular group, 1.5 standard polygon, 2.3 Stokes' theorem, 9.2

surface Laplacian 0', 9.3

orbit of a point, 2.3 orthogonal group O(n), 9.3

symmetric power L-function, 13.8 symmetric square L-function, 13.8

parabolic vertex, 2.3 periodic function, 1.1 Poincark differential, 2.1 Poincar6 series, 3.1 Poisson summation formula, 1.1 positive definite quadratic form, 9.1 primary integer, 8.3

Tchebyshev polynomials, 4.4

prime discriminant, 12.5 principal congruence group, 2.4

theta function 8(z), 10.1 theta function 8x(z), 10.5 theta multiplier, 2.8 theta-function, 2.8 total mass, 11.2 trace of motion, 2.2 tridiagonal metrix, 10.4 twisted sutomorphic form, 7.3

Index

unimodular transformation, 1.3 unit group, 12.2 upper half-plane, 1.3

volume element, 9.1

Weierstrass equation, 1.2 Weierstrass equation for an elliptic curve, 8.1

Weierstrass p-function, 1.2

Weil's bound for Kloosterman sums, 4.3 Whittaker function, 13.7 width of a cusp, 2.4 zonal spherical function, 9.4

259

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