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Anatoli Andrianov

Introduction to Siegel Modular Forms and Dirichlet Series

ABC

Anatoli Andrianov Russian Academy of Sciences Steklov Institute of Mathematics Fontanka 27 191023 St. Petersburg Russia [email protected]

Editorial board: Sheldon Axler, San Francisco State University, San Francisco, CA, USA Vincenzo Capasso, University of Milan, Milan, Italy Carles Casacuberta, Universitat de Barcelona, Barcelona, Spain Angus MacIntyre, Queen Mary, University of London, London, UK Kenneth Ribet, University of California, Berkeley, CA, USA Claude Sabbah, Ecole Polytechnique, Palaiseau, France Endre Süli, Oxford University, Oxford, UK Wojbor Woyczynski, Case Western Reserve University, Cleveland, OH, USA

ISBN 978-0-387-78752-7 DOI 10.1007/978-0-387-78753-4

e-ISBN 978-0-387-78753-4

Library of Congress Control Number: 2008938066 Mathematics Subject Classiﬁcation (2000): 11Fxx, 11F66 This is a translation of the Dutch, Meetkunde, originally published by Epsilon–Uitgaven, 2000. c Springer Science+Business Media, LLC 2009 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identiﬁed as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed on acid-free paper springer.com

To Goro Shimura and to my granddaughter Sasha

Preface

Several years ago I was invited to an American university to give one-term graduate course on Siegel modular forms, Hecke operators, and related zeta functions. The idea to present in a concise but basically complete and self-contained form an introduction to an important and developing area based partly on my own work attracted me. I accepted the invitation and started to prepare the course. Unfortunately, the visit was not realized. But the idea of such a course continued to be alive till after a number of years this book was finally completed. I hope that this short book will serve to attract young researchers to this beautiful field, and that it will simplify and make more pleasant the initial steps. No special knowledge is presupposed for reading this book beyond standard courses in algebra and calculus (one and several variables), although some skill in working with mathematical texts would be helpful. The reader will judge whether the result was worth the effort. Dedications. The ideas of Goro Shimura exerted a deep influence on the number theory of the second half of the twentieth century in general and on the author’s formation in particular. When Andr`e Weil was signing a copy of his “Basic Number Theory” to my son, he wrote in Russian, ”To Fedor Anatolievich hoping that he will become a number theoretist”. Fedor has chosen computer science. Now I pass on the idea to Fedor’s daughter, Alexandra Fedorovna. Contents. The main objective of this book is to give a concise but basically complete and self-contained introduction to the multiplicative theory of Siegel modular forms, Hecke operators, and zeta functions, including the classical case of modular forms in one variable. Chapter 1 contains a compressed exposition of essential features of the theory of Siegel modular forms of integral weight for congruence subgroups of the symplectic modular group Spn (Z) of arbitrary genus n. Chapter 2 treats analytical properties of radial Dirichlet series attached to modular forms of genera 1 and 2. Chapter 3 is dealing with the abstract theory of Hecke–Shimura rings for symplectic and related group. Action of Hecke operators on Siegel modular forms is considered in Chapter 4. In Chapter 5, we examine applications of

vii

viii

Preface

Hecke operators to a study of multiplicative properties of Fourier coefficients of modular forms and the related Euler product factorization of radial Dirichlet series attached to eigenfunctions. This leads us to Hecke zeta functions of modular forms in one variable and to spinor (or Andrianov) zeta functions of Siegel modular forms of genus two. At the end of this chapter we arrive at the proof of the analytic continuation and functional equation (under certain assumptions) of Euler products associated with modular forms of genus two. The book contains a number of exercises that usually consider some interesting points not included in the main text, partly for the reasons of space and partly because of their special character. References. I try to present the proofs in full detail whenever it is reasonable and possible. The main text contains no references. Essential references are included in the Notes at the end of the book. It should be noted that the author does not pretend to give an encyclopedic survey of modular forms or a complete bibliography, but rather to hint at certain principal points of the theory and illustrate how they have been reached. Acknowledgments. I would like to express my deep gratitude to Fedor Andrianov, the assumed co–author, who carefully read the manuscript and suggested a number of improvements. I am very grateful to Mark Spencer, of Springer New York for continued interest in this book and highly effective cooperation. Great acknowledgment must also go to Charlene Cruz Cedras, the editorial assistant of Mark Spencer, who has given me valuable assistance and support. Finally, I must acknowledge my grateful thanks to Dr. David Kramer for highly careful editing of the manuscript, that I have never encountered. St Petersburg January/April 2008

Anatoli Andrianov

Contents

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii Introduction: The Two Features of Arithmetic Zeta Functions . . . . . . . . . . .

1

1

Modular Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.1 The Symplectic Group and the Upper Half-Plane . . . . . . . . . . . . . . . . 7 1.2 Fundamental Domains for the Modular Group . . . . . . . . . . . . . . . . . . 11 1.3 Modular Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2

Dirichlet Series of Modular Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Radial Dirichlet Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Mellin Transform of Cusp Forms in One Variable . . . . . . . . . . . . . . . . 2.3 Transformations of Lobachevsky Half-Spaces . . . . . . . . . . . . . . . . . . . 2.4 Radial Series and Eisenstein Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Properties of Radial Series for Sums of Two Squares . . . . . . . . . . . . .

3

Hecke–Shimura Rings of Double Cosets . . . . . . . . . . . . . . . . . . . . . . . . . . 63 3.1 An Approach to Multiplicativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 3.2 Abstract Rings of Double Cosets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 3.3 Rings of Double Cosets of the General Linear Group . . . . . . . . . . . . . 75 3.4 Rings of Double Cosets of the Symplectic Group . . . . . . . . . . . . . . . . 85 3.5 Rings of Triangular-Symplectic Double Cosets . . . . . . . . . . . . . . . . . . 106

4

Hecke Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 4.1 Hecke Operators for Congruence Subgroups . . . . . . . . . . . . . . . . . . . . 119 4.2 Action of Hecke Operators for Γn0 (q) . . . . . . . . . . . . . . . . . . . . . . . . . . 125 4.3 Hecke Operators and Siegel Operator . . . . . . . . . . . . . . . . . . . . . . . . . . 130

5

Euler Factorization of Radial Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 5.1 Radial Series of Genus One and Zeta Functions . . . . . . . . . . . . . . . . . 138 5.2 Binary Radial Series and Gaussian Composition . . . . . . . . . . . . . . . . . 145 5.3 Zeta Functions of Eigenforms for Genus 2 . . . . . . . . . . . . . . . . . . . . . . 165

41 41 44 47 55 58

ix

x

Contents

Conclusion: Other Groups, Other Horizons . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179

General Notation

The letters N, P, Z, Q, R, and C are reserved for the set of positive rational integers, the set of positive rational prime numbers, the ring of rational integers, the field of rational numbers, the field of real numbers, and the field of complex numbers, respectively. K[x1 , . . . , xn ] is the ring of polynomials in x1 , . . . , xn with coefficients in K, and Am n is the set of all m × n−matrices with entries in a set A. If M is a matrix, tM always denotes the transpose of M, σ (M) for a square M is the trace of M, and M for a complex matrix M means the matrix with conjugate entries. If Y is a real symmetric matrix, then Y > 0 (resp., Y 0) means that Y is positive definite (resp., positive semidefinite). For two matrices A and B of suitable dimensions we write A[B] = tBAB.

xi

Introduction: The Two Features of Arithmetic Zeta Functions

A zeta function in arithmetic is, generally speaking, a generating function for an arithmetic problem written in the form of a Dirichlet series. A well stated zeta function must have at least two principal features: an Euler product factorization and an analytic continuation over the whole complex plane satisfying functional equations. The first reflects relations between the global arithmetic problem and its localizations, while the second provides a kind of reciprocity between the localizations. Let us illustrate this with some examples Riemann Zeta Function. The Riemann zeta function, ∞

1

∑ ns

ζ (s) =

(Re s > 1),

n=1

is the generating function for the numbers of ideals of given norm in the ring Z of rational integers. It has an Euler product factorization of the form

ζ (s) = ∏ p

1 1− s p

−1 (Re s > 1),

the product being taken over all rational prime numbers p. It was Bernhard Riemann who proved in the middle of the nineteenth century that ζ (s) has an analytic continuation over the whole complex s-plane, is holomorphic except for a simple pole of residue 1 at s = 1, and satisfies the functional equation that the function π −s/2Γ (s/2)ζ (s), where Γ is the gamma function, is invariant under the substitution s → 1 − s. He also discovered that the problem of distribution of prime numbers is closely connected with the location of complex zeros of the zeta function in the vertical strip 0 ≤ ℜs ≤ 1. At the end of the century, J. H´adamard and Ch. de la Vall´ee Poussin proved that ζ (s) has no zeros on the line ℜs = 1, which implied the famous asymptotic formula for the number π (x) of prime numbers not exceeding x, x (x → ∞). π (x) ∼ log x

1

2

Introduction

Zeta Function of Algebraic Varieties. The global zeta function of a nonsingular algebraic variety V over the field Q of rational numbers is defined by an Euler product ζ ∗ (V, s) = ∏ ζ Vp , p−s p

p−s ),

where p runs over all prime numbers such that V of local zeta functions ζ (Vp , has a “good” and, in particular, nonsingular reduction Vp modulo p, i.e., the variety over the finite field F(p) of p elements obtained by replacing equations defining V with corresponding congruences modulo p. The local zeta function is the zeta function of Vp defined by

ζ (Vp , t) = exp

∞

∑ N(pδ )t δ /δ

,

δ =1

where N(pδ ) is the number of points on Vp with coordinates in the finite field F(pδ ) of pδ elements. According to Bernhard Dwork, the local zeta functions ζ (Vp , t) of nonsingular varieties Vp are rational fractions in t. It follows that the global zeta function ζ ∗ (V, s) can be written as a Dirichlet series convergent in a right halfplane of the complex variable s. It is generally believed that the zeta function can be analytically continued over the whole s−plane as a meromorphic function and satisfies functional equations, but it is doubtful that a human being living now will see a complete proof. Nevertheless, even particular cases present considerable interest (all genuine number theory consists of particular cases; everything else is algebra). Let us consider a (projective) elliptic curve E:

y2 z = x3 + axz2 + bz3

(a, b ∈ Z).

The points on E with coordinates in Q form an abelian group, which we denote by EQ ; a theorem of L.J. Mordell tells us that the group EQ is finitely generated, i.e., is a product of a finite group by a lattice of finite rank g. A principal problem of the theory is to determine the group EQ , and, in particular, to determine the rank g. In the mid 1960’s B.J. Birch and H.P.F. Swinnerton-Dyer put forward revolutionary conjectures connecting the group EQ with the zeta function ζ ∗ (E, s) of the curve. Let us recall some details. A prime number p is said to be good if it does not divide 6(27b2 + 4a3 ). For such a prime p, the reduction Ep :

y2 z ≡ x3 + axz2 + bz3

(mod p)

of E modulo p is an elliptic curve over F(p). It is well known that the zeta function of E p over F(p) has the form

ζ (E p , t) =

1 − (1 + p − N(p))t + pt 2 . (1 − t)(1 − pt)

Introduction

3

Then we may define a zeta function of E by −1 ζ ∗ (E, s) = ∏ 1 − (1 + p − N(p))p−s + p1−2s , p

where the product is taken over all good primes. It converges for ℜs > 3/2. Then the main Birch–Swinnerton-Dyer conjecture says that ζ ∗ (E, s) has a zero of order g at s = 1. Generally, it is still open. Zeta Functions of Automorphic Forms. Despite the clear importance of zeta functions of algebraic varieties, algebraic geometry provides no means for their investigation. The only hope is to relate them to techniques coming from an analytic background, probably with zeta functions of automorphic forms, which, in contrast, usually have vast means for analytic investigation, but often lack clear arithmetic motivation. Let us consider the simplest case of a Dirichlet series of modular forms of integral weight for congruence subgroups K of the modular group SL2 (Z). Let us recall the corresponding definitions. A function F on the upper half-plane H = {x + iy ∈ C y > 0} is said to be a modular cusp form of weight k for the group K if it is holomorphic on H, equals zero at all cusps of K, and satisfies az + b ab = F(z) for each ∈ K. (cz + d)−k F cd ac + d All such functions form a finite dimensional space N = Nk (K) over the field C of complex numbers. If the group K contains the matrix 10 11 , then every F ∈ N can be presented by a Fourier series absolutely convergent on H of the form ∞

F(z) =

∑ f(m)e2π imz

m=1

with constant Fourier coefficients f(m). Let us associate with F the Dirichlet series of F defined by ∞ f(m) Z(F, s) = ∑ . s m=1 m The series converges absolutely in a right half-plane of the variable s, and can be presented there by means of a Mellin integral

Φ (s) = Φ (F, s) =

∞ 0

F(iy)ys−1 dy = (2π )−sΓ (s)Z(F, s).

Suppose that the group K satisfies −1 0 −1 0 −1 K =K q 0 q 0

4

Introduction

for a positive integer q. Then it is easy to check that for each cusp form F ∈ N, the function (F ω )(z) = q−k/2 z−k F(−1/qz) again belongs to N and satisfies F ω ω = (−1)k F. It follows that one can write the direct sum decomposition N = N+ + N− , where for F ∈ N± , F ω = ±ik F. Exercise. Prove the above assertions. If F ∈ N± , then F(i/qy) = ±(−1)k qk/2 yk F(iy)

(y > 0),

and we can write, for ℜs sufficiently large, that

Φ (s) =

q−1/2 0

∞

=

F(iy)ys−1 dy +

q−1/2

∞ q−1/2

F(iy)ys−1 dy

F(i/qy)(1/qy)s−1 (1/qy2 )dy +

= ±(−1)k qk/2−s

∞ q−1/2

∞

F(iy)yk−s−1 dy +

q−1/2 ∞

F(iy)ys−1 dy

q−1/2

F(iy)ys−1 dy.

Both of the last integrals are holomorphic for all s, and so is the function Φ (s). Moreover, the last expression implies that

Φ (k − s) = ±(−1)k qs−k/2 Φ (s), which is the functional equation for the Dirichlet series Z(F, s). Note that the simplest of the groups K satisfying the given conditions is the group

ab Γ0 (q) = ∈ SL2 (Z) c ≡ 0 (mod q) . cd The problem of Euler product factorization of the Dirichlet series corresponding to modular forms of integral weight for the groups of the type Γ0 (q) was essentially solved by Erich Hecke in 1937 and completed by A.O.L. Atkin and J. Lehner in 1970. In particular, it was found that although the Dirichlet series of a cusp form does not necessarily have an Euler product factorization, the space of cusp forms has a basis consisting of forms with Dirichlet series decomposable into Euler products, which we call the zeta functions of modular forms. Such forms can be characterized as eigenfunctions of certain rings of linear operators, the Hecke operators, acting on the spaces Nk (K). Since the nineteenth century, the main arithmetic application of modular forms had been the analytical theory of integral quadratic forms. The reason is that the generating Fourier series with coefficients equal to numbers of integral representations

Introduction

5

of positive integers by a positive definite integral quadratic form is a modular (not cusp) form. But in the middle of the twentieth century Goro Shimura and Yutaka Taniyama proposed famous conjectures relating modular forms and elliptic curves over Q. The Shimura–Taniyama conjecture includes the conjecture that the zeta function ζ ∗ (E, s) of every elliptic curve E over Q completed by appropriate p-factors for bad primes is, in fact, the zeta function of a cusp form of weight 2 for the group Γ0 (q), where q is the product of some degrees of bad primes. In 1985, G. Frey made the remarkable observation that this conjecture would imply Fermat’s last theorem. The precise relation of the two was established later by K.A. Ribet, which allowed A. Wiles in 1995 to prove the Fermat’s last theorem, one of the brightest achievements of mathematics of the twentieth century. One can hardly doubt that the relation between zeta functions of elliptic curves and zeta functions of modular forms in one variable described by the Shimura–Taniyama conjecture is only a particular case of some general links between global zeta functions of algebraic varieties and zeta functions of automorphic forms. Speaking of abelian varieties in place of elliptic curves, one can expect that modular forms in one variable should be replaced by Siegel modular forms for congruence subgroups of the symplectic modular group Γ n = Spn (Z), and this expectation is supported by some numerical evidence.

Chapter 1

Modular Forms

1.1 The Symplectic Group and the Upper Half-Plane Symplectic Matrices. A matrix M ∈ C2n 2n is said to be symplectic if it satisfies the relation 0 1n t (1.0) MJM = µ (M)J with J = Jn = −1n 0 and a nonzero scalar µ (M) called the multiplier of M, where 0 = 0n and 1n are the zero matrix and the unit matrix of order n, respectively. It is clear that the product of two symplectic matrices of the same order is again a symplectic matrix, and the multiplier of a product is the product of multipliers of factors. Lemma 1.1. Let M = CA DB , where the blocks A, B, C, and D are complex square matrices of order n, and let µ be a nonzero complex number. Then the following conditions are equivalent: (1) M is symplectic with the multiplier µ (M) = µ ; (2) tM is symplectic with the multiplier µ (tM) = µ ; (3) M is invertible, and t D −tB −1 ; µM = −tC tA

(1.1)

(4) the blocks A, B, C, and D satisfy the conditions t

AC = tCA,

BD = tDB,

and

t

AD −t CB = µ 1n

(1.2)

C tD = D tC,

and

A tD − B tC = µ 1n .

(1.3)

t

or the conditions A tB = B tA,

Proof. This is an easy but useful exercise on multiplication of block–matrices, which we leave to the reader.

A. Andrianov, Introduction to Siegel Modular Forms and Dirichlet Series, Universitext, DOI 10.1007/978-0-387-78753-4 1, c Springer Science+Business Media LLC 2009

7

8

1 Modular Forms

Exercise 1.2. Prove the lemma. In the course of our arithmetic considerations we shall be interested in discrete subgroups and subsemigroups of the general real positive symplectic group of genus n consisting of all real symplectic matrices of order 2n with positive multipliers: 2n t G = Gn = GSp+ MJM = µ (M)J, µ (M) > 0 . (1.4) n (R) = M ∈ R2n Action on Upper Half-Plane. The group G is a real Lie group acting as a group of analytic automorphisms on the n(n + 1)/2−dimensional open complex variety (1.5) H = Hn = Z = X + iY ∈ Cnn tZ = Z, Y > 0 , called the upper half-plane of genus n, with the action defined by the rule AB GM= : Z → M Z = (AZ + B)(CZ + D)−1 (Z ∈ H). CD

(1.6)

In order to verify this, we have to check that the mapping (1.6) is always defined, maps the upper half-plane into itself, and satisfies (MM ) Z = M M Z

(M, M ∈ G, Z ∈ H).

(1.7)

The relations (1.7) follow from the definition by a formal comparison of both parts, provided that both parts are defined. So it would be sufficient to prove the following two lemmas. Lemma 1.3. For every matrices M = CA DB ∈ G and Z ∈ H, the matrices J(M, Z) = CZ + D

(1.8)

are nonsingular and satisfy the rule J(MM , Z) = J(M, M Z )J(M , Z)

(M, M ∈ G, Z ∈ H).

(1.9)

Proof. The relation (1.9) formally follows from the definitions if the matrix J(M , Z) is nonsingular. As to nonsingularity, we note first that the matrix (1.8) is nonsingular for every M = CA DB ∈ G and Z = i1n . Indeed, otherwise, the matrix (Ci + D) t(Ci + D) = C tC + D tD + i(C tD − D tC) = C tC + D tD (see (1.3)) is singular, and so since it is symmetric and semidefinite, there is a nonzero real n-column T such that tT (C tC + D tD)T = 0, whence tTC tCT = 0 and tT D tDT = 0, and so tTC = tT D = 0. The last relations imply that the rank of the matrix (C, D) is less than n, which is impossible, since the matrix M is nonsingular.

1.1 The Symplectic Group and the Upper Half-Plane

9

Now note that each matrix Z = X +iY ∈ H can be written in the form Z = M i1n

with a matrix M ∈ G. It is sufficient to take M =

A1 X tA−1 1 0

tA−1 1

, if Y = A1 tA1 . Finally,

by (1.9) with Z = i1n , we get J(MM , i1n ) = J(M, M i1n )J(M , i1n ) = J(M, Z)J(M , i1n ), which implies that J(M, Z) is nonsingular. Lemma 1.4. Let M = CA DB ∈ G and Z = X +iY ∈ H then the matrix Z = X +iY = M Z is symmetric, and Y = µ (M) t(CZ + D)−1Y (CZ + D)−1 .

(1.10)

In particular, Z ∈ H. Proof. As follows easily from the relations (1.2), the matrix t

(CZ + D)M Z (CZ + D) = (Z tC + tD)(AZ + B) = Z tCAZ + tDAZ + Z tCB + tDB

is symmetric, and so the matrix Z = M Z is symmetric too. Further, by (1.2), we have

(CZ + D)(Z − Z )(CZ + D) = (Z tC + tD)(AZ + B) − (Z tA + tB)(CZ + D)

t

= Z(tCA −t AC)Z + (tDA −t BC)Z + Z(tCB −t AD) = µ (M)(Z − Z). The formula (1.10) and the lemma follow. The above formulas will allow us to find a G-invariant element of volume on H. The upper half-plane H is clearly an open subset of the n(n + 1)-dimensional real affine space, and we can consider the Euclidean element of volume on H, dZ =

∏

1≤α ≤β ≤n

dxαβ dyαβ

(Z = (xαβ + iyαβ ) ∈ H).

(1.11)

Lemma 1.5. For each matrix M = CA DB ∈ G, the element of volume (1.11) satisfies the relation dM Z = µ (M)n(n+1) | det(CZ + D)|−2n−2 dZ. + iy ) = M Z . Proof. For Z = (zαβ ) = (xαβ + iyαβ ) ∈ H, we set Z = (zγδ ) = (xγδ γδ , y with We have to compute the absolute value of the Jacobian of the variables xγδ γδ respect to the variables xαβ , yαβ . First, we shall consider the transformation of the differentials of the complex variables zαβ . For Z1 , Z2 ∈ H, since Z2 is symmetric, we get

Z2 − Z1 = (Z2 tC + tD)−1 (Z2 tA + tB) − (AZ1 + B)(CZ1 + D)−1 = µ (M)(Z2 tC + tD)−1 (Z2 − Z1 )(CZ1 + D)−1 ,

10

1 Modular Forms

where we have also used the relations (1.2). It follows that D Z = µ (M)(Z tC + tD)−1D Z(CZ + D)−1 , where D Z = (dzαβ ) and D Z = (dzγδ ) are the matrices of differentials of the variables zαβ and zγδ , respectively. Note that if ρ (U) with U ∈ GLn (C) is the transformation (vαβ ) → U(vαβ ) tU of variables vαβ = vβ α with 1 ≤ α , β ≤ n, then det ρ (U) = (detU)n+1 . This can be easily checked by ordering the variables vαβ lexicographically and replacing U by an upper triangular matrix of the form W −1UW . Let d Z and d Z be the columns with entries dzαβ (1 ≤ α , β ≤ n) and dzγδ (1 ≤ γ , ≤ δ ≤ n) arranged in a fixed order. Then the above considerations imply the relation

d Z = ρ ( µ (M) t(CZ + D)−1 )dd Z.

Taking d Z = d X + iddY , d Z = d X + iddY , and ρ ( µ (M) t(CZ + D)−1 ) = R + iS, we obtain that d X = Rdd X − SddY, d Y = Sdd X + RddY. Thus, the Jacobian equals R −S 1n −i1n R −S 1n i1n det = det 0 1n 0 1n S R S R R + iS 0 = det = µ (M)n(n+1) | det(CZ + D)|−2n−2 . 0 R − iS By combining the above lemma and formula (1.10), we get the following result. Proposition 1.6. The element of volume on H given by d ∗ Z = detY −(n+1) dZ

(Z = X + iY ∈ H),

(1.12)

where dZ = dXdY is the Euclidean element of volume (1.11), is invariant under all transformations of the group G : d ∗ M Z = d ∗ Z

(M ∈ G).

It is easy to see that two matrices M, M of G have the same action (1.6) on an open subset of H if and only if M = λ M with λ in the set R∗ of nonzero real numbers. It follows that the group of all transformations of the upper half-plane of the form (1.6) is isomorphic to the factor groups G/{R∗ 12n } S/{±12n }, where

S = Sn = Spn (R) = M ∈ Gn µ (M) = 1

(1.13)

1.2 Fundamental Domains for the Modular Group

11

is the (real) symplectic group of genus n. We have already seen in the proof of Lemma 1.3 that each matrix Z ∈ H can be written in the form Z = M iE with a matrix M ∈ S. Therefore, the upper half-plane can be identified with the homogeneous space of the symplectic group by the stabilizer U of the point iE in S. More precisely, we have the following lemma. Lemma 1.7. The map M → M i1n defines a one-to-one correspondence S/U ↔ H, which is compatible with the actions of the group S, where on the left side it acts by multiplication from the left; the stabilizer U has the form

A B n n ∈S ; U=U = −B A A B the map −B A → A + iB is an isomorphism of U onto the unitary group of order n; in particular, the group U is compact. Exercise 1.8. Prove the lemma and preceding assertions. Exercise 1.9. Show that the Cayley mapping Z → W = (Z − i1n )(Z + i1n )−1

(Z ∈ H),

is an analytic isomorphism of H onto the bounded domain W ∈ Cnn tW = W, WW < E , where the inequality is understood in the sense of Hermitian matrices. Show that the inverse mapping is given by W → Z = i(1n +W )(1n −W )−1 .

1.2 Fundamental Domains for the Modular Group Modular Group. The modular (symplectic) group or the Siegel modular group of genus n, i.e., the group of all integral symplectic matrices of order 2n with unit multiplier, (1.14) Γ = Γn = Spn (Z) = Sn Z2n 2n , is clearly a discrete subgroup of the symplectic group S. The same is true for each subgroup K of S commensurable with the group Γ, i.e., such that the intersection K ∩ Γ is of finite index both in K and Γ. Lemma 1.7 implies then that each such group K acts discretely on the upper half-plane. Automorphic forms for subgroups K of the symplectic group, which we are going to consider in this chapter, are functions on the upper half-plane having certain analytic properties and satisfying functional equations connecting its values at points of each K-orbit

12

1 Modular Forms

K Z = M Z M ∈ K

(Z ∈ H)

of K on H, so that such a function is uniquely determined by its restriction to any subset of H that meets each K-orbit. We recall that a closed subset D of a topological space X is called a fundamental domain for a discrete transformation group G acting on X if it meets each of the G-orbits G(x) = {g(x)|g ∈ G} with x ∈ X and has no distinct inner points belonging to the same orbit. It follows from the definition that the decomposition g(D) with G = g ∈ G g(x) = x, ∀x ∈ X (1.15) X= g∈G/G

holds, and its components pairwise have no common inner points. Fundamental domains do not necessarily exist. Minkowski Reduction Domain. The construction of fundamental domains for the modular symplectic group is essentially based on the Minkowski reduction theory of positive definite quadratic forms. In matrix language, the problem of reduction of positive definite quadratic forms relative to unimodular equivalence is that of construction of a fundamental domain for the unimodular group Λ = Λn = GLn (Z)

(1.16)

acting on the cone P = Pn = Y ∈ Rnn tY = Y,

Y >0

(1.17)

of real positive definite matrices of order n by ΛV :

Y → Y [V ] = tVYV.

The columns of a matrix U ∈ Λ will be denoted by u1 , . . . , un , so that U = (u1 , . . . , un ). In order to choose a special representative Y [U] of the orbit Λ(Y ) = Y [V ] V ∈ Λ of a point Y ∈ P, we determine the matrix U column by column with the help of some minimal conditions. Let Λr = Λnr = (u1 , . . . , ur ) ∈ Znr (u1 , . . . , ur , ∗, . . . , ∗) ∈ Λ be the set of all integral n × r−matrices composed of the first r columns of matrices in Λ. For a given Y ∈ P , we choose u1 ∈ Λ1 such that the value Y [u1 ] = tu1Y u1 of the quadratic form with the matrix Y on the column u1 is minimal; this can be done, since the form is positive definite. Next, we determine u2 such that (u1 , u2 ) ∈ Λ2 and the value Y [u2 ] is minimal. On replacing u2 by −u2 if necessary, one may assume

1.2 Fundamental Domains for the Modular Group

13

that tu1Y u2 ≥ 0. Proceeding in the same way, at the r−th step we choose ur such that (u1 , . . . , ur ) ∈ Λr , Y [uur ] is minimal, and tu r−1Yuur ≥ 0. Finally, when r = n, we get a unimodular matrix U = (u1 , . . . , un ) ∈ Λ and a matrix T = (tαβ ) = Y [U] ∈ Λ(Y ), which is called Minkowski reduced, or just reduced. Let us determine the conditions for a positive definite matrix to be reduced in terms of its entries. First of all, by induction on n based on the Euclidean algorithm, one can easily prove the following. Lemma 1.10. An integral n-column u belongs to Λ1 if and only if its entries are coprime. Also, as an easy exercise on multiplication of block matrices, we get the following lemma. Lemma 1.11. Two matrices U, U of Λn have the same first r columns if and only if 1 B with D ∈ Λn−r and B ∈ Zn−r U = U r r . 0 D Let U = (u1 , . . . , un ) ∈ Λn . By Lemma 1.11, the set of all r−th columns of all matrices U ∈ Λn with the first columns u1 , . . . , ur−1 coincides with the set of columns of the form Uv, where v is an integral n-column whose last n−r +1 entries vr , . . . , vn form the first column of a matrix D ∈ Λn−r+1 . By Lemma 1.10, the last condition means that the numbers vr , . . . , vn are coprime. Thus, if U = (u1 , . . . , un ) ∈ Λn and 1 ≤ r ≤ n, then (1.18) u ∈ Zn (u1 , . . . , ur−1 , u) ∈ Λnr = UVr,n , where

Vr,n = v =t (v1 , . . . , vn ) ∈ Zn gcd(vr , . . . , vn ) = 1 .

By the definition and (1.18), we conclude that a matrix T = (tαβ ) = Y [U] is reduced if and only if it satisfies the conditions Y [Uv] ≥ Y [ur ],

for all v ∈ Vr, n and 1 ≤ r ≤ n,

and ur−1Y ur ≥ 0,

t

for 1 < r ≤ n,

where U = (u1 , . . . , un ). Since Y [U] = T = (tαβ ), we have Y [ur ] = trr and = tr−1,r . Hence, the above conditions mean exactly that T belongs to the set M = Mn = {T = (tαβ ) ∈ Pn trr ≤ T [v] for all v ∈ Vr,n (1 ≤ r ≤ n);

tu r−1Y ur

tr−1,r ≥ 0 (1 < r ≤ n)}, called the Minkowski reduction domain.

(1.19)

14

1 Modular Forms

Theorem 1.12. Every orbit Λ(Y ) of the group Λ on P contains at least one and not more than finitely many points of the Minkowski domain M. If T , T are two inner points of M, and T = T [U] with U ∈ Λ, then U = ±1n ; in particular, no different inner points of M belong to the same orbit. In other words, M is a fundamental domain of Λ on P. Proof. The above consideration shows that for a given Y ∈ P, there exists U ∈ Λ such that Y [U] ∈ M, and every column of such U can be chosen in finitely many ways. Let us set M = Mn = {T ∈ Pn trr < T [v], v ∈ Vr,n , v = ±er (1 ≤ r ≤ n); tr−1,r > 0 (1 < r ≤ n)}, where e1 , . . . , en are the columns of the unit matrix 1n . It is clear that M ∈ M and ) belong to M each inner point of M is contained in M . If T = (tαβ ) and T = (tαβ = T [u ] ≥ t , and T = T [U] with U = (u1 , . . . , un ) ∈ Λ, then u1 ∈ V1, n , whence t11 1 11 and similarly, t11 ≥ t11 . It follows that t11 = t11 = T [u1 ], and so u1 = ±e1 . Then u2 ∈ V2, n , and in the same way we conclude that u2 = ±e2 . By repeating the same arguments, we see that ur = ±er for all r = 1, . . . , n. Now the conditions tr−1, r > 0,

t tr−1, r = ur−1 T ur > 0 (1 < r ≤ n)

imply that ur = er or ur = −er for r = 1, . . . , n, and T = T . The entries of reduced matrices T = (tαβ ) satisfy some useful inequalities. First of all, since trr ≤ T [er+1 ] = tr+1,r+1 , it follows that t11 ≤ t22 ≤ · · · ≤ tnn .

(1.20)

Then by tll ≤ T [er ± el ] = trr ± 2trl + tll , where 1 ≤ r < l ≤ n, we obtain |2trl | ≤ trr

if r = l.

(1.21)

Finally, Minkowski have proved a deeper inequality for reduced matrices, which we cite without proof: t11t22 · · ·tnn ≤ cn det T

(T = (tαβ ) ∈ Mn ),

(1.22)

where cn is a positive constant depending only on n. The Minkowski inequality implies that every T = (tαβ ) ∈ Mn satisfies the inequality T≥

1 nn−1 cn

diag(t11 ,t22 , . . . ,tnn ).

Indeed, let ρ1 , . . . , ρn be the eigenvalues of the matrix −1/2

T = T [diag(t11

−1/2

,t22

−1/2

, . . . ,tnn

)],

(1.23)

1.2 Fundamental Domains for the Modular Group

15

then ρ1 + · · · + ρn = n, and by (1.22),

ρ1 · · · ρn = (t11t22 · · ·tnn )−1 det T ≥ 1/cn ; it follows that ρα ≤ n and ρα ≥ 1/nn−1 cn for α = 1, . . . , n, which implies (1.23). Exercise 1.13. Show that

t11 t12 ∈ P2 0 ≤ 2t12 ≤ t11 ≤ t22 . M2 = t12 t22 Show that in the inequalities (1.22) one can take c1 = 1 and c2 = 4/3, and the values are minimal. 1 1/2 [Hint: For minimality of c2 , consider T = 1/2 1 ]. Exercise 1.14. Two binary quadratic forms f(x, y)and f (x, y) are said to be equivalent if f (x, y) = f (α x + β y, γ x + δ y) with αγ βδ ∈ Λ2 . Show that the number of classes of equivalent positive definite quadratic forms f (x, y) = ax2 + bxy + cy2 with integral coefficients a, b, c and a fixed discriminant d = b2 − 4ac is finite. Construction of Fundamental Domains. Let us return to the action of the modular group (1.14) on the upper half-plane. We consider orbits Γ Z of the modular group on H. For Z = X + iY ∈ H = Hn , we shall call the positive real number detY the height of the point Z and denote it by h(Z). By (1.10), we have AB −2 Z ∈ H, M = ∈Γ . (1.24) h(M Z ) = | det(CZ + D)| h(Z) CD Lemma 1.15. Each orbit of the group Γ on H contains points Z of maximal height. These points can be characterized by the inequalities ∗ ∗ | det(CZ + D)| ≥ 1 for every ∈ Γ. CD Proof. In view of (1.24) we have to show that | det(CZ + D)| takes a minimum on each orbit. Note that for any M = CA DB ∈ Γ and V ∈ Λ = Λn , the product t −1 t −1 t −1 V 0 V A V B M= 0 V VC V D also belongs to Γ. It follows that if (C, D) is the ”second row” of a matrix of Λ, then the same is true for (VC, V D). Replacing M in M X + iY = X + iY

16

1 Modular Forms

by the above product does not change the value | det(CZ + B)| and replaces the matrix (Y )−1 by the matrix tV (Y )−1V . Therefore, we may assume that the positive definite matrix (Y )−1 is Minkowski reduced. Let us denote by cr and dr (r = 1, . . . , n) the columns of the matrices tC and X tC + tD, respectively, and by t1 , . . . ,tn the diagonal entries of (Y )−1 . Then, by (1.10), we can write (Y )−1 = (CZ + D)Y −1 (Z tC + tD) = (CX + D)Y −1 (X tC + tD) +CY tC, whence for r = 1, . . . , n, we get tr = Y

−1

Y [cr ] [dr ] +Y [cr ] ≥ Y −1 [dr ] .

(1.25)

If for some r, the columns cr and dr are both zero, then the r−th column of the matrix (C D) is also zero, which is impossible, since M is nonsingular. Since Y > 0, the value Y [cr ] assumes a positive minimum when Z is fixed and cr is an arbitrary nonzero integral column. On the other hand, if cr = 0, then dr is the r−th column of tD and so is a nonzero integral column. Since Y −1 > 0, the value Y −1 [dr ] also assumes a positive minimum. It follows then from (1.25) that the numbers u1 , . . . , un have a positive lower bound independent of M. The relations (1.22) and (1.24) imply the inequality t1t2 · · ·tn ≤ cn (detY )−1 = cn (detY )−1 | det(CZ + D)|2 . If we now assume that a condition | det(CZ + D)| ≤ h is satisfied for an arbitrarily large number h, then it implies upper bounds for t1 ,t2 , . . . ,tn . Then from (1.25) we obtain upper bounds for entries of the columns cr and dr and hence for the entries of the matrices C and D. Therefore, the condition is satisfied only for finitely many pairs (C, D) if Z is fixed and h is a given large number. This proves the lemma. Theorem 1.16. Let D = Dn be the subset of matrices Z = X + iY ∈ Hn satisfying the following conditions: (1) | det(CZ + D)| ≥ 1 for every ( C∗ D∗ ) ∈ Γ = Γn ; reduction domain (1.19); (2) Y ∈ Mn , where Mn is the Minkowski t n (3) X ∈ Xn = X = (xαβ ) ∈ Rn X = X, |xαβ | ≤ 1/2 (1 ≤ α , β ≤ n) . Then D meets each Λ-orbit on H, and Z = M Z for two inner points Z, Z ∈ D with M ∈ Γ if and only if M = ±E, i.e., D is a fundamental domain of Γ on H. If Z = X = iY ∈ D, then Y ≥ bn E

and

σ (Y −1 ) ≤ n/bn ,

(1.26)

where bn is a positive constant depending only on n, and σ denotes the trace. The volume of D with respect to the invariant element of volume (1.12) is finite.

1.2 Fundamental Domains for the Modular Group

17

Proof. Let us consider the orbit Γ Z of a point Z ∈ H. By Lemma 1.15, the orbit contains a point Z = X +iY of maximal height, and the point satisfies the condition (1). Every transformation of the form t V SV −1 = tV Z V + S = X [V ] + S + iY [V ] Z → 0 V −1 with V ∈ Λ = Λn and an integral symmetric matrix S of order n corresponds to a matrix of Γ and does not change the height of Z . By Theorem 1.12, there is a matrix V ∈ Λ such that the matrix Y = Y [V ] belongs to M n . Also, clearly, there is S such that X [V ] + S ∈ Xn . Then Z = X + iY ∈ Γ Z

D.

Suppose now that Z = M Z for two points Z, Z ∈ D with M = CA DB ∈ Γ. Then h(Z ) = h(Z), by Lemma 1.15. It follows from (1.24) that | det(CZ + D)| = 1. On the other hand, since Z = M −1 Z , we conclude that | det(− tCZ + tA)| = 1 (see (1.1)). If C = 0, then the equations are nontrivial, and so the points Z, Z belong to the boundary of D. If C = 0, then M has the form t V SV −1 with V ∈ Λ and S = tS ∈ Znn . M= 0 V −1 So we have Z = X + iY = X[V ] + S + iY [V ],

X + iY = Z,

where

and in particular, Y = Y [V ]. Since Y and Y are both in Mn , it follows from Theorem 1.12 that Y and Y are boundary points of Mn or V = ±E. In the last case we have X = X + S, whence S = 0, unless X and X belong to the boundary of Xn . We conclude that M = ±12n , unless Z and Z are boundary points of D. Let Z = (zαβ ) = (xαβ + iyαβ ) ∈ D. the inequality | det(CZ + D)| ≥ 1 for the pair 10 0 0 (C, D) = , 00 0 1n−1 2 + y2 ≥ 1. Since |x | ≤ 1/2, it follows that implies the inequality |z11 | = x11 11 11 √ i.e., y11 ≥ 3/2. The last inequality and the inequalities (1.20) imply y211 ≥ 3/4, √ for α = 1, . . . , n. The first inequality of (1.26) follows then from that yαα ≥ 3/2 √ (1.23) with bn = 3/2nn−1 cn . The inequality implies that each eigenvalue of Y −1 is not greater than 1/bn , which proves the second inequality. Finally, by (1.12), Theorem 1.16, and (1.26) we have

v(Dn ) =

Dn

(detY )−n−1 dXdY ≤

Y ∈Mn ,Y ≥bn E

(detY )−n−1 dY,

18

1 Modular Forms

which, by (1.20), (1.21), and (1.22), can be estimated as ≤ ≤

−1 −n−1 dY bn ≤y11 ≤y22 ≤···≤ynn ; (cn y11 y22 · · · ynn ) |2yαβ |≤yαα (α =β )

y11 ,y22 ,··· ,ynn ≥bn

=c

n

∏

∞

α =1 cn

−n−1 (c−1 n y11 y22 · · · ynn )

n

α ∏ yn− αα

α =1

dy11 dy22 · · · dynn

y−α −1 dy < ∞.

Exercise 1.17. Prove that D1 is the so-called modular triangle,

1 2 2 D1 = z = x + iy ∈ H |x| ≤ , |z| = x + y ≥ 1 , 2 and one can take b1 =

√ 3 2 .

Draw the modular triangle.

Two binary quadratic forms Q(x, y) = q11 x2 + q12 xy + q22 y2 and Q (x, y) = q11 x2 + q12 xy + q22 y2 are said to be properly equivalent (over Z) if

Q (x, y) = Q(ax + by, cx + dy)

with

ab ∈ Γ1 = SL2 (Z). cd

(1.27)

Exercise 1.18. Show that any real positive definite quadratic form Q is properly equivalent to a form Q whose coefficients satisfy the inequalities |q12 | ≤ q11 ≤ q22 . Show that the cone given by the inequalities in the space of the coefficients of the form contains no distinct inner points corresponding to properly equivalent forms. [Hint: Let ω and ω be the roots belonging to H of the equations Q(x, 1) = 0 and respectively. Show that (1.27) is equivalent with the conditions ab ∈ Λ; (q12 )2 − 4q11 q22 = (q12 )2 − 4q11 q22 and ω = M −1 ω with cd

Q (x, 1) = 0,

then use Theorem 1.16 and Exercise 1.17.] Theorem 1.19. Each subgroup of the symplectic group S = Sn of the form KM = M −1 KM, where K is a subgroup of finite index in the modular group Γ = Γn , and M belongs to the general symplectic group G = Gn , has a fundamental domain D(KM ) on H = Hn , and one can take D(KM ) =

ˇ γ ∈K\Γ

(M −1 γ ) D ,

(1.28)

1.3 Modular Forms

19

where D = D(Γ) is a fundamental domain of Γ , and γ ranges over a system of representatives of different left cosets of Γ modulo the subgroup Kˇ = K ∪ (−12n )K. The invariant volume of the fundamental domain is finite. Proof. First of all, the set (1.28) is closed, as a finite union of closed subsets. Next, if (M −1 γ ) Z and (M −1 γ ) Z are two inner points of the set belonging to the same orbit of KM , i.e., (M −1 γ ) Z = (M −1 δ M) (M −1 γ ) Z

= (M −1 δ γ ) Z

with δ ∈ K, one can assume that Z and Z are inner points of D. Then, by (1.7), γ Z = (δ γ ) Z , and so Z = Z , and γ = ±δ γ , by the definition of D, which implies that γ = γ and δ = ±12n . Finally, since D meets each Γ -orbit, we have

H=

δ ∈Γ/{±12n }

whence

δ D =

(δ γ ) D ,

ˇ ˇ δ ∈K/{±1 2n } γ ∈K\Γ

H = M −1 H =

((M −1 δ M)(M −1 γ )) D .

ˇ ˇ δ ∈K/{±1 2n } γ ∈K\Γ

This proves the decomposition (1.15), whence the set D(KM ) really meets each KM -orbit. Finally, the set (1.28) is a finite union of images of the domain D under symplectic transformations and hence together with this domain has finite volume.

1.3 Modular Forms Definition of Modular Forms. Acting on the upper half-plane H = Hn , the general symplectic group G = Gn operates also on complex-valued functions F on H by Petersson operators of integral weights k: n(n+1) AB GM= : F → F|k M = µ (M)nk− 2 j(M, Z)−k F(M Z ) CD = µ (M)nk−

n(n+1) 2

det(CZ + D)−k F((AZ + B)(CZ + D)−1 ),

(1.29)

where j(M, Z) = det J(M, Z) = det(CZ + D). n(n+1)

Note that the factor µ (M)nk− 2 is introduced for aesthetic reasons, because it simplifies a number of forthcoming formulas. In fact, it can be replaced by an arbitrary quasicharacter of G or just omitted without affecting essential properties of the operators. By Lemma 1.3, the Petersson operators map holomorphic functions to holomorphic, moreover, it follows from the definition and relations (1.7), (1.9) that the operators satisfy the rules F|k rM = rn(k−n−1) F|k M

for all r ∈ R, r = 0

(1.30)

20

1 Modular Forms

and for M, M ∈ Gn F|k MM = µ (MM )nk−

n(n+1) 2

j(MM , Z)−k F((MM ) Z )

= (µ (M)µ (M ))nk− = µ (M )nk−

n(n+1) 2

n(n+1) 2

( j(M, M Z ) j(M , Z))−k F(M M Z

)

j(M , Z)−k (F|k M)(M Z ) = (F|k M)|k M .

(1.31)

Let K be a subgroup of G commensurable with the modular group Γ = Γn , χ a character of K, that is, a multiplicative homomorphism of K into the nonzero complex numbers with kernel of finite index in K, and k an integer. A complexvalued function F on H is called a (Siegel) modular form of weight k and character χ for the group K if the following conditions are satisfied: (i) F is a holomorphic function in n(n + 1)/2 complex variables on H; (ii) for every matrix M ∈ K, the function F satisfies the functional equation F|k M = χ (M)F,

(1.32)

where |k is the Petersson operator of weight k; (iii) if n = 1, then every function F|k M with M ∈ Γ1 is bounded on each subset of H1 of the form H1ε = {x + iy ∈ H1 | y ≥ ε } with ε > 0. The set Mk (K, χ ) of all modular forms of weight k and character χ for the group K is clearly a linear space over the field C. Exercise 1.20. Let k > 2 be an integer. Show that the Eisenstein series Ek (z) =

∑

c,d∈Z (c,d)=(0,0)

1 (cz + d)k

(z ∈ H1 )

converges absolutely and defines a modular form of weight k and the unit character for Γ1 . [Hint: If S is a compact subset of H1 , then |α z+ β | ≥ b(|α |+|β |) for all α , β ∈ R and z ∈ S with a positive constant b. Since there are only 4r pairs of integers (c, d) with |c| + |d| = r, it follows that Ek (z) is dominated term by term on S by the series k b−k ∑∞ r=1 4r/r .] For q ∈ N, we shall denote by Γ (q) = Γn (q) = M ∈ Γn M ≡ 12n (mod q) the principal congruence subgroup of level q of the modular group. Considering matrices of Γ modulo q, we get a homomorphism of the modular group into the finite group of symplectic matrices of order 2n over the residue class ring Z/qZ with kernel Γ (q). It follows that Γ (q) is a normal subgroup of the modular group of finite index.

1.3 Modular Forms

21

A subgroup K of the symplectic group (1.13) is called a congruence subgroup if it contains a principal congruence subgroup as a subgroup of finite index. A character of such K is said to be a congruence character if it is trivial on a principal congruence subgroup contained in K. Proposition 1.21. Let K be a congruence subgroup of Sn , χ a congruence character of K, and M a matrix of Gn with rational entries then the group KM = M −1 KM

(1.33)

is again a congruence subgroup of the symplectic group, the character M −1 KM M → χM (M ) = χ (MM M −1 )

(1.34)

of this group is a congruence character, and the image F|k M of each modular form F ∈ Mk (K, χ ) under the Petersson operator |k M is a modular form of weight k and character χM for the group KM : F|k M ∈ Mk (KM , χM ). Proof. Let q be a positive integer such that Γn (q) ⊂ K and χ is trivial on Γn (q). It can be assumed that the matrix M is integral. Let µ (M) = q then it is easy to see that the group Γn (qq ) is contained in KM and the character χM is trivial on this group. It follows from definitions and Lemma 1.3 that the function F = F|k M is holomorphic on Hn . If M ∈ KM , then MM M −1 ∈ K, and by (1.31), we have F |k M = F|k M|k M = F|k MM M −1 M = F|k MM M −1 |k M = χ (MM M −1 )F|k M = χM (M )F . Finally, if n = 1 and V ∈ Γ1 , we can write F |kV = F|k MV and present the matrix MV in the form MV = W M1 with W ∈ Γ1 = SL2 (Z) and an upper triangular matrix M1 . Then the function F |kV = F|kW M1 = (F|kW )|k M1 together with F|kW is clearly bounded on each subset of H1 of the form H1ε with ε > 0. Exercise 1.22. Let k > 2 be an integer, a, b ∈ Z, and q ∈ N. Show that the Eisenstein series 1 Ek (z; (a, b)|q) = (z ∈ H1 ) ∑ (cz + d)k c,d∈Z, (c,d)=(0,0), (c,d)≡(a,b) (mod q)

converges absolutely and defines a modular form of weight k and the unit character for Γ1 (q) satisfying Ek (z; (a, b)|q)|k M = Ek (z; (a, b)M|q) for each M ∈ Γ1 . [Hint: Use Exercise 1.20.]

22

1 Modular Forms

Fourier Expansion of Modular Forms. Theorem 1.23. Let K be a congruence subgroup of Sn , and χ a congruence character of K. Then each modular form F ∈ Mk (K, χ ) has an expansion of the form F(Z) =

πi

f (A)e q σ (AZ) ,

∑ n

(1.35)

A∈E , A≥0

with constant coefficients f (A), where En = A = (aαβ ) ∈ Znn tA = A,

a11 , a22 , . . . , ann ∈ 2Z

is the set of “even” matrices of order n, σ denotes the trace, and where q = q(K, χ ) is a positive integer such that the group K contains a subgroup of the form

1n qB n t n (1.36) T(q) = T (q) = B = B ∈ Zn . 0 1n The series (1.35) converges absolutely on H and uniformly on each subset of H of the form with ε > 0; Hε = Hnε = X + iY ∈ Hn Y ≥ ε 12n (1.37) in particular, F is bounded on each of the subsets. The coefficients f (A) satisfy the relations πi

f ( tVAV ) = (detV )k χ (M)e− q σ (AVU) f (A) for every matrix M of the group K of the form −1 V U . M = M(U, V ) = 0 tV

(∀ A ∈ En ),

(1.38)

(1.39)

The expansion (1.35) is called the Fourier expansion of F, and the numbers f (A) with A ∈ En , A ≥ 0 are the Fourier coefficients of F. Proof. The functional equations (1.32) for matrices of the subgroup T(q) ⊂ K become F(Z + qB) = F(Z) (Z = (zαβ ) ∈ H, B = tB ∈ Znn ). This means that F is periodic of period q in each of the variables zαβ = zβ α . Since F is also holomorphic, it can be expanded in a Fourier series of the form 2π i aαβ xαβ , F(Z) = ∑ f (A ; Y ) exp q 1≤α∑ A 0. Now let n ≥ 2. It was found by Max Koecher that in this case, an analogue of condition (iii) of the definition of modular forms is fulfilled automatically (Koecher’s effect). We use the same arguments. Let Λ be the group of all matrices V ∈ Λ = SLn (Z) such that the matrix M(0, V ) of the form (1.39) with U = 0 belongs to the

24

1 Modular Forms

kernel of the character χ . Since χ is a congruence character, the group Λ has finite index in Λ. By (1.38), we have f ( tVAV ) = f (A), for all A ∈ En and V ∈ Λ . Hence the expansion (1.40) can be rewritten in the form F(Z) =

∑

A∈E/Λ

f (A)η (Z, {A}),

where the sum is extended over a system of representatives for the classes {A} = { tVAV |V ∈ Λ } of the set E modulo the equivalence A ∼ tVAV with V ∈ Λ , and where πi η (Z, {A}) = ∑ e q σ (A Z) . A ∈{A}

If f (A) = 0, then the series f (A)η (Z, {A}) converges absolutely for every Z ∈ H, because it is a partial sum of an absolutely convergent series; in particular, the series

η (i1n , {A}) =

∑

π

e− q σ (Ai )

Ai ∈{A}

is convergent. Since the trace σ (Ai ) of every Ai ∈ {A} is a rational integer, it follows that the class {A} cannot contain more than a finite number of matrices Ai with σ (Ai ) < 0. On the other hand, we shall show now that the trace σ (Ai ) assumes infinitely many negative values on the class {A} of any integral symmetric matrix A of order n ≥ 2, unless A ≥ 0. If A does not satisfy A ≥ 0, then there is an integral n-column h such that thAh < 0. For r ∈ Z we set Vr = 1n + rH

with H = (t1 h, . . . ,tn h) ∈ Znn ,

where t1 , . . . ,tn are some integers. Since the rank of the matrix rH is equal to 1 or 0, it follows that detVr = 1 + σ (rH) = 1 + r(t1 h1 + · · · + tn hn ), where hα are entries of h. Since n ≥ 2, there are integers t1 , . . . ,tn not all equal to 0 such that t1 h1 + · · · + tn hn = 0. Then we have Vr ∈ Λ,

H 2 = ((t1 h1 + · · · + tn hn )hα tβ ) = 0,

and Vr = V1r .

Since the index of Λ in Λ is finite, it follows that the matrix Va = V1a belongs to Λ for some a ∈ N, and so do the matrices Vab = Vab for every integer b, hence the matrices tVab AVab belong to the class {A}, and σ tVab AVab = σ (A) + 2abσ (AH) + a2 b2 σ ( tVAV ) = σ (A) + 2abσ (AH) + a2 b2 ( thAh)(t12 + · · · + tn2 ). The last expressions is a polynomial in b of degree 2 with negative leading coefficient. Hence it takes infinitely many negative values on Z.

1.3 Modular Forms

25

The above considerations show that a coefficient f (A) in (1.30) equals zero, unless A ≥ 0, which proves the expansion (1.35). Finally, it is easy to see that the series (1.35) is majorized on each set (1.37) by a convergent series with nonnegative constant coefficients, and so converges there uniformly. Exercise 1.24. Show that if k > 2 is even, then the Fourier expansion of the Eisenstein series Ek (z) defined in Exercise 1.20 has the form ∞ (2π i)k ∞ 1 k−1 e2π inz . Ek (z) = 2 ∑ k + 2 ∑ ∑d (k − 1)! n n=1 n=1 d|n [Hint: Show first that 1

π2

∞

(2π )2t

∑ (z + d)2 = sin2 π z = (1 − t)2 = (2π i)2 ∑ dt d ,

d∈Z

d=1

where t = e2π iz . Then differentiate both parts k − 2 times.] Cusp Forms. If K is a congruence subgroup, χ is a congruence character of K, and M is a matrix of the general symplectic group (1.4) with rational entries, then by Proposition 1.21 and Theorem 1.23, the function F|k M has a Fourier expansion of the form πi (1.42) (F|k M)(Z) = ∑ fM (A)e q σ (AZ) A∈En , A≥0

with a positive integer q depending on K, χ , and M, which converges absolutely on H and uniformly on the subsets (1.37). The modular form F is called a cusp form if the coefficients fM (A) of the decomposition (1.42) satisfy the conditions fM (A) = 0 for all M ∈ G(Q) and A ∈ E with det A = 0,

(1.43)

where G(Q) = Gn (Q) = Gn ∩ Q2n 2n . The subspace of cusp forms of Mk (K, χ ) will be denoted by Nk (K, χ ). Proposition 1.25. Let K be a congruence subgroup of S = Sn and χ a congruence character of K. Then for each cusp form F ∈ Nk (K, χ ) and each matrix M ∈ Gn (Q), the modular form F|k M ∈ Mk (KM , χM ), where KM = M −1 KM and χM is the character (1.34), is a cusp form with a Fourier expansion of the form (F|k M)(Z) =

∑ n

πi

fM (A)e q σ (AZ) ,

(1.44)

A∈E , A>0

where q is a positive integer. If k ≥ 0, then the form F|k M satisfies |(F|k M)(X + iY )| ≤ c(detY )k/2

(X + iY ∈ H),

(1.45)

26

1 Modular Forms

and its Fourier coefficients satisfy | fM (A)| ≤ c (det A)k/2

for all A ∈ E

with A > 0,

(1.46)

where c and c are constants depending only on F and M. First we shall prove the following simple lemma. Lemma 1.26. Let

φ (Y ) =

∑

ϕ (A)e−ησ (AY ) ,

A∈En , A>0

where Y belongs to the cone P = Pn of positive definite matrices of order n, and η > 0, be a series with nonnegative coefficients ϕ (A) convergent for all Y ∈ P. Then for every Minkowski reduced matrix Y satisfying Y ≥ d · 1n with d > 0, the estimate

φ (Y ) ≤ d e−d

σ (Y )

holds with positive constants d and d . Proof of the lemma. If Y = (yαβ ) ∈ P and A = (aαβ ) ∈ E satisfies A > 0, then by (1.23),

σ (AY ) ≥ bσ (A diag(y11 , y22 , . . . , ynn )) = b

n

∑ aαα yαα ≥ 2bσ (Y ),

α =1

where b is a positive constant depending only on n. On the other hand, if Y > dE, then for A = (aαβ ) ∈ E with A > 0, we obtain σ (AY ) ≥ d σ (A). (Note that we have used twice the obvious inequality σ (AR) ≥ σ (BR) valid if matrices A − B and R are positive semidefinite.) It follows then from the above inequalities that 1 σ (AY ) ≥ bσ (Y ) + d σ (A), 2 hence

φ (Y ) ≤

∑

ϕ (A)e−η (bσ (Y )+ 2 d σ (A)) = e−η bσ (Y ) φ 1

A∈E, A>0

1 dE , 2

which proves the estimate. Proof of Proposition 1.25. The Fourier expansion (1.44) and the inclusion F|k M ∈ Nk (KM , χM ) follow from Proposition 1.21 and the definition of cusp forms. By Proposition 1.21 there exists a level q ∈ N such that the principal congruence subgroup Γ (q) is contained both in KM and the kernel of character χ . Let us consider the function (1.47) G = ∑ |F |k γi |, where F = F|k M. γi ∈Γ (q)\Γ

1.3 Modular Forms

27

It follows from (1.31) and (1.32) that G is independent of the choice of representatives in the cosets Γ (q)\Γ and satisfies G|k γ =

∑

|F |k γi γ | = G

for all γ ∈ Γ ,

(1.48)

γi ∈Γ (q)\Γ

because γi γ runs through a system of representatives for Γ (q)\Γ when γi does. Hence by (1.10), we see that the function H(Z) = H(X + iY ) = (detY )k/2 G(Z) = (detY )k/2

∑

|F |k γi |

(1.49)

γi ∈Γ (q)\Γ

satisfies H(γ Z ) = H(Z)

for all Z ∈ H

and γ ∈ Γ.

(1.50)

Since by Theorem 1.21, each of the functions F |k γi = F|k M γi is bounded on the sets Hε with ε > 0, it follows from (1.26) that the functions are bounded on the fundamental domain D of Γ , defined in Theorem 1.16. Furthermore, each of the these functions satisfies π |F |k γi | ≤ ∑ | fMγi (A)|e− q σ (AY ) , A∈E, A>0

and the last series converges on the cone P = Pn . If Z ∈ D, then Y is Minkowski reduced and satisfies Y ≥ bn 1n ; hence by Lemma 1.24, each of the series is dominated by a function of the form d exp(−d σ (Y )) with constants d and d depending on F and M. Therefore, since k ≥ 0, we obtain H(X + iY ) ≤ d [Γ : Γ(q)](detY )k/2 exp(−d σ (Y )) ≤δ

n

∏ yαα exp(−d yαα ), k/2

α =1

where yαα are the diagonal entries of Y , and we have also used the following consequence of the Hadamard’s determinant theorem: detY ≤ y11 y22 · · · ynn

(Y ∈ Pn ).

(1.51)

Since the last expression is clearly bounded on P, it follows that H is bounded on D. Thus by (1.50), H is bounded on H, which implies the estimate (1.45). The estimate for Fourier coefficients follows from (1.45). One can prove that the coefficients of the Fourier expansion (1.35) of an arbitrary modular form of nonnegative weight k and congruence character for a congruence subgroup of Sn satisfy | f (A)| ≤ c(det A)k where c depends only on the form.

(A ∈ En , A ≥ 0),

(1.52)

28

1 Modular Forms

The Siegel Operator. The general philosophy of modular forms emanates from the idea that consideration of arbitrary modular forms can usually be reduced to the case of cusp forms and the cases of modular forms of smaller genera. The reduction is ensured by the so-called Siegel operator and its iterations. Let F = F(Z) be a modular form of weight k for a congruence subgroup K of the symplectic group S = Sn and a congruence character χ of K. Since the Fourier series (1.25) for F converges uniformly on subsets of Hn of the form (1.27), the limit πi Z 0 (F|Φ)(Z ) = lim F f (A) lim e q σ (AZy ) , (1.53) = ∑ 0 iy y→+∞ y→+∞ A∈En , A≥O where Zy =

Z 0 0 iy

, exists for every Z ∈ Hn−1 . If A =

σ (A Z ) + iyann , hence lim e

π i σ (AZ ) y q

y→+∞

− πq yann πqi σ (A Z )

= lim e

e

y→+∞

A ∗ ∗ ann

πi e q σ (A Z ) , = 0,

Since A ≥ 0, the equality ann = 0 implies that A has the form

∑

(F|Φ)(Z ) =

f

A ∈E(n−1) , A ≥0

, then σ (AZy ) =

if ann = 0, if ann > 0. A 0 0 0

. Thus we have

πi A 0 e q σ (A Z ) , 0 0

(1.54)

for all Z ∈ H(n−1) . The last series is a partial series for the Fourier expansion of (n−1) F, and so it converges absolutely on H(n−1) and uniformly on Hε with ε > 0. If n = 1, we set (1.55) F|Φ = lim F(iy) ∈ C. y→+∞

As above, the limit exists and is equal to the constant term of the Fourier expansion of F. The linear operator (F ∈ Mk (K, χ ))

Φ : F → F|Φ

(1.56)

is called the Siegel operator. (n−1) , we shall In order to show that the function F|Φ is a modular form on H need new notation. Let n > 1. For a matrix M = CA DB with square blocks A , B ,

C , and D of order n − 1, we set − → AB M = CD where A =

A 0 0 1

,B=

B 0 0 0

,C=

− → A B = M, φ (M ) = C D

and

← − denote by S the set of all matrices to S:

(1.57)

C 0 , D = D 0 . If S is a subgroup of Gn , we 0 0 0 1 − → M = CA DB such that the matrix M belongs

1.3 Modular Forms

29

− → ← − A B 2n−2 ∈ R2n−2 M ∈ S . S = M = C D ← − Then it is clear that S is a subgroup of S(n−1) , and the map ψ given by

− → ← − S M → ψ (M ) = M ∈ S

(1.58)

← − is a homomorphic embedding of the group S into S. Lemma 1.27. Let K be a congruence subgroup of Sn with n > 1, and χ a congru← − ence character of K. Then the group K is a congruence subgroup of G(n−1) , and − given by χ the map ← → ← − −(M ) = χ (− K M → ← χ M) (1.59) ← − is a congruence character of the group K . Proof. By the assumptions, K contains a principal congruence subgroup Γ (q) = ←−− Γ n (q) of finite index belonging to kernel of χ . Then the group Γ (q) is clearly a ← − −. It principal congruence subgroup of K belonging to the kernel of the character ← χ ←−− ←−− ←−− ← − remains to prove that Γ (q) has finite index in K . Indeed, if Γ (q)Mα = Γ (q)Mβ , −→ −→ ← − where Mα , Mβ belong to K , then Γ (q)Mα = Γ(q)Mβ , since otherwise, we would −→ −→ ← − have Mα (Mβ )−1 ∈ Γ (q), and so Mα (Mβ )−1 ∈ Γ (q). Now we can prove our next result. Theorem 1.28. Let K be a congruence subgroup of Sn , and χ a congruence character of K. Then the Siegel operator Φ maps the space Mk (K, χ ) into the space ← − − Mk ( K , ← χ ): ← − − Φ : Mk (K, χ ) → Mk ( K , ← χ ), (1.60) ← − − χ ) = C. where for n = 1, we set M ( K , ← k

Proof. One can assume that n > 1. Let F ∈ Mk (K, χ ), Z ∈ Hn−1 , M = CA DB ∈ ← − S , Zλ = Z0 i0λ , and M = CA DB = ψ (M ) ∈ S, where ψ is the embedding (1.58). Then we have −1 C Z + D 0 A Z + B 0 = M Z λ , M Zλ = 0 iλ 0 1 −(M ). χ det(CZ + D) = det(C Z + D ), and χ (M) = ← λ

It follows that (F|Φ|k M )(Z ) = det(C Z + D )−k lim F(M Z λ ) λ →+∞ −k

= lim det(CZλ + D) F(M Zλ ) = (F|k M|Φ)(Z ). λ →+∞

(1.61)

30

1 Modular Forms

← − In particular, if M ∈ K , then we have −(M )F|Φ. χ F|Φ|k M = χ (M)FΦ = ← Furthermore, it follows from (1.54) and (1.61) that the function (F|Φ|k M )(Z ) is with ε > 0. holomorphic on Hn−1 , and is bounded on each subset Hn−1 ε The next lemma gives useful characterizations of cusp forms in terms of Siegel operators. Lemma 1.29. Let K be a congruence subgroup of Sn and χ a congruence character of K and let F ∈ Mk (K, χ ). Then the following three conditions are equivalent: (1) The function F is a cusp form; (2) F satisfies (F|k M)|Φ = 0

for all M ∈ Gn (Q) = Gn

where Φ is the Siegel operator; (3) F satisfies (F|k γ )|Φ = 0

Q2n 2n ,

for all γ ∈ Γn .

Proof. It follows from definition of cusp forms and formula (1.54) that condition 1 implies condition 2. Condition 3 is a particular case of condition 2. Hence it remains to prove that each function F ∈ Mk (K, χ ) satisfying condition 3 is a cusp form. It easily follows by an induction based on the Euclidean algorithm that each matrix n M ∈ Gn (Q) can be written in the form M = γ M1 , where γ ∈ Γ and M1 has the form A1 B1 M1 = 0 D1 with the zero n × n−block 0 = 0n (see Proposition 3.35(1) below). Since F|k M = F|k γ M1 = F|k γ |k M1 , the Fourier expansion (1.42) of the function F|k M can be rewritten in the form

∑ n

πi

−1 fM (A)e q σ (AZ) = (F|k M)(Z) = (det D1 )−k (F|k γ )(A1 ZD−1 1 + B1 D1 )

A∈E , A≥0

= (det D1 )−k

∑ n

A ∈E , A ≥0

π i σ (A B D−1 ) 1 1

fγ (A )e q1

π i σ (D−1 A A Z) 1 1

e q1

,

where fγ (A ) are the coefficients of the Fourier expansion (1.42) of the function F|k γ . Hence if fγ (A ) = 0 for all A ∈ En with det A = 0, then fM (A) = 0 for all A ∈ En with det A = 0. Spaces of Modular Forms. We can now prove the following important theorem. Theorem 1.30. Let K be a congruence subgroup of Sn , χ a congruence character of K, and k a nonnegative integer. Then the space Mk (K, χ ) of modular forms of weight k and character χ for the group K is finite-dimensional over the field C. The proof of the theorem is based on the following key lemma.

1.3 Modular Forms

31

Lemma 1.31. Let F(Z) =

∑

f (A)eπ iσ (AZ)

A∈E, A>0

be a cusp form of a nonnegative integral weight k and the unit character χ for the full modular group Γ = Γn . Suppose that the Fourier coefficients satisfy the conditions f (A) = 0

if

σ (A) ≤

kn , 2π bn

(1.62)

where bn is a constant satisfying the inequalities (1.26) of Theorem 1.16. Then the form F is identically equal to zero. Proof of the lemma. By Lemma 1.4 and the definition of modular forms, the function H(Z) = H(X + iY ) = (detY )k/2 |F(Z)| satisfies H(γ Z ) = H(Z) for all γ ∈ Γ and, as we have seen in the proof of Proposition 1.25, is bounded on the fundamental domain Dn of Γ defined in Theorem 1.16. Moreover, it follows from the estimates (1.51) and (1.52) that H(X + iY ) → 0, with detY → +∞ remaining in Dn . It follows from Theorem 1.16 that any subset of Dn of the form {X + iY ∈ Dn | detY ≤ c} with c > 0 is bounded and closed, and therefore is compact. Hence the function H(Z) attains its maximum µ at some point Z0 = X0 + iY0 of Dn . Since H is Γ -invariant, we conclude that µ is the maximum of H on H, that is, H(Z) ≤ H(Z0 ) = µ for all Z ∈ H. Let us set Zt = Z0 + tE, where t = u + iv is a complex parameter, and consider the function h(t) = F(Zt )e−π iλ σ (Zt ) =

∑

f (A)eπ i(σ (AZ0 )+t σ (A))−λ σ (Z0 +tE))

∑

f (A)eπ i(σ (AZ0 −λ Z0 )) eπ it(σ (A)−λ n) = h (w),

A∈E, A>0

=

A∈E, A>0

where w = eπ it and λ satisfies λ n = 1 + [kn/2π bn ], with [α ] denoting the greatest integer not exceeding α . By the assumption of the lemma, we have f (A) = 0 if σ (A) − λ n < 0, and so the expansion of the function h (w) does not contain negative powers of w. If ε > 0 is so small that Zt ∈ H for v ≥ −ε , then the expansion converges absolutely and uniformly on the half-plane v ≥ −ε , and so the function h (w) is holomorphic in the disk |w| ≤ eπε = τ . Since τ > 1, it follows, by the maximum-modulus principle, that there is a point w0 = eπ it0 satisfying |w0 | = τ and h (1) ≤ h (w0 ). Coming back to the function h, we can rewrite the last inequality in the form |F(Z0 )eπλ σ (Y0 ) | ≤ |Ft0 |eπλ σ (Y0 ) eπλ nv0 , where t0 = u0 + v0 , hence (detY0 )−k/2 H(Z0 ) ≤ (detYt0 )−k/2 H(Zt0 )eπλ nv0 .

32

1 Modular Forms

Since H(Z0 ) = µ and H(Zt0 ) ≤ µ , the last inequality implies the inequality

µ ≤ µ (detY0 )k/2 (detYt0 )−k/2 eπλ nv0 = µψ (v0 ), where ψ (v) = det(E + vY0−1 )−k/2 eπλ nv . We have ψ (0) = 1. Let us show that the derivative of ψ is positive at v = 0. We can write n

ψ (v) = eπλ nv ∏ (1 + vλ j )−k/2 , j=1

where λ1 , . . . , λn are the eigenvalues of Y0−1 , hence the value of the derivative at v = 0 is k k πλ n − (λ1 + · · · + λn ) = πλ n − σ (Y0−1 ) 2 2 kn kn kn − = π 1+ > 0, ≥ πλ n − 2bn 2π bn 2bn by (1.26), since X0 + iY0 ∈ D n . It follows that for small ε > 0, we have ψ (v0 ) = ψ (−ε ) < 1 (we recall that eπε = τ = |eπ i(u0 +iv0 ) | = e−π v0 ). Then the above inequality shows that µ = 0, and so F is identically equal to zero. Proof of Theorem 1.30. Note, first of all, that if the character χ is trivial on a principal congruence subgroup Γ (q) = Γn (q) contained in the group K, then the space Mk (K, χ ) is contained in the space M = Mk (Γ (q)) of modular forms of weight k and the unit character for the group Γ (q). Therefore, it will be sufficient to prove that each of these spaces is finite-dimensional. We recall that Γ (q) is a normal subgroup of finite index ν = [Γ : Γ (q)] in Γ . Let γ1 , . . . , γν be a system of representatives for cosets of Γ modulo Γ (q). For a function F ∈ M, let us consider the functions F1 = F|k γ1 , . . . , Fν = F|k γν .

(1.63)

By (1.31) and (1.32), the functions do not depend on the choice of the representatives γ j . Since for any γ ∈ Γ , the set γ1 γ , . . . , γν γ is again a set of representatives for the cosets, it follows that the functions F1 |k γ , . . . , Fν |k γ coincide up to a permutation with the functions (1.63). By Lemma 1.22, each of the functions (1.63) belongs to M, and by Proposition 1.25, is a cusp form if F is a cusp form. Let us now derive from Lemma 1.31 its generalization to the subspace N = Nk (Γ(q)) of cusp forms of M in the following form: if F(Z) =

∑

f (A)eπ iσ (AZ) ∈ N,

A∈E, A>0

and the Fourier coefficients f (A) satisfy f (A) = 0

if σ (A) ≤

knqν , 2π bn

(1.64)

1.3 Modular Forms

33

then F is identically equal to zero. For that we shall consider the product ν

G(Z) = ∏ Fj (Z) j=1

of the functions (1.63). By (1.29) and the above considerations, we have, for every M ∈ Γ, ν

G|kν M = j(M, Z)−kν G(M Z ) = ∏ ( j(M, Z)−k Fj ( Z )) j=1

ν

ν

j=1

j=1

= ∏ Fj |k M = ∏ Fj = G. Since G obviously satisfies the analytic conditions of the definition of cusp forms, we conclude that G is a cusp form of weight kν for the group Γ. Let f j (A) be the Fourier coefficients of the function Fj , so that Fj =

∑

πi

f j (A)e q σ (AZ) .

A∈E, A>0

Then the Fourier coefficients g(A) of G can be written in the form g(A) =

∑

f1 (A1 ) · · · fν (Aν ).

A1 +···+Aν =qA

Let A be a positive definite matrix of E satisfying σ (A) ≤ knν /2π b, where b = bn . Then the inequality 1 kν n σ (A) = (σ (A1 ) + · · · + σ (Aν )) ≤ q 2π b for positive definite even matrices A1 , . . . , Aν implies that σ (A j ) < knqν /2π b for each j = 1, . . . , ν , since the trace of any positive definite matrix is positive. If, for example, F1 = F and so f1 = f , then the condition (1.64) implies that each of the terms of the sum for g(A) has a factor of the form f1 (A1 ) = f (A1 ) with σ (A1 ) < knqν /2π b, which is zero. Then by Lemma 1.29, G = 0, and so F = 0. Now we can prove that the subspace N of cusp forms of M is finite-dimensional. Since entries of positive semidefinite matrices A = (aαβ ) satisfy the inequalities aαα ± 2aαβ + aβ β ≥ 0, it follows that the number of positive semidefinite even matrices A of order n with σ (A) ≤ 2N does not exceed the bound (N + 1)n (2N + 1)n(n−1)/2 .

(1.65)

Therefore, the number of positive definite even matrices A satisfying the inequality in (1.64) is bounded by a number of the form dn (kqν )n(n+1)/2 , where dn depends only on n. Taking dn to be integral, we see that d + 1 arbitrary functions

34

1 Modular Forms

F1 , . . . , Fd+1 of N are linearly dependent, since one can always find complex numbers c1 , . . . , cd+1 not all equal zero and such that the function F = c1 F1 + · · · + cd+1 Fd+1 satisfies the condition (1.64) and therefore is identically equal to zero. Finally, we use induction on n to prove the theorem for the entire spaces Mn = Mk (Γn (q)). Let us define for n ≥ 1 the linear map

Φ = Φn :

Mn → Mn−1 × · · · × Mn−1 ν times

by F|Φ = (F1 |Φ, . . . , Fν |Φ)

(F ∈ Mn ),

where ν is the index of Γn (q) in Γn , Φ the Siegel operator (1.56), Fj the functions (1.63), and where we set M0 = C. By Lemma 1.29, the kernel of Φn coincides with the subspace Nn of cusp forms of Mn and so is finite-dimensional. If the theorem is already proved for n − 1 ≥ 1, then the image of Φn is finite-dimensional and so is the space Mn . The image of Φ1 is obviously finite-dimensional, which proves the theorem for n = 1. One can show that M(S, χ ) = {0} if k is a negative integer, but we consider only modular forms of nonnegative integral weights and do not use this result. Exercise 1.32. Show that Mk (Γn ) = {0} if nk is odd. Exercise 1.33. Show that the spaces Mk (Γ1 ) for k = 0, 2, 4, 6, 8, 10 contain no cusp form, and there is not more than one linearly independent cusp form of weight 12 for Γ1 . [Hint: Use Lemma 1.31 for n = 1 with b1 given in Exercise 1.17.] Exercise 1.34. let k be a positive even integer. Show that the Fourier coefficients of any modular form ∞

F(z) =

∑ f(a)e2π iaz ∈ Mk (Γ1 )

a=0

satisfy

f(a) = f(0)ζ (k)−1

(2π i)k d k−1 + f (a), (k − 1)! ∑ d|n

where |f (a)| ≤ cF ak/2 ,

where ζ (s) is the Riemann zeta function. [Hint: Show first that the function F(z) − f(0)(2ζ (k))−1 Ek (z) is a cusp form. Then use Exercise 1.33 and (1.46).] Exercise 1.35. Show that the spaces Mk (Γ1 ) for k = 0, 4, 6, 8, 10 are spanned respectively by 1, E4 , E6 , E8 , and E10 , where Ek are the Eisenstein series of Exercise 1.20.

1.3 Modular Forms

35

Exercise 1.36. Show that the function ∆ (z) = ((2ζ (4))−1 E4 (z))3 − ((2ζ (6))−1 E6 (z))2 is a nonzero cusp form of the space M12 (Γ1 ) and that ∆ (z) and E12 (z) span the space. [Hint: With the help of classical formulas for ζ (4) and ζ (6) and formulas for Fourier coefficients of E4 (z) and E6 (z) show that the coefficient of e2π iz in the Fourier expansion of ∆ (z) is equal to 1728.] Petersson Scalar Product. Every space Nk (K, χ ) of cusp forms of an integral weight k and a congruence character χ for a congruence subgroup K of the symplectic group can be endowed with the structure of a Hilbert space by means of the scalar product. For two functions F and F on H = Hn , we consider the differential form on H defined by

ωk (F, F ) = F(Z)F (Z)h(Z)k d ∗ Z,

(1.66)

where h(Z) = h(X + iY ) = detY is the height of Z, d ∗ Z is the invariant element of volume (1.12), and as usual, bar means complex conjugation. It follows from Lemma 1.4 and Proposition 1.6 that for each matrix M = CA DB ∈ G = Gn , the form satisfies the relation

ωk (F, F )(M Z ) = F(M Z )F (M Z )h(M Z )k d ∗ M Z

= µ (M)nk det(CZ + D)−k F(M Z ) × det(CZ + D)−k F (M Z )h(Z)k d ∗ Z = µ (M)−nk+n(n+1) (F|k M) (Z) F |k M (Z)h(Z)k d ∗ Z = µ (M)n(n+1−k) ωk F|k M, F |k M (Z), (1.67) where |k M is the Petersson operator (1.29) of weight k. In particular, if F, F ∈ Mk (K, χ ) and M ∈ K, then by (1.32), we have

ωk (F, F )(M Z ) = ωk (χ (M)F, χ (M)F )(Z) = ωk (F, F )(Z). It follows that the integral

D(K)

ωk (F, F )(Z)

(1.68)

(1.69)

on a fundamental domain D(K) of K on H does not depend on the choice of the fundamental domain, provided that it converges absolutely. Lemma 1.37. Let K be a congruence subgroup of S = Sn , χ a congruence character of K, and k an integer. Suppose that at least one of the forms F, F ∈ Mk (K, χ ) is a cusp form. Then the integral (1.69) converges absolutely. Proof. By increasing the space Mk (K, χ ), one may assume that K is a subgroup of finite index in Γ and the character χ is trivial on K. Let us take then as D(K) a

36

1 Modular Forms

fundamental domain of the form (1.28) with M = 12n , that is, a finite union of the sets γ j D with γ j ∈ Γ, where D = Dn is the fundamental domain of Γ described in Theorem 1.16. Then, by (1.67), it is sufficient to show that each integral γ D

ωk (F, F )(Z) =

D

ωk F|k γ , F |k γ (Z) with γ ∈ Γ

converges absolutely. Assuming that F is a cusp form, by (1.42) and (1.43), we can write absolutely convergent Fourier expansions F|k γ =

∑

πi

fγ (A)e q σ (AZ) ,

A∈E,A≥0

πi

fγ (A)e q σ (AZ) ,

∑

F |k γ =

A∈E,A>.0

where E = En and q is such that Γ (q) ⊂ K. It follows that

∑

|(F|k γ )(Z)(F |k γ )(Z)| ≤

π

c(A)e− q σ (AZ) ,

A∈E,A>0

where Z = X + iY , with nonnegative coefficients c(A), and the last series converges on H. Then, by Theorem 1.16 and Lemma 1.24, we get the inequality |(F|k γ )(Z)(F |k γ )(Z)| ≤ d e−d

σ (Y )

(Z = X + iY ∈ D),

with positive constants d and d . Thus, in order to prove the lemma, it suffices to show that the integral

e−d σ (Y ) (detY )k−n−1

D

∏

1≤α ≤β ≤n

dxαβ dyαβ

with d > 0 converges. If (xαβ ) + i(yαβ ) ∈ D, then it follows from the definition of D and (1.22) that |xαβ | ≤ 1/2 for 1 ≤ α , β ≤ n and yαβ ≤ yαα /2 for α = β . In the √ proof of Theorem 1.16 we have seen that yαα ≥ 3/2 for α = 1, . . . , n. Then the inequality (1.22), if k < n, and the inequality (1.51), if k ≥ n + 1, imply that the last integral is majorized by

c

n √ yk−n−1 e−δ yαα dxαβ dyαβ αα |xαβ |≤1/2,yαα ≥ 3/2, α =1 1≤α ≤β ≤n |yαβ |≤yαα /2(α =β )

∏

∞

=c∏

α =1

∞ √

3/2

∏

k−n−1+n−α −δ yαα yαα e dyαα < ∞.

The above lemma justifies the following definition. For two modular forms F, F ∈ M(K, χ ) of an integral weight k and a congruence character χ for a congruence subgroup K ⊂ S such that at least one of the forms is a cusp form, the integral

1.3 Modular Forms

37

ˇ −1 (F, F ) = ν (G)

D(G)

ωk (F, F )(Z),

(1.70)

ˇ the where G is a congruence subgroup of Γ contained in K, Gˇ = G ∪ (−12n )G, ν (G) index of Gˇ in Γ , and D(G) a fundamental domain of G on H, is called the Petersson scalar product of these forms. Theorem 1.38. Under the above assumption, the Petersson scalar product has the following properties: (1) It converges absolutely and does not depend on the choice of fundamental domain D(G). (2) The scalar product is independent of the choice of the subgroup G of Γ contained in K. (3) It is linear in F and conjugate linear in F . (4) It satisfies (F, F ) = (F , F). (5) If F is a cusp form, then (F, F) ≥ 0, and (F, F) = 0 only if F = 0. (6) If M ∈ Gn ∩ Q2n 2n is a symplectic matrix with entries in the field Q of rational numbers and positive multiplier, then F|k M, F |k M = µ (M)n(k−n−1) (F, F ), (1.71) where functions the F|k M and F |k M are considered as elements of the space −1 Mk M KM, χM with the character χM defined by (1.34).

Proof. The first property follows from (1.68) and Lemma 1.37. In order to prove the second property, let us assume that G is another congruence subgroup contained in K. Then on replacing G by G ∩ G, one can assume that G ⊂ G. Let Γ = Gˇ γi and Gˇ = Gˇ δ j i

j

be decompositions into different left cosets. Then we have Γ=

Gˇ δ j γi ,

i, j

and the left cosets are distinct. It follows from Theorem 1.19 that one can take D(G ) =

(δ j γi ) Dn ,

i, j

hence we obtain

ν (Gˇ )−1

D(G )

ωk (F, F )(Z) = ν (Gˇ )−1 ∑ ∑ i

j

δ j γi Dn

= ν (Gˇ )−1 [Gˇ : Gˇ ] ∑ i

ωk (F, F )(Z)

γi Dn

ωk (F, F )(Z),

38

1 Modular Forms

where we have also used relations (1.68). Again by Theorem 1.19, the last expression is equal to ˇ −1 ν (G) ωk (F, F )(Z), D(G)

which proves the property (2). Properties (3), (4), and (5) follow directly from the definition. In order to prove relation (1.71), we note that since all entries of M are rational, the group M −1 KM is again a congruence subgroup of S, and so the group G(M) = G M −1 GM is again a congruence subgroup of Γ and, in particular, has finite index in Γ . It follows from Lemma 1.20, part (2), and (1.67) that −1 F|k M, F |k M = ν Gˇ (M)

D(G(M) )

ωk F|k M, F |k M (Z)

−1 = µ (M)n(k−n−1) ν Gˇ (M)

ωk (F, F ) (M Z )

−1 = µ (M)n(k−n−1) ν Gˇ (M)

ωk (F, F )(Z),

D(G(M) )

M D(G(M) )

where D Gˇ (M) is a fundamental domain for G(M) . It is clear that the set M D G(M) is a fundamental domain for the group MG(M) M −1 = MGM −1 ∩ G = G(M−1 ) . Hence, again by property (2), we can rewrite the last expression in the form −1 −1 µ (M)n(k−n−1) ν Gˇ (M) ν Gˇ (M−1 ) ν Gˇ (M−1 ) −1 ν Gˇ (M−1 ) (F, F ), = µ (M)n(k−n−1) ν Gˇ (M)

ωk (F, F D G(M −1 )

In order to prove the relation (1.71), it is sufficient to prove the equality ν Gˇ (M) = ν Gˇ (M−1 ) ,

)(Z)

(1.72)

which follows from the following more general lemma. Lemma 1.39. Let K be a congruence subgroup of Γn , and let M be a matrix of Gn with rational entries. Then the groups K(M) = K ∩ M −1 KM and K(M−1 ) = K ∩ MKM −1 are congruence subgroups of Γn with equal indices in K: K : K(M) = K : K(M−1 ) . (1.73) Proof of the lemma. By Proposition 1.21 these groups are intersections of congruence subgroups of Sn and hence themselves are congruence subgroups. Let D be a fundamental domain for the group K(M) . Since K(M−1 ) = MK(M) M −1 , it follows D as a fundamental domain for K(M−1 ) . Then Theorem that one can take the set M D 1.19 implies the following relations for the invariant volumes of the domains D and D : M D

1.3 Modular Forms

39

D) = Kˇ : Kˇ (M) v(D(K)), v(D

D ) = Kˇ : Kˇ (M−1 ) v(D(K)), v(M D

D) = v(M D D ), this relation implies both the where Kˇ = K ∪ (−12n )K. Since v(D equalities (1.72) and (1.73), We have proved the lemma and the theorem. Exercise 1.40. Show that the Eisenstein series Ek (z) defined in Exercise 1.32 is orthogonal to the space Nk (Γ1 ) of cusp forms of weight k for Γ1 . [Hint: Note that Ek (z) = (1 + (−1)k )ζ (k)

∑

j(M, z)−k ,

M∈Γ10 \Γ1

where j(M, z) is defined by (1.30) and Γ10 = a0 db ∈ Γ1 , which allows one to rewrite the scalar product of Ek (z) on a cusp form as integral over a fundamental domain for the group Γ10 , say {z = x + iy ∈ H1 | −1/2 ≤ x ≤ 1/2}.]

Chapter 2

Dirichlet Series of Modular Forms

2.1 Radial Dirichlet Series A natural way to approach zeta functions of modular forms is based on consideration of Dirichlet series constructed by means of Fourier coefficients of the forms. As was indicated in the introduction, a right zeta function must have certain analytic properties and an Euler product factorization. Therefore, the choice of appropriate Dirichlet series is motivated by a possibility of their analytic investigation plus a close relation with Euler products. These two features do not necessarily go together. Radial Series of Cusp Forms. Let us consider modular forms of the spaces Nnk (q, χ ) = Nk (Γn0 (q), χ ) of cusp forms of nonnegative integral weights k for the congruence groups

AB ∈ Γn C ≡ 0 (mod q) Γn0 (q) = CD of integral levels q with characters χ of the form AB AB = χ (det D) χ ∈ Γn0 (q) , CD CD

(2.1)

(2.2)

(2.3)

where χ is a Dirichlet character modulo q. According to Theorem 1.21 and Proposition 1.23, each such form F has the Fourier expansion F(Z) =

∑ n

f (A)eπ iσ (AZ) ,

(2.4)

A∈E ,A>0

which converges absolutely on Hn and uniformly on subsets (1.37) and has Fourier coefficients f (A) satisfying the relations f ( tVAV ) = χ (detV )(detV )k f (A)

(for all A ∈ En and V ∈ Λn )

A. Andrianov, Introduction to Siegel Modular Forms and Dirichlet Series, Universitext, DOI 10.1007/978-0-387-78753-4 2, c Springer Science+Business Media LLC 2009

(2.5) 41

42

2 Dirichlet Series of Modular Forms

and the estimate | f (A)| ≤ c(det A)k/2 ,

(2.6)

where c = cF depends only on F. If n = 1, the Fourier expansion can be written the form ∞

F(z) =

∑ f(m)e2π imz ,

m=1

and the simplest Dirichlet series associated to F is ∞

R(s) = R(s; F) =

f(m) , s m=1 m

∑

(2.7)

It converges absolutely and uniformly in right half-planes ℜs ≥ k/2 + 1 + ε with ε > 0 and uniquely determines the form F. Using the Euler formula ∞ 0

ys−1 e−α y dy = Γ(s)α −s

(α > 0, Re s > 0),

(2.8)

where Γ(s) is the gamma function, we can see that the series (2.7) can be presented in the half-planes ℜs ≥ k/2 + 1 + ε by the absolutely and uniformly convergent Mellin integral Φ(s; F) = =

∞

F(iy)ys−1 dy

0 ∞

∞

m=0

0

∑ f(m)

ys−1 e−2π my dy = (2π )−s Γ(s)R(s; F).

(2.9)

In the next section we shall see that the integral representation allows one to show that the function (2.9) has an analytic continuation over the whole s-plane as a holomorphic function and satisfies a functional equation. In Chapter 5 we shall see that the series (2.7) has close ties with Euler products. If n > 1, there is no unique natural way to introduce Dirichlet series corresponding to modular forms. The simplest generalizations of series (2.7) to genera n > 1 are given by so-called radial Dirichlet series (corresponding to the matrix “ray” A, 2A, . . . ) ∞ f (mA) (2.10) R(s) = R(s; F, A) = ∑ s m=1 m of a cusp form F with the Fourier expansion (2.4), where A is a fixed even positive definite matrix of order n. The estimate (2.6) implies that each radial series converges absolutely and uniformly in the right half-planes ℜs > nk/2 + 1 + ε

(2.11)

2.1 Radial Dirichlet Series

43

with ε > 0. For n > 1, the radial Dirichlet series does not necessarily determine the initial modular form F, since it includes only Fourier coefficients depending on matrices of the given ray A, 2A, . . . of positive definite even matrices and gives no information on Fourier coefficients with arguments belonging to other rays. Integral Representation of Radial Series. In order to derive an analytic expression of a radial series through original modular forms, we consider, for a fixed nonzero even matrix A of order n, the hyperplane in the space of real symmetric matrices of order n given by (2.12) X(A) = X = tX ∈ Rnn σ (AX) = 0 and the lattice in the hyperplane defined by X0 (A) = X(A)

Znn .

(2.13)

For every even matrix A of order n, the mapping

X → eπ iσ (A X)

(X ∈ X(A))

defines clearly a character of the compact quotient group X(A)/X0 (A), and the character is trivial if and only if A is a multiple of A, A = mA (note that the corresponding hyperplanes and lattices coincide if A = mA and have nontrivial intersections otherwise). Thus, 1 if A = mA, π iσ (A X) e [dX] = 0 otherwise, X(A)/X0 (A) where [dX] is the normalized Haar measure on X(A)/X0 (A). Integrating termwise the Fourier expansion (2.4) of the restriction of F on subsets X(A) + iY ⊂ Hn over a fundamental domain X(A)/X0 (A) of the translation group X0 (A) on X(A), we get

F(X + iY )[dX] = X(A)/X0 (A)

∑ n

f (A )e−πσ (A Y )

eπ iσ (A X) [dX]

X(A)/X0 (A)

A ∈E , A >0

=

∑n

f (mA)e−π mσ (AY ) .

m∈Q, mA∈E , mA>0

We recall that the a divisor of nonzero even matrix A is the largest number d ∈ N such that the matrix d −1 A is even. Nonzero even matrices of divisor d = 1 are called primitive. If A is even and primitive, then its rational multiple mA is even if and only if the coefficient m is integral. Since A and mA are both positive definite, m is positive. Thus, the last relation in the case of an even primitive positive definite matrix A turns into

∞

F(X + iY )[dX] = X(A)/X0 (A)

∑

m=1

f (mA)e−π mσ (AY ) .

(2.14)

44

2 Dirichlet Series of Modular Forms

There are many ways to transform the power series on √ the right to corresponding Dirichlet series. For example, if we put in (2.14) Y = v det AA−1 with v > 0, then, by using again the Euler integral (2.8), we obtain the identity ∞ √ vs−1 F(X + iv det AA−1 )[dX] dv 0 ∞

=

∑

m=1

X(A)/X0 (A) ∞

√ det Av

vs−1 e−π nm

f (mA)

0

√ dv = (π n det A)−s Γ(s)

√ = (π n det A)−s Γ(s)R(s; F, A)

∞

∑

m=1

f (mA) ms (2.15)

In Section 2.2 we briefly recall the classical theory of Mellin transforms of cusp forms in one variable. The rest of this chapter is devoted to analytic properties of radial Dirichlet series of cusp forms for genus 2. In order to illustrate the main features of the problem, we consider in detail the case of cusp forms of level q = 1 and radial series corresponding to the matrix A = 20 02 of the quadratic form x2 +y2 . Some features of the situation for other A are presented in the exercises. At present, the radial series of cusp forms for genus 2 are sufficiently investigated only in the case of level q = 1 (see Notes). The problem of Euler factorization of the radial series for genera n = 1 and n = 2 will be treated in Chapter 5. If n > 2, both the problem of analytic investigation and the problem of Euler factorization of the radial series are still essentially open.

2.2 Mellin Transform of Cusp Forms in One Variable Inversion Mapping. In this section we assume that n = 1. For a cusp form F ∈ N = N1k (q, χ ) we consider the simplest radial Dirichlet series R(s; F) defined by (2.7). As we have seen, in each of the right half-planes ℜs ≥ k/2 + 1 + ε , the series can be presented by the absolutely and uniformly convergent Mellin integral (2.9). Let us set 0 −1 1 . (2.16) Ω = Ω (q) = q 0 Since

d −c/q ab −1 , Ω Ω = −qb a cd

it follows that ΩK0 Ω−1 = K0 , where K0 = Γ10 (q), and the function F ∗ (z) = F|k Ω = q−1 z−k F(−1/qz) satisfies F ∗ |k M = F|k ΩMΩ−1 |k Ω = χ (a)F|k Ω = χ¯ (d)F ∗ , for each M =

(2.17) ab ∈ K0 , cd

2.2 Mellin Transform of Cusp Forms in One Variable

45

where χ¯ = χ −1 is the conjugate character. It is easy to see with the help of the Euclidean algorithm that among left multiples of any integral matrix of order 2 by matrices of Γ = Γ1 = SL2 (Z) there are always upper triangular matrices. In particular, for every matrix M ∈ Γ there exists a matrix M ∈ Γ such that ∗∗ . ΩM = M 0∗ Since F is a cusp form, it follows that for every M ∈ Γ the function F ∗ |k M has a Fourier expansion of the form (1.44). We conclude that F ∗ ∈ N(q, χ¯ ). Since −q 0 = (−1)k qk−2 F, F|k Ω2 = F|k 0 −q the inversion mapping F → F ∗ satisfies (F ∗ )∗ = (−1)k qk−2 F

(2.18)

and so determines an isomorphism of the space N(q, χ ) = N1k (q, χ ) onto N(q, χ¯ ) = N1k (q, χ¯ ). In particular, if χ = χ¯ , i.e., the character χ is real and takes only values ±1, the mapping is an automorphism of the space N(q, χ ). In this case the inversion mapping √ with i = −1 F → (−i)k q−k/2+1 F ∗ is also an automorphism of the space, and its square is the identity map. It follows that in the case of real χ , one can write a direct sum decomposition N(q, χ ) = N+ (q, χ ) + N− (q, χ ), where

(−i)k q−k/2 z−k F(−1/qz) = ±F

if F ∈ N± (q, χ ).

(2.19) (2.20)

Analytic Continuation and Functional Equation. Let us return to the integral (2.9). Theorem 2.1. For a nonnegative integer k, a positive integer q, and a Dirichlet character χ modulo q, let F ∈ N(q, χ ) = N1k (q, χ ) be a cusp form of weight k and character χ of the form (2.3) for the group K0 = Γ10 (q). Then the function Φ(s; F) = (2π )−s Γ(s)R(s; F), where R(s; F) is the Dirichlet series (2.7), has the following properties: (1) The function Φ(s; F) has an analytic continuation over the whole s-plane as a holomorphic function. (2) The function Φ(s; F) satisfies the functional equation Φ(k − s; F) = ik qs−k+1 Φ(s; F ∗ ), where F ∗ ∈ N(q, χ¯ ) is the function (2.17).

(2.21)

46

2 Dirichlet Series of Modular Forms

(3) If the character χ is real, and the form F belongs to one of the subspaces N± (q, χ ) of the decomposition (2.19), then the function Φ(s; F) satisfies Φ(k − s; F) = ±(−1)k qs−k/2 Φ(s; F).

(2.22)

Proof. Similarly to the case χ = 1 considered in the introduction, we can write, for ℜs sufficiently large, the identity Φ(s; F) = =

q−1/2

F(iy)ys−1 dy +

0

∞ q−1/2

∞ q−1/2

F(iy)ys−1 dy

F(i/qy)(1/qy)s−1 (1/qy2 )dy +

∞ q−1/2

F(iy)ys−1 dy,

where in the first integral we have changed the variable y → 1/qy. By substituting F(i/qy) = q(iy)k F ∗ (iy) in the first integral, we can rewrite the last relation in the form Φ(s; F) = ik q1−s

∞ q−1/2

F ∗ (iy)yk−s−1 dy +

∞ q−1/2

F(iy)ys−1 dy.

(2.23)

Since F and F ∗ are both cusp forms, the functions |F(iy)| and |F ∗ (iy)| approach exponentially zero as y → ∞ (see Lemma 1.24). It follows that the integrals in (2.23) both converge absolutely and uniformly on any compact subset of the s-plane. This proves part (1). By (2.23) with k − s in place of s and (2.18), we obtain Φ(k − s; F) = ik qs−k+1 = ik qs−k+1

∞ q−1/2 ∞

F ∗ (iy)ys−1 dy +

q−1/2

∞ q−1/2

F(iy)yk−s−1 dy

F ∗ (iy)ys−1 dy k k−s−1

∞

+ (−i) q

q−1/2

k 2−k

(−1) q

∗ ∗

k−s−1

(F ) (iy)y

dy

= ik qs−k+1 Φ(s; F ∗ ), which proves the functional equation (2.21). Finally, if, say, F ∈ N− (q, χ ), then it satisfies (−i)k q−k/2+1 F ∗ = −F, and so the functional equation turns into the equation Φ(k − s; F) = ik qs−k+1 (−1)ik qk/2−1 Φ(s; F) = −(−1)k qs−k/2 Φ(s; F). Exercise 2.2. Let F ∈ Mk (q, χ ) be an arbitrary modular form with Fourier expansion ∞

F(z) = f(0) +

∑ f(m)e2π imz .

m=1

2.3 Transformations of Lobachevsky Half-Spaces

47

Show that the Dirichlet series ∞

R(s; F) =

f(m) s m=1 m

∑

converge absolutely and uniformly in a right half-plane of the variable s; show that the functions Φ(s; F) = (2π )−s Γ(s)R(s; F) can be continued analytically to the whole s-plane as a meromorphic function having at most two simple poles at points s = 0 and s = k and satisfying the functional equation Φ(k − s; F) = ik qs−k+1 Φ(s; F ∗ ), where F ∗ ∈ N(q, χ¯ ) is the function (2.17); find the residues at the poles of Φ(s; F). [Hint: Use the integral representation Φ(s; F) =

!∞ 0

(F(it) − f(0))ys−1 dy.]

Exercise 2.3. Compute explicitly in the terms of the Riemann zeta function the Dirichlet series R(s; Ek ) for the Eisenstein series Ek (z) of even weight k > 2 defined in Exercise 1.32. [Hint: Use Exercise 1.33.]

2.3 Transformations of Lobachevsky Half-Spaces Transformation Group. The integral representation (2.15) for n = 2 shows that the radial Dirichlet series R(s; F, A) can be obtained by integration of the restriction of F to a real three-dimensional domain √ (2.24) H(A) = X(A) + iv det AA−1 ⊂ H2 . Further transformation of this representation is based on the remarkable circumstance that the real symplectic group S2 = Sp2 (R) defined by (1.13) has a rather large subgroup S(A) acting as a transitive group of analytic automorphisms of the domain H(A). By the end of this chapter the genus n is equal to 2, and the corresponding index is usually omitted. Let us consider now in detail the particular case A = 2E = 212 , where we write E in place of 12 . In this case one can take & ⎧ ⎪ x y ⎪ ⎪ X = X(2E) = x, y ∈ R , ⎪ ⎪ ⎪ y −x ⎪ ⎪ ⎪ & ⎪ ⎨ x y (2.25) X0 = X0 (2E) = x, y ∈ Z , ⎪ y −x ⎪ ⎪ ⎪ & ⎪ ⎪ ⎪ x y ⎪ ⎪ ⎪ |x| ≤ 1/2, |y| ≤ 1/2 . ⎩X/X0 = X(2E)/X0 (2E) = y −x

48

2 Dirichlet Series of Modular Forms

Since clearly, one can take [dX] = dx dy, the identity (2.15) for A = 2E turns into the identity (4π )−s Γ(s)R(s; F, 2E) & ∞ x + iv y F dx dy vs−1 dv. = |x|≤1/2, y −x + iv 0

(2.26)

|y|≤1/2

The domain of integration of this double integral is contained in the open subset

x + iv y (2.27) H = H(2E) = x, y, v ∈ R, v > 0 ⊂ H2 . y −x + iv Let us consider the set AB S = S(2E) = A, D ∈ Y, CD

B,C ∈ X,

A tD − B tC = E ,

where Y = Y (2E) =

x y x, y ∈ R . −y x

(2.28)

(2.29)

Proposition 2.4. The set S is a subgroup of the group S = Sp2 (R). For every matrix M = CA DB ∈ S, the transformation Z → M Z = (AZ + B)(CZ + D)−1

(Z ∈ H)

is a real analytic automorphism of the domain H. The action of S on H is transitive. x y y is always symmetric, it follows Proof. Since a matrix of the form −y x xy −x from (1.3) and (1.13) that S ⊂ S. Let us set

AB R= A, D ∈ Y, B,C ∈ X . CD Then S = S ∩ R. If M, M ∈ S, then M, M ∈ S and so MM ∈ S. From the obvious fact that X and Y are abelian groups under the addition of matrices, and from easily verified inclusions YY ⊂ Y,

XX ⊂ X,

XY ⊂ X,

YX ⊂ X

we conclude that MM ∈ R. Thus, MM ∈ S ∩ R = S. Further, the formula (1.1) implies that M −1 ∈ S if M ∈ S. Hence S is a subgroup of S. Now we shall show that H = M iE M ∈ S . (2.30)

2.3 Transformations of Lobachevsky Half-Spaces

Let Z =

x

y y −x

+ iv ∈ H. It is clear that the matrix ⎞ ⎛ √ x y 1 √ vE v y −x ⎠ is in S MZ = ⎝ √1 0 v

49

(2.31)

and MZ iE = Z. Thus, the left-hand side of (2.30) is contained in the right-hand side. In order to prove the reverse inclusion we note that if M = CA DB ∈ S with x1 y1 x2 y2 x3 y3 x4 y4 A= , B= , C= , D= , −y1 x1 y2 −x2 y3 −x3 −y4 x4 and M iE = X + iY , then, by (1.10) and (1.3), we have Y = t(D − iC)−1 (D + iC)−1 = (D tD +C tC)−1 −1 = x32 + x42 + y23 + y24 E.

(2.32)

Thus, it is sufficient to check that σ (X ) = 0. Since det(D + iC) = x32 + y23 + x42 + y24 , we have −1 M iE = x32 + y23 + x42 + y24 x4 − ix3 −y4 − iy3 x2 + ix1 y2 + iy1 , (2.33) × y2 − iy1 −x2 + ix1 y4 − iy3 x4 + ix3 hence

x32 + y23 + x42 + y24 σ (X ) = x2 x4 + x1 x3 + y2 y4 + y1 y3 − y2 y4 − y1 y3 − x2 x4 − x1 x3 = 0.

We have proved the relation (2.30), which implies the assertions of the proposition on the action of S on H. The above proposition shows that the domain H ⊂ H2 can be considered as a homogeneous space for the Lie group S ⊂ S2 . Next, we shall show that the pair (S, H) is naturally isomorphic to the pair (G, L), where G = SL2 (C), L is the 3-dimensional hyperbolic half-space (the Lobachevsky space) L = u = (z, v) z = x + iy ∈ C, v > 0 , and the action of G on L is given by the rule αβ Gg= : γ δ (α z + β )(γ¯z¯ + δ¯ ) + α γ¯v2 v u = (z, v) → g(u) = , , ∆(g, u) ∆(g, u)

(2.34)

50

2 Dirichlet Series of Modular Forms

where the bar means complex conjugation and where ∆(g, u) = |γ z + δ |2 + |γ |2 v2 .

(2.35)

In order to establish the relation of the pairs (S, H) and (G, L) for a real matrix of the form ⎞ ⎛ x1 y1 x2 y2 ⎜−y1 x1 y2 −x2 ⎟ AB ⎟ (2.36) M= =⎜ ⎝ x3 y3 x4 y4 ⎠ ∈ S, CD y3 −x3 −y4 x4 we set

x1 + iy1 y2 + ix2 αβ = , g(M) = y3 − ix3 x4 − iy4 γ δ

(2.37)

x y + ivE ∈ H y −x

(2.38)

u(Z) = (y + ix, v) ∈ L.

(2.39)

and for Z= we write

Proposition 2.5. In the above notation the following assertions hold: (1) g (S) = G, and the map g : S → G is an isomorphism of real Lie groups; the map u : H → L is a real analytic isomorphism. (2) The maps g and u transform the action of S on H into the action (2.34) of G on L in the sense that u (M Z ) = g(M) (u(Z))

(M ∈ S, Z ∈ H) ;

(2.40)

in particular, the action (2.34) satisfies (g, g ∈ G, u ∈ L). (gg )(u) = g(g (u)) AB (3) For all M = ∈ S and Z ∈ H, the identity CD det(CZ + D) = ∆(g(M), u(Z))

(2.41)

(2.42)

holds; in particular, the function (g, u) → ∆(g, u) is a factor of automorphy of the pair (G, L), i.e., it does not vanish on G × L and satisfies ∆(gg , u) = ∆(g, g (u))∆(g , u) (g, g ∈ G, u ∈ L).

(2.43)

Proof. The condition M ∈ S for a real matrix of the form (2.36) is equivalent to the relation A tD − B tC = E, that is, the relations x1 x4 + y1 y4 − x2 x3 − y2 y3 = 1,

−y1 x4 + x1 y4 − y2 x3 + x2 y3 = 0,

2.3 Transformations of Lobachevsky Half-Spaces

51

which can be rewritten in the form (x1 + iy1 )(x4 − iy4 ) − (y2 + ix2 )(y3 − ix3 ) = 1. The last relation is equivalent to the inclusion g(M) ∈ SL2 (C) = G. The relations g(MM ) = g(M)g(M ) for matrices M and M of the form (2.36) follow from the definitions by direct multiplication of the matrices, which we leave to the reader. The rest of part (1) is clear. In order to prove the relations (2.40), let us set A B , MMZ = M = C D where MZ is the matrix (2.31). By multiplying the matrices, we get √ x1 y1 x1 y1 = v , A = −y1 x1 −y1 x1 1 x1 x + y1 y + x2 x1 y − y1 x + y2 x2 y2 =√ , B = y2 −x2 v −y1 x + x1 y + y2 −y1 y − x1 x − x2 √ x y x y C = 3 3 = v 3 3 , y3 −x3 y3 −x3 1 x3 x + y3 y + x4 x3 y − y3 x + y4 x4 y4 √ = . D = −y4 x4 v y3 x − x3 y − y4 y3 y + x3 x + x4 For the matrix

X + iY = M Z = M MZ iE

= M iE ,

by the identities (2.32) and (2.33) with M in place of M, we obtain, respectively, −1 E Y = (x3 )2 + (y3 )2 + (x4 )2 + (y4 )2 and −1 X + iY = (x3 )2 + (y3 )2 + (x4 )2 + (y4 )2 x4 − ix3 −y4 − iy3 x2 + ix1 y2 + iy1 , × y2 − iy1 −x2 + ix1 y4 − iy3 x4 + ix3 hence

2 2 2 2 −1 x y X = (x3 ) + (y3 ) + (x4 ) + (y4 ) y −x

with x = x2 x4 + x1 x3 + y2 y4 + y1 y3 and y = −x2 y4 + x1 y3 + y2 x4 − y1 x3 . By substituting, we get 1 (x3 x + y3 y + x4 )2 + (x3 y − y3 x + y4 )2 v ∆(g, u) 1 2 2 , = v |γ | + |γ (y + ix) + δ |2 = v v

(x3 )2 + (y3 )2 + (x4 )2 + (y4 )2 = v(x32 + y23 ) +

52

2 Dirichlet Series of Modular Forms

where g = g(M) and u = u(Z) are given by (2.37) and (2.39), and y + ix = −x2 y4 + x1 y3 + y2 x4 − y1 x3 + i x2 x4 + x1 x3 + y2 y4 + y1 y3 1 = − (x1 x + y1 y + x2 )(x3 y − y3 x + y4 ) + vx1 y3 v 1 + (x1 y − y1 x + y2 )(x3 x + y3 y + x4 ) − vy1 x3 v 1 + i (x1 x + y1 y + x2 )(x3 x + y3 y + x4 ) + vx1 x3 v 1 + (x1 y − y1 x + y2 )(x3 y − y3 x + y4 ) + vy1 y3 v 1 = ((x1 + iy1 )(y + ix) + (y2 + ix2 ))((y3 + ix3 )(y − ix) + (x4 + iy4 )) v 1 + v(x1 + iy1 )(y3 + ix3 ) = (α z + β )(γ¯z¯ + δ¯ ) + α γ¯v. v The above computations imply the formula (2.40): v v x y E u (M Z ) = u(X + iY ) = u + i ∆(g, u) y −x ∆(g, u) v v (y + ix ), = g(M) (u(Z)) . = ∆(g, u) ∆(g, u) Formula (2.41) follows from (2.40) and (1.7). Direct computation shows that x + iv y x4 y4 x3 y3 + det(CZ + D) = det y3 −x3 −y4 x4 y −x + iv = (x3 x + y3 y + x4 )2 − (ivx3 )2 + (x3 y − y3 x + y4 )2 − (ivy3 )2 = ∆(g, u), as we have seen above. The relation (2.43) follows from (2.42) and (1.9). For further integration on L we shall need a G-invariant volume element. Proposition 2.6. The volume element on L given by du =

dx dy dv v3

(u = (x + iy, v) ∈ L)

(2.44)

is invariant under every transformation (2.34). Proof. It is easy to see that the group G = SL2 (C) is generated by the matrices α 0 1β 0 1 , , and (2.45) 01 −1 0 0 α −1

2.3 Transformations of Lobachevsky Half-Spaces

53

with α , β ∈ C, α = 0. For each of the matrices (2.45), the check of the assertion is an easy exercise, which we leave to the reader. Exercise 2.7. Let A be a real symmetric positive definite matrix of order 2, and sup√ pose τ ∈ GL2 (R) satisfies τ A tτ = det AE. Show that for every matrix M contained in the group t τ O −1 , with Mτ = S(A) = Mτ SMτ O τ −1 the transformation Z → M Z defines a real analytic automorphism of the domain H(A) given by (2.24) and (2.12), and the action of S(A) on H(A) is transitive. [Hint: Show first that H(A) = Mτ H .] Exercise 2.8. In the notation and assumptions of the previous exercise show that Proposition 2.5 remains true if one replaces the pair S, H by the pair S(A), H(A), and the maps g, u by the maps gτ : S(A) → G, uτ : H(A) → L defined by (M ∈ S(A)) and uτ (Z) = u Mτ−1 Z

(Z ∈ H(A)). gτ (M) = g Mτ−1 MMτ Discrete Subgroup. We turn now to integration of restrictions of modular forms to Lobachevsky subspaces of H2 . By the end of this chapter we shall restrict our consideration to the case of cusp forms for the group Γ2 = Γ20 (1) with trivial character. Each function F ∈ N2k = N2k (1, 1) satisfies the functional equations AB −k for all M = ∈ Γ2 . (2.46) det(CZ + D) F (M Z ) = F CD Let F- be the function on the Lobachevsky half-space L corresponding via (2.39) to the restriction of F on H ⊂ H2 , i.e., - = F(Z) F(u) and let g=

αβ = g(M) γ δ

if u = u(Z) with Z ∈ H, with

M=

AB ∈ S Γ2 . CD

(2.47)

(2.48)

Then the relations (2.46) and Proposition 2.6 imply the relations = ∆(g(M), u(Z))−k F(g(M)(u(Z))) ∆(g, u)−k F(g(u)) = det(CZ + D)−k F (M Z ) = F(Z) = F(u).

(2.49)

Lemma 2.9. The restriction of the map M → g(M) to the subgroup S ∩ Γ2 is an isomorphism of the subgroup onto the special linear group Λ = Λ(O) = SL2 (O)

(2.50)

of order 2 over the ring of Gaussian integers √ O = Z + iZ, i.e., the ring of integers of the imaginary quadratic field Q[i] (i = −1).

54

2 Dirichlet Series of Modular Forms

Proof. Since the map g is isomorphism of S on SL2 (C), it is sufficient to note that a matrix M of the form (2.36) contained in S belongs to the group Γ2 if and only ifall of its entries belong to Z. This is equivalent to the conditions that g(M) = αγ βδ ∈ SL2 (C) with α , β , γ , δ belonging to the ring Z + iZ = O. Analogously to the case of the symplectic modular group considered in Section 1.2, the group Λ is a discrete subgroups of G discretely acting on L. Similarly to Theorem 1.16, we have the following theorem. Theorem 2.10. The closed subset of L given by D = D(O) = (x + iy, v) ∈ L 0 ≤ x + y, x ≤ 1/2, y ≤ 1/2, 1 ≤ x2 + y2 + v2

(2.51)

is a fundamental domain of Λ on L, i.e., it meets each of the Λ -orbits and has no distinct inner points belonging to the same orbit. The fundamental domain has finite volume with respect to the invariant volume element (2.44). Proof. By analogy with the symplectic case, let us call the positive real number v2 = h(u) the height of a point u = (z, v) ∈ L. If M = CA DB ∈ S with g(M) = αγ βδ and Z ∈ H with u(Z) = (z, v), then, by (1.24), (2.42), (2.40), and (2.34), the heights of the points M Z and g(u) satisfy the relation −2

h(M Z ) = | det(CZ + D)| h(Z) =

v ∆(g, u)

2 = h(g(u)).

It follows that the values of the height of points of the orbit Λ (u) are among the values of the heights of points of the orbit Γ2 Z , where u(Z) = u. Hence by Lemma 1.15, we conclude that each orbit of Λ on L contains points u of maximal height and the points can be characterized by inequalities ∆(g, u) ≥ 1

for every g ∈ Λ .

(2.52)

Similarly to the proof of Theorem 1.16, in order to construct a fundamental domain for Λ on L, first of all, we shall choose on the Λ -orbitof a point u ∈ L a point u = α β with α , α −1 , β ∈ O (z , v ) of maximal hight. Further, the matrices g = 0 α −1 belong to Λ , and the corresponding transformations do not change the height: g(u ) = g((z , v )) = α α¯ −1 z + β α¯ −1 , v = (±z + β , v ) with β = β α¯ −1 ∈ O, since the inclusions α , α −1 ∈ O mean that α is a unit of O, i.e., α = ±1, ±i. It is clear that the sign and the number β ∈ O can be chosen such that the point u = (x + iy, v) = (±z + β , v ) satisfies the conditions 0 ≤ x + y, x ≤ 1/2, y ≤ 1/2. Since the height of u is the same as thatof u and so is again maximal 0 1 shows that for the considered orbit, the inequality (2.52) for g = −1 0

2.4 Radial Series and Eisenstein Series

55

∆(g, u) = |z|2 + v2 = x2 + y2 + v2 ≥ 1. Thus, u ∈ D. now that u = σ (u) for two points u = (z, v), u = (z , v ) ∈ D and g = Suppose α β ∈ Λ . Then by the above, the points u, u have the same height v = v, whence γ δ

∆(g, u) = 1. Similarly, ∆(g−1 , u ) = 1. If γ = 0, the equations are both nontrivial, and so points u, u belong to the boundary of D. But if γ = 0, then, as we have seen above, z = ±z + β . This implies that z = z, unless the complex numbers z = x + iy and z = x + iy belong to the boundary of the triangle 0 ≤ x + y, x ≤ 1/2, y ≤ 1/2. Finiteness of the volume is obvious. b Exercise 2.11. Let A = 2a b 2c be a positive definite even matrix. Check that the matrix 1 2c √−b ∈ SL2 (R) τ = τ (A) = √ 2c det A 0 det A √ satisfies tτ Aτ = det AE. Assuming that gcd(a, b, c) = 1, show then that the corresponding mapping gτ = gτ (A) of Exercise 2.8 maps the group Γ(A) = S(A)

Γ2

isomorphically onto the group

αβ −1 ∈ G α , δ ∈ O(A), β ∈ A(A), γ ∈ A(A) , Γ(O(A), A(A)) = γ δ where O(A) is the subring of the discriminant − det A of the ring of integers of the √ imaginary quadratic field K = Q( − det A), A(A) = ωc {a, ω }2 , and {a, ω } is the module in K of the form aZ + ω Z. [Hint: Check first that AB ˆ Γ(A) = A ∈ Y0 (A), CD

ˆ D ∈ Y0 (A), A tD − B tC = E , B ∈ X0 (A),C ∈ X0 (A),

where X0 (A) = X(A) ∩ Z22 , X(A) is the set (2.12), Y0 (A) = Y (A) ∩ Z22 , Y (A) = {Y ∈ R22 |YA tY = det Y · A, det Y ≥ 0}, and Aˆ = det A · A−1 .]

2.4 Radial Series and Eisenstein Series In this section we shall show that the integral in the representation (2.26) of the radial series R(s; F, A) for the matrix A = 2E can be interpreted as an integral convolution of the restriction of F on H considered via (2.39)√as a function on L with an Eisenstein series for the discrete subgroup Λ = SL2 (Z[ −1]) ⊂ G.

56

2 Dirichlet Series of Modular Forms

Let F be a cusp form of the space N2k = N2k (1, 1) and F- the corresponding function (2.47) on the Lobachevsky half-space L. By (2.49), the function F- satisfies the functional equations = F(u) ∆(g, u)−k F(g(u))

(u ∈ L, g ∈ Λ ) .

(2.53)

s+2 du, F(u)v

(2.54)

The identity (2.26) can be rewritten in the form (4π )−s Γ(s)R(s; F, 2E) =

P

P = {u = (x + iy, v) ∈ L |x| ≤ 1/2, |y| ≤ 1/2}

where

and du is the invariant element of volume (2.44). Note first of all that the set P can be interpreted as a fundamental domain for the group

±1 β Λ Λ0 = ∈ 0 ±1 acting on L. Let

Λ=

Λ0g j

j

be a decomposition into disjoint left cosets, and let D be a fundamental domain of the group Λ on L. Then the set P =

g j (D)

j

can also be taken as a fundamental domain of the group Λ 0 . Since the integrand in (2.54) is invariant under every transformation of Λ 0 , it follows that the integral does not change when the domain P is replaced by another sufficiently good fundamental domain of Λ0 , for example by P , provided that the integral converges absolutely. Then we obtain

s+2 du = F(u)v

P

P

s+2 du = ∑ F(u)v j

s+2 du. F(u)v

g j (D)

In each of the last integrals we make the change of variables u → g−1 j u . Then, by invariance of the volume element du and (2.53), we get P

s+2 du = ∑ F(u)v

j

=

D

D

F(u)

vs+2 du ∆(g j , u)s−k+2

∗ (u, s − k + 2)vk du, F(u)E

2.4 Radial Series and Eisenstein Series

57

where

1 . ∆(g, u)s Λ0 \Λ Λ g∈Λ We note now that two matrices αγ βδ and αγ βδ of Λ belong to the same left coset modulo Λ 0 if and only if their second rows satisfy {γ , δ } = ±{γ , δ }. Moreover, since the Euclidean algorithm holds in the ring of Gaussian integers, for every pair of relatively prime numbers γ , δ ∈ O one can find numbers α , β ∈ O such that αδ − γβ = 1. It follows that E ∗ (u, s) = vs

vs 2

E ∗ (u, s) =

∑

∑

γ ,δ ∈O γ O+δ O=O

1 s . 2 |γ z + δ | + |γ |2 v2

(2.55)

The series (2.55) is a so-called Eisenstein series for the group Λ . Lemma 2.12. The Eisenstein series (2.55) converges absolutely and uniformly on compact subsets of L in each half-plane ℜs > 2 + ε with ε > 0 and satisfies the relations for all g ∈ Λ . (2.56) E ∗ (g(u), s) = E ∗ (u, s) Proof. For u = (z, v) in a compact subset of L, we clearly have |γ z + δ |2 + |γ |2 v2 > c(|γ |2 + |δ |2 ),

(2.57)

where c is a positive constant. It follows that the series for 2c−1 v−s E ∗ (u, s) in the half-plane ℜs > 2 + ε is majorized termwise by the convergent series ∞

1

∑

(γ ,δ )=(0,0)

(|γ |2 + |δ |2 )2+ε

=

r4 (n) , 2+ε n=1 n

∑

where r4 (n) is the number of integral solutions of the equation x12 + x22 + x32 + x42 = n (for convergence, see the Jacobi formula for r4 (n)). By (2.34) and (2.43), we obtain E ∗ (g(u), s) = (v(g(u))s

∑

1

Λ0 \Λ Λ g ∈Λ −s

= v ∆(g, u) s

∑

Λ0 g ∈Λ

∆(g , g(u))s

∆(g, u)s = E ∗ (u, s). g, u)s ∆(g Λ \Λ

We summarize the results of the above considerations in the following lemma. Lemma 2.13. For every cusp form F ∈ N2k with Fourier expansion (2.4), the radial Dirichlet series ∞ f (2mE) RF (s) = ∑ ms a=1

58

2 Dirichlet Series of Modular Forms

satisfies in a right half-plane of the variable s the identity (4π )−s Γ(s)RF (s) =

D

∗ (u, s − k + 2)vk du, F(u)E

(2.58)

where F(u) is the restriction of F on H considered as a function on L, E ∗ (u, s) the Eisenstein series (2.55), du the invariant measure (2.44), and where D is a fundamental domain for the group Λ on L. b be a positive definite even matrix satisfying Exercise 2.14. Let A = 2a b 2c gcd(a, b, c) = 1. In the notation of Exercises 2.8 and 2.11, let F-A (u) = F(h−1 τ (u)) be the function on the space L corresponding via the map hτ : H(A) → L to the restriction on H(A) of a cusp form F ∈ N2k . Assuming that there exists a fundamental domain D(A) of finite volume for the group Γ(O(A), A(A)) on L, show that the radial Dirichlet series R(s; F, A) satisfies in a right half-plane of the variable s the identity 1−s 1 (2π )−s (det A) 2 Γ(s)R(s; F, A) = 2

where EA∗ (u, s) = vs

σ ∈Γ0

D(A)

F-A (u)EA∗ (u, s − k + 2)vk du,

1 , ∆( σ , u)s (O(A),A(A))\Γ(O(A),A(A))

∑

and Γ0 (O(A), A(A)) =

∗∗ ∈ Γ(O(A), A(A)) . 0∗

2.5 Properties of Radial Series for Sums of Two Squares In this section, in order to apply the integral representation (2.58), we represent the Eisenstein series E ∗ (u, s) by means of suitable theta series of a positive definite quadratic form in four variables, recall standard properties of the theta series, and finally, prove that the radial Dirichlet series R(s; F) = R(s; F, 2E) has a meromorphic analytic continuation over the whole s-plane and satisfies a functional equation with two gamma–factors. Eisenstein Series and Theta Series. Let us introduce the theta series πt (t > 0, u = (z, v) ∈ L). Θ(t, u) = ∑ exp − (|γ z + δ |2 + |γ |2 v2 ) v γ ,δ ∈O If t > η > 0 and u is in a compact subset of L, then v < b, and by (2.57), the theta series is majorized termwise by the series πη c πη c 4 ∞ 2 2 (| n2 , exp − γ | + | δ | ) = exp − ∑ ∑ b b n=−∞ γ ,δ ∈O

2.5 Properties of Radial Series for Sums of Two Squares

59

and so it converges absolutely and uniformly. If ℜs is sufficiently big, then by using the Euler integral (2.8), we get ∞ 0

t s−1 (Θ(t, u) − 1)dt = Γ(s)

v s

π

∑

1

γ ,δ ∈O, (γ ,δ )=(0,0)

(|γ z + δ |2 + |γ |2 v2 )s

.

√ Since each ideal A of the ring O = Z[ −1] of Gaussian integers is principal, A = α O, and its norm N(A) is α α¯ = |α |2 , for every nonzero pair γ , δ ∈ O with γ O + δ O = A, we have γ = αγ , δ = αδ with γ O + δ O = O, hence the last series is equal to Γ(s)

v s

π

∑ A

∑

1

γ ,δ ∈O γ O+δ O=O

N(a)s (|γ z + δ |2 + |γ |2 v2 )s

= 2π −s Γ(s)ZO (s)E ∗ (u, s),

where A ranges through all nonzero integral ideals of the ring O, ZO (s) = ∑ A

1 N(A)s

(2.59)

√ is the Dedekind zeta function of the ring Z[ −1], and E ∗ (u, s) is the Eisenstein series (2.55). We have proved the identity ∞

t s−1 (Θ(t, u) − 1)dt = 2π −s Γ(s)ZO (s)E ∗ (u, s),

0

(2.60)

valid in a right half-plane of the variable s. Lemma 2.15. (Inversion f ormula) The theta series Θ(t, u) satisfies the identity 1 1 ,u (t > 0, u ∈ L). Θ(t, u) = 2 Θ t t Proof. First we shall prove that for each symmetric positive definite matrix Q of order r, the theta series

θ (t, Q) =

∑

e−π tQ[N]

(Q[N] = tNQN)

(2.61)

N∈Zr

satisfies the inversion formula 1 θ (t, Q) = √ r θ t det Q

1 −1 ,Q . t

Indeed, the function

θ (t, X, Q) =

∑

N∈Zr

e−π tQ[N+X]

(X ∈ Rr )

(2.62)

60

2 Dirichlet Series of Modular Forms

is a periodic function of X and is equal to its Fourier series

θ (t, X, Q) =

∑ r a(M)e2π i NX , t

(2.63)

M∈Z

where 1

a(M) = 0

···

+∞

=

−∞

1 0

···

θ (t, X, Q)e−2π i MX dx1 · · · dxr t

+∞ −∞

e−π tQ[X]−2π i MX dx1 · · · dxr . t

Completing the square, we have i 1 tQ[X] + 2i tMX = tQ[X + Q−1 M] + Q−1 [M] t t

(i =

√ −1),

whence π

−1 [M]

a(M) = e− t Q

Rr

−1 [M]

e−π tQ[X+ t Q i

dX

(dX = dx1 · · · dxr ).

By Cauchy’s theorem, we can write − πt Q−1 [M]

a(M) = e

Rr

e−π tQ[X] dX

π −1 1 e− t Q [M] =√r t det Q

Rr

π −1 t 1 e− t Q [M] , e−π YY dY = √ r t det Q

where Y = SX with tSS = tQ and dY = | det S|dX, which together with (2.63) for X = 0 proves the formula (2.62). Coming back to the theta function Θ(t, u), we note that an easy straightforward calculation allows us to interpret it as a special case of the theta function (2.61), namely, (2.64) Θ(t, u) = θ (t, Q), where

⎛

⎞ [u] 0 x y 1 ⎜ 0 [u] −y x⎟ ⎟ Q= ⎜ v ⎝ x −y 1 0⎠ y x 0 1

if u = (x + iy, v) ∈ L

and [u] = x2 + y2 + v2 .

It is easy to check that det Q = 1 and ⎛ ⎛ ⎞ ⎞ 0 0 01 1 0 −x −y ⎜ ⎟ 1 ⎜ 0 1 y −x⎟ ⎟ = tUQU, U = ⎜ 0 0 1 0⎟ ∈ GL4 (Z). Q−1 = ⎜ ⎝ ⎝ ⎠ 0 −1 0 0⎠ v −x y [u] 0 −1 0 0 0 −y −x 0 [u]

2.5 Properties of Radial Series for Sums of Two Squares

61

Then by (2.64) and (2.62), we obtain 1 −1 1 ,Q Θ(t, u) = θ (t, Q) = 2 θ t t 1 1 1 1 1 1 , Q[U] = 2 θ , Q = 2Θ ,u . = 2θ t t t t t t Analytic Continuation and Functional Equation. Now we are finally able to prove the following result. Proposition 2.16. Let F be a cusp form of integral weight k for the group Γ2 . Then the function Ψ(s; F) = (2π )−2s Γ(s)Γ(s − k + 2)ZO (s − k + 2)R(s; F), where Γ(s) is the √ gamma function, ZO (s) is the Dedekind zeta function (2.59) of the ring O = Z[ −1], and R(s; F) = R(s; F, 2E) is the radial Dirichlet series corresponding to the ray of the matrix A = 2E = 2 · 12 of the sum of two squares, can be continued analytically to the whole s-plane as a meromorphic function having at most two poles at the points s = k − 2 and s = k, and satisfying the functional equation Ψ(2k − 2 − s; F) = Ψ(s; F). Proof. By (2.58), in a right half-plane of the variable s we have the identity Ψ(s; F) = π −s Γ(s − k + 2)ZO (s − k + 2)

D

∗ (u, s − k + 2)vk du, F(u)E

which, by (2.60), can be rewritten in the form ∞ 1 2−k k s−k+1 Ψ(s; F) = π t (Θ(t, u) − 1)dt du. F(u)v 2 D 0

(2.65)

In view of Lemma 2.15, we obtain 1 0

t s−k+1 (Θ(t, u) − 1)dt =

1 0

1

t s−k+1 Θ(t, u)dt −

1

t s−k+1 dt

0

1 1 t s−k−1 Θ( , u)dt − t s − k +2 0 ∞ 1 t k−s−1 Θ(t, u)dt − = s − k +2 1 ∞ 1 1 − . = t k−s−1 (Θ(t, u) − 1)dt + s−k s−k+2 1 =

62

2 Dirichlet Series of Modular Forms

Thus, the inner integral in (2.65) can be written in a right half-plane in the form ∞ 1

=

t s−k+1 (Θ(t, u) − 1)dt +

∞ 1

1 0

t s−k+1 (Θ(t, u) − 1)dt

(t s−k+1 + t k−s−1 )(Θ(t, u) − 1)dt +

1 1 − . s−k s−k+2

(2.66)

Substituting this expression in (2.65), we get ∞ k 2π k−2 Ψ(s; F) = F(u)v (t s−k+1 + t k−s−1 )(Θ(t, u) − 1)dt du D 1 1 1 k − du. (2.67) + F(u)v s−k s−k+2 D Let us take now the domain D to be the set D described in Theorem 2.10, which has a single cusp at the point v = +∞. Since F is a cusp form, the function F(u) exponentially tends to zero as u = (z, v) tends from within D to the cusp. The same is obviously true with respect to the function Θ(t, u) − 1 under the inner integral in the last formula. Thus, the integral representation (2.67) gives a meromorphic analytic continuation of Ψ(s; F) to the whole complex plane. This function is regular at all points, except possibly for simple poles at s = k and s = k − 2, when ! k D F(u)v du = 0. Furthermore, the right-hand side of (2.67) is invariant under the substitution s → 2k − 2 − s, which proves the functional equation. Exercise 2.17. Prove that the function Φ(s) = π −s Γ(s)ZO (s)E ∗ (u, s), where Γ(s) is the gamma function, ZO (s) is the zeta function (2.59), and E ∗ (u, s) is the Eisenstein series (2.55), can be continued analytically to the whole s-plane as a meromorphic function having only two simple poles at the points s = 0 and s = 2, and satisfying the functional equation Φ(2 − s) = Φ(s). [Hint: Use the integral representation (2.60) and identity (2.66) with s + k − 2 in place of s.]

Chapter 3

Hecke–Shimura Rings of Double Cosets

3.1 An Approach to Multiplicativity Numerous classical formulas for the numbers of integral representations of integers by integral quadratic forms often show remarkable multiplicative features. Thus, for example, Jacobi’s formula for the number of representations r4 (m) of an odd integer m as the sum of four of integer squares, r4 (m) = 8 ∑ d, d|m

expresses the number through the multiplicative function ς (m) = ∑d|m d with the rule of multiplication mn , ς (m)ς (n) = ∑ d ς d2 d|m,n whereas Ramanujan’s formula for the number of representations of integers as sum of 24 integral squares expresses the number as a linear combination of two multiplicative functions. The numbers of integral representations of integers by an integral positive definite quadratic form can be considered as Fourier coefficients of a modular form, the theta series of the quadratic form. In order to reveal multiplicative properties of the numbers of representations, it is reasonable to consider a more general but formulated in more invariant form problem of multiplicative properties of Fourier coefficients of modular forms. Let us go into some detail. One can argue as follows: let, for example, ∞

F(z) =

∑ f(m)e2π imz ∈ Mk = Mk (Γ )

m=0

be a modular form of integral weight k for the group Γ = Γ1 . Generally speaking, a “multiplicativity” of the function m → f(m) should at least imply that there are

A. Andrianov, Introduction to Siegel Modular Forms and Dirichlet Series, Universitext, DOI 10.1007/978-0-387-78753-4 3, c Springer Science+Business Media LLC 2009

63

64

3 Hecke–Shimura Rings of Double Cosets

regular relations between this function and functions m → f(pm) for some prime numbers p. The values f(pm) are the Fourier coefficients of the function ∞

Fp (z) =

∑

m=0

f(pm)e2π imz =

1 p−1 ∑F p b=0

z+b p

p−1

=

∑ F|k

b=0

1b , 0p

where |k M are the Petersson operators (1.29) for genus n = 1. If the linear operator F → Fp maps the space Mk into itself, then one could hope to find its eigenfunctions in Mk , i.e, functions satisfying Fp = λ (p)F. For each such function one gets relations f(pm) = λ (p)f(m) for all m = 1, 2, . . . , which provide a kind of multiplicativity. Now, the function Fp belongs to Mk if and only if Fp |k γ = Fp for all γ ∈ Γ . Since F|k γ = F if γ ∈ Γ , and |k M1 M2 = |k M1 |k M2 for every two real matrices of order two with determinant, the inclusion Fp ∈ Mk would be true if the set of positive coincides up to an order with the set left cosets Γ 10 0p , Γ 10 1p , . . . Γ 10 p−1 p Γ 10 0p γ , Γ 10 1p γ , . . . Γ 10 p−1 γ for every γ ∈ Γ . But this is not true, since, p for example, the left coset 10 0 1 0 1 p0 p0 =Γ =Γ Γ 0p −1 0 −1 0 01 01 differs from any left coset Γ 10 bp . On the other hand, we note that all the left cosets 1 b 1 01 b Γ 0 p γ = Γ 0 p 0 1 γ with γ ∈ Γ belong to the double coset Γ 10 0p Γ , and we may replace the sum Fp by the sum over all representatives of left cosets contained in that double coset. Thus, in place of the operator F → Fp , we arrive at the famous Hecke operator T (p) : F → F|T (p) = ∑ F|k M. γ ∈Γ \Γ 10 0p Γ

Since every right on an element of Γ only rearranges left Γ -cosets multiplication contained in Γ 10 0p Γ , it easily follows that T (p) maps the space Mk into itself. Since p is a prime number, one can easily check that the set

10 10 1 p−1 p0 , ,..., , 0p 1p 0 p 01 is a set of representatives of all different left Γ -classes contained in this double coset. Thus, we conclude that ∞ ∞ p0 F|T (p) = Fp + F|k = ∑ f(pm)e2π imz + pk−1 ∑ f(m)e2π ipmz 01 m=0 m=0 ∞ m e2π imz , = ∑ f(pm) + pk−1 c p m=0 where f( mp ) = 0 if p m. If F ∈ Mk is an eigenfunction for the operator T (p), F|T (p) = λ (p)F, then, by comparing Fourier coefficients, we have

3.1 An Approach to Multiplicativity

65

m k−1 = λ (p)f(m) f (pm) + p f p

(m = 0, 1, 2, . . . ),

and in particular, f (p) = λ (p)f(1). Hence we obtain m k−1 = f (p)f (m) f (1) f (pm) + p f p

(m = 0, 1, 2, . . . ),

which is not very bad kind of multiplicativity with respect to the prime number p. The above consideration points to an approach to the problem of multiplicativity of Fourier coefficients of modular forms for subgroups of modular groups as well as the problem of Euler factorization of Dirichlet series formed by the Fourier coefficients, namely the approach based on the study of linear operators on spaces of modular forms associated to certain double cosets of the corresponding subgroup. The double cosets form associative rings, the rings of double cosets or the Hecke–Shimura rings, which we consider in this chapter first in an abstract situation and then for discrete subgroups of the general linear and symplectic groups. In the next chapter we shall study the linear operators on spaces of modular forms corresponding to the rings of double cosets. Exercise 3.1. (1) Show that the subspace of cusp forms Nk (Γ ) ⊂ Mk (Γ ) is invariant with respect to all of the operators T (p). (2) Show that the function ∆ (z) ∈ N12 (Γ ) defined in Exercise 1.36 is an eigenfunction for all operators T (p). (3) (Ramanujan/Mordell) Let ∆ (z) =

∞

∑ f(m)e2π imz .

m=1

Show that the corresponding radial Dirichlet series has the Euler factorization ∞

f(m) 1 = f(1) ∏ , s −s + p11−2s m 1 − τ (p)p m=1 p∈P

∑

where τ (p) are the eigenvalues of T (p). [Hint for (3): using the multiplicative relations m 11 = τ (p)f (m) f (pm) + p f p

(m = 0, 1, 2, . . . ),

prove first that for every prime p and every m not divisible by p, we have the identity ∞

(1 − τ (p)t + p11t 2 ) ∑ f(mpδ )t δ = f(m).] δ =0

66

3 Hecke–Shimura Rings of Double Cosets

3.2 Abstract Rings of Double Cosets Definition of Hecke–Shimura rings. In this section we assign to each pair of a multiplicative semigroup with unit and its subgroup, satisfying a simple finiteness condition, an associative ring constructed of cosets of the semigroup modulo the subgroup and consider their basic properties. The simplest situation in which a ring of cosets can be naturally defined is the case of a group Σ and its normal subgroup Λ . In this case each double coset Λ gΛ of Σ modulo Λ consists of a single left coset, Λ gΛ = Λ g; hence one can define the product of double cosets just by (Λ gΛ )(Λ gΛ ) = (Λ g)(Λ g ) = (Λ gg ) = (Λ ggΛ ) and then extend the multiplication by linearity on finite formal linear combinations of the double cosets with coefficients, say, in the ring Z. As a result, we arrive at the group ring of the factor group Λ \Σ over Z. The definition of Hecke–Shimura rings of double cosets just transfers this approach to the situation in which double cosets are unions of finite numbers of left cosets. Let Σ be a multiplicative semigroup with identity element and Λ a subgroup of Σ . We shall say that the pair Λ , Σ is l(eft)-finite if each double coset Λ gΛ ⊂ Σ is a finite union of left cosets modulo Λ : ν (g)

Λ gΛ =

Λ gi

(ν (g) = #(Λ \Λ gΛ )).

(3.1)

i=1

The pair is said to be d(ouble)-finite if each of the double cosets is both a finite union of left cosets and a finite union of right cosets modulo Λ . For an l-finite pair Λ , Σ we denote by L = L(Λ , Σ ) the free Z-module consisting of all formal finite linear combinations with coefficients in Z of the symbols (Λ g) with g ∈ Σ , which are in one-to-one correspondence with the left cosets Λ g of Σ relative to the group Λ . The semigroup Σ naturally operates on L by multiplication from the right:

Σ g : τ = ∑ ai (Λ gi ) → τ g = ∑ ai (Λ gi g). i

i

Let us consider the submodule D of L consisting of all elements, that are invariant under the right multiplication by elements in Λ : (3.2) D = D(Λ , Σ ) = τ ∈ L τλ = τ for all λ ∈ Λ . If

τ = ∑ ai (Λ gi ) and τ = ∑ ai (Λ gj ) i

j

are two elements in D, then the linear combination

ττ = ∑ ai aj (Λ gi gj ) i, j

(3.3)

3.2 Abstract Rings of Double Cosets

67

is independent of the choice of the representatives gi ∈ Λ gi and gi ∈ Λ gi and again belong to D. To see this, we first note that ττ is independent of the choice of representatives gi ∈ Λ gi . Now, if we replace the representatives gj by λ j gj with λ j ∈ Λ , by definition of D, we have

∑ ai (Λ gi λ j ) = τλ j = τ = ∑ ai (Λ gi ) i

i

for each j. Hence

∑ i, j

∑

aj

=∑

aj

ai aj (Λ gi λ j gj ) =

j

j

∑ ai (Λ gi λ j )

i

∑ ai (Λ gi ) i

gj

gj = ∑ ai aj (Λ gi gj ), i, j

and so the product is independent of the choice of gj ∈ Λ gj . Finally, if λ ∈ Λ , we have (ττ )λ = ∑ ai aj (Λ gi gj λ ) = τ · (τ λ ) = ττ , i, j

whence ττ

∈ D. Since the multiplication (3.3) is clearly bilinear and associative, we see that it defines on D(Λ , Σ ) a structure of an associative ring, the Hecke–Shimura ring of the pair Λ , Σ (over Z). Basic Rings. The ring Z in the definition of the Hecke–Shimura ring of an l-finite pair Λ , Σ can be replaced by an arbitrary associative and commutative ring A with identity. It leads us to the Hecke–Shimura ring DA (Λ , Σ ) of the pair Λ , Σ over A. The choice of a basic ring A is insignificant for a majority of the properties of the Hecke–Shimura rings that we are going to consider. As a rule, we take A = Z, except for the consideration of spherical representations, when we take A to be the field Q of rational numbers, and the study of relations between Hecke operators and the Siegel operator, when it is convenient to take A = C, the field of complex numbers. If the basic ring is not indicated, it means that A = Z. Standard Bases and Multiplication Rules. Let Λ , Σ be an l-finite pair. If g1 , . . . , gν is a complete set of representatives of different left cosets modulo Λ contained in a double coset Λ gΛ ⊂ Σ , then the set g1 λ , . . . , gν λ with any λ ∈ Λ is clearly still a complete set of representatives of different left cosets modulo Λ contained in a double coset Λ gΛ . It follows that each linear combination of left cosets of the form τ (g) = τΛ (g) = ∑ (Λ gi ) (g ∈ Σ ) (3.4) gi ∈Λ \Λ gΛ

satisfies τ (g)λ = τ (g) and so belongs to the ring D = D(Λ , Σ ). By an easy induction, left to the reader, on the number of distinct left cosets that actually occur in a linear combination contained in D, one gets that each such combination is a linear combination of elements of the form τ (g) with integral coefficients.

68

3 Hecke–Shimura Rings of Double Cosets

Moreover, the elements τ (g) corresponding to different double cosets Λ gΛ are obviously linearly independent over Z. We conclude that the elements τ (g) corresponding to different double cosets Λ gΛ form a Z-basis of the ring D. The elements τ (g) will be sometimes referred as double cosets (modulo Λ ) and the ring D = D(Λ , Σ ) as the (Hecke–Shimura) ring of double cosets or just dc-ring of the pair Λ , Σ . Note that the element

τ (e) = (Λ e) = (Λ ),

(3.5)

where e is the identity of Σ , is the identity of the ring D. The following lemma describes the product of double cosets in terms of double cosets. Lemma 3.2. The product of double cosets in a dc-ring D = D(Λ , Σ ) can be computed by the following formulas: if

∑

τ (g) =

τ (g ) =

(Λ gi ) and

gi ∈Λ \Λ gΛ

∑

gj ∈Λ \Λ g Λ

(Λ gj )

are two elements of the form (3.4) in D, then

τ (g)τ (g ) =

∑

Λ hΛ ⊂Λ gΛ g Λ

c(g, g ; h)τ (h),

where h ranges over a set of representatives for the double cosets modulo Λ contained in Λ gΛ gΛ , and a coefficient c(g, g ; h) can be defined as the number of pairs i, j such that gi gj ∈ Λ h; the coefficients c(g, g ; h) can also be written in the form c(g, g ; h) = d(g, g ; h)ν (g )ν (h)−1 , where d(g, g ; h) is the number of representatives gi satisfying gi g ∈ Λ hΛ , and ν (g ), ν (h) are the numbers of left cosets in the double cosets Λ gΛ and Λ hΛ , respectively. Proof. By definition, we get

τ (g)τ (g ) =

∑

gi ∈Λ \Λ gΛ

(Λ gi )

∑

gj ∈Λ \Λ g Λ

(Λ gj ) = ∑(Λ gi gj ). i, j

Since each of the products gi gj is clearly contained in Λ gΛ gΛ , it follows that the product can be written in the form

τ (g)τ (g ) =

∑

Λ hΛ ⊂Λ gΛ g Λ

c(g, g ; h)τ (h) =

∑

Λ hΛ ⊂Λ gΛ g Λ

c(g, g ; h)

∑

(Λ hk )

hk ∈Λ \Λ hΛ

with some integral coefficients c(g, g ; h). By comparing the coefficients of (Λ h) in the two decompositions of the product into left cosets, we conclude that c(g, g ; h) is equal to the number of pairs i, j such that Λ gi gj = Λ h, i.e. gi gj ∈ Λ h. Further, it is easy to see that the number c(g, g ; h) depends only on the double cosets of g, g , and h. Thus, the sum

3.2 Abstract Rings of Double Cosets

∑

69

c(g, g ; h) = ν (h)c(g, g ; h)

Λ hk ⊂Λ \Λ hΛ

is equal to the number of pairs i, j such that gi gj ∈ Λ hΛ . Since each representative gj can obviously be taken in the form gj = g λ j with λ j ∈ Λ , we conclude that the last number equals the number d(g, g ; h) of representatives gi with gi g ∈ Λ hΛ multiplied by ν (g ). The following lemma often allows one to simplify the decomposition of double cosets into left cosets. Lemma 3.3. Let G be a group, Λ a subgroup of G, and g ∈ G. Let

Λ=

(Λ ∩ g−1Λ g)λi

(resp., Λ =

λ j (Λ ∩ gΛ g−1 ))

λ j ∈Λ /Λ ∩gΛ g−1

λi ∈Λ ∩g−1 Λ g\Λ

(3.6) be decompositions into different cosets. Then

Λ gΛ =

Λ gλi

(resp., Λ gΛ =

λi ∈Λ ∩g−1 Λ g\Λ

λ j gΛ ),

(3.7)

λ j ∈Λ /Λ ∩gΛ g−1

where the cosets are disjoint; in particular,

ν (g) = #(Λ \Λ gΛ ) = [Λ : Λ ∩ g−1Λ g] (resp., ν (g) = #(Λ gΛ /Λ ) = [Λ : Λ ∩ gΛ g−1 ]).

(3.8)

Proof. Let us consider, for example, the case of left cosets. It is clear that the right side of (3.7) is contained in the left side. Let g = λ gλ ∈ Λ gΛ . By (3.6), the element λ belongs to a left coset (Λ ∩g−1Λ g)λi , i.e. λ = αλi , where α ∈ Λ and gα g−1 ∈ Λ . Then we have g = λ gαλi = λ gα g−1 gλi ∈ Λ gλi , and so the left side of (3.7) is contained in the right side. Now, if a coset Λ gλi meets Λ gλ j , then we have λ gλi = δ gλ j with λ , δ ∈ Λ , whence g−1 δ −1 λ gλi = λ j , and so (Λ ∩ g−1Λ g)λi = (Λ ∩ g−1Λ g)λ j , because the element g−1 δ −1 λ g = λ j λi−1 is obviously contained in Λ ∩ g−1Λ g. Left and Right Rings of Double Cosets. Suppose now that Λ , Σ is a d-finite pair, i.e., such that each double coset Λ gΛ in Σ is both a finite union of left cosets and a finite union of right cosets modulo Λ . In this case, the product of double cosets of Σ modulo Λ can be defined not only by a product of right Λ -invariant linear combinations of left cosets modulo Λ , as above, but also by a similar product of left Λ -invariant linear combinations of right cosets modulo Λ . The natural question is whether we get the same product of the double cosets. The solution is not quite trivial, since a double coset can perfectly well consist of different numbers of left and right costs. Nevertheless, the answer is as simple as “yes”, at least if Σ is contained in a group.

70

3 Hecke–Shimura Rings of Double Cosets

Theorem 3.4. Let Λ , Σ be a d-finite pair contained in a group G. Then the product of any double cosets of Σ modulo Λ defined by multiplication of right Λ -invariant linear combinations of left cosets modulo Λ coincides with their product defined by a similar multiplication of left Λ -invariant linear combinations of right cosets modulo Λ . The proof of the theorem is a rather technical one; however we shall outline it, because the result has not only philosophical but quite practical value. First we have to recall definitions and simple properties of commensurability. We say that two subgroups Γ1 and Γ2 of a multiplicative group G are commensurable if their intersection is of finite index in both Γ1 and Γ2 . Then we write Γ1 ∼ Γ2 . The relation of commensurability is clearly reflexive and symmetric. It is not hard to see that the relation is also transitive. Lemma 3.5. Let G be a group, and Γ a subgroup of G. Then the commensurator of Γ in G, Γ- = Γ-G = {g ∈ G|g−1Γ g ∼ Γ }, is a group. Proof. If g ∈ Γ-, then Γ = g(g−1Γ g)g−1 ∼ gΓ g−1 , and so g−1 ∈ Γ-. Now, if g1 , g2 ∈ −1 −1 −1 Γ-, then g−1 1 Γ g1 ∼ Γ , hence g2 g1 Γ g1 g2 ∼ g2 Γ g2 ∼ Γ , and so by transitivity, −1 (g1 g2 ) Γ g1 g2 ∼ Γ . Lemma 3.6. Let G be a group, Γ a subgroup of G, and Γ- = Γ-G the commensurator of Γ in G. Then the function

ν (g) Γ- g → η (g) = ν (g−1 )

with ν (h) = [Γ : Γ ∩ h−1Γ h]

is a homomorphism of the group Γ- into the multiplicative group of nonzero rational numbers, which is trivial on the subgroup Γ . Proof. Let us denote by X the set of all subgroups of G that are commensurable with Γ . By transitivity, if Γ1 , Γ2 ∈ X , there exists a group Γ ∈ X of finite index both in Γ1 and Γ2 , for example, Γ1 ∩ Γ2 . We then set

η (Γ1 /Γ2 ) = [Γ1 : Γ ][Γ2 : Γ ]−1 .

(3.9)

It is easy to see that η is independent of the choice of Γ and satisfies the relations

η (Γ1 /Γ2 )η (Γ2 /Γ3 ) = η (Γ1 /Γ3 ),

η (g−1Γ1 g/g−1Γ2 g) = η (Γ1 /Γ2 ) (g ∈ Γ-). (3.10)

For g ∈ Γ- and Γ ∈ X , we set η (g) = η (Γ /g−1Γ g). It is readily verified that η (g) does not depend of the choice of Γ . Hence, by (3.10), we obtain

η (g1 g2 ) = η (Γ /(g1 g2 )−1Γ g1 g2 ) −1 −1 −1 = η (Γ /g−1 1 Γ g1 )η (g1 Γ g1 /(g2 g1 Γ g1 g2 )) = η (g1 )η (g2 )

3.2 Abstract Rings of Double Cosets

71

if g1 , g2 ∈ Γ-. On the other hand, if g ∈ Γ-, then

η (g) = η (Γ /Γ ∩ g−1Γ g)η (Γ ∩ g−1Γ g/g−1Γ g) = ν (g)η (gΓ g−1 ∩ Γ /Γ ) = ν (g)ν (g−1 )−1 = η (g). Proof of Theorem 3.4. It suffices to consider the products of two double cosets. By Lemmas 3.2 and 3.3, we can write

τ (g)τ (g) ´ =

∑

Λ hΛ ∈Λ gΛ g´Λ

d(g, g; ´ h)[Λ : Λ ∩ g´−1Λ g][ ´ Λ : Λ ∩ h−1Λ h]−1 τ (h),

where d(g, g; ´ h) is the number of representatives gi ∈ Λ \Λ gΛ satisfying gi g´ ∈ Λ hΛ . On the other hand, by similar reasoning for corresponding formal sums of right cosets, we get

∑

(gi Λ ) ×

=

∑

∑

g´ j ∈Λ g Λ /Λ

g-i ∈Λ gΛ /Λ

Λ hΛ ∈Λ gΛ g´Λ

(g´ j Λ )

- g; d(g, ´ h)[Λ : Λ ∩ gΛ g−1 ][Λ : Λ ∩ hΛ h−1 ]−1

∑

(hk Λ ),

hk ∈Λ hΛ /Λ

- g; where d(g, ´ h) is the number of representatives g´ j ∈ Λ g´Λ /Λ satisfying gg´ j ∈ Λ hΛ . In order to prove the equality of the two products, we have only to show that ´ Λ : Λ ∩ h−1Λ h]−1 d(g, g; ´ h)[Λ : Λ ∩ g´−1Λ g][ - g; = d(g, ´ h)[Λ : Λ ∩ gΛ g−1 ][Λ : Λ ∩ hΛ h−1 ]−1

(3.11)

if h ∈ Λ gΛ g´Λ . By Lemma 3.6 for Γ = Λ , we have ´ [Λ : Λ ∩ h−1Λ h][Λ : Λ ∩ hΛ h−1 ]−1 = η (h) = η (g)η (g) ´ Λ : Λ ∩ g´Λ g´−1 ]−1 . = [Λ : Λ ∩ g−1Λ g][Λ : Λ ∩ gΛ g−1 ]−1 [Λ : Λ ∩ g´−1Λ g][ Thus, the relation (3.11) is equivalent to - g; ´ h)[Λ : Λ ∩ g−1Λ g]. d(g, g; ´ h)[Λ : Λ ∩ g´Λ g´−1 ] = d(g,

(3.12)

By the definition and Lemma 3.3, we see that the number d(g, g; ´ h) can be written in the form d(g, g; ´ h) = #{λ ∈ Λ(g) \Λ | gλ g´ ∈ Λ hΛ }

with Λ(g) = Λ ∩ g−1Λ g.

(3.13)

For λ ∈ Λ , it is easy to see that the double coset Λ gλ g´Λ depends only on the coset Λ(g) λΛ(g´−1 ) , and the relation Λ(g) λ δ = Λ(g) λ with δ ∈ Λ(g´−1 ) holds if and only if δ ∈ λ −1Λ(g) λ . Thus we can rewrite the number (3.13) in the form d(g, g; ´ h) =

∑

[Λ(g´−1 ) : (Λ(g´−1 ) ∩ λ −1Λ(g) λ )].

λ ∈Λ(g) \Λ /Λ(g´−1 ) , gλ g∈ ´ Λ hΛ

72

3 Hecke–Shimura Rings of Double Cosets

Quite similarly, - g; d(g, ´ h) =

∑

[Λ(g) : (Λ(g) ∩ λΛ(g´−1 ) λ −1 )].

λ ∈Λ(g) \Λ /Λ(g´−1 ) , gλ g∈ ´ Λ hΛ

Using the symbol (3.9) and its properties (3.10), we obtain - g; d(g, ´ h) =

∑

λ ∈Λ(g) \Λ /Λ(g´−1 ) , gλ g∈ ´ Λ hΛ

η (Λ(g) /(Λ(g) ∩ λΛ(g´−1 ) λ −1 ))

= ∑ η (λ −1Λ(g) λ /λ −1Λ(g) λ ∩ Λ(g´−1 ) ) λ

= ∑ η (λ −1Λ(g) λ /Λ(g) )η (Λ(g) /Λ )η (Λ /Λ(g´−1 ) ) λ

× η (Λ(g´−1 ) /Λ(g´−1 ) ∩ λ −1Λ(g) λ ) = ∑ η (λ )ν (g)−1 ν (g´−1 )η (Λ(g´−1 ) /Λ(g´−1 ) ∩ λ −1Λ(g) λ ) λ

´ h), = ν (g)−1 ν (g´−1 )d(g, g; where ν is defined by (3.8). Extension of Antiautomorphisms. For a semigroup Σ and its subgroup Λ , we say that a mapping g → g∗ of Σ into itself is an antiautomorphism of the pair Λ , Σ if it is one-to-one on Σ and on Λ and satisfies (gg) ´ ∗ = g´∗ g∗

for all g, g´ ∈ Σ .

(3.14)

The main application of Theorem 3.4 is the following proposition on the extension of antiautomorphisms of d-finite pairs to the corresponding dc-rings. Proposition 3.7. Let Λ , Σ be a d-finite pair contained in a group, and let g → g∗ be an antiautomorphism of the second order of this pair. Then the linear mapping of the dc-ring D = D(Λ , Σ ) into itself defined on the double cosets (3.4) by

τ (g) → τ (g)∗ = τ (g∗ )

(g ∈ Σ )

(3.15)

is an antiautomorphism of the second order of the ring D. If, in addition, the mapping is identical on all of the double cosets, then the ring D is commutative. Proof. For the first statement it is sufficient to prove that ´ ∗ = τ (g´∗ )τ (g∗ ) (g, g´ ∈ Σ ). (τ (g)τ (g)) By Lemma 3.2, we can write the left hand side of (3.16) in the form

∑

Λ hΛ ⊂Λ gΛ g´Λ

c(g, g; ´ h)τ (h)∗ =

∑

Λ hΛ ⊂Λ gΛ g´Λ

c(g, g; ´ h)τ (h∗ ),

(3.16)

3.2 Abstract Rings of Double Cosets

73

where

∑

c(g, g; ´ h) =

1.

gi ∈Λ \Λ gΛ , g´ j ∈Λ \Λ g´Λ , gi g´ j ∈Λ h

By Theorem 3.6, we can compute the right-hand side of (3.16) by considering the double cosets τ (φ (g )) and τ (φ (g)) as sums of the right cosets modulo Λ . By similar reasoning, since the map g → g∗ is an antiautomorphism of the second order carrying Λ onto itself, we obtain

∑

g´∗j ∈Λ g´∗ Λ /Λ

where

(g´∗j Λ )

∑

g∗i ∈Λ g∗ Λ /Λ

(g∗i Λ ) =

∑

c (g´∗ , g∗ ; h∗ )

Λ h∗ Λ ⊂Λ g´∗ Λ g∗ Λ

∑

c (g´∗ , g∗ ; h∗ ) =

∑

h∗k ∈Λ h∗ Λ /Λ

(h∗k Λ ),

1.

g´∗j ∈Λ g´∗ Λ /Λ ; g∗i ∈Λ g∗ Λ /Λ ; g´∗j g∗i ∈h∗ Λ

Furthermore, the conditions g´∗j ∈ Λ g´∗Λ /Λ , g∗i ∈ Λ g∗Λ /Λ and Λ h∗Λ ⊂ Λ g´∗Λ g∗Λ , h∗k ∈ Λ h∗Λ /Λ , g´∗j g∗i ∈ h∗Λ are equivalent to the conditions g´ j ∈ Λ \Λ g´Λ , gi ∈ Λ \Λ gΛ

and Λ hΛ ⊂ Λ gΛ g´Λ , hk ∈ Λ \Λ hΛ , gi g´ j ∈ Λ h,

´ h), which proves (3.16). respectively. It follows that c (g´∗ , g∗ ; h∗ ) = c(g, g; As to the second statement, the assumption implies that τ ∗ = τ for all τ ∈ D; hence we have ´ τ τ´ = (τ τ´ )∗ = τ´ ∗ τ ∗ = ττ

for all τ , τ´ ∈ D.

Extension of Embeddings. Next we consider embeddings of Hecke–Shimura rings based on inclusions of corresponding pairs. The embeddings turn out to be useful when initial rings seem to be too narrow and we are looking for suitable extensions. Suppose that Λ , Σ and Λ0 , Σ0 are two l-finite pairs contained in the same group and satisfying the following conditions:

Λ0 ⊂ Λ ,

Σ ⊂ Λ Σ0 ,

and

Λ ∩ Σ0 Σ0−1 ⊂ Λ0 ,

(3.17)

where Σ0−1 = {g−1 |g ∈ Σ0 }. Since Σ ⊂ Λ Σ0 , each linear combination of left cosets τ ∈ L(Λ , Σ ) can be written in the form

τ = ∑ ai (Λ gi )

with gi ∈ Σ0 .

i

Let us consider then the formal linear combination of left cosets

ι (τ ) = ∑ ai (Λ0 gi ). i

74

3 Hecke–Shimura Rings of Double Cosets

The element ι (τ ) is independent of the choice of representatives gi ∈ Σ0 in Λ gi , since if gi ∈ Λ gi ∩ Σ0 are other representatives, the last condition of (3.17) implies −1 that λi = gi g−1 i ∈ Λ ∩ Σ 0 Σ 0 ⊂ Λ0 , whence

ι (τ ) = ∑ ai (Λ0 gi ) = ∑ ai (Λ0 λi gi ) = ∑ ai (Λ0 gi ) = ι (τ ). i

i

i

The condition Λ0 ⊂ Λ implies that the map ι carries different left cosets modulo Λ to different left cosets modulo Λ0 . Thus we obtain an additive embedding of the modules of left cosets L(Λ , Σ ) → L(Λ0 , Σ0 ). Proposition 3.8. Let Λ , Σ and Λ0 , Σ0 be two l-finite pairs contained in a group and satisfying conditions (3.17). Then the restriction of the map ι to the Hecke– Shimura ring D(Λ , Σ ) ⊂ L(Λ , Σ ) is a ring monomorphism of D(Λ , Σ ) into the ring D(Λ0 , Σ0 ): ι : D(Λ , Σ ) → D(Λ0 , Σ0 ). (3.18) If, moreover,

Σ0 ⊂ Σ

and

νΛ (g) = νΛ0 (g) for all g ∈ Σ0 ,

(3.19)

where ν stands for the indices (3.8), then the mapping (3.18) is an isomorphism of the rings. Proof. The first assertion follows directly from definitions and the inclusion Λ0 ⊂ Λ . As to the second, it suffices to show that ι (τΛ (g)) = τΛ0 (g) for all g ∈ Σ0 and double cosets (3.4) if the conditions (3.19) are fulfilled. Let λ1 , . . . , λν with ν = νΛ0 (g) be a set of representatives of the left cosets of Λ0 modulo Λ0 ∩ g−1Λ0 g. By Lemma 3.3, we have ν

τΛ0 (g) = ∑ (Λ0 gλi ). i=1

On the other hand, the elements gλ1 , . . . , gλν belong to Λ gΛ and are contained in different left cosets modulo Λ , since a relation gλi = δ λ j with δ ∈ Λ would imply that δ = (gλi λ j−1 )g−1 ⊂ Λ ∩ Σ0 Σ0−1 ⊂ Λ0 . By (3.19) and (3.7), we obtain ν

τΛ (g) = ∑ (Λ gλi ), i=1

and so ι (τΛ (g)) = τΛ0 (g) . Exercise 3.9. Let Λ , Σ and Λ0 , Σ0 be two l-finite pairs contained in a group and satisfying conditions (3.17). Let g → g∗ be an antiautomorphism of the second order of the both pairs. Show then that

ι (τ )∗ = ι (τ ∗ ) for all τ ∈ D(Λ , Σ ).

3.3 Rings of Double Cosets of the General Linear Group

75

Exercise 3.10. Let Λ , Σ be an l-finite pair. Show that the index map D(Λ , Σ ) ∑ ai τΛ (gi ) → ν (∑ ai τ (gi )) = ∑ ai ν (gi ), i

i

i

where ν (gi ) are the indices (3.8), is a ring homomorphism into Z.

3.3 Rings of Double Cosets of the General Linear Group Although our main objective is the study of dc-rings of the symplectic group, we shall start our consideration of concrete rings with the case of the general linear group because this case is easier, it coincides with the symplectic case for matrices of order two, and, as we shall see later, in some aspects the symplectic case can be reduced to the general linear case. Global Rings. Our basic pair here will be the pair Λ = Λn = GLn (Z) = g ∈ Znn det g = ±1 , Σ = Σn = g ∈ Znn det g = 0 with n = 1, 2, . . . . Lemma 3.11. The pair Λn , Σn is d-finite for every n. Proof. By considering the entries of matrices in Λ modulo a positive integer q, we get a homomorphism of Λ into the finite group GLn (Z/qZ). The kernel of this homomorphism is the principal congruence subgroup of level q of Λ , Λ (q) = Λn (q) = λ ∈ Λ λ ≡ 1n (mod q) , (3.20) which is therefore a normal subgroup of finite index in Λ . If g ∈ Λ , then clearly the groups gΛ (d)g−1 and g−1Λ (d)g, where d = | det g|, are both contained in Λ . It follows that the groups Λ ∩ g−1Λ g and Λ ∩ gΛ g−1 both contain the group Λ (d), and so have finite indices in Λ . The lemma follows then from Lemma 3.3. The lemma allows us to define the Hecke–Shimura ring (3.1), H = Hn = D(Λn , Σn ),

(3.21)

of the pair Λn , Σn . As we have seen in the previous section, elements of the form (3.4), which, for the group Λ = Λn , will be denoted by t(g) = τΛ (g)

(g ∈ Σn ),

(3.22)

corresponding bijectively to the double cosets Λ gΛ contained in Σ , can be taken as a (free) basis of the ring H (over Z). The following theorem gives a convenient parametrization of the double cosets.

76

3 Hecke–Shimura Rings of Double Cosets

Theorem 3.12. (The theorem on elementary divisors). Each of the double cosets Λ gΛ with Λ = Λn and g ∈ Σ = Σn contains a unique representative of the form ed(g) = diag(d1 , . . . , dn ) with d1 , . . . , dn ∈ N and d1 |d2 | · · · |dn .

(3.23)

First we cite a simple lemma, which like Lemma 1.10 is an easy consequence of the Euclidean algorithm. Lemma 3.13. Let l be a nonzero integral n-row (resp.,n-column) and d the greatest common divisor of the entries of l. Then there exists a matrix λ ∈ Λn such that lλ = (d, 0, . . . , 0) (resp., λ l = t(d, 0, . . . , 0)) . Proof of Theorem 3.12. First we note that elementary integral operations on rows of matrices g ∈ Σ such as a permutation of rows and addition to one row of another row multiplied by an integer (resp., similar operations on columns) can be derived by multiplication of the matrix from the left (resp., from the right) by a matrix of Λ and so does not change the coset Λ gΛ . In order to prove that a representative of the form (3.23) exists, we apply induction to n. If n = 1, there is nothing to prove. Suppose that n > 1, and the theorem has already been proved for matrices of order n − 1. Denote by δ = δ (g) the minimum of the greatest common divisor of entries of rows of a matrix g ∈ Σn . By rearranging rows and applying Lemma 3.13, we may replace g by a matrix g = (gi j ) ∈ Λ gΛ with δ (g ) = δ (g) and the first row (g11 , . . . , g1n ) = (δ , 0, . . . , 0). Now we apply induction to δ . If δ = 1, then by adding to other rows suitable multiples of the first row, we can replace g by a matrix g = (gi j ) in the same double coset with g21 = 0, . . . , gn1 = 0, and the theorem is true by the induction assumption on n. But if δ > 1, then by similar reasoning, we can replace g by a the matrix g = (gi j ) in the same double coset with 1 ≤ g21 ≤ δ , . . . , 1 ≤ gn1 ≤ δ . If the greatest common divisor of entries of each row of the matrix g is equal to δ , then g21 = · · · = gn1 = δ , all entries of g are divisible by δ , and we clearly can replace g by a matrix of the form

δ 0 0 δ g1

with g1 ∈ Σ(n−1) , which is again contained in Λ gΛ , and we can use the induction assumption on n. Otherwise, δ (g ) < δ , and we may apply the induction assumption on δ . This proves the existence of a representative of the form (3.23). If D = diag(d1 , . . . , dn ) and D = diag(d1 , . . . , dn ) are two matrices of the form (3.23) satisfying D = λ Dλ with λ , λ ∈ Λ , then the Binet–Cauchy formula implies that each r-minor of D with 1 ≤ r ≤ n is divisible by d1 · · · dr ; in particular, d1 · · · dr |d1 · · · dr . Similarly, d1 · · · dr |d1 · · · dr . Hence D = D . The matrix ed(g) = diag(d1 , . . . , dn ) is called the matrix of elementary divisors of g, and the numbers dr = dr (g) are the elementary divisors of g. The Binet–Cauchy formula implies that the product d1 · · · dr is equal to the greatest common divisor of the minors of order r of g; in particular d1 · · · dn = | det g|.

(3.24)

It follows that the additive structure of the ring H is very simple: it is just a lattice spanned by the generators

3.3 Rings of Double Cosets of the General Linear Group

77

t[d1 , . . . , dn ] = t(diag(d1 , . . . , dn )) with d1 , . . . , dn ∈ N and d1 |d2 | · · · |dn , (3.25) where t(g) are the double cosets (3.22). Let us now turn to the multiplicative structure. First of all, we have the following theorem. Theorem 3.14. The dc-ring Hn = D(Λn , Σn ) is commutative for every n. Proof. The mapping g → tg, where tg is the transpose of g, is clearly an antiautomorphism of order two of the pair Λn , Σn that does not change diagonal matrices. Theorem 3.12 and Proposition 3.7 complete the proof. As to explicit rules of multiplication, we have the following basic theorem. Theorem 3.15. The double cosets (3.25) satisfy the rules t[d1 , . . . , dn ]t[d1 , . . . , dn ] = t[d1 d1 , . . . , dn dn ] if gcd(dn /d1 , dn /d1 ) = 1.

(3.26)

In particular, [d]t[d1 , . . . , dn ] = t[d1 , . . . , dn ][d] = t[dd1 , . . . , ddn ],

(3.27)

[d] = [d]n = t(d · 1n ) = (Λ d · 1n ).

(3.28)

where Proof. Since the double coset (3.28) consists of a single left coset, the relation (3.27) follows directly from the definition of multiplication in dc-rings. It follows from (3.27) that it is sufficient to prove the general relations (3.26) only when d1 = d1 = 1 and gcd(dn , dn ) = 1, or under the more general assumption that the numbers d1 · · · dn and d1 · · · dn are coprime. In order to prove it, we shall use the following lemma. Lemma 3.16. The elements

∑

t(a) = t n (a) =

t[d1 , . . . , dn ] ∈ Hn

(3.29)

1≤d1 |···|dn , d1 ···dn =a

with a ∈ N have decompositions into left cosets of the form t(a) =

∑

(Λ g)

(3.30)

g∈Λ \Σ , | det g|=a

and satisfy the relations t(a)t(a ) = t(aa)

if gcd(a, a ) = 1.

(3.31)

Proof of the lemma. The decompositions (3.30) follow from (3.24) and Theorem 3.12. Let g1 , . . . , gν and g1 , . . . , gµ be systems of representatives for the left cosets modulo Λ of matrices in Σ with determinants ±a and ±a , respectively. We have to show that the product of these systems g1 g1 , . . . , dν dµ is a system of representatives for

78

3 Hecke–Shimura Rings of Double Cosets

the left cosets modulo Λ of matrices in Σ with determinant ±aa , provided that a and a are coprime. First, all of the products have determinant ±aa . Further, the products belong to different left cosets modulo Λ : if λ gi gj = gk gl with λ ∈ Λ , then −1 has determinant ±1 and is integral, since its prodthe matrix γ = g−1 k λ gi = gl (g j ) ucts by coprime numbers det gk = ±a and det gj = ±a are both integral, and hence γ ∈ Λ and γ gj = gl , so gj = gl and λ gi = gk , whence gi = gk . Finally, if h ∈ Σ and det h = ±aa , then by Theorem 3.12, it follows easily that h can be written in the form h = h1 h2 with h1 , h2 ∈ Σ satisfying det h1 = ±a and det h2 = ±a . We have h2 = δ gj and h1 δ = γ gi with δ , γ ∈ Λ ; hence h = h1 h2 = γ gi gj ∈ Λ gi gj . We can now return to the proof of the theorem. Let g = ed(g) and g = ed(g ) be two matrices of elementary divisors satisfying det g = a and det g = a . Since clearly gg ∈ Λ gΛ gΛ , it follows from Lemma 3.2 that the product of corresponding double cosets (3.22) has the form t(g)t(g ) = t(gg ) + s(g, g ), where s(g, g ) is a linear combination of double cosets with nonnegative coefficients. Hence, we obtain t(a)t(a ) =

∑

g=ed(g), det g=a

=

∑

g=ed(g), det g=a, g =ed(g ), det g =a

t(g)

∑

t(g )

∑

s(g, g ).

g =ed(g ), det g =a

t(gg ) +

g=ed(g), det g=a, g =ed(g ), det g =a

Since the numbers a and a are coprime, it follows easily that the first sum on the right is equal to t(aa ), which equals t(a)t(a ) by Lemma 3.14. Hence, the second sum on the right is zero, and so each of the sums s(g, g ) is zero too. Exercise 3.17. For g, g ∈ Σn satisfying gcd(dn (g)/d1 (g), dn (g )/d1 (g )) = 1, prove that the elementary divisors satisfy the relations di (gg ) = di (g)di (g ) (i = 1, 2, . . . , n). The following useful lemma gives a convenient description of representatives for the left cosets of Σn modulo Λn . A matrix g = (gi j ) ∈ Σn will be called reduced if its entries satisfy ⎧ ⎪g11 > 0, ⎨ (3.32) if 1 ≤ i < j ≤ n, 0 ≤ gi j < g j j ⎪ ⎩ if i > j. gi j = 0 Lemma 3.18. Each left coset Λn g ⊂ Σn contains its unique reduced representative. Proof. We shall apply induction to n. If n = 1 there is nothing to prove. Let n > 1 and suppose that the lemma has already been proved for n − 1. By Lemma 3.13 applied

3.3 Rings of Double Cosets of the General Linear Group

79

to the first column of g, we conclude that the coset Λ g contains a representative g d1 ∗ n−1 of the form 0 g1 with d1 > 0 and g1 ∈ Σ . By the inductive hypothesis, we can

write g1 = λ g0 , where λ ∈ Λn−1 and g0 is reduced. It follows from the Euclidean such that the matrix algorithm that there exists a row l ∈ Zn−1 1 1 0 1 l g ∈ Λ g 0 1n−1 0 λ −1 is reduced. The uniqueness follows easily by contradiction. Exercise 3.19. Show that for every l ∈ N, the element t(l) = t 2 (l) of H2 has the following decomposition into left cosets ab Λ2 . t(l) = ∑ 0d a,b∈N, ad=l, 0≤b 0. An element of H p is called primitive (resp., imprimitive) if it is a linear combination of primitive (resp., imprimitive) elements. It is clear that each element t ∈ H p has a unique decomposition t = t pr + t im , where t pr is primitive and t im is imprimitive. It follows from (3.27) that the set J of all imprimitive elements of H p is the principal ideal generated by the element [p] = [p]n , J = [p]H p . satisfying Lemma 3.21. The linear mapping φ : Hnp → Hn−1 p φ t[pδ1 , . . . , pδn ] =

t[pδ2 , . . . , pδn ] 0

if 0 = δ1 ≤ δ2 ≤ · · · ≤ δn , if 0 < δ1 ≤ δ2 ≤ · · · ≤ δn ,

is a ring epimorphism with kernel J . Proof. It is obvious that this mapping is an epimorphism of Z-modules. It follows from (3.27) that the kernel φ equals J . Thus, we have only to check that it is a ring homomorphism. For this, it suffices to verify the relations

3.3 Rings of Double Cosets of the General Linear Group

φ t[1, pδ 2 , . . . , pδ n ]t[1, pδ 2 , . . . , pδ n ] = t[pδ 2 , . . . , pδ n ]t[pδ 2 , . . . , pδ n ],

81

(3.39)

where 0 ≤ δ2 ≤ · · · ≤ δn and 0 ≤ δ2 ≤ · · · ≤ δn . Let us set 1 0 1 0 , g = diag(1, pδ 2 , . . . , pδ n ) = . g = diag(1, pδ 2 , . . . , pδ n ) = 0 g0 0 g0 By Lemma 3.2 and the definition of φ , we obtain

φ (t(g)t(g )) = ∑ c(g, g ; h)t(h0 ), h

where h ranges over the matrices of the form 10 h00 , with h0 = diag(pγ2 , . . . , pγn ), 0 ≤ γ2 ≤ · · · ≤ γn , and γ2 + · · · + γn = δ2 + · · · + δn + δ2 + · · · + δn . Similarly, in the we have the relation ring Hn−1 p t(g0 )t(g0 ) = ∑ c(g0 , g0 ; h0 )t(h0 ), h0

where h0 ranges over the matrices indicated above. Thus, in order to prove (3.39), we have only to show that c(g, g ; h) = c(g0 , g0 ; h0 ).

(3.40)

Since the coefficient c(g, g ; h) depends only on the double cosets of matrices g, g , h h0 0 modulo Λ , we may replace h by h = 0 1 . By Lemmas 3.2 and 3.18, the coefficient c(g, g ; h) is equal to the number of pairs d, d of reduced matrices satisfying the conditions d ∈ Λ gΛ , d ∈ Λ gΛ , and dd = λ h . Since λ = dd (h )−1 , the matrix λ is an upper triangular matrix; in particular, it can be written in the form λ = λ00 λ∗ with λ0 ∈ Λn−1 and λnn = ±1. We can write d = d00 dvnn , nn d v d = 00 d , where d0 and d0 are reduced matrices of order n − 1, dnn > 0, and nn

all of the entries of the columns v and v are nonnegative integers smaller than , respectively. Then the relation dd = λ h implies that d d = λ dnn and dnn nn nn nn = 1, and hence and d0 d0 = λ0 h0 . By the first of the relations we have d = d nn nn v = v = 0. Thus, d = d00 01 , d = d00 01 , where d0 and d0 are reduced matrices of

order n − 1 satisfying d0 d0 = λ0 h0 with λ0 ∈ Λn−1 . It is clear that d0 ∈ Λn−1 g0 Λn−1 and d0 ∈ Λn−1 g0 Λn−1 . Conversely, arbitrary matrices d and d satisfying these conditions are reduced, belong to the double cosets Λ gΛ and Λ gΛ , respectively, and satisfy dd = λ h with λ ∈ Λ . This proves the equality (3.40). Theorem 3.22. For every n = 1, 2, . . . and every prime number p, the local ring Hnp is generated by the elements 1n−i 0 n n (1 ≤ i ≤ n); (3.41) πi (p) = πi (p) = t(Di ) with Di = Di = 0 p · 1i

82

3 Hecke–Shimura Rings of Double Cosets

the elements π1 (p), . . . , πn (p) are algebraically independent over Z, where Z is identified with the isomorphic subring Z [1]n ⊂ Hnp . Proof. For the first assertion it suffices to show that each element t of the form (3.38) is a polynomial in π1 (p), . . . , πn (p) with coefficients in Z. We shall use induction on n and N = δ1 + · · · + δn . For n = 1 the assertion is clear, since t[pδ ] = π11 (p)δ . Let us assume that n > 1 and the assertion has already been proved for n − 1. If N = 1, then t = π1 (p). Let N > 1 and the assertion is true for N < N. If δ1 ≥ 1, by (3.27) we have t = πn (p)δ1 t , where t = t[1, pδ2 −δ1 , . . . , pδ2 −δ1 ] with N = N − nδ1 < N, and hence t is an integral polynomial in elements (3.41). But if δ1 = 0, then the element t is primitive, and so by the inductive assumption on n, the image t = t[pδ2 , . . . , pδn ] = φ (t) of t under the mapping of Lemma 3.21 has the form t = n−1 (p)), where P(π1n−1 (p), . . . , πn−1 P(x1 , . . . , xn−1 ) =

∑

i

n−1 ai1 ,...,in−1 x1i1 · · · xn−1 ,

i1 +2i2 +···+(n−1)in−1 =N n−1 (p)in−1 since each double coset (h) entering into the product π1n−1 (p)i1 · · · πn−1 i +2i +···+(n−1)i δ +···+ δ N n n−1 = p 2 must satisfy | det h| = p 1 2 = p . By the definition of n (p)) bethe mapping φ , we conclude that the element t1 = t − P(π1n (p), . . . , πn−1 longs to the kernel of φ , and hence, by Lemma 3.21, is divisible by [p] = πnn (p) in the ring H p : t1 = πnn (p)t1 . By the above, t1 is a linear combination of ele ments t[pδ1 , . . . , pδn ] with δ1 + · · · + δn = N, then t1 is a linear combination of elements t[pδ1 , . . . , pδn ] with δ1 + · · · + δn = N − n. The first assertion follows. Now by induction on n we shall prove that the elements (3.41) are algebraically independent over Z. It is clear if n = 1, since the elements π11 (p)δ = t1 (pδ ) with δ = 0, 1, 2, . . . correspond to distinct double cosets modulo Λ1 = {±1} and so are linearly independent. Let us assume now that n > 1 and that the assertion has already been proved for n − 1. If π1n (p), . . . , πnn (p) are algebraically dependent and P(x1 , . . . , xn ) is a polynomial of minimal total degree such that P(π1n (p), . . . , πnn (p)) = 0, then, by applying the homomorphism φ of Lemma 3.21, we have

φ (P(π1n (p), . . . , πnn (p)) = P(φ (π1n (p)), . . . , φ (πnn (p))) = P(π1n−1 (p), . . . , πnn−1 (p), 0) = 0. By the inductive hypothesis, we obtain that P(x1 , . . . , xn ) = xn P (x1 , . . . , xn ), where P is a polynomial of smaller degree. By (3.27), the element πnn (p) = [p]n is not a zero divisor in Hn . Thus, P (π1n (p), . . . , πnn (p)) = 0, a contradiction. Spherical Mapping. By Theorem 3.22, every element of a local Hecke–Shimura ring of the general linear group can be written as a polynomial in a finite number of generators. Sometimes one needs to find these polynomials explicitly. However, a direct calculation proves to be rather complicated. To simplify the calculation, it turns out to be convenient to introduce certain mappings of the local Hecke–Shimura rings into rings of symmetric polynomials. Since the polynomials that appear do

3.3 Rings of Double Cosets of the General Linear Group

83

not necessarily have integral coefficients, it will be reasonable to consider Hecke– Shimura rings over the field Q of rational numbers in place of the ring Z. We shall denote the rings of double cosets over Q by the same symbol as before, but with an upper tilde, for example, - Λ , Σ ) = DQ (Λ , Σ ) D(

-p = H - n = DQ (Λn , Σn ). H p p

or

A matrix g = (gi j ) ∈ Σn is called (upper) triangular if gi j = 0 for all i > j if, moreover, all diagonal entries gii are positive, the matrix is called triangular-plus. By Lemma 3.18, each left coset Λn g ⊂ Σn contains a triangular-plus representative, which is not unique, in contrast to the reduced representative. On the other hand, a product of triangular-plus matrices is again triangular-plus, which is generally not true for reduced matrices. We fix an order n ∈ N and a prime number p. Diagonal entries of a triangular-plus representative contained in a left coset Λ g ⊂ Σ p = Σnp are nonnegative powers of p, say, pδ1 , pδ2 , . . . , pδn . It is clear that such a diagonal depends only on the left coset. Let x1 , x2 , . . . , xn be algebraically independent commuting variables. We set n

ω pn ((Λ g)) = ∏(xi p−i )δi ,

(3.42)

i=1

-p = H - n with a j ∈ Q we define the and for a given element t = ∑ j a j (Λ g j ) ∈ H p polynomial ω pn (t) = (ω pn (t))(x1 , . . . , xn ) = ∑ a j ω pn ((Λ g j )). j

Theorem 3.23. (1) The mapping - n → Q[x1 , . . . , xn ] ω = ω pn : H p

(3.43)

- n into the ring of polynomials in is a Q-linear homomorphism of the ring H p x1 , . . . , xn over Q. (2) The ω -images of the generators (3.41) are given by ω (πin (p)) = p−i(i+1)/2 si (x1 , . . . , xn ) (1 ≤ i ≤ n), where si (x1 , . . . , xn ) =

∑

1≤α1 0, is a ring epimorphism with kernel I. Exercise 3.39. Prove the lemma. From this lemma, similarly to the proof of Theorem 3.22, using induction on n, we obtain the following structural theorem. Theorem 3.40. For every n = 1, 2, . . . and every prime number p, the ring Lnp is generated over the subring Z Z 1 n by the element T (p) = T n (p) = T (1, . . . , 1, p, . . . , p) n

(3.64)

n

and the elements Ti (p2 ) = Tin (p2 ) = T (1, . . . , 1, p, . . . , p, p2 , . . . , p2 , p, . . . , p) n−i

i

n−i

(3.65)

i

for i = 1, . . . , n; the elements T (p), T1 (p2 ), . . . , Tn (p2 ) are algebraically independent over Z. Exercise 3.41. Provide details of the proof of the theorem. Regular Hecke–Shimura Rings for Congruence Subgroups. We shall see here that the basic structure theorems established above for Hecke–Shimura rings of the full modular group remain essentially true in the case of congruence subgroups.

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3 Hecke–Shimura Rings of Double Cosets

We recall that a subgroup K of the modular group Γ = Γn is called a congruence subgroup if it contains a principal congruence subgroup Γ (q) = Γn (q) defined above (1.33). It is easy to check by induction on n that the natural map modulo q, t (3.66) Γn → Spn (Z/qZ) = M ∈ (Z/qZ)2n 2n MJn M ≡ Jn (mod q) , is an epimorphism with kernel Γn (q), whence Γn (q) is a normal subgroup of the modular group with the factor group isomorphic to Spn (Z/qZ). In particular, we have the following equality for the index of Γn (q):

ν (Γn (q)) = [Γn : Γn (q)] = #(Spn (Z/qZ)).

(3.67)

Lemma 3.42. If q and q are coprime integers, then Γn (q)Γn (q ) = Γn . Proof. The relation (3.67) shows that the index ν (Γn (q)) is the number of solutions of a system of polynomial congruences modulo q. Then the Chinese remainder theorem implies that the index is multiplicative in q, i.e., ν (Γn (qq )) = ν (Γn (q))ν (Γn (q )), provided that q and q are coprime. On the other hand, if q and q are coprime, then clearly Γn (q) ∩ Γn (q ) = Γn (qq ), which together with the multiplicativity of indexes implies that the embedding Γn (q) ⊂ Γn defines an isomorphism of the factor groups Γn (q)/Γn (qq ) and Γn /Γn (q ). Thus, for every γ ∈ Γn there is γ ∈ Γn (q ) such that γγ ∈ Γ(q). Then γ ∈ Γn (q)(γ )−1 ⊂ Γn (q)Γn (q ). Let us turn now directly to Hecke–Shimura rings of congruence subgroups. Let K = K n be a congruence subgroup of Γ = Γn . It is easy to see that for every M ∈ ∆ = ∆n , the intersection M −1 KM ∩ K is again a congruence subgroup of Γ . Hence, it has finite index in K. By Lemma 3.3, the double coset KMK is a finite union of left cosets modulo K. Similarly, KMK is a finite union of right cosets modulo K. Thus, the pair K, ∆ is d-finite, and we can define the Hecke–Shimura ring D(K, ∆ ) of the pair. However, this ring is generally too big to have good properties and is usually replaced by appropriate subrings corresponding to certain subsemigroups of ∆ . First of all, for positive integers n and q, we define the subsemigroup

1n 0 n (mod q) gcd(µ (M), q) = 1 ∆ (q) = ∆ (q) = M ∈ ∆ M ≡ 0 µ (M) · 1n (3.68) of ∆ = ∆n and define the dc-ring L(q) = Ln (q) = D(Γn (q), ∆ n (q))

(3.69)

as the (regular) Hecke–Shimura ring of the principal congruence subgroup Γn (q). We shall restrict ourselves to consideration of congruence subgroups K of Γ containing a principal congruence subgroup Γ (q) that satisfies the condition K ∆ (q) = ∆ (q)K.

(3.70)

3.4 Rings of Double Cosets of the Symplectic Group

93

Such group K will be called a q-symmetric group of genus n. For a q-symmetric group of genus n, we set

∆ (K) = ∆ (K, q) = K ∆ (q)K = K ∆ (q) = ∆ (q)K.

(3.71)

It is clearly a subsemigroup of ∆ , and the pair K, ∆ (K) is d-finite. The Hecke– Shimura ring (3.72) L(K) = D(K, ∆ (K)) will be called the (regular) Hecke–Shimura ring of the group K. By TK (M) = τK (M)

(M ∈ ∆ )

(3.73)

we shall denote the basic elements of the ring L(K) of the form (3.4) corresponding to double cosets KMK ⊂ ∆ (K) and refer to these elements as double classes. The following important theorem discloses relations between regular Hecke–Shimura rings for various q-symmetric groups of given genus. Theorem 3.43. Let K and K be two q-symmetric groups of a given genus n. Suppose that K ⊂ K. Then the following assertions are true: (1) ∆ (K) = K ∆ (K ) = ∆ (K )K. (2) Γ ∩ ∆ (K) = K. (3) If M, M ∈ ∆ (K ) and M ∈ KMK, then M ∈ Γ (q)MK . (4) Let M ∈ ∆ (K ) and let TK (M) and TK (M) be two elements of the form (3.73) then a decomposition TK (M) = ∑i (KMi ) with Mi ∈ ∆ (K ) implies the decomposition TK (M) = ∑i (K Mi ) and vice versa. (5) The linear mappings ι : L(K) → L(K ) and ι : L(K ) → L(K) defined on the double classes by

ι (TK (M)) = TK (M ) if M ∈ ∆ (K) and M ∈ KMK ∩ ∆ (K ) and

ι (TK (M )) = TK (M ) if M ∈ ∆ (K ),

respectively, are mutually inverse isomorphisms of the rings. Proof. By (3.71), we get ∆ (K) = K ∆ (q) = KK ∆ (q) = K ∆ (K ). Similarly, ∆ (K) = ∆ (K )K. This proves part (1). If γ ∈ Γ ∩ ∆ (K), then γ = δ M, where δ ∈ K and M ∈ ∆ (K). Hence M = δ −1 γ ∈ Γ ∩ ∆ (q) = Γ (q). Thus, γ ∈ KΓ (q) ⊂ K, and part (2) is proved. Let M = γ M γ with γ , γ ∈ K. Let us choose an integer q prime to q such that the matrix q M −1 is integral. By Lemma 3.42, the matrix γ can be written in the form γ = γ1 γ2 , where γ1 ∈ Γ (q) and γ2 ∈ Γ (q ). Then we can write M = γ1 M γ3 , where γ3 = M −1 γ2 M γ . Since γ2 ∈ Γ (q ), it follows that the matrix γ3 is integral, and so γ3 ∈ Γ . On the other hand, γ3 = M −1 γ1−1 M . If M = M0 δ and M = M0 δ , where M0 , M0 ∈ ∆ (q) and δ , δ ∈ K , then γ3 ≡ δ −1 M0−1 M0 δ ≡ δ −1 δ (mod q). Hence γ3 ∈ K . This proves part (3).

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3 Hecke–Shimura Rings of Double Cosets

In order to prove part (4), we note that all of the left cosets K Mi are distinct and their union contains the double coset K MK . On the other hand, it follows from part (3) that all of the matrices Mi are contained in the double coset K MK , which proves the direct assertion of part (4). The inverse assertion follows from the direct one and part (1). Finally, it follows from parts (1), (2), and (4) that the pairs K, ∆ (K) and K , ∆ (K ) satisfy the conditions (3.17) and (3.19). Thus, by Proposition 3.8, the corresponding mapping (3.18) is an isomorphism. It follows from (4) that this mapping coincides with the mapping ι defined in part (5). The mapping ι is just the inverse of ι . The above theorem shows that the Hecke–Shimura rings L(K) of various q-symmetric groups K are all naturally isomorphic to each other and to the ring L(q). Thus, any of the groups K can be taken to study the ring. Moreover, the isomorphisms not only establish one-to-one correspondences between the basic elements (3.73) of the rings, but also between the left cosets entering in their decompositions. The last circumstance plays an essential part in the theory of representations of these rings by Hecke operators. An important example of q-symmetric groups apart from the group Γ (q) = Γ n (q) is presented by the group Γ0 (q) = Γn0 (q) defined by (2.2). Lemma 3.44. Let

∆0 (q) = ∆n0 (q)

AB = M= ∈ ∆ gcd(det M, q) = 1, C ≡ 0 (mod q) . CD

(3.74)

Then Proposition 3.35 remains true for the pair Γ0 (q), ∆0 (q) in place of Γ , ∆ , i.e., each left coset of ∆0 (q) modulo Γ0 (q) contains a “triangular” representative of the form (3.57). Moreover,

∆0 (q) = Γ0 (q)∆ (q) = ∆ (q)Γ0 (q);

(3.75)

in particular, the group Γ0 (q) is q-symmetric, and ∆ (Γ0 (q)) = ∆0 (q). Proof. By Proposition 3.35, there is γ ∈ Γ such that the matrix γ M has the “triangular” form Then obviously, γ ∈ Γ0 (q). This proves the first asser (3.57). tion. Let M = CA DB ∈ ∆0 (q). In order to prove that M ∈ Γ0 (q)∆ (q), by the first part, we can assume that C = 0. Thus by (1.3), we get the relations A tD = µ (M)1n and A tB = B tA; hence the matrices A and D are invertible modulo q, and the matrix A−1 B is symmetric modulo q. It follows that the matrix −1 −1 A A −A−1 BD−1 0 1n −µ −1 A−1 B = , γ= 0 1n 0 µ D−1 0 µ D−1 where µ = µ (M), is a rational q-integral symplectic matrix with multiplier 1. Since the mapping (3.66) is epimorphic, there exists a matrix γ ∈ Γ such that γ ≡ γ

3.4 Rings of Double Cosets of the Symplectic Group

95

(mod q). Then it follows from the definitions that γ ∈ Γ0 and γ M ∈ ∆ (q). Hence, Γ0 (q) ⊂∈ Γ0 (q)∆ (q). The inverse inclusion is obvious. The second of the decompositions (3.75) follows from the first one by application of the antiautomorphism M → µ (M)M −1 of the semigroup ∆0 (q). For positive integers n and q, we semigroup define the ∆ q = ∆n q = M ∈ ∆n gcd(q, det M) = 1

(3.76)

and the Hecke–Shimura ring L{q} = Ln q = D(Γn , ∆n q ).

(3.77)

Since ∆ q ⊂ ∆ , the ring L q can be naturally considered as a subring of the ring L. Theorem 3.45. Let K be a q-symmetric subgroup of the modular group Γ = Γn . Then the Z-linear mapping ιK : L(K) → L q

(3.78) defined on the elements of the form (3.73) by

ιK (TK (M)) = T (d1 , . . . , dn , e1 , . . . , en )

(M ∈ ∆ (K)),

where di = di (M) and e j = e j (M) are the symplectic divisors of M, is a ring isomorphism. In particular, the ring L(K) is commutative. Proof. It follows from Theorem 3.43 and Lemma 3.44 that each of the rings L(K) is isomorphic to the ring L0 (q) = Ln0 (q) = D(Γn0 (q), ∆n0 (q)),

(3.79)

and in addition, the isomorphism links elements (3.73) of the rings corresponding to matrices from the same double cosets modulo Γ , and hence, with the same symplectic divisors. Therefore, it suffices to prove the theorem only for the group Γ0 (q). Note that the group Γ is also q-symmetric and

∆ (Γ , q) = Γ ∆ (q) = ∆ (q)Γ = ∆ q .

(3.80)

Thus, Theorem 3.43 can be applied to the groups K = Γ and K = Γ0 (q). Since the matrix sd(M ) of symplectic divisors of a matrix M ∈ ∆0 (q) again belongs to ∆0 (q), it follows from part (3) of this theorem that sd(M ) ∈ Γ0 (q)M Γ0 (q). Hence in part (5) of the theorem we can take M = M = sd(M ), which completes the proof. The above theorem implies that for every q-symmetric subgroup K of Γ , the double class TK (M) of a matrix M ∈ ∆ (K) depends only on the symplectic divisors d1 , . . . , dn , e1 , . . . , en of M, which can be arbitrary positive integers satisfying the conditions (3.47) and gcd(di ei , q) = 1. This justifies the notation

96

3 Hecke–Shimura Rings of Double Cosets

TK (M) = TK (sd(M)) = TK (d1 , . . . , dn , e1 , . . . , en ) (M ∈ ∆ (K))

(3.81)

and TK (m) =

∑

TK (M)

∑

TK (d1 , . . . , dn , e1 , . . . , en ),

M∈K\∆ (K)/K, µ (M)=m

=

(3.82)

d1 e1 =···=dn en =m, d1 |d2 |···|dn |en |en−1 |···|e1

where m is prime to q. From Lemma 3.32, Theorem 3.31, and Theorem 3.45 we obtain the following theorem. Theorem 3.46. The elements (3.81) of the Hecke–Shimura ring L(K) of a qsymmetric group K ⊂ Γ = Γn satisfy the relations TK (d1 , . . . , dn , e1 , . . . , en )TK (d1 , . . . , dn , e1 , . . . , en ) = TK (d1 d1 , . . . , dn dn , e1 e1 , . . . , en en ), (3.83) provided that gcd(e1 /d1 , e1 /d1 ) = 1; in particular, d K TK (d1 , . . . , dn , e1 , . . . , en ) = TK (d1 , . . . , dn , e1 , . . . , en ) d K = TK (dd1 , . . . , ddn , de1 , . . . , den ),

(3.84)

where d K = T (d · 12n ) = (Γn d · 12n ).

(3.85)

The elements (3.82) satisfy the relations TK (m)TK (m ) = TK (mm ) if gcd(m, m ) = gcd(mm , q) = 1.

(3.86)

Exercise 3.47. Show that the elements (3.82) for n = 1 satisfy the relations TK (m)TK (m ) =

∑ d · d K TK (mm /d 2 )

if gcd(mm , q) = 1,

d|m,m

where d K are the elements (3.85). Use the relations to show that the formal Dirichlet series with coefficients TK (m) for m prime to q has a formal Euler factorization of the form −1 TK (m) = ∏ TK (1) − TK (p)p−s + p K p1−2s . s m p∈P, pq m∈N, gcd(m,q)=1

∑

[Hint: For K = Γ10 (q), show first that the linear map L10 (q) → H1 = D(Λ2 , Σ2 ) transforming elements TK (D) to tΛ (D) is an embedding of the rings and then use Exercise 3.19.]

3.4 Rings of Double Cosets of the Symplectic Group

97

In the same way as was done for the full modular group, consideration of global Hecke–Shimura rings L(K) for q-symmetric subgroups K of Γ reduces to the study of local subrings. For a prime number p not dividing q, let us introduce the subsemigroup of ∆ (K) given by (3.87) ∆ p (K) = M ∈ ∆ (K) µ (M)|p∞ and the local subring L p (K) of L(K) defined by L p (K) = D(K, ∆ p (K)).

(3.88)

Similarly to the case of full modular group, using corresponding results and Theorem 3.45, we get the following result. Theorem 3.48. The ring L(K) of each q-symmetric group K of genus n is generated by the local subrings L p (K) for all prime numbers p not dividing q; each of the local subrings L p (K) is naturally isomorphic to the local subring L p = Lnp of the ring L = Ln and generated over Z Z 1 K by the n + 1 algebraically independent elements (3.89) TK (p) = TK (1, . . . , 1, p, . . . , p) n

n

and Ti, K (p2 ) = TK (1, . . . , 1, p, . . . , p, p2 , . . . , p2 , p, . . . , p) (i = 1, . . . , n). n−i

i

n−i

(3.90)

i

The following lemma describes explicit decompositions of certain generators in terms of left cosets, which will be needed below. Lemma 3.49. The following decompositions hold for the group K = Γn0 (q) and each prime number p not dividing q: n .B pD K , (3.91) TK (p) = ∑ ∑ 0 D i=0 D∈Λ \Λ D (p)Λ i

B∈B(D)/mod D

where Λ = Λn = GLn (Z), B(D) are the sets (3.58), residue classes taken modulo the relation (3.59), Di (p) = Dni (p) = diag(1, . . . , 1, p, . . . , p), n−i

and we set

. = tD−1 ; D

mod D are (3.92)

i

(3.93)

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3 Hecke–Shimura Rings of Double Cosets

in the same notation, Tn−1,K (p2 ) =

2 .B p D K ∑ 0 D D∈Λ \Λ Dn−1 (p)Λ , B 2 .B p D K + ∑ 0 D D∈Λ \Λ pD1 (p)Λ , B 2 .B p D K , + ∑ 0 D D∈Λ \Λ D (p)Λ , B

(3.94)

n

where the matrix B in each of the three sums on the right ranges over all representa2 tives of residue classes B(D)/mod D such that the rank of the matrix p 0D. DB over the field F p = Z/pZ of residues modulo p is equal to 1; finally, Tn,K (p2 ) = p K = K(p · 12n ) . (3.95) Proof. Without loss of generality it suffices to consider the case K = Γ = Γn . It follows from Theorem 3.28 that the set ∆ (p) = {M ∈ ∆n |µ (M) = p} consists of the single double coset modulo Γ of the diagonal matrix generating the element (3.89). By (1.2), an integral matrix of the form (3.57) belongs to ∆ (p) if . and B ∈ B(D). According to Theorem 3.12, the matrix pD . and only if A = pD with an integral matrix D is integral if and only if D belongs to one of the double cosets Λ Di (p)Λ for i = 0, 1, . . . , n. Thus, the decomposition (3.91) follows from Proposition 3.35(2). Further, it follows from Theorem 3.28 or formula (3.54) that the set ∆ (p2 ) = {M ∈ ∆n |µ (M) = p2 } is the union of the double cosets modulo Γ of the matrices generating elements (3.90), and the double coset corresponding to an element Ti (p2 ) consists of all matrices in ∆ (p2 ) whose rank over the field F p is equal to n − i. It is easy to see that the rank over F p of a matrix in ∆ (p2 ) of the form (3.57) can be equal to 1 only if D ∈ Λ Di (p)D j (p)Λ , where 0 ≤ j ≤ i ≤ n and j + n − i ≤ 1, which implies that (i, j) = (n, 0), (n − 1, 0), or (n, 1). Then the decomposition (3.94) follows from Proposition 3.35(2). The decomposition (3.95) is obvious. Exercise 3.50. Show that for K = Γn0 (q) the decompositions of the elements (3.90) with i = 1, . . . , n into left cosets can be written in the form 2 .B p D 2 K Ti (p ) = , ∑ ∑ 0 D α +β ≤n,α ≥i D∈Λ \Λ D (p)D (p)Λ , B α

β

where the matrix B in each of the inner sums ranges overall representatives of 2D . B p residue classes B(D)/mod D such that the rank of the matrix 0 D over the field F p is equal to n − i. Spherical Mapping. Calculations in local Hecke–Shimura rings of the symplectic group can be considerably simplified if one uses a realization of the rings given

3.4 Rings of Double Cosets of the Symplectic Group

99

by spherical polynomials. Since these polynomials have, generally speaking, not integral but rational coefficients, it will be convenient to extend the rings of double cosets under consideration to the corresponding rings over the field Q of rational numbers, i.e., the rings L(K) = DQ (K, ∆(K)),

L-p (K) = DQ (K, ∆ p (K)), and

L-p = L-np = DQ (Γn , ∆np ).

It follows from Theorem 3.48 that each of the local rings L-p (K), for a q−symmetric group K and a prime p not dividing q, is naturally isomorphic to the ring L-p . Hence, one can restrict oneself to consideration of spherical mappings of the rings L-p . Let M ∈ ∆ p = ∆np . According to Lemma 3.35 and (1.3), the left coset Γ M, where Γ = Γn , contains a representative of the form δ .B p D (3.96) M = , where pδ = µ (M). 0 D It is clear that the left coset Λ D modulo Λ = Λn depends only on the left coset Γ M. Let x0 , x1 , . . . , xn be independent (commuting) variables. We assign to the left coset (Γ M) the monomial Ω((Γ M)) = x0δ ω ((Λ D)), where ω ((Λ D)) is the monomial in the variables x1 , . . . , xn assigned to the left coset (Λ D) by (3.42), and to every element T = ∑ j a j (Γ M j ) ∈ L-p = L-np with a j ∈ Q we assign the polynomial Ω(T ) = (Ωnp (T ))(x0 , x1 , . . . , xn ) = ∑ a j Ω((Γ M j )) ∈ Q[x0 , x1 , . . . , xn ].

(3.97)

j

We shall call the map Ω = Ωnp : L-np → Q[x0 , x1 , . . . , xn ] the spherical mapping of the ring L-np . Theorem 3.51. The spherical mapping Ωnp is a Q-linear isomorphism of the ring L-np onto the subring Q[x0 , x1 , . . . , xn ]W of Q[x0 , x1 , . . . , xn ] consisting of all polynomials invariant under the group W = W n of transformations of rational fractions in x0 , x1 , . . . , xn generated by all permutations in variables x1 , . . . , xn and by the transformations ρ1 , . . . , ρn of the variables x0 , x1 , . . . , xn defined by

ρi (x0 ) = x0 xi ,

ρi (xi ) = xi−1 , and ρi (x j ) = x j if j = 0, i.

(3.98)

Proof. In order to prove the theorem, it is sufficient, by Theorem 3.40, to show that the Ω-images of the generators (3.64)–(3.65) are algebraically independent and generate the ring Q[x0 , x1 , . . . , xn ]W . Here we shall restrict ourselves to small genera n = 1 and n = 2, since these are the only cases we use below. The general proof is based on similar ideas but demands more computations and space. For n = 1 it follows from the decompositions of Lemma 3.49 that Ω(T (p)) = x0 + x0 x1 and Ω(T1 (p2 )) = Ω( p ) = p−1 x02 x1 . These polynomials are clearly algebraically independent. Thus, it is sufficient to prove that

100

3 Hecke–Shimura Rings of Double Cosets

Q[x0 , x1 ]W = Q[x0 + x0 x1 , x02 x1 ]. The substitution x0 = z1 , x0 x1 = z2 transforms the ring Q[x0 + x0 x1 , x02 x1 ] into Q[z1 + z2 , z1 z2 ] and the ring Q[x0 , x1 ]W into the ring Q[z1 , z2 ]S of all symmetric polynomials in z1 , z2 over Q (note that if a polynomial P(x0 , x1 ) = ∑i, j≥0 ci j x0i x1j is W -invariant, i.e., P(x0 x1 , x1−1 ) = P(x0 , x1 ), then ci j = 0 implies that i ≥ j, and so the image P(z1 , z2 /z1 ) of P(x0 , x1 ) is a polynomial in z1 , z2 ). Hence, the theorem on symmetric polynomials proves the assertion for n = 1. Standard computations based on the definitions and Proposition 3.35 show that the following formulas hold for n = 2:

Ω(T1 (p

2

Ω(T (p)) = x0 (1 + x1 )(1 + x2 ), 2 )) = x0 ω (t[1, p]) + x02 p3 ω (t[p, p2 ]) + x02 (p2 − 1)ω (t[p, p]) 1 p2 − 1 2 x x1 x2 + x02 (x1 + x2 )(1 + x1 x2 ), = p3 0 p

(3.99)

(3.100)

where t[d, d ] are the elements (3.25) (note that the number of symmetric matrices of order 2 with rank 1 over the field F p is p2 − 1), and Ω(T2 (p2 )) = Ω( p ) =

1 2 x x1 x2 . p3 0

(3.101)

According to these formulas, it suffices to show that the polynomials t = x0 (1 + x1 )(1 + x2 ),

u = x02 x1 x2 ,

and v = x02 (x1 + x2 )(1 + x1 x2 )

are algebraically independent and generate the ring Q[x0 , x1 , x2 ]W . Suppose that these polynomials are algebraically dependent, and let G(y, y1 , y2 ) be the polynomial of minimal degree such that G(t, u, v) = 0. Let us expand G in powers of y1 : G = ∑i gi (y, y2 )yi1 . If g0 = 0, then G is divisible by y1 , and y−1 1 G is a polynomial of smaller degree with the same property as G. Hence, g0 = 0. By the assumption, we have ∑i gi (t, v)ui = 0. Since it is an identity in x0 , x1 , x2 , we can set here x2 = 0. The substitution transfers the polynomials t, v into x0 (1 + x1 ), x02 x1 , respectively, and u turns to zero. Hence, we get the identity g0 (x0 (1 + x1 ), x02 x1 ) = 0, a contradiction, because the polynomials x0 + x0 x1 and x02 x1 are algebraically independent. Now we shall prove by induction on m that each W −invariant polynomial F(x0 , x1 , x2 ) of degree m relative to x0 is a polynomial in t, u, v. If m = 0, then F = F(x1 , x2 ) is a symmetric polynomial satisfying F(x1−1 , x2 ) = F(x1 , x2 ), and therefore it must be equal to a constant. Suppose that the assertion has already been proved for the polynomials whose degree in x0 is less than m with m ≥ 1. Let F(x0 , x1 , x2 ) =

∑

0≤i≤m

x0i fi (x1 , x2 )

3.4 Rings of Double Cosets of the Symplectic Group

101

be W −invariant polynomial of degree m relative to x0 . Since the variables x0 , x1 , x2 are algebraically independent, it follows from the definition of the group W that each of the polynomials x0i fi (x1 , x2 ) is also W −invariant. Thus, we may assume that F = x0m f (x1 , x2 ). Since F is W 2 −invariant, it is also W 1 −invariant, where W 1 acts only on x0 , x1 . In particular, it is true for the polynomial F(x0 , x1 , 0). Since we have already proved the assertion for n = 1, we can write F(x0 , x1 , 0) = P(x0 (1+x1 ), x02 x1 ), where P(y1 , y2 ) = ∑i, j ai j yi1 y2j is a polynomial over Q. Since F is of the form x0m f (x1 , x2 ), and the variables x0 , x1 are algebraically independent, it follows that ai j = 0 if i+2 j = m. Thus, the polynomial F (x0 , x1 , x2 ) = F(x0 , x1 , x2 )−P(t, v) can be written in the form F (x0 , x1 , x2 ) = x0m f (x1 , x2 ) −

∑

ai j t i v j = x0m G(x1 , x2 ),

i+2 j=m

where G is a polynomial. Since the polynomials t, v turn at x2 = 0 into x0 (1 + x1 ) and x02 x1 , respectively, it follows that the polynomial F is identically zero at x2 = 0. Thus, it is divisible by x2 . Since F is symmetric in x1 , x2 , it is divisible by x1 x2 . Hence F (x0 , x1 , x2 ) = x0m x1 x2 G (x1 , x2 ), where G is a polynomial. Since F is W -invariant, it is invariant under the transformation ρ1 , i.e., it satisfies F (x0 x1 , x1−1 , x2 ) = F (x0 , x1 , x2 ). It follows that G satisfies x1m−2 G (x1−1 , x2 ) = G (x1 , x2 ). If m = 0 or m = 1, than G is not a polynomial in x1 . Thus, m ≥ 2, and we can write F(x0 , x1 , x2 ) = F (x0 , x1 , x2 ) − P(t, v) = x02 x1 x2 F (x0 , x1 , x2 ) − P(t, v), where F is a W −invariant polynomial of degree m − 2 in x0 . The theorem will allow us to reduce calculations in local Hecke–Shimura rings of the symplectic group to calculations with invariant polynomials. In order to illustrate this approach, we shall consider generating formal power series for the sums T (pδ ) ∈ Lnp of all double cosets of multiplier pδ . It follows from (3.54) and Proposition 3.35 that δ .B p D Γ , (3.102) T (pδ ) = ∑ 0 D 0≤δ ≤···≤δ ≤δ , 1

n

D∈Λ \Λ diag(pδ1 ,...,pδn )Λ , B∈B(D)/mod D

where Λ = Λn and Γ = Γn . It is easy to check that the number of elements of the set B(D)/mod D for a matrix D ∈ Σn with elementary divisors d1 , . . . , dn depends only on the elementary divisors and is equal to d1n d2n−1 · · · dn . Therefore, by the definition of Ω, we obtain the following formula for Ω-images of the elements T (pδ ): Ω(T (pδ )) = x0δ

∑

0≤δ1 ≤···≤δn ≤δ

pnδ1 +(n−1)δ2 +···+bn ω (t(pδ1 , . . . , pδn )),

(3.103)

102

3 Hecke–Shimura Rings of Double Cosets

where ω is defined by (3.42) and (3.43). Hence, the Ω-image of the generating formal power series of genus n with coefficients T (pδ ), ∞

Z p (v) =

∑ T (pδ )vδ ,

(3.104)

δ =0

can be written in the form Ω(Z p )(v) =

∞

∑ Ω(T (pδ ))vδ

δ =0

∑

=

pnδ1 +(n−1)δ2 +···+bn ω (t(pδ1 , . . . , pδn ))(x0 v)δ .

(3.105)

0≤δ1 ≤···≤δn ≤δ

For the genus n = 1, we get Ω(Z p )(v) =

∑

pδ1 (x1 p−1 )δ1 (x0 v)δ

0≤δ1 ≤δ

= (1 − x0 v)−1 (1 − x0 x1 v)−1 = (1 − Ω(T (p))v + pΩ((p))v2 )−1 , (3.106) where (p) = (p)1 = T1 (p2 ) is the double coset of the form (3.53) (formulas for Ω(T (p)) and Ω(T1 (p2 )) follow from Proposition 3.35). Hence, since the spherical mapping is monomorphic, we obtain summation formulas for the series Z p (v). Proposition 3.52. For every prime number p, the following identity holds in the ring of formal power series over L1p : ∞

∑ T (pδ )vδ = (1 − T (p)v + p p v2 )−1 .

(3.107)

δ =0

Similarly, for genus n = 2 we obtain Ω(Z p )(v) =

∑

p2δ1 +δ2 ω (t(pδ1 , pδ2 ))(x0 v)δ .

0≤δ1 ≤δ2 ≤δ

Since by (3.27) and Theorem 3.23,

ω (t(pδ1 , pδ2 )) = ω ([pδ1 ]t(1, pδ2 −δ1 )) = (p−3 x1 x2 )δ1 ω (t(1, pδ2 −δ1 ), where [d] = [d]2 are elements (3.28), the sum for Ω(Z p )(v) after the replacement of δ2 and δ by δ1 + α and δ2 + β , respectively, can be rewritten in the form Ω(Z p )(v) =

∑

p2δ1 +δ1 +α (p−3 x1 x2 )δ1 ω (t(1, pα ))(x0 v)δ1 +α +β

α ,β ,δ1 ≥0

= (1 − x0 v)−1 (1 − x0 x1 x2 v)−1

∑ ω (t(1, pα ))(px0 v)α .

α ≥0

3.4 Rings of Double Cosets of the Symplectic Group

103

In order to compute the last sum, we note that with the help of Proposition 3.35 it can be easily seen that any set of the form α α −β

p 0 1 a p b α β (a mod p ), (0 < β < α , b mod p , p b) , 0 pα 0 1 0 pβ can be taken as a set of of the left cosets modulo Λ = Λ2 contained representatives in the double coset Λ 10 p0α Λ . Hence, by the definition of ω , we see that

∑ ω (t(1, pα ))(px0 v)α

α ≥0

=

∑

α

−2 α

β

−1 γ

−1 α

p (x2 p ) + (x1 p ) +

α ≥0

=

∑

∑

β

β −1

(p − p

−1 α −β

)(x1 p )

−2 β

(x2 p )

(px0 v)α

0 1 is an element of the form (3.19), and Ω(T) = Ωnp (T)(x0 , x1 , . . . , xn ) is the spherical polynomial (3.140), then Ωn−1 p (Ψ(T(M)))(x0 , x1 , . . . , xn−1 ) = Ωnp (T(M))(χ (p)pk−n x0 , x1 , . . . , xn−1 , χ (p)pn−k ),

(4.32)

where we assume that the spherical mapping is extended to complexifications of Hecke–Shimura rings by linearity. Proof. All assertions of thetheorem, δ except for the formula (4.32), have already . i Bi n D p , then by formula (4.30) and the definibeen proved. If T (M) = ∑i Γ0 0 D i tion of spherical mapping (see (3.140) and (3.43)), we obtain (k−n) δ ) (χ (p)p−k )di x0δ ω pn−1 (Λn−1 Di )(x1 , . . . , xn−1 ) Ωn−1 p (Ψ(T(M))) = ∑( χ (p)p i

Di ∗ = ∑(x0 χ (p)pk−n )δ ω pn Λn (x1 , . . . , xn−1 , χ (p)pn−k ), di 0 p i

which proves the formula (4.32). ´p Let us look now at the restriction of the mapping Ψ to the complexification L L of the isomorphic image (3.123) in p of the local symplectic Hecke–Shimura ring L p (Γ0 (q)). Proposition 4.20. In the notation and under the assumption of Theorem 4.19, the ´ np into the ring mapping Ψ = Ψnk,χ with n > 1 transforms the complexification L L´ n−1 p ; the mapping ´ np → L ´ n−1 (4.33) Ψ = Ψnk,χ : L p is epimorphic, except for the case n = k and χ (p) = −1, in which case the image ´ n−1 ´ np ) is a proper subring of L Ψ(L p . Proof. It follows from Theorem 3.51 that the extended spherical mapping Ω = ´ np → C[x0 , x1 , . . . , xn ] is an isomorphism of the ring L ´ np onto the subring Ωnp : L n C[x0 , x1 , . . . , xn ]W of W = W -invariant polynomials. Relations (4.32) imply that n−1 −invariant ´n the spherical polynomial Ωn−1 p (Ψ(T)) of an element T ∈ L p is W ´ np ) ⊂ L´ n−1 if Ωnp (T) is W n -invariant. Thus, Ψ(L p .

134

4 Hecke Operators

´ np ) is generated over C by Ψ-images of a system of generators of The ring Ψ(L n ´ the ring L p or the ring Lnp . The relations (4.32) allow one to find spherical polynomials corresponding to the images if spherical polynomials for the generators are known. Computation of spherical polynomials corresponding to generators for an arbitrary genus n > 1 is not hard, but it requires a lot of space. In order to illustrate the computations, we restrict ourselves here to the simplest case n = 2. It was shown in the course of the proof of Theorem 3.51 that the sets {x02 x1 , x0 (1 + x1 )} and {x02 x1 x2 , x0 (1 + x1 )(1 + x2 ), x02 (x1 + x2 )(1 + x1 x2 )} consist of spherical polynomials corresponding to certain systems of generators (over Q) of the rings L1p ´ 2p ) is generated and L2p , respectively. It follows from (4.32) that the image Ψ(L over C by the elements whose spherical polynomials are equal to the polynomials χ (p)pk−2 x02 x1 , χ (p)pk−2 x0 (1 + x1 )(1 + χ (p)p2−k ), and χ (p)2 p2(k−2) x02 (x1 + χ (p)p2−k )(1 + χ (p)p2−k x1 ). If 1 + χ (p)p2−k = 0, then clearly these elements gen´ 1p . But if 1 + χ (p)p2−k = 0, i.e., k = 2 and χ (p) = χ (p) = −1, erate the whole ring L then the image is generated by the elements with spherical polynomials x02 x1 and 2 ´ 1p . x02 (1 − x1 )2 = x0 (1 + x1 ) − 4x02 x1 and so cannot be equal to L Diagonalization of Hecke Operators for the Full Modular Group. The Zharkovskaya relations allow one to reduce certain problems relating to Hecke operators on modular forms of a given genus to similar problems for cusp forms and modular forms of smaller genera. In particular, the problem of simultaneous diagonalization of Hecke operators on invariant spaces of modular forms is a problem of this kind. Theorem 4.7 solves the problem of diagonalization for cusp forms. Here we shall consider the problem for invariant subspaces of the space Mnk = Mk (Γn ) = Mk (Γn , 1)

(4.34)

of modular forms of integral nonnegative weight k for the full modular group Γn with the trivial character, but first we shall prove a useful lemma on cusp forms contained in these spaces. Lemma 4.21. Let F=

∑ n

f (A)eπ iσ (AZ) ∈ Mnk

(4.35)

A∈E , A≥0

be a modular form of weight k for the full modular group Γn then the following three conditions are equivalent: (a) F is a cusp form. (b) F|Φ = 0, where Φ is the Siegel operator (1.56). (c) The Fourier coefficients of F satisfy the rule f (A) = 0 if det A = 0. Proof. Condition (a) implies (b), by Lemma 1.29. According to formula (1.54), condition (b) implies (c). Let M ∈ Gn (Q) be a symplectic matrix of genus n with rational entries and positive multiplier µ = µ (M). It follows from Proposition 3.35(1) B n A that there is a matrix γ ∈ Γ such that M = γ M with triangular M = 0 D . Then, if the condition (c) is fulfilled, we have

4.3 Hecke Operators and Siegel Operator

135

F|k M = F|k γ M = F|k γ |k M = F|k M = µ nk−

n(n+1) 2

(det D )−k

−1 )

∑ n

f (A)eπ iσ (A(A Z+B )(D )

∑ n

f (A)eπ iσ (AB (D ) eπ iσ ((D )

A∈E , A>0

=µ

n(n+1) nk− 2

(det D )−k

−1

−1 AA Z)

.

A∈E , A>0

Each of the matrices (D )−1 AA = µ −1 tA AA is symmetric and positive definite together with A, whence F|k M|Φ = 0, by the definition of Φ. Thus, by Lemma 1.29, F is a cusp form. Theorem 4.22. Every subspace V ⊂ Mnk invariant with respect to all Hecke operators |k T = |k,1 T with T ∈ Ln = L(Γn ) has a basis of common eigenfunctions for all of the operators. Proof. The same arguments that were used to prove Theorem 4.7 show that the theorem is true for invariant subspaces of the space Nnk = Nk (Γn ) of cusp forms. Now let V ⊂ Mnk be an arbitrary invariant subspace. We set V1 = V ∩ Nnk and denote by V2 = V⊥ 1 = { f ∈ V|(F, G) = 0 for all G ∈ V1 } the orthogonal complement of V1 in V with respect to the Petersson scalar product (1.70). It follows in a standard way from the properties of scalar product listed in Theorem 1.38 that the space V is the orthogonal direct sum of the subspaces V1 and V2 : V = V1 ⊕ V2 .

(4.36)

By Proposition 4.4 and our assumption, the subspace V1 is the intersection of two invariant subspaces and so is also invariant with respect to all of the Hecke operators. By Lemma 4.6 with q = 1, each of the Hecke operators |k T with T ∈ Ln satisfies (F|k T, F ) = (F, F |k T )

(4.37)

if one of the forms F, F ∈ Mnk is a cusp form. It follows that the subspace V2 is again invariant under all of the Hecke operators, and it suffices to prove that this subspace has a basis of common eigenfunctions for all of the operators. Since the subspace V2 does not contain nonzero cusp forms, it follows from Lemma 4.21 that the Siegel operator Φ maps this subspace isomorphically onto its image V = V2 |Φ in Mn−1 k . Since the mapping (4.33) with χ = 1 is epimorphic for every prime p, it follows is invariant with respect to all of from Theorem 4.19 that the subspace V ⊂ Mn−1 k for all prime p and hence, by Theorem 3.37, the Hecke operators |k T with T ∈ Ln−1 p n−1 with all T ∈ L . Assuming that the theorem has already been proved for invariant subspaces of Mn−1 k , we see that there is a basis F1 , . . . , Fd of the space V consisting −1 of common eigenfunctions. The inverse images F1 = F1 |Φ , . . . , Fd = Fd |Φ−1 of the functions F1 , . . . , Fd in V2 clearly form a basis of V2 , and each of the functions is a common eigenfunctions for all of the Hecke operators on Mnk . In order to see the latter, it is sufficient to consider only the operators corresponding to elements of a local Hecke–Shimura ring Lnp . For T ∈ Lnp , by (4.31), we obtain

136

4 Hecke Operators

(Fi |k T )|Φ = Fi |Ψ(T) = λ (Ψ(T))Fi = (λ (Ψ(T))Fi )|Φ, where |k = |k,1 , T is the image of T in Lnp , and λ (Ψ(T)) is a scalar. Hence Fi |k T − λ (Ψ(T))Fi |Φ = 0, and so Fi |k T − λ (Ψ(T))Fi = 0. In order to complete the induction, it remains to prove the theorem for n = 1. If we again present an invariant subspace V of M1k in the form (4.35) of the direct sum of two invariant subspaces, then since M0k = C, the dimension of V2 is 0 or 1. If dim V2 = 0, then V ⊂ N1k and the assertion has already been proved. But if dim V2 = 1, then every function of V2 is a common eigenfunction. Exercise 4.23. Show that all eigenvalues of all Hecke operators on Mnk are real numbers. Exercise 4.24. Let p be a prime number and F ∈ Mnk a nonzero common eigenfunction of all Hecke operators corresponding to elements T of Lnp with the eigenvalues λ (T ) = λF (T ). Let Q p,F (v) = ∑ (−1)n λ (qni (p))vi 0≤i≤2n

be the spinor p-polynomial of F resulting from the replacement of coefficients of the spinor p-polynomial (3.112) with eigenvalues of corresponding Hecke operators acting on the eigenfunction F, and let α0 (p), α1 (p), . . . , αn (p) be parameters of the linear extension on L-np of the homomorphism T → λ (T ) defined in Proposition 3.55. Prove the following formulas: (1) Q p,F (v) = (1 − α0 (p)v) ∏nr=1 ∏1≤i1 1 variables, since generally, there is no multiplication of the classes, except for some isolated cases such as Gauss composition of binary forms. A possible way out of the situation is to consider suitable generating series for values of the functions instead of the values themselves. Traditionally, starting from Euler, multiplicativity of arithmetic sequences is customarily expressed in the form of an Euler product factorization of the generating Dirichlet series. It turns out that in the situation of modular forms, suitable Dirichlet series constructed by Fourier coefficients of eigenfunctions of Hecke operators can be expressed through Dirichlet series formed by the corresponding eigenvalues. The latter, according to the multiplicative properties of the relevant Hecke–Shimura rings, may have an Euler product factorization. Such relations between Dirichlet series constructed by Fourier coefficients of the eigenfunctions and the Euler product formed with the help of the corresponding eigenvalues prove to be useful in both directions: on the one hand, they establish the multiplicativity of the Fourier coefficients, and on the other hand, the relations allow one to express the Euler products, called zeta functions of modular forms, in terms of underlying modular forms and so to relate their analytic properties with those of the forms, as has been noted in the introduction. A. Andrianov, Introduction to Siegel Modular Forms and Dirichlet Series, Universitext, DOI 10.1007/978-0-387-78753-4 5, c Springer Science+Business Media LLC 2009

137

138

5 Euler Factorization of Radial Series

5.1 Radial Series of Genus One and Zeta Functions In this section,

F = F(z) ∈ Mk (q, χ ) = Mk (Γ0 (q), χ )

(5.1)

is a modular form in one complex variable z belonging to the upper half-plane H = H1 = {z = x + iy ∈ C|y > 0} of a positive integral weight k and a character χ of the form (2.3) for the group Γ0 (q) = Γ10 (q), where q ∈ N and χ is a Dirichlet character modulo q. We shall write the Fourier expansion (1.35) of F in the form F(z) =

∑

f (2a)eπ i2az =

∞

∑ f(a)e2π iaz

(5.2)

a=0

2a∈E1 , a≥0

with Fourier coefficients f(a) = f (2a) indexed by all nonnegative integers. Fourier Coefficients of Eigenfunctions and Eigenvalues. It turns out that for modular forms in one variable, there are close relations between individual Fourier coefficients of the eigenfunctions and the corresponding eigenvalues. Proposition 5.1. Let F be a modular form (5.1) with Fourier expansion (5.2). Suppose that F is an eigenfunction for the Hecke operator |T (m) = |k,χ T (m), where T (m) ∈ L0 (q) = L10 (q) with m prime to q is an element of the form (3.82) for K = Γ0 (q): (5.3) F|T (m) = λ (m)F. Then the Fourier coefficients f(a) of F and the eigenvalue λ (m) = λF (m) are linked by the relations ma for all a ≥ 0. (5.4) λ (m)f(a) = ∑ χ (d)d k−1 f 2 d d|m, a Proof. By Lemma 4.12, Proposition 3.61, and Lemma 4.14, the Fourier coefficients (f | T (m))(a) of the function F|T (m) can be written in the form (f |T (m))(a) = (f |T(m))(a) = =

∑

f|

∑

d,d1 ∈N,dd1 =m

Π1+ (d)Π1− (d1 ) (a)

((f |Π1+ (d))|Π1− (d1 ))(a)

d, d1 ∈N,dd1 =m

=

∑

χ (d1 )d1k−1 (f |Π1+ (d))(a/d1 )

∑

χ (d1 )d1k−1 f(da/d1 ) =

dd1 =m, d|a

=

dd1 =m, d|a

∑

χ (d)d k−1 f(ma/d 2 ).

d|m, a

On the other hand, by (5.3), this Fourier coefficient is equal to λ (m)f(a). The relation (5.4) follows.

5.1 Radial Series of Genus One and Zeta Functions

139

First of all, the relations (5.4) are useful for studying properties of the eigenvalues. Theorem 5.2. The eigenvalues λ (m) = λF (m) of the Hecke operators |T (m) = |k,χ T (m) corresponding to a nonzero common eigenfunction F ∈ Mk (q, χ ) of the operators with gcd(m, q) = 1 have the following properties: (1) For every m and m prime to q, the eigenvalues satisfy the multiplicative relations mm . (5.5) λ (m)λ (m ) = ∑ χ (d)d k−1 λ d2 d|m, m (2) If F is a cusp form, then the eigenvalues satisfy the inequalities |λ (m)| ≤ cmk/2 ,

(5.6)

where the constants c = cF depend only on F. Proof. Let f(0), f(1), . . . be the Fourier coefficients of F. Since k > 0, the form F is not a constant, and there are positive integers a with f(a) = 0. Let κ = κ (F) be the smallest of such integers, and let δ be the greatest divisor of κ that is coprime to q. Since the numbers δ and κ /δ are coprime, it follows from (5.4) with m = δ and a = κ /δ that f(κ ) = f(δ · κ /δ ) = χ (δ )λ (δ )f(κ /δ ). Thus f(κ /δ ) = 0, and so δ = 1, which means that

κ (F)|q∞ , in particular, κ (F) = 1 if q = 1.

(5.7)

Hence by (5.4) with a = κ , we obtain the relations

λ (m) = f(mκ )/f(κ ) for all m prime to q.

(5.8)

By (5.8) and (5.4), we get

λ (m)λ (m ) = λ (m)f(m κ )/f(κ ) =

∑

d|m, m κ

χ (d)d

k−1

mm · κ f d2

/f(κ ) =

∑

d|m, m

χ (d)d

k−1

λ

mm d2

,

since common divisors of m and m κ must divide m . The estimates (5.6) follow from (5.8) and (1.46). It follows from the estimate (1.52) of the Fourier coefficients and relations (5.8) that, for, the eigenvalues of an arbitrary eigenfunction satisfy the inequalities |λ (m)| ≤ cF mk

(m ∈ N, gcd(m, q) = 1).

(5.9)

Exercise 5.3. Prove the relations (5.5), using the relations of Exercise 3.47 and relations (4.26).

140

5 Euler Factorization of Radial Series

Exercise 5.4. In the notation of Theorem 5.2, show that if f(0) = 0, then

λ (m) = ∑ χ (d)d k−1

(m ∈ N, gcd(m, q) = 1).

d|m

Exercise 5.5. In the notation of Theorem 5.2, prove that the eigenvalues λ (m) = λF (m) corresponding to the Eisenstein series F = Ek (z) ∈ Mk (see Exercise 4.9) are given by the formula

λ (m) = ∑ d k−1

(m ∈ N);

d|m

use the formula to show that the Fourier coefficients f(a) of Ek satisfy the relations

f(a) = f(1) ∑ d k−1

(a ∈ N)

d|a

(compare Exercise 1.24). Dirichlet Series and Zeta Functions of Modular Forms. Here we reformulate the relations (5.4) and (5.5) in the terms of generating Dirichlet series. Let us look first at a formal consequence of relations (5.5) in terms of Dirichlet series. Lemma 5.6. Let q ∈ N then the relations (5.5) for a nonzero function

λ : {m ∈ N| gcd(m, q) = 1} → C are equivalent to the following formal Euler factorization of the Dirichlet series with coefficients λ (m): −1 λ (m) λ (p) χ (p)pk−1 1 − = + . ∑ ∏ ms ps p2s p∈P, pq m∈N, gcd(m,q)=1

(5.10)

Proof. The relations (5.5) are equivalent to the relations

λ (m)λ (m ) = λ (mm ) if gcd(mm , q) = 1 and gcd(m, m ) = 1

(5.11)

together with the relations

λ (p)λ (pδ ) = λ (pδ +1 ) + χ (p)pk−1 λ (pδ −1 ) for each p ∈ P, p q and δ ≥ 1. (5.12) The relations (5.11) are obviously equivalent to the formal factorization ∞ λ (m) λ (pδ ) = ∏ ∑ ∑ δs , ms p∈P, pq δ =0 p m∈N, gcd(m, q)=1 whereas the recursion relations (5.12) can be written as the identities for formal power series of the form

5.1 Radial Series of Genus One and Zeta Functions

141

(1 − λ (p)t + χ (p)pk−1t 2 )

∞

∑ λ (pδ )t δ

=1

δ =0

or, by substituting t = p−s , as the summation formulas −1 λ (pδ ) λ (p) χ (p)pk−1 . ∑ δ s = 1 − ps + p2s δ =0 p ∞

Exercise 5.7. Under the assumptions of Lemma 5.6 prove the following formal Euler factorization of the formal Rankin Dirichlet series:

λ (m2 ) ms m∈N, gcd(m,q)=1 −1 χ (p)pk−1 λ (p2 ) − χ (p)pk−1 χ (p2 )p2k−2 1+ 1− = ∏ + . ps ps p2s p∈P, pq

∑

[Hint: First sum the formal power series ∑δ ≥0 λ (p2δ )t δ .] Let F ∈ Mk (q, χ ) be a nonzero common eigenfunction for all regular Hecke operators, i.e., operators |T = |k,χ T with T ∈ L0 (q) and in particular, F|T (m) = λ (m)F for all m prime to q. Then we call the Dirichlet series

ζr (s, F) =

λ (m) ms m∈N, gcd(m,q)=1

∑

(5.13)

the regular zeta function of the eigenfunction F. If q = 1, we omit the adjective “regular” and the lower index r and speak just about the zeta function ζ (s, F) of F. From Lemma 5.6 and the estimate (5.6), by standard calculus, we have the following result. Proposition 5.8. Let a nonzero cusp form F ∈ Nk (q, χ ) be a common eigenfunction for all the regular Hecke operators. Then the regular zeta function ζr (s, F) of F converges absolutely and uniformly in each right half-plane ℜs ≥ 2k + 1 + ε with ε > 0 and has the factorization into an absolutely and uniformly convergent Euler product of the form

ζr (s, F) =

∏

1−

p∈P, pq

λ (p) χ (p)pk−1 + ps p2s

−1 .

(5.14)

For an arbitrary nonzero eigenform F ∈ Mk (q, χ ), by Lemma 5.6 and (5.9), the factorization (5.14) holds in every half-plane ℜs ≥ k + 1 + ε with ε > 0. Exercise 5.9. Show that the zeta function of the Eisenstein series Ek (z) has the form

ζ (s, Ek ) = ζ (s)ζ (s − k + 1), where ζ (s) is the Riemann zeta function.

142

5 Euler Factorization of Radial Series

In Section 2.1 we associated with each cusp form of the space Nk (q, χ ) = N1k (q, χ ) the radial Dirichlet series of the form (2.7) constructed by means of Fourier coefficients of F, which, according to Theorem 2.1, have nice analytic properties. On the other hand, by Theorem 4.7, the space Nk (q, χ ) is spanned by eigenfunctions for all of the regular Hecke operators, and according to Proposition 5.8, the regular zeta function of every such eigenfunction is a Dirichlet series with Euler product factorization. A natural question arises whether radial Dirichlet series and Dirichlet series with Euler product associated with an eigenform are related to each other, or in other words, whether the Dirichlet series constructed by Fourier coefficients of an eigenform have an Euler product factorization. The following theorem gives a partly positive answer. Theorem 5.10. Let a nonzero cusp form F ∈ Nk (q, χ ) with Fourier coefficients f(a) be a common eigenfunction for all regular Hecke operators, in particular, F|k,χ T (m) = λ (m)F for every m prime to q. Then the identity −1 ∞ f(a) f(a) λ (p) χ (p)pk−1 1 − = + (5.15) ∑ s ∑ as ∏ ps p2s a=1 a a|q∞ p∈P, pq is valid in the half-plane ℜs > 2k + 1, where the series and product are absolutely convergent. In particular, if q = 1, then −1 f(a) λ (p) χ (p)pk−1 ∑ s = f(1) ∏ 1 − ps + p2s a=1 a p∈P ∞

(ℜs >

k + 1). 2

(5.16)

Proof. By Proposition 5.1, and we have ∞

f(a) f(am) = ∑ as = ∞ ∑ (am)s a=1 a|q , gcd(m, q)=1

f(a) ∑∞ as a|q

λ (m) , ms m, gcd(m,q)=1

∑

whence, by Proposition 5.8, we obtain the identity (5.15). For an arbitrary eigenform of all regular Hecke operators on the space Mk (q, χ ), for the same reasons and by the corresponding estimates, the identity (5.15) holds in the half-plane ℜs > k + 1. We shall use the notation Zq (s, F) =

∑∞

a|q

f(a) as

(5.17)

for the q-part of the Dirichlet series Z(s, F) of a modular form F ∈ Mk (q, χ ) whose coefficients are the Fourier coefficients of F indexed only by positive integers dividing a power of q. In this and the above notation, the identity (5.15) takes the form Z(s, F) = Zq (s, F)ζr (s, F) (resp., Z(s, F) = f(1)ζ (s, F) if q = 1).

(5.18)

5.1 Radial Series of Genus One and Zeta Functions

143

Exercise 5.11. Let F1 , . . . , Fh be a basis of an invariant subspace of the space Mk (q, χ ) under all regular Hecke operators, in particular, Fi |k,χ T (m) =

∑

λi j (m)Fj

(i = 1, . . . , h; gcd(m, q) = 1).

1≤ j≤h

Let f(a) with a = 0, 1, . . . be the h-column with i-entry for i = 1, . . . , h equal to the Fourier coefficient fi (a) of Fi , and let Λ(m) = λi j (m) . Prove the following relations: ma (a = 0, 1, . . . ; gcd(m, q) = 1); Λ(m)f(a) = ∑ χ (d)d k−1 f d2 d|m,a mm (gcd(mm , q) = 1); Λ(m)Λ(m ) = ∑ χ (d)d k−1 Λ 2 d d|m,m −1 ∞ f(a) −s −s k−1−2s 1h ∑ a f(a) = ∏ 1h − p Λ(p) + χ (p)p ∑ as . a=1 a|q∞ p∈P, pq Exercise 5.12. Let F ∈ Mk be a nonzero eigenform with F|k T (m) = λ (m)F for all m ∈ N. Prove that if the eigenvalues satisfy the Ramanujan–Petersson inequality k−1 |λ (p)| ≤ 2p 2 for each prime number p, then for arbitrary m ∈ N, the eigenvalues satisfy k−1 |λ (m)| ≤ τ (m)m 2 , where τ (m) is the number of positive divisors of m. [Hint: use Exercise 4.23.] Functional Equation for the Zeta Function. According to Theorem 2.1, the radial Dirichlet series Z(s, F) of a cusp form F ∈ Mk (q, χ ) has an analytic continuation over the whole s-plane such that the function Φ(s, F) = (2π )−s Γ(s)Z(s, F), where Γ(s) is the gamma function, is holomorphic everywhere and satisfies the functional equation 0 −1 ∈ Mk (q, χ ). (5.19) Φ(k − s, F) = ik qs−k+1 Φ(s, F ∗ ), with F ∗ = F|k 0 q The following proposition complements properties of the Dirichlet series entering into the functional equation. Proposition 5.13. If a modular form F ∈ Mk (q, χ ) is an eigenfunction for all the Hecke operators |k,χ T (m) with m prime to q, and λ (m) are the corresponding eigenvalues, then the form F ∗ ∈ Mk (q, χ ) is an eigenfunction for all Hecke operators |k,χ T (m) with m prime to q, and the corresponding eigenvalues are equal to χ (m)λ (m); the regular zeta function of the eigenfunction F ∗ has properties similar

144

5 Euler Factorization of Radial Series

to those of the regular zeta function of F listed in Theorem 5.10; in particular, it has an Euler factorization of the form

χ (m)λ (m) ms m∈N, gcd(m,q)=1 −1 χ (p)λ (p) χ (p)pk−1 1− = ∏ + ps p2s p∈P, pq

ζr (s, F ∗ ) =

∑

Proof. To prove the first part it is sufficient to show that 0 −1 0 −1 |k,χ T (p) = χ (p)F|k,χ T (p)|k F|k 0 q 0 q

(5.20)

(5.21)

p not dividing q. By Lemma 3.49, the for every F ∈ Mk (q, χ ) and primenumber matrix smap, 0, 0, 1. and matrices 10 bp with b = 0, 1, . . . , p − 1 form a system of representatives of left cosets modulo K = Γ10 (q) contained in the double coset T (p). If p b, then there is b ∈ {1, . . . , p − 1} such that 1 + qbb ≡ 0 (mod p). Hence for b = 1, 2, . . . , p − 1, we get the relation 1b 0 −1 0 −1 1b p −b , = −qb (1 + qbb )/p 0 p q 0 q 0 0p where the first factor on the right belongs to the group K. Hence by (4.13), (4.14), and (1.31), we obtain 0 −1 0 −1 p0 |k,χ T (p) = χ (p)F|k F|k 0 q 0 q 01 p−1 0 −1 10 0 −1 1b + F|k + ∑ F|k 0 q 0p 0 q 0p b=1 10 0 −1 p0 0 −1 + F|k = χ (p)F|k 0p q 0 01 q 0 p−1 1 b 0 −1 p −b + ∑ F|k −qb (1 + qbb )/p 0 p q 0 b=1 10 p0 + F|k,χ = χ (p) F|k,χ 0p 01 p−1 1b 0 −1 + ∑ F|k,χ |k , 0 p q 0 b=1 which proves the relation (5.21). The Euler factorization (5.20) follows from Proposition 5.8 with F ∗ in place of F. It follows from the proposition and Theorem 5.10 that the function Φ(s, F) and Φ(s, F ∗ ) in the functional equation (5.19), for a nonzero eigenfunction of all regular

5.2 Binary Radial Series and Gaussian Composition

145

Hecke operators on the space Nk (q, χ ), in the half-plane ℜs > 2k + 1, can be written in the form Φ(s, F) = (2π )−s Γ(s)Zq (s, F)ζr (s, F) and

Φ(s, F ∗ ) = (2π )−s Γ(s)Zq (s, F ∗ )ζr (s, F ∗ ),

where ζr (s, F) and ζr (s, F ∗ ) are the regular zeta functions of the eigenforms F and F ∗ , and Zq (s, F), Zq (s, F ∗ ) are the corresponding series (5.17). Thus, the functional equation can be considered as a functional equation for the zeta functions. In particular, since for q = 1 clearly F ∗ = F and Z1 (s, F) = Z1 (s, F ∗ ) = f(1) = 0, we get the following theorem. Theorem 5.14. The zeta function ζ (s, F) of a nonzero eigenform F ∈ Nk has a meromorphic continuation to the whole s-plane; the function Φ(s, F) = (2π )−s Γ(s)ζ (s, F) is holomorphic and satisfies the functional equation Φ(k − s, F) = ik Φ(s, F).

(5.22)

Exercise 5.15. Show that the zeta function ζ (s, Ek ) of the Eisenstein series Ek with even k has a meromorphic continuation over the whole s-plane, and that the function Φ(s, Ek ) = (2π )−s Γ(s)ζ (s, Ek ) is analytic except for two simples poles at s = 0 and s = k and satisfies the functional equation Φ(k − s, F) = ik Φ(s, F). [Hint: Use Exercise 2.2, or Exercise 5.9 and properties of the Riemann zeta function.] In framework of the theory of old and new forms initiated by A.O.L. Atkin and J. Lehner, which we do not touch on in this book, it turns out that under certain conditions, the q-partial Dirichlet series Zq (s, F) is a finite Euler product extending only over prime divisors of q.

5.2 Binary Radial Series and Gaussian Composition In this section we fix a nonzero modular form F = F(Z) ∈ M2k (q, χ ) = Mk (Γ20 (q), χ )

(5.23)

of positive integral weight k and character χ modulo q ∈ N for the group Γ20 (q) defined on the upper half-plane H2 of genus 2. The Fourier expansion (1.35) of such an F takes the form F(Z) =

∑

A∈E2 ,

A≥0

f (A)eπ iσ (AZ)

(5.24)

146

5 Euler Factorization of Radial Series

with Fourier coefficients indexed by matrices A of integral nonnegative semidefinite binary quadratic forms 1 2a b x X= . (5.25) a(x, y) = tXAX = ax2 + bxy + cy2 ⇔ A = b 2c y 2 Depending on circumstances, we shall use the language of even matrices or of integral quadratic forms with similarly defined basic notation. So we say that a number d ∈ N divides the form or matrix (5.25) if it divides the coefficients a, b, c, and the greatest divisor e = e(a) = e(A) = gcd(a, b, c) (5.26) is the divisor of a nonzero form or matrix. If the divisor is equal to 1, the form and matrix are called primitive. The number det a = det A is the determinant of the form, and the number δ = δ (a) = δ (A) = − det a = b2 − 4ac (5.27) is the discriminant of the form and the matrix. If A and A are even matrices and A = tλ Aλ with λ ∈ Λ2+ = SL2 (Z), then matrices A and A and the corresponding quadratic forms are called (properly) equivalent, and we write then A ∼ A

and

a ∼ a.

Equivalent even matrices or integral forms make up a (proper) class. We denote by {A}+ and {a}+ the proper classes of A and a, respectively. All even matrices and integral forms from a fixed class have the same signature, divisor, and discriminant. According to (4.16), the Fourier coefficients f (A) of every modular form for Γ20 (q) depend only on the proper class of the matrix A. Fourier Coefficients of Eigenfunctions and Eigenvalues. As in the case of modular forms in one variable, Fourier coefficients of the eigenfunctions of genus 2 are closely related to the corresponding eigenvalues, but the relations are not so simple. Proposition 5.16. Let F be a modular form (5.23) with Fourier expansion (5.24). Suppose that F is an eigenfunction for the Hecke operator |T (m) = |k,χ T (m), where T (m) ∈ L0 (q) = L20 (q) with m prime to q is an element of the form (3.82) for K = Γ20 (q): (5.28) F|T (m) = λ (m)F. Then the Fourier coefficients f (A) of F and the eigenvalue λ (m) = λF (m) are linked by the relations

∑

λ (m) f (A) =

χ (d1 )χ (d22 )d1k−2 d22k−3

d,d1 ,d2 ∈N; A∈d2 E2 , dd1 d2 =m

×

∑

D∈Λ+ \Λ+ diag(1,d1 )Λ+ ; d2−1 A[ tD]∈d1 E2

where Λ+ = Λ+2 .

f

d A[ tD] , d1 d2

(5.29)

5.2 Binary Radial Series and Gaussian Composition

147

Proof. For the Fourier coefficient of the function F|T (m) indexed by a matrix A ∈ E2 , by formula (3.132) of Proposition 3.61, we obtain the formula

λ (m) f (A) = ( f |T (m))(A) =

∑

( f |Π+ (d)Π(d1 )Π− (d2 ))(A),

d,d1 ,d2 ∈N, dd1 d2 =m

where Π± = Π2± . By formula (4.24) of Lemma 4.14, we get ( f |Π+ (d)Π(d1 )Π− (d2 ))(A) = ( f |Π+ (d)|Π(d1 ))|Π− (d2 ) (A) χ (d2 )2 d22k−3 ( f |Π+ (d)|Π(d1 ))(d2−1 A) if A ∈ d2 E2 , = 0 if A ∈ / d2 E2 . Further, by formulas (4.27) and (4.25) of Lemma 4.14, we conclude that ( f |Π+ (d))|Π(d1 ) (A ) = χ (d1 )d1k−2 ( f |Π(d)) d1−1 A [ tD] ∑ D∈Λ+ \Λ+ diag(1, d1 )Λ+ ; A [ tD]∈d1 E2

= χ (d1 )d1k−2

∑

f

D∈Λ+ \Λ+ diag(1, d1 )Λ+ ; A [ tD]∈d1 E2

d t A [ D] . d1

These formulas imply the formulas (4.29). Although formulas (5.29) look rather complicated, we shall need only certain particular cases. Corollary 5.17. Let F ∈ M2k (q, χ ) be a nonzero modular form with Fourier expansion (5.24) satisfying F|T (m) = λ (m)F for all of the Hecke operators |T (m) = |k,χ T (m) with m prime to q. Then for every primitive matrix A ∈ E2 and integer ν coprime to q, the Fourier coefficients of F and the eigenvalues λ (m) = λF (m) are linked by the relation λ (m) f (ν A) = f dd1−1 ν A[ tD] . ∑ χ (d1 )d1k−2 ∑ d,d1 ∈N, dd1 =m

D∈Λ+ \Λ+ diag(1, d1 )Λ+ , A[ tD]∈d1 E2

(5.30) Proof. With ν A in place of A, where A and ν satisfy the stated conditions, the formula (5.29) turns into (5.30), since for primitive A, the index d2 must be equal to 1, and the conditions ν A[ tD] ∈ d1 E2 and A[ tD] ∈ d1 E2 are obviously equivalent. Let us look at some attractive formal implementations of these last relations. Note that for every matrix D entering the inner sum on the right, the matrix d1−1 A[ tD] is even and has the same (positive) determinant as the matrix A. Let us suppose for a moment that the discriminant of a primitive nonsingular matrix A is a so-called single-class discriminant, i.e., such that each even primitive matrix A of order 2 with

148

5 Euler Factorization of Radial Series

the same discriminant is (properly) equivalent to A. Then, since Fourier coefficients depend only on the classes of indexing matrices, the last formula takes the shape

λ (m) f (ν A) =

∑

d,d1 ∈N, dd1 =m

f (d ν A)χ (d1 )d1k−2 ρ (d1 , A),

ρ (d, A) = # D ∈ Λ+ \Λ+ diag(1, d)Λ+ A[ tD] ∈ dE2 .

where

(5.31)

Now if we divide both parts of the last relation by ms = d s d1s , considered just a formal quasicharacter of the semigroup N, and then sum up these relations over all m prime to q, then as a result we get the following identity for formal Dirichlet series: f (ν A)

λ (m) ms m∈N, gcd(m, q)=1

∑

=

∑

d,d1 ∈N, gcd(dd1 , q)=1

=

∑

d∈N, gcd(d, q)=1

f (d ν A) χ (d1 )d1k−2 ρ (d1 , A) ds d1s

f (d ν A) ds d

χ (d1 )ρ (d1 , A) . d1s−k+2 1 ∈N, gcd(d1 , q)=1

∑

(5.32)

Therefore, by dividing both sides by ν s and summing up over all ν ∈ N with ν |q∞ , since n = d ν ranges over all of N, we obtain for every primitive matrix A of singleclass discriminant the relation

∑

ν ∈N, ν |q∞

f (ν A) λ (m) ∑ ν s m∈N, gcd(m, ms q)=1 ∞

=

∑

n=1

f (nA) χ (d)ρ (d, A) . ∑ ns d∈N, gcd(d, d s−k+2 q)=1

(5.33)

It follows from Theorem 3.46 and Proposition 3.53 that the Dirichlet series on the left with coefficients λ (m) has an Euler product factorization. Thus, an identity of the form (5.33), on the one hand, expresses this Euler product by means of a radial Dirichlet series on the right and so opens a way to derive analytic properties of the product from those of the radial series, and on the other hand, it reveals multiplicative properties of Fourier coefficients of the eigenfunction in the form of a (partial) Euler product factorization of radial series (we shall see later that the second Dirichlet series on the right also has an Euler product factorization). Note that in order to reveal multiplicativity of the Fourier coefficients, one needs similar identities for a variety of radial series, whereas in order to study analytic properties of the Euler product, it suffices to take such an identity only for a single nonzero radial series. Both of these problems will be considered below. Exercise 5.18. In the notation and under assumptions of Proposition 5.16, if, moreover, f (02 ) = 0, prove that

5.2 Binary Radial Series and Gaussian Composition

λ (m) =

∑

d,d1 ,d2 ∈N; dd1 d2 =m

149

χ (d1 )χ (d22 )d1k−2 d22k−3 #{Λ+ \Λ+ diag(1, d1 )Λ+ }.

Π -Operators and Multiplication of Quadratic Modules. Along with the simple operations of multiplication and division of even matrices or corresponding quadratic forms by positive integers, the above formulas for the action of Hecke operators on Fourier coefficients of modular forms include more complicated operators |Π(d) of the form (4.27). It turns out that these operators can also be interpreted with the help of a multiplication of quadratic forms, namely the multiplication in the sense of Gaussian composition. We shall treat the Gaussian composition of quadratic forms using the equivalent language of multiplication of quadratic modules. First we have to recall some common definitions related to modules in a quadratic number field. For details see any textbook on algebraic numbers. Let K be a quadratic extension of the field Q of rational numbers. An additive subgroup A of K finitely generated over Z is called a module in K. Two modules A and A are said to be similar or belong to one class {A } = {A} if A = α A with a nonzero α of K; in this case one writes A ∼ A. The set of all modules in K is a disjoint union of the classes of similar modules. A module A in K is said to be full if QA = K. Each full module in K has a Z−basis of two elements. A full module in K that contains 1 and is a ring is called an order in K. Every order in K is contained in the maximal order O = O(K), the ring of all (algebraic) integers of the field K over Q. Let O be an order and ω1 , ω2 a basis of O . Then the number δ (O ) = det(tr(ωi ω j )), where tr means the trace from K to Q, is independent of the choice of the basis and called the discriminant of O . The discriminant of the maximal order O is called the discriminant of the field K. If A is a module in K, then the ring O(A) = α ∈ K α A ⊂ A is called the ring of multipliers of the module. Similar modules have equal rings of multipliers. The ring of multipliers of a full module is an order in K. For each full module, there is a similar module contained in its ring of multipliers. For a full module A with O(A) = O , let α1 , α2 and ω1 , ω2 be bases of A and O , respectively. Then the absolute value of the determinant of the transition matrix from the first basis to the second, N(A) = | det(ai j )|,

where αi = ∑ ai j ω j ,

is independent of the choice of bases, and is called the norm of A. If A ⊂ O , then N(A) = [O : A] (the index of A in O ). Let A and A be two full modules in K. Then the set AA is again a full module in K, which is called the product of A and A . The norm of the product is equal to the product of norms: N(AA ) = N(A)N(A ).

150

5 Euler Factorization of Radial Series

For a full module A in K we denote by A the module consisting of the conjugates α¯ over Q of the elements α ∈ A. The conjugate module A is again a full module with the same ring of multipliers as that of A, and the following formula holds: AA = N(M)O(A). All full modules in K with a fixed ring of multipliers O form a commutative group under multiplication of modules. The quotient group of this group by the subgroup of modules similar to O is a finite group H(O ) called the class group of O . The order h(O ) = #H(O ) of the class group is the class number of O . Full modules A in K contained in its ring of multipliers O(A) = O are called regular ideals of the ring O . Each regular ideal of an order is uniquely (up to order) decomposable into a product of regular prime ideals. √ Every quadratic extension of Q has the form K = Q( d0 ), where d0 = 0, 1 is a square-free integer. As a basis of the maximal order O of such K one can take the numbers 1 and ω , where √ (1 + d0 )/2 if d0 ≡ 1 (mod 4), ω= √ d0 if d0 ≡ 2 or 3 (mod 4), and the discriminant of the field is equal to if d0 ≡ 1 (mod 4), d0 d = δ (K) = δ (O) = 4d0 if d0 ≡ 2 or 3 (mod 4). √ In both cases one can write K = Q( d). Any order O of K has the form

Ol = Z + l ω Z, where l is the index [O : O ]. The discriminant of Ol is equal to dl 2 . For α , β , . . . ∈ K we shall denote by {α , β , . . . } the full module in K generated over Z by α , β , . . . : {α , β , . . . } = Zα + Zβ + · · · . The following lemma is often useful in computations with quadratic modules. / Q, and let aγ 2 + bγ + c = 0, Lemma 5.19. Let K be a quadratic field, γ ∈ K, γ ∈ where a, b, and c are the rational integers satisfying gcd(a, b, c) = 1 and a > 0. Then the module A = {1, γ } satisfies the conditions

O(A) = {1, aγ },

N(A) = 1/a,

δ (O(A)) = b2 − 4ac.

Proof. A number α = r + r γ ∈ K with r, r ∈ Q satisfies α A ⊂ A if and only if α = r + r γ ∈ A and αγ = −r c/a + (r − r b/a)γ ∈ A, which is equivalent to the inclusions r, r , r c/a, r b/a ∈ Z. Since the numbers a, b, c are coprime, these inclusions are equivalent to the inclusions r ∈ Z, r ∈ aZ, whence O(A) = {1, aγ }. For the basis 1, γ of A and the basis 1, aγ of O(A), we have N(A) = | det(diag(1, 1/a))| = 1/a. The formula for the discriminant follows from the definition.

5.2 Binary Radial Series and Gaussian Composition

151

The relation between quadratic modules and prime numbers is described by the following proposition. √ Proposition 5.20. Let K = Q( d) be the quadratic field with discriminant d, and Ol the order of K with discriminant dl 2 . Suppose that p is a rational prime number not dividing l. Then a full module A in K satisfying the conditions

O(A) = Ol ,

A ⊂ Ol ,

N(A) = p,

(5.34)

exists if and only if the congruence x2 ≡ d (mod 4p) is solvable. If the congruence is solvable and p does not divide d, then there are precisely two different modules P and P satisfying (5.34), and P = P. If the congruence has a solution and p divides d, then there is exactly one module P satisfying (5.34), and P = P. Finally, if the congruence has no solutions, then there is the unique complete module P satisfying

O(P) = Ol ,

P ⊂ Ol ,

N(P) = p2 ,

(5.35)

and this module is P = pOl . All of the listed modules A = P are regular prime ideals of the ring Ol . Proof. Suppose that the module A satisfies conditions (5.34). Then the index of A in Ol is p, and so the least positive integer contained in A equals p. Hence, the / Q. module A has a basis of the form p, pγ , A = {p, pγ }, where γ ∈ K and γ ∈ The number γ satisfies the equation aγ 2 + bγ + c = 0, where a, b, and c are rational integers with gcd(a, b, c) = 1 and a > 0. Then, by Lemma 5.19, we have p = N(A) = N(pOl )N({1, γ }) = p2 /a, whence a = p. On the other hand, b2 − 4ac = δ (O(A)) = dl 2 , and so b2 ≡ dl 2 (mod 4p). Since d ≡ 0 or 1 (mod 4) and p l, we conclude that the congruence x2 ≡ d (mod 4p) has a solution. Conversely, if this congruence has a solution, then there are rational integers b, c such that b2 − 4ac = d. It follows from the definition of d that gcd(p, b, c) = 1, and so gcd(p, bl, cl 2 ) = 1. Let γ be a root of the equation px2 + blx + cl 2 = 0. Then the module A = {p, pγ } satisfies (5.34). Let Ai = {p, pγi }, where i = 1, 2, be two modules satisfying (5.34). As has been shown, each of the numbers γi satisfies a relation pγi2 + bi γi + ci = 0, where bi , ci are rational integers, gcd(p, bi , ci ) = 1, and b2i − 4pci = dl 2 . In particular, b2i ≡ dl 2 (mod 4p), whence b1 ≡ b2 (mod 2p) or b1 ≡ −b2 (mod 2p). In the first case we obtain b2 = b1 + 2t p and √ √ √ A2 = {p, (−b2 ± l d)/2} = {p, (−b1 ± l d)/2 − t p} = {p, (−b1 ± l d)/2}. The last module is equal to A1 or A1 . Similarly, in the second case, b2 = −b1 + 2t p and √ √ A2 = {p, (b1 ± l d)/2 − t p} = {p, (−b1 ∓ l d)/2} = A1 or A1 . If p|d, then b1 ≡ −b1 (mod 2p); hence A1 = A1 . Let us assume now that the congruence x2 ≡ d (mod 4p) has no solutions. If A is a complete module satisfying (5.35), then the least positive integer contained in A

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5 Euler Factorization of Radial Series

is equal to pδ with δ = 1 or δ = 2, and A = {pδ , pδ γ }. If aγ 2 + bγ + c = 0, where a, b, c ∈ Z, a > 0, and gcd(a, b, c) = 1, then by Lemma 5.19, we conclude that p2 = N(A)N(pδ )N({1, γ }) = p2δ /a, O(A) = {1, aγ }, and δ (O(A)) = b2 − 4ac. If δ = 2, then a = p2 and dl 2 = b2 − 4ac ≡ b2 (mod 4p2 ), which, since p l, contradicts the assumption. Thus δ = 1, a = 1, O(A) = Ol = {1, γ }, and A = p{1, γ } = pOl . On the other hand, the module pOl clearly satisfies (5.35). Every ideal A ⊂ Ol with prime norm is clearly a prime ideal. Let N(A) = p2 . / A, then the module A = αOl + A is complete and If α ∈ O(A) = Ol and α ∈ N(A) = p or 1, since A = A. By the above, the equality N(A ) = p in this case is impossible; hence N(A ) = 1, A = Ol , and α is invertible modulo A. Thus, the quotient Ol /A is a field, and the ideal A is prime. In view the above proposition it will be convenient to define the sign ε p (δ ) of δ modulo a prime p by ⎧ ⎪ if the congruence x2 ≡ d (mod 4p) is solvable and p d, ⎨1 ε p (d) = 0 if the congruence x2 ≡ d (mod 4p) is solvable and p | d, ⎪ ⎩ −1 if the congruence x2 ≡ d (mod 4p) has no solutions. (5.36) We conclude the survey part of this subsection with a description of the correspondence between modules in quadratic fields of negative discriminant and integral positive definite binary quadratic forms. For the quadratic forms, their matrices, and related notion, we shall use the notation √ (3.25)–(3.27) and the definitions of the beginning of this section. Let K = Q( d) be an imaginary quadratic field with discriminant d < 0, and let A be a full module in K with the ring of multipliers of the form O(A) = Ol . With every Z basis α , β of A ordered by the condition ℑ(β /α ) > 0 we associate the binary quadratic form a = a(A) =

1 (α x + β y)(α x + β y) = ax2 + bxy + cy2 . N(A)

The form a is clearly positive definite. It follows easily from Lemma 5.1 that the form is integral and primitive with discriminant δ (a) = b2 − 4ac = dl 2 equal to the discriminant of the ring of multipliers O(A). Conversely, if a(X) = ax2 +bxy+cy2 is a positive definite integral primitive quadratic form of discriminant b2 − 4ac = dl 2 , then the module √ & −b + l d A = A(a) = a, 2 is a full module in K with ring of multipliers Ol contained in the ring. It is not difficult to check that the indicated correspondences define a bijection between the set of all classes of equivalent full modules in K with the ring of multipliers Ol and the set of all classes of properly equivalent positive definite integral primitive binary quadratic forms of discriminant dl 2 . Let a and a be two integral primitive positive definite binary quadratic forms of the same negative discriminant δ , and let A and A be the matrices of the forms. Since √ the modules A = A(a) and A = A(a ) are full modules in the field K = Q( δ )

5.2 Binary Radial Series and Gaussian Composition

153

with the same ring of multipliers of discriminant δ = dl 2 , the same is true for their product AA . We define the Gaussian composition a × a of the forms a and a as the form corresponding to the product AA , a × a = a(AA ), and denote by A × A the matrix of a × a . For any full module A in K with the same ring of multipliers, we denote by A × A the matrix of the quadratic form a × a(A), A × A = A × A(A),

(5.37)

where A(A) is the matrix of the form a(A). The proper equivalence classes of the products depend only on proper classes of the quadratic forms or their matrices and the equivalence class of A. Thus, the above formulas define the products of corresponding classes. Since the set of equivalence classes of full modules with the ring of multipliers of discriminant δ forms an abelian group under multiplication of the classes, the set of classes with respect to proper equivalence of integral primitive positive definite binary quadratic forms of discriminant δ and the set of proper classes of their matrices can also be endowed with a group structure. We shall denote by H(δ ) all three of these naturally isomorphic class groups and by h(δ ) = #{H(δ )} the class number. We can now return to the action of Π-operators. First we shall examine the summation conditions in the formula (4.27) defining Π-operators. By Lemma 3.3, one can take Λ+ \Λ+ diag(1, d)Λ+ = diag(1, d)U U ∈ Λd \Λ+ , where Λd = Λ+

αβ ∈ Λ+ β ≡ 0 diag(1, d) Λ+ diag(1, d) = γ δ −1

(mod d) .

It is easy to check that two matrices of Λ+ with first rows (u1 , u2 ) and (u1 , u2 ) belong to the same left coset modulo Λd if and only if (u1 , u2 ) ∼ = (u1 , u2 ) (mod d) ⇐⇒ u1 u2 − u2 u1 ≡ 0

(mod d).

(5.38)

Thus, we have proved the following lemma. Lemma 5.21. One can take Λ+ \Λ+ diag(1, d)Λ+

u1 u2 1 , (u1 , u2 ) ∈ P (Z/dZ) , = diag(1, d)U U = v1 v2

(5.39)

where P1 (Z/dZ) is the “projective line modulo d”, i.e., a set of representatives for classes of all coprime pairs of integers (u1 , u2 ) modulo the equivalence (5.38).

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5 Euler Factorization of Radial Series

Exercise 5.22. Prove that the cardinal numbers κ (d) = #(P1 (Z/dZ)) satisfy the following rules: κ (dd ) = κ (d)κ (d ) if gcd(d, d ) = 1, and κ (pδ ) = pδ −1 (p + 1) if p is a prime number. Lemma 5.21 will allow us to rewrite formula (4.27) for the action of Hecke operators |k,χ Π(d) on Fourier coefficients of functions of Fε in terms of points of the projective line modulo d. Let a = a(u) with u = (u1 , u2 ) be the quadratic form (5.25) with the matrix A, and b(u, v) = uA tv = b(v, u) the corresponding bilinear form. Then for D of the form u 10 u1 u2 = , D= dv v1 v2 0d we have

A[ tD] = DA tD =

2a(u) db(u,v) uA tu duA tv = . db(v,u) 2d 2 a(v) dvA tu d 2 vA tv

By Lemma 5.21, in this notation we can rewrite formula (4.27) in the form ( f |k,χ Π(d))(A) = χ (d)d k−2 ( f |P(d))(A), where ( f |P(d))(A) =

∑

u∈P1 (Z/dZ), a(u)≡0 (mod d)

2a(u)/d b(u,v) f , b(v,u) 2da(v)

(5.40)

(5.41)

and where for each u ∈ P1 (Z/dZ), by v is denoted a 2-row such that ( uv ) ∈ Λ+ . Since the definition of operators |P(d) is independent of the choice of k and χ , according to (5.40) we can express the operators |P(d) on Fourier coefficients of functions of Fε through the operators |k,χ Π(d), for arbitrary k and χ satisfying (−1)k χ (−1) = ε (−1) and χ (d) = 0, by the formula |P(d) = χ (d)d 2−k |k,χ Π(d).

(5.42)

The next key lemma interprets values of the images ( f |P(p))(A) for prime p and primitive A in terms of multiplication of quadratic modules. b Lemma 5.23. Let A = 2a b 2c be the matrix of an integral primitive quadratic form a of negative discriminant δ = b2√ − 4ac = dl 2 , where d is the discriminant of the imaginary quadratic field K = Q( δ ). Then for each function f on E2 depending only on classes of even matrices relative to proper equivalence and each prime number p not dividing the number l, we have ⎧ ⎪ ⎨ f (A × P) + f (A × P) if ε p (d) = 1, ( f |P(p))(A) = f (A × P) if ε p (d) = 0, ⎪ ⎩ 0 if ε p (d) = −1,

5.2 Binary Radial Series and Gaussian Composition

155

where ε p (d) is the sign (5.36), P a regular prime ideal of norm p of the order Ol ⊂ K with discriminant dl 2 , P the conjugate ideal, and where × is the multiplication (5.37). Proof. Let us consider the congruence a(x1 , x2 ) = ax12 + bx1 x2 + cx22 ≡ 0

(mod p).

(5.43)

It is easy to check in the framework of the elementary theory of quadratic congruences that for each prime p not dividing l, the number of different points of the projective line P1 (Z/pZ) modulo p, i.e., the number of inequivalent coprime pairs of integers with respect to the equivalence (5.38) modulo p, satisfying this congruence is equal to 1 + ε p (d). Let ε p (d) = 1, and let u = (u1 , u2 ) and u = (u1 , u2 ) be two inequivalent modulo p primitive solutions of (5.43). Since the discriminant δ = b2 − 4ac is not divisible by p, we may assume that a(u) = pa1 , a(u ) = pa2 , where a1 and a2 are not divisible by p. We can find integers v1 , v2 , v1 , v2 such that u1 u2 u u , U2 = 1 2 ∈ Λ+ . U1 = v1 v2 v1 v2 We set

2pa1 b1 A1 = U1 A U1 = , b1 c1 2pa2 b2 , A2 = U2 A tU2 = b2 c2

t

2a1 b1 , = b1 2pc1 2a2 b2 A2 = . b2 2pc2 A1

Then by (5.41), we get (P(p) f )(A) = f (A1 ) + f (A2 ), and it is sufficient to prove that with the appropriate choice of regular prime ideal P of norm p in Ol , one has A1 ∼ A × P and

A2 ∼ A × P,

(5.44)

where the symbol ∼ stands for proper equivalence. Let us consider the modules √ & √ & −b2 + δ −b1 + δ , A = p, A = p, 2 2 of the field K. It follows from Lemma 5.19 that the ring of multipliers of each of these modules is Ol , that they are both contained in Ol , and that they have the norm N(A) = N(A ) = p. Let us show that A + A = Ol . To prove this it is obviously sufficient to check that the number (b2 − b1 )/2 is not divisible by p. We set t 1 t2 −1 ∈ Λ+ . U2U1 = T = t3 t4

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5 Euler Factorization of Radial Series

Since the solutions (u1 , u2 ) and (u1 , u2 ) are not equivalent modulo p, we have t2 = u1 u2 − u2 u1 ≡ 0 (mod p). It follows from the relation TA1 tT = A2 that pa1t12 + b1t1t2 + c1t22 = pa2 and 2pa1t1t3 + b1 (t1t4 + t2t3 ) + 2c1t2t4 = b2 . From the first equality we obtain the congruence b1t1 + c1t2 ≡ 0 (mod p), and it follows from the second that b1 (t1t4 + t2t3 ) + 2c1t2t4 − b1 b2 − b1 ≡ = t2 (b1t3 + c1t4 ) (mod p). 2 2 If (b2 − b1 )/2 were divisible by p, we would get the system of congruences b1t1 + c1t2 ≡ b1t3 + c1t4 ≡ 0 (mod p), which would imply that b1 ≡ c1 ≡ 0 (mod p), a contradiction, because δ = b21 −4pa1 c1 is not divisible by p. It follows from what we have proved and Proposition 5.20 that one can set A = P and A = P, where P, P are the unique pair of regular conjugate distinct prime ideals in Ol with norm p. In order to prove (5.43), let us denote by a1 , a2 , a1 , and a2 the quadratic forms with matrices A1 , A2 , A1 , and A2 , respectively, and √ by A1 , A2 , A1 , and A2 the correspond2 ing modules in K. If we set γ1 = (−b1 + δ )/2, then clearly γ1 = −pa1 c1 − b1 γ , and since a1 and p are coprime, we obtain A1 P = {a1 , γ1 }{p, γ1 } = {pa1 , pγ1 , a1 γ1 , −pa1 c1 − b1 γ1 } = {pa1 , γ1 } = A1 . Hence A1 P = A1 PP = A1 N(P) ∼ A1 , and A1 ∼ A1 × P ∼ A × P. Similarly, A2 ∼ A × P. This proves (5.44) and the first formula of the lemma. Now let ε p (d) = 0 and let u = (u1 , u2 ) be the unique solution of (5.43) modulo equivalence (5.38). We can find integers v1 , v2 such that U1 = ( uv11 uv22 ) ∈ Λ+ and set 2pa1 b1 2a1 b1 , A1 = . A1 = U1 A tU1 = b1 c1 b1 2pc1 Then the right-hand side of (5.41) is equal to f (A1 ), and it is sufficient to prove that A1 ∼ A × P,

(5.45)

where P is the unique regular prime ideal of norm p in Ol . By our assumption, the discriminant δ = dl 2 = b2 − 4ac = b21 − 4pa1 c1 is divisible by p and p l. It follows that b1 ≡ 0 (mod p) and dl 2 ≡ −4pa1 c1 (mod p2 ). If p = 2, since d is not divisible by p2 , the last congruence implies that a1 c1 is not divisible by p. If p = 2, then d = 4d0 with d0 ≡ 2, 3 (mod 4). Let us set b1 = 2b0 . Then δ = 4d0 l 2 = 4(b20 − 2a1 c1 ), whence 2a1 c1 = b20 − d0 l 2 ≡ b20 − d0

(mod 4).

The number b20 − d0 is not divisible by 4, since b20 ≡ 0, 1 (mod 4) and d0 ≡ 3, 3 (mod 4). Hence, a1 c1 is not divisible by p also for p = 2. Further, since the quadratic form a1 with matrix A1 together with a is primitive, and p a1 , it follows that the form a1 with matrix A1 is also primitive. Let us consider the module A = {p, γ },

5.2 Binary Radial Series and Gaussian Composition

157

√ where γ = (−b1 + δ )/2 satisfies γ 2 + b1 γ + pa1 c1 = 0. By Lemma 5.19 and Proposition 5.20, the module A = P is the unique regular prime ideal of Ol of norm p. Let us denote by a1 and a1 the quadratic forms with matrices A1 and A1 , respectively, and by A1 and A1 the corresponding modules in K. Since γ 2 = −b1 γ − pa1 c1 and p a1 , we have A1 P = {a1 , γ }{p, γ } = {pa1 , pγ , a1 γ , −pa1 c1 − b1 γ1 } = {pa1 , γ } = A1 . Hence, since P = P, we have A1 P = A1 PP ∼ A1 , and the corresponding matrices satisfy (5.45). Finally, if ε p (d) = −1, then, as was noted at the beginning of the proof, the congruence (5.43) has no solutions on the projective line modulo p, which proves the last formula of the lemma. Exercise 5.24. In the notation of Lemma 5.23, assuming that ε p (d) = 0 and p|l, prove the formula ( f |P(p))(A) = f (A × Ol/p ). Along with operators corresponding to elements Π(p) we shall also need operators associated with elements Ψ(p) ∈ L 2p defined by formula (3.145). The action of these operators can be naturally reduced to computation of simple trigonometric sums, which will be done in the next lemma. Lemma 5.25. In the notation and under the assumptions of the previous lemma and Lemma 3.68, the following formulas hold ⎧ ⎪ if ε p (d) = 1, ⎨p − 1 π iσ (AB)/p (5.46) e = S p (A) = ε p (d) = 0, −1 if ∑ ⎪ ⎩ B= tB∈Z22 /pZ22 , −(p + 1) if ε p (d) = −1, r p (B)=1

Proof. The set of matrices of the form

2 α0 γβ γβ , α , γ = 1, . . . , p − 1, β = 0, 1, . . . , p − 1 0 0 γβ γ can clearly be taken as a summation set in the sum S p (A). Thus, since A = we obtain 2 S p (A) = ∑ e2π iaα /p + ∑ e2π iγ (aβ +bβ +c)/p . α

2a

b b 2c

,

γ ,β

The sum on α is obviously equal to p − 1 if p|a and −1 if p a; similarly, the sum on γ is p − 1 if p|aβ 2 + bβ + c and −1 if p aβ 2 + bβ + c. Hence if, for example, ε p (d) = −1, then the second of the alternatives is realized in all cases, and we have S p (A) = −1 + p(−1) = −(p + 1). The other cases are similar, and we leave their consideration to the reader as a useful exercise on quadratic congruences. The above lemmas together with Lemma 3.71 will allow us to discover the action of the operators |P(d) with composite d on Fourier coefficients of functions of Fε in terms of characters of class groups.

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5 Euler Factorization of Radial Series

2 Theorem 5.26. Let δ be a negative integer written in the form √ δ = dl , where d is the discriminant of the imaginary quadratic field K = Q( δ ) and l is a positive integer. Let Ol be the order in K of the discriminant δ , A1 , . . . , Ah with h = h(δ ) a full system of representatives of classes of properly equivalent even positive definite primitive matrices of order two and the discriminant δ , and let ψ be a character of the class group H(δ ). Then for every function f on E2 whose values are Fourier coefficients of a function of Fε and positive integer d prime to l, the following formulas hold for the action of operators (5.41) on averaging

ψ (A1 ) f (A1 ) + · · · + ψ (Ah ) f (Ah ) of f over the group H(δ ) with character ψ : h

∑ ψ (Ai ) f (Ai )

|P(d)

i=1

= λ (P(d); ψ , δ )

h

= ∑ ψ (Ai )( f |P(d))(Ai ) i=1

h

∑ ψ (Ai ) f (Ai )

,

(5.47)

i=1

where the numbers λ (P(d); ψ , δ ) satisfy the formal identity λ (P(d); ψ , δ ) ψ (P) −1 1 1 − 2s ∏ 1 − = ∏ , ∑ ds p N(P)s P| p p∈P, p l d∈N, gcd(d,l)=1

(5.48)

where P in the inner product on the right runs through all regular prime ideals of the ring Ol dividing the principal ideal pOl . Proof. Since the function f depends only on proper classes of even matrices, by Lemma 5.23 in the notation of the lemma, for a prime p not dividing l, we obtain h

∑ ψ (Ai ) f (Ai )

i=1

h

|P(p) = ∑ ψ (Ai )( f |P(p))(Ai ) i=1

⎧ h ⎪ ⎨∑i=1 ψ (Ai ) f (Ai × P) + f (Ai × P) if ε p (d) = 1, = ∑hi=1 ψ (Ai ) f (Ai × P) if ε p (d) = 0, ⎪ ⎩ 0 if ε p (d) = −1,

which clearly implies the formula (5.47) with d = p and ⎧ ⎪ ⎨ψ (P) + ψ (P) if ε p (d) = 1, λ (P(p); ψ , δ ) = ψ (P) = ψ (P) if ε p (d) = 0, ⎪ ⎩ 0 if ε p (d) = −1.

(5.49)

In order to reduce the action of operators |P(d) for composite d to the cases of prime divisors of d, we shall use the relation (5.42) between operators |P(d) and

5.2 Binary Radial Series and Gaussian Composition

159

|k Π = |k,1 Π(d) with χ = 1 (the unit character modulo 1) and relations for elements Π(d) obtained in Section 3.5. First of all, by Lemma 3.62, we have |P(d)|P(d1 ) = d k−2 d1k−2 |Π(d)|Π(d1 ) = (dd1 )k−2 |Π(dd1 ) = |P(dd1 ) if gcd(d, d1 ) = 1.

(5.50)

Then, by Lemma 3.71, for each prime p we get the following summation formula for the formal power series with operators |P(pν ) as coefficients:

∑ |P(pν )vν = ∑ |k Π(pν )(p2−k v)ν

ν ≥0

ν ≥0

= (1 − |k p2 p 2 (p2−k v)2 ) −1 × 1 − |k Π(p)p2−k v + (|k p p 2 + |k pΨ(p))(p2−k v)2 −1 = (1 − |k p 2 p6−2k v2 ) 1 − |P(p)v + (|k p 2 + |k Ψ(p))p5−2k v2 . (5.51) Let us consider now the action of operators |k p 2 and |k Ψ(pi ) on the average. By formula (4.26) for n = 2 and χ = 1, we have h

∑ ψ (Ai ) f (Ai )

|k p 2 = p2k−6

i=1

h

∑ ψ (Ai ) f (Ai )

.

(5.52)

i=1

By formula (4.22) of Lemma 4.13 for χ = 1 applied to the double cosets entering in the decomposition (3.45) of the element Ψ(p), for each A ∈ E2 , one can write ( f |k Ψ(p))(A) = p2k−6 S p (A) f (A), where S p (A) is the trigonometric sum (5.46). Hence, by Lemma 5.25, we derive the formula h

∑ ψ (Ai ) f (Ai )

|k Ψ(p) = p2k−6 λ (Ψ(p); ψ , δ )

i=1

where

⎧ ⎪ ⎨(p − 1) λ (Ψ(p); ψ , δ ) = −1 ⎪ ⎩ −(p + 1)

h

∑ ψ (Ai ) f (Ai )

,

(5.53)

i=1

if ε p (d) = 1, if ε p (d) = 0, if ε p (d) = −1.

It follows from the relations (5.50) and (5.51) that each of the operators |P(d) with d = ∏i pνi i , where pi are different prime divisors of d, is a polynomial in operators |P(pi ), |k pi 2 , and |k Ψ(pi ) that commute with each other. Then the relations (5.47) with some coefficients λ (P(d); ψ , δ ) follow for every d prime to l from the formulas for primes proved above. In order to compute the coefficients, we note first that by (5.50), we have

λ (P(dd1 ); ψ , δ ) = λ (P(d); ψ , δ )λ (P(d1 ); ψ , δ )

if gcd(d, d1 ) = 1,

(5.54)

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5 Euler Factorization of Radial Series

which implies the formal Euler product factorization λ (P(d); ψ , δ ) 1 ν = ∏ ∑ ∑ λ (P(p ); ψ , δ ) pν s . ds p∈P, pl ν ≥0 d∈N, gcd(d,l)=1

(5.55)

It follows from the summation formula (5.51) and formulas (5.47) with d = p, (5.52), and (5.53) that each of the p-factors of the last Euler product can be summed in the form 1 1 ν λ (P(p ); ψ , δ ) = 1 − ∑ pν s p2s ν ≥0 −1 λ (P(p); ψ , δ ) (1 + λ (Ψ(p); ψ , δ ))p−1 × 1− + . ps p2s By substituting here the values for λ (P(p); ψ , δ ) and λ (|k Ψ(p); ψ , δ ) computed above, we obtain that the fraction on the right is equal to −1 (ψ (P) + ψ (P)) (1 + p − 1)p−1 1 1− + 1 − 2s p ps p2s −1 ψ (P) ψ (P) 1 1− 1 − = 1 − 2s p N(P)s N(P)s if ε p (d) = 1, where P and P is a pair of conjugate regular prime ideals of Ol of norm p, 1−

1 p2s

1−

ψ (P) (1 − 1)p−1 + ps p2s

−1

ψ (P) −1 1 1− = 1 − 2s p N(P)s

if ε p (d) = 0, where P is a regular prime ideal of Ol of norm p, and 1−

1 p2s

−1 (1 − (p + 1))p−1 1 ψ (P) −1 1+ 1 − = 1 − p2s p2s N(P)s

if ε p (d) = −1, where P = pOl is a regular prime ideal of Ol of norm p2 (with ψ (P) = 1). Hence, according to Proposition 5.20, we conclude that in all three cases, the p-factor of the factorization (5.55) can be written in the form 1 1 ψ (P) −1 ∑ λ (P(p ); ψ , δ ) pν s = 1 − p2s ∏ 1 − N(P)s , ν ≥0 P|p ν

(5.56)

where P ranges over all regular prime ideals of the ring Ol dividing the principal ideal pOl . The formal Euler factorization (5.48) follows from (5.55) and (5.56). Exercise 5.27. Let ρ (d, A) be the number (5.31) of solutions of the congruence a(u) ≡ 0 (mod d), where a is the quadratic form with matrix A, on the projective

5.2 Binary Radial Series and Gaussian Composition

161

line modulo d. In the notation and under the assumptions of Theorem 5.26, prove the formal identity −1 ρ (d, A) 1 1 1 − 2s ∏ 1 − = ∏ . ∑ ds p N(P)s P| p p∈P, p l d∈N, gcd(d,l)=1 Multiplicative Properties of Fourier Coefficients. We return to a nonzero modular form F of weight k > 0 and character χ modulo q for the group Γ20 (q) with Fourier expansion (5.24) and assume that F is an eigenfunction for Hecke operators |T (m) = |k,χ T (m) corresponding to all elements T (m) ∈ L0 (q) = L20 (q) of the form (3.82), where K = Γ20 (q), with eigenvalues λ (m) = λF (m): F|T (m) = λ (m)F

for all m prime to q.

(5.57)

Then by Corollary 5.17, for every primitive matrix A ∈ E2 and positive integer ν coprime to q, the Fourier coefficients of F and the eigenvalues are linked by the relation (5.30), which we shall rewrite in the form

λ (m) f (ν A) =

∑

d,d1 ∈N, dd1 =m

χ (d1 )d1k−2 (( f |Π+ (ν d))|P(d1 ))(A),

(5.58)

where |Π+ = |k,χ Π+ are the operators (4.25) and the operators P are defined by 2 (5.41). Let, as in Theorem 5.26, δ be a negative integer written in the form √ δ = dl , where d is the discriminant of the imaginary quadratic field K = Q( δ ) and l is a positive integer. Let Ol be the order in K of discriminant δ , A1 , . . . , Ah with h = h(δ ) a full system of representatives of classes of properly equivalent even positive definite primitive matrices of order two with discriminant δ , and let ψ be a character of the class group H(δ ). Let us multiply both sides of the relation (5.58) with A = Ai by ψ (Ai ) and sum these relations over i = 1, . . . , h. We get the relation h

λ (m) ∑ ψ (Ai ) f (ν Ai ) = i=1

∑

d, d1 ∈N, dd1 =m

h

χ (d1 )d1k−2 ∑ ψ (Ai )(( f |Π+ (d ν ))|P(d1 ))(Ai ). i=1

If m is also coprime to l, then, since values of the functions f |Π+ (ν d) together with values of f can be considered as Fourier coefficients of functions of Fε , we can apply relations (5.47) of Theorem 5.26 to the inner sums on the right of the last formula, which leads us the relations h

λ (m) ∑ ψ (Ai ) f (ν Ai ) i=1

=

χ (d1 )d1k−2 λ (P(d1 ); ψ , δ ) ∑ ψ (Ai )( f |Π+ (d ν )))(Ai )

∑

χ (d1 )d1k−2 λ (P(d1 ); ψ , δ ) ∑ ψ (Ai ) f (d ν Ai ).

d, d1 ∈N, dd1 =m

=

h

∑

d, d1 ∈N, dd1 =m

i=1 h

i=1

162

5 Euler Factorization of Radial Series

Hence, dividing both parts by ms = d s d1s and summing over all m ∈ N with gcd(m, lq) = 1, we come to the following generalization of the relations (5.32), obtained under the assumption that h(δ ) = 1, to the case of arbitrary class number: h

∑ ψ (Ai ) f (ν Ai )

i=1

∑

=

d1 ∈N, gcd(d1

λ (m) ms m∈N, gcd(m, lq)=1

∑

χ (d1 )d1k−2 λ (P(d1 ); ψ , δ ) h f (d ν Ai ) , ∑ ψ (Ai ) ∑ s d ds 1 i=1 , lq)=1 d∈N, gcd(d, lq)=1 (5.59)

where ν is an arbitrary positive integer prime to all integers m ∈ N satisfying gcd(m, lq) = 1, in other words, ν satisfies ν |(lq)∞ , i.e., divides a power of lq. Ultimately, we come to the following theorem on multiplicative properties of Fourier coefficients of eigenfunctions. Theorem 5.28. Let F be a nonzero modular form of weight k ∈ N and character χ modulo q for the group Γ20 (q). Suppose that F is an eigenfunction satisfying (5.57). Let√δ be a negative integer, δ = dl 2 , where d is the discriminant of the field K = Q( δ ) and l ∈ N, A1 , . . . , Ah with h = h(δ ) a full system of representatives of classes of properly equivalent even positive definite primitive matrices of order two with discriminant δ , and let ψ be a character of the group H(δ ) of the classes of matrices with respect to the Gaussian composition (5.37). Then for each positive integer ν dividing a power of lq, the following formal identity holds: h

∑ ψ (Ai ) f (ν Ai ) ∏

i=1

=

∏

P lq

p∈P, p lq

−s Q−1 p, F (p )

ψ (P)χ (N(P)) 1− N(P)s−k+2

−1

h

∑ ψ (Ai )

i=1

∑

d∈N, gcd(d, lq)=1

f (d ν Ai ) , ds

(5.60)

where Q p, F (v) =1 − λ (p)v + (λ (p)2 − λ (p2 ) − χ (p2 )p2k−4 )v2 − χ (p2 )p2k−3 λ (p)v3 + χ (p4 )p4k−6 v4 ,

(5.61)

and P on the right ranges over all regular prime ideals of the ring Ol not dividing the principal ideal lqOl . Proof. By Theorem 3.46, we have

λ (mm ) = λ (m)λ (m )

if

gcd(m, m ) = gcd(mm , q) = 1.

By Proposition 3.53, for each prime number p not dividing q, we get the formal identity

5.2 Binary Radial Series and Gaussian Composition ∞

163

∞

∑ λ (pβ )vβ = ∑ λ (T (pβ ))vβ

β =0

β =0

= (1 − p λ ( p )v2 )(1 − λ (q1 (p))v + λ (q2 (p))v2 − λ (q3 (p))v3 + λ (q4 (p))v4 )−1 , 2

where λ (q1 (p)) = λ (T (p)) = λ (p),

λ (q2 (p)) = λ (T (p)2 ) − λ (T (p2 )) − p2 λ ( p ) = λ (p)2 − λ (p2 ) − p2 λ ( p ), λ (q3 (p)) = p3 λ ( p )λ (p), and λ (q4 (p)) = p6 λ ( p )2 . Since by (4.26), we can write λ ( p ) = λ ( p ) = χ (p2 )p2k−6 , it follows that ∞

∑ λ (pβ )vβ = (1 − χ (p2 )p2k−4 v2 )Q−1 p, F (v),

β =0

where Q p, F (v) is the polynomial (5.61). Thus, we obtain the formal Euler product factorization ∞ λ (m) λ (pβ ) = ∏ ∑ s βs m p∈P, p lq β =0 p m∈N, gcd(m, lq)=1

∑

=

∏

−s (1 − χ (p2 )p−2(s−k+2) )Q−1 p, F (p ).

(5.62)

p∈P, p lq

On the other hand, by (5.48) we can write

χ (d)d k−2 λ (P(d); ψ , δ ) χ (d) = λ (P(d); ψ , δ ) s−k+2 ∑ s d d d∈N, gcd(d,l)=1 d∈N, gcd(d,l)=1

∑

=

∏

p∈P, p l

1−

χ (p2 ) p2(s−k+2)

∏

P| p

χ (N(P))ψ (P) 1− N(P)s−k+2

−1 .

(5.63)

Let us return now to the identity (5.59). On replacing the Dirichlet series on both sides of this identity by their product factorizations obtained above and dividing both sides by ∏ p∈P, p lq (1 − χ (p2 )p−2(s−k+2) ), we obtain to the relation (5.58). Let us reformulate the theorem in the terms of the multiplicativity of functions of integral argument. For an integer r, we say that a nonzero function

ϕ : N r = {m ∈ N| gcd(m, r) = 1} → C is r −multiplicative if it satisfies the following two conditions: (1) ϕ (m m ) = ϕ (m)ϕ (m ) if m and m are coprime (and prime to r); (2) for each prime number p not dividing r, the formal power series ∑ν ≥0 ϕ (pν )vν is formally equal to a rational fraction Np (v)D p (v)−1 , where Np (v) and D p (v) are polynomials with Np (0) = D p (0) = 1.

164

5 Euler Factorization of Radial Series

Alternatively, these conditions are equivalent to the condition that the following formal Euler factorization holds

ϕ (m) = ∏ Np (p−s )D p (p−s )−1 , s m p∈P, p r m∈N, gcd(m, r)=1

∑

where Np (v) and D p (v) are polynomials with Np (0) = D p (0) = 1. The function m → λ (m), where λ (m) are eigenvalues of a nonzero modular form F ∈ M2k (q, χ ) satisfying (5.57), gives an example of q -multiplicative functions. Other examples are given by the following corollary. Corollary 5.29. In the notation of Theorem 5.28, each of the functions d → f (d ν A), where A is a positive definite even primitive matrix of order two with discriminant δ = dl 2 and ν divides a power of lq, is a linear combination with constant coefficients of at most h = h(δ ) lq -multiplicative functions. Proof. It follows from Theorem 5.28 that each of the functions d →

∑

ψ (Ai ) f (d ν Ai )

1≤i≤h

is proportional to an lq -multiplicative function. If, say, A ∼ A j , then 1 f (d ν A) = f (d ν A j ) = ∑ ψ (A j ) ∑ ψ (Ai ) f (d ν Ai ) , h ψ 1≤i≤h by the orthogonality relations for the characters. Exercise 5.30. In the notation of Theorem 5.28 prove the following generalization of the relations (5.33): h

∑ ψ (Ai )

i=1

=

∑

ν ∈N, ν | (lq)∞

f (ν Ai )

∏

p∈P, p lq

−s Q−1 p, F (p )

∞ ψ (P)χ (N(P)) −1 h f (n Ai ) 1 − ψ (Ai ) ∑ . ∑ ∏ s−k+2 ds N(P) i=1 n=1 P lq

Exercise 5.31. Prove that for each prime number p not dividing q, the polynomial (5.61) has a factorization of the form Q p, F (v) = 1− α0 (p)v 1− α0 (p)α1 (p)v 1− α0 (p)α2 (p)v 1− α0 (p)α1 (p)α2 (p)v , where α0 (p)2 α1 (p)α2 (p) = χ (p2 )p2k−3 . [Hint: Use Proposition 3.55.]

5.3 Zeta Functions of Eigenforms for Genus 2

165

5.3 Zeta Functions of Eigenforms for Genus 2 According to Theorem 4.7, each invariant subspace of cusp forms for the group Γ20 (q) has a basis of eigenfunctions for all regular Hecke operators. The Zharkovskaya commutation relations (Theorem 4.19) allow one to prove that all regular Hecke operators can be simultaneously diagonalized on certain invariant spaces of modular forms consisting not only of cusp forms, but, for example, on the whole space M2k of all modular forms of weight k for the full modular group Γ2 (Theorem 4.22). This justifies a look at the eigenforms. Regular Zeta Functions of Eigenforms. As in Section 5.2, we fix a nonzero modular form F = F(Z) ∈ M2k (q, χ ) = Mk (Γ2o (q), χ ) of positive integral weight k and character χ modulo q ∈ N for the group Γ20 (q) and suppose that F is an eigenfunction for all Hecke operators |T (m) = |k,χ T (m), where T (m) ∈ L0 (q) = L20 (q) are the elements of the form (3.82), with the eigenvalues λ (m), (gcd(m, q) = 1). F|T (m) = λ (m)F It follows from Theorem 3.46 and Proposition 3.53 that one can write the formal identities ∞ λ (m) λ (pβ ) = ∏ ∑ s βs m p∈P, p q β =0 p m∈N, gcd(m, q)=1 χ (p2 ) −s 1 − 2(s−k+2) = ∏ ∏ Q−1 p, F (p ), p p∈P, p q p∈P, p q

∑

(5.64)

where Q p, F (v) =1 − λ (p)v + (λ (p)2 − λ (p2 ) − χ (p2 )p2k−4 )v2 − χ (p2 )p2k−3 λ (p)v3 + χ (p4 )p4k−6 v4 . We call the infinite product

ζr (s, F) =

∏

p∈P, p q

−s Q−1 p, F (p )

(5.65)

the regular zeta function of the eigenform F; if q = 1, we omit the adjective “regular” as well as the lower subscript r and get the zeta function ζ (s, F) of F. Lemma 5.32. The regular zeta function ζr (s, F) of a nonzero cusp form F ∈ N2k (q, χ ) converges absolutely and uniformly in each right half-plane ℜs > k +1+ ε with ε > 0.

166

5 Euler Factorization of Radial Series

Proof. Let f (A) be a fixed nonzero Fourier coefficient of F. By relation (2.29) of Proposition 5.16 and estimate (1.46) of Proposition 1.25, for each m prime to q, we obtain the inequalities c |λ (m)| ≤ | f (A)| d, d =

∑

1 , d2 ∈N, dd1 d2 =m

c(det A)k/2 | f (A)| d, d

∑

d1k−2 d22k−3 κ (d1 ) det

1 , d2 ∈N, dd1 d2 =m

k/2 d A[diag(1, d1 )] d1 d2

d k d1k−2 d2k−3 κ (d1 ) < c τ3 (m)mk ,

where c is a constant, κ (d1 ) = #(Λ+ \Λ+ diag(1, d1 )Λ+ ) ≤ d12 is the number of points on the projective line modulo d1 , and τ3 (m) is the number of factorizations of m into products of three positive integers. Hence using the elementary estimate τ3 (m) < c(ε )mε with arbitrary ε > 0, it follows that |λ (m)| < c mk+ε

(gcd(m, q) = 1, ε > 0),

(5.66)

where c = c (ε ) is a constant depending only on ε . It follows from the last estimate that the Dirichlet series on the left in (5.54) converges absolutely and uniformly on each of the indicated right half-planes. On the other hand, by (5.54), we can write the product (5.65) in the form −1 ∞ χ (p2 ) λ (pβ ) ζr (s, F) = ∏ 1 − p2(s−k+2) ∏ ∑ βs p∈P, p q p∈P, p q β =0 p ∞ ∞ χ (p2α ) λ (pβ ) = ∏ ∑ 2α (s−k+2) ∏ ∑ βs , p∈P, p q α =0 p p∈P, p q β =0 p where absolute and uniform convergence of both products in the half-planes of the stated form follows from estimates (5.66) by a well-known classical test. For example, |λ (pβ )| |λ (m)| 1 1 < ∑ < c ∑ ℜs−k−ε < c ∑ 1+ε −ε ℜs β s| m m m |p m>1 m>1 m>1 p, β ≥1

∑

if ℜs > k + 1 + ε with ε > ε , and the last series with positive constant terms is convergent. It follows from the lemma that the regular zeta function ζr (s, F) of a cusp form F ∈ N2k (q, χ ) is a holomorphic function of the complex variable s in each of the right half-planes ℜs > k + 1 + ε with ε > 0. The question of analytic continuation of the zeta functions of cusp forms of level q = 1 will be considered in the next subsection.

5.3 Zeta Functions of Eigenforms for Genus 2

167

Exercise 5.33. Prove that the regular zeta function of an arbitrary eigenform F ∈ M2k (q, χ ) converges absolutely and uniformly in each right half-plane ℜs > 2k + 1 + ε with ε > 0. [Hint: Use the estimate (1.52).] Exercise 5.34. Let F ∈ M2k (q, χ ) be an eigenfunction for all regular Hecke operators. Prove that if k = 2 or the character χ is trivial, then the image F|Φ ∈ M1k (q, χ ) of F under the Siegel operator (1.56) is again an eigenfunction for all regular Hecke operators, and if F|Φ = 0, the regular zeta functions of F and F|Φ are linked by the relation ζr (s, F) = ζr (s, F|Φ)ζr (s − k + 2, χ ; F|Φ), where

ζr (s, χ ; F|Φ) =

∏

p∈P, p q

−s Q−1 p, F|Φ ( χ (p)p ).

Analytic Continuation and Functional Equation. Here we shall prove that under certain conditions, the zeta function of a cusp eigenform of level one has a meromorphic continuation over the whole s-plane and satisfies a functional equation with two gamma factors. eigenform for all Hecke operators with Theorem 5.35. Let F ∈ N2k be a cusp Fourier coefficients f (A). Suppose that f 20 02 = 0. Then the zeta function ζ (s, F) of F can be continued to the whole s-plane as a meromorphic function. More precisely, the function Ψ(s, F) = (2π )−2s Γ(s)Γ(s − k + 2)ζ (s, F),

(5.67)

where Γ(s) is the gamma function, has a continuation to the whole s-plane as a meromorphic function with at most two simple poles at the points s = k − 2 and s = k and satisfies the functional equation Ψ(2k − 2 − s, F) = Ψ(s, F).

(5.68)

Proof. Let us use Theorem 5.28 √of the discriminant δ = d = −4 of the √ in the case Gaussian number field K = Q( −1) = Q( −4) with class number h = h(−4) = 1, the representative A1 = 2 · 12 of proper classes of discriminant −4, and the unit characters χ and ψ . Then the identity (5.60) takes the form ∞

f (A1 )ζ (s, F) = ZO (s − k + 2) ∑

d=1

where ZO (s) = ∏ P

1 1− N(P)s

−1

f (dA1 ) , ds

=∑ A

(5.69)

1 , N(A)s

with P √ and A ranging over all prime ideals and nonzero integral ideals of the ring O = Q[ −1] of the Gaussian integers, respectively, is the Dedekind zeta function

168

5 Euler Factorization of Radial Series

of the ring. It follows from Lemma 5.32 and estimate (1.46) that both sides of (5.69) define functions holomorphic in the half-plane ℜs > k + 1. Since f (A1 ) = 0, in this half-plane we can write the functional identity Ψ(s, F) = (2π )−2s Γ(s)Γ(s − k + 2)ζ (s, F) ∞

= f (A1 )−1 (2π )−2s Γ(s)Γ(s − k + 2)ZO (s − k + 2) ∑

d=1

f (dA1 ) . ds

According to Proposition 2.16, the function on the right has a meromorphic continuation to the whole s-plane with at most two simple poles at the points s = k − 2 and s = k and satisfies the functional equation (5.68). Exercise 5.36. Let F ∈ M2k be an eigenfunction for all Hecke operators. Suppose that F is not a cusp form, i.e., F = f |Φ = 0. Prove that in this case, the function (5.67) is defined and holomorphic in the half-plane ℜs > 2k + 1, has an analytic continuation to the whole s-plane as a holomorphic function if F ∈ M1k is a cusp form, and as a meromorphic function with four simple poles at the points s = 0, k − 2, k, 2k − 2, if F is an Eisenstein series. In all cases it satisfies the functional equation (5.68).

Conclusion: Other Groups, Other Horizons

Although the theory of zeta functions of Siegel modular forms is far from being complete, and many fundamental questions (such as analogues of the Shimura– Taniyama conjecture for abelian varieties) have not even been touched, one must keep in mind that the symplectic group is but one example from a big family of arithmetically significant groups. The most natural situation, when it is possible to approach both analytic properties and factorization into an Euler product of arithmetic zeta functions, is the case in which zeta functions can be associated with representations of suitable arithmetic discrete subgroups of Lie groups on related function spaces of automorphic forms. The typical example of this kind is provided by “zeta functions of bilinear forms.” Let, for example, ⎛ ⎛ ⎞ ⎛ ⎞⎞ x1 y1 ⎜ ⎜ .. ⎟ ⎜ .. ⎟⎟ t q = q(X, Y ) = ∑ qi j xi y j = XQY ⎝X = ⎝ . ⎠ , Y = ⎝ . ⎠⎠ i, j=1,...,m

xm

ym

be a bilinear form of order m with integral nonsingular matrix Q = (qi j ). With the form q we associate the automorph semigroup of q, A(q) = D ∈ Zm m q(DX, DY ) = µ q(X, Y ) with µ = µ (D) > 0 , and the group of units of q, E(q) = {D ∈ A(q)| µ (D) = 1} . The pair (E(q), A(q)) is quite often left-finite in the sense of Section 3.2, and we may define the Hecke–Shimura ring D = D(E(q), A(q)) of the pair (over Z), which is also called the automorph class ring of q and is denoted by H(q). Generally speaking, one has little to say about the ring H(q), and it should be replaced by the more complicated construction of a matrix Hecke–Shimura ring. Each bilinear form

169

170

Conclusion: Other Groups, Other Horizons

is the sum of uniquely defined symmetric and skew-symmetric forms, i.e., the forms q satisfying q(X, Y ) = q(Y, X) or q(X, Y ) = −q(Y, X), respectively, and corresponding groups and semigroups of automorphs are intersections of those for the components. In the case of either a pure symmetric or skewsymmetric form q, each double coset of the semigroup A(q) modulo the group E(q) is a finite union of left cosets. In such a case, a theory can be developed that would resemble the Hecke–Shimura theory outlined above for the most interesting skewsymmetric case of the bilinear form with matrix Q = Jn of order 2n defined in (0.1) (with the integral symplectic group Γn = Spn (Z) as the group of units and Siegel modular forms as representation spaces for symplectic Hecke–Shimura rings). In the symmetric case, when the form q is symmetric and the group of units E(q) coincides with the proper integral orthogonal group of the quadratic form q(X, X), the theory of orthogonal matrix Hecke–Shimura rings is similar in many respects to the theory of symplectic rings, including the formal Euler factorization of generating Dirichlet series. In the orthogonal case, Hecke–Shimura rings operate on spaces of harmonic polynomials related to basic quadratic forms, and one can define zeta functions corresponding to these representations. It was revealed recently that in the case of positive definite quadratic forms in two and four variables, the relevant (orthogonal) zeta functions coincide with zeta functions corresponding to representations of symplectic Hecke–Shimura rings on spaces of theta series of genus 1 and 2 with harmonic coefficients, respectively. The coincidence is even more striking because Hecke–Shimura rings of quite different groups of units are involved: finite orthogonal groups and infinite symplectic groups. Other examples of relations between zeta functions of different arithmetic groups are provided by various lifting of automorphic forms and related zeta functions to similar groups of higher orders such as Saito–Kurokawa and Ikeda lifts, and various liftings of zeta functions of the general linear group to their symmetric degrees. There is no doubt that further progress in number theory will be closely connected with the investigation of relations among zeta functions of representations of Hecke–Shimura rings of various arithmetic discrete subgroups of Lie groups on automorphic functions.

Notes

Introduction For the Riemann zeta function, see, e.g., [Ti86]. On zeta functions of algebraic varieties one can read [Shi71, Chapter 7]. For the Birch–Swinnerton-Dyer conjectures see [BSD63/65]. For modular forms in one variable and corresponding Dirichlet series see [Ogg69]. The Hecke theory of the Euler product factorization of Dirichlet series of modular forms is stated in [He37]. The original Atkin–Lehner theory was set forth in [AtL70]. For the history of the Shimura–Taniyama conjecture see [Shi89]. For the proof of Fermat’s last theorem and related questions see [Wil95].

Chapter 1 The general references on this chapter, where one can find all omitted details and much more, are [An87, Chapters 1 and 2] and [AnZ90/95]. The later book also treats modular forms of half-integer weight. For a classical introduction to Siegel modular forms and Dirichlet series see [Kl90]. More details on the theory of Siegel modular forms and functions can be found in [Fr83], [Ma55], and [Si39]. For the treatment of automorphic forms on Lie groups see [SC58]. For modular forms in one variable see the books [Ogg69] and [Ma64]; for elementary presentations see [Ser70] and [Ga62]; relations to algebraic geometry are considered in [Fr83] and [Shi71]. Relations of modular forms to integral quadratic forms are considered in [Ki86] (numbers of representations of quadratic forms by quadratic forms) and in [An87] (multiplicative properties of the representations). Section 1.2. For details on the reduction theory of positive definite quadratic forms see, e.g., [Ca78, Chapter 12]. For the construction of the fundamental domain of the modular group see also [Si73].

171

172

Notes

Section 1.3. The notation (1.29) is a variation of the notation introduced by H. Petersson. The Koecher effect was discovered in [Ko54/55]. For the rather complicated proof of the estimate (1.52) see, e.g., [An87, Theorem 2.3.4]. The scalar product of modular forms in one variable was introduced by H. Petersson in [Pe39/41]; the consideration of the general case is based on a similar idea and was initiated by H. Maass in [Ma51].

Chapter 2 Section 2.1. Radial Dirichlet series (2.10) are not the only kind of Dirichlet series constructed with the help of Fourier coefficients of common eigenfunctions for regular Hecke operators that have an Euler product factorization and good analytic properties. Another kind of such Dirichlet series is given by series of the form

∑

M∈SLn (Z)\{M∈Znn | det M>0}

ψ (det M) f (MA tM) , (det M)s

where n ≥ 1, f are Fourier coefficients of a common eigenfunction for regular Hecke operators on the space Mk (Γn0 (q), χ ) with integral or half-integral k, ψ is a Dirichlet character, and A is a fixed even positive definite matrix of order n. For n = 1 these series were considered in [Ra39], [Se40], and [Shi75]; Euler product factorizations for an arbitrary n and integral k were discussed in [An87, §4.3.3], whereas both integral and half-integral k were considered in [AnZ90/95, §3.3]; analytic properties of corresponding Euler products (the standard zeta functions of eigenforms F) of level q = 1 with ψ = 1 were considered in [AnK78] with certain restrictions, and in [Bo85] without restrictions. Section 2.3. Our exposition of the theory of radial Dirichlet series corresponding to the ray mA1 = 2m · 12 (m = 1, 2, . . . ) of the matrix of a quadratic form x12 + x22 was outlined first in [An71, §3]; the general case of radial series corresponding to rays of matrices of arbitrary positive definite integral primitive binary quadratic forms was considered in detail in [An74, §§3.3–3.7].

On Chapter 3 Section 3.1. The multiplicative properties of the Fourier coefficients of the modular form ∆ (z) ∈ N12 (Γ ) cited in Exercise 3.1(3) were observed (!) and conjectured by Ramanujan and a little later proved by Mordell in [Mo17]. Mordell’s proof actually contains the idea of Hecke operators for the particular case of the space N12 (Γ ), an idea that had to wait twenty more years to be reborn in the case of more general spaces of modular forms by Hecke in [He37]. Section 3.2. Our definition of an abstract ring of double cosets by means of multiplication of right-invariant linear combinations of left cosets is equivalent to the

Notes

173

definition given by Shimura in [Shi63] with the help of the direct multiplication of double cosets, but in some respects it turns out to be more convenient. Section 3.3. The rings of double cosets of the general linear group are discussed here and in [An87, §3.2] in the spirit of the fundamental paper [Ta63] of Tamagawa. For the explicit formulas for spherical functions on GLn see [An70]. Section 3.4. The symplectic case is described as in [An87, §3.3]. The structure of Hecke–Shimura rings for Γn was indicated first in [Sa63] and [Shi63]. I cannot judge who was the first. The passage to congruence subgroups goes back to Hecke’s ideas in [He37]. Our presentation uses common sense and an analogy with the case of the general linear group rather than any historical reminiscences. For the theory of singular Hecke–Shimura rings see [An99]. The theory of spherical mappings in the form of a general theory of zonal spherical functions on reductive algebraic groups over p-adic fields is due to Satake; see [Sa63]. We use an elementary approach based on explicit formulas. The series (3.107) and (3.109) were summed up in [Shi63]; similar series for arbitrary n were computed in [An69] and [An70]. Section 3.5. The embedding of symplectic Hecke–Shimura rings into rings of triangular-symplectic double cosets allows one to split their elements into elementary components naturally related to the general linear group. This gives impetus to the theory of factorization of standard symplectic polynomials in triangular extensions, presented in detail in [An87, §§3.4–3.5]. Here we use only the elementary aspects of the theory.

Chapter 4 General references are [An87, §§4.1–4.2] or [AnZ90/95]. Section 4.1. Hecke operators for the group Γ1 = SL2 (Z) and some congruence subgroups were introduced in [He37]. Hecke operators on Siegel modular forms were first introduced by Sugawara in [Su37] and [Su38] and after the war were studied by Maass in [Ma51]. The existence of a basis of common eigenfunctions for Hecke operators acting on cusp forms in one variable was proved by Petersson in [Pe39/41]. For Siegel modular forms, Petersson’s ideas were developed by Maass in [Ma51]. For diagonalization of singular Hecke operators see [AtL70] in the case of genus n = 1, and [An99], [An03] in the general case. Section 4.3. Zharkovskaya commutation relations for the group Γn and the unit character are due to Natasha Zharkovskaya [Zha74], who generalized the Maass relation for Γ2 obtained in [Ma51]. Our consideration follows the same idea.

Chapter 5 Section 5.1. Relations of Fourier coefficients of eigenforms with eigenvalues can be found in [He37].

174

Notes

Section 5.2. For more details on the relations of binary quadratic forms and modules in quadratic number fields see, e.g., [An87, Appendix 3], or the excellent book on algebraic numbers [BSh63, Chapter 2]. Section 5.3. It is proved in [An74, Theorem 3.1.1] without any restrictions that the zeta function ζ (s, F) of an arbitrary eigenfunction for all Hecke operators F ∈ M2k of integral weight k ≥ 0 can be continued to the whole s-plane as a meromorphic function; the function Ψ(s, F) = (2π )−2s Γ(s)Γ(s − k + 2)ζ (s, F), where Γ is the gamma function, is meromorphic on the s-plane with the only possible poles at s = 0, k − 2, k, 2k − 2 and satisfies the functional equation Ψ(2k − 2 − s, F) = (−1)k Ψ(s, F). This does not contradict the functional equation (5.67), because the assumptions of Theorem 5.35 imply that k there must be even. The relation between the eigenvalues of Hecke operators and Fourier coefficients of eigenfunctions for Siegel modular forms of genus n discovered by Zharkovskaya in [Zha75] can possibly provide the first step to approach the still open problem of analytic properties of similar zeta functions of cusp form for genera n > 2.

Conclusion On relations of zeta functions of orthogonal and symplectic groups see [An06]. For a general outlook on problems of Euler products attached to automorphic forms see Langlands’ lectures [La67] and [La69].

References

[An69]

A.N. Andrianov, Rationality theorems for Hecke series and zeta-functions of the groups GLn and Spn over local fields, Izv. Akad. Nauk Ser. Mat. 33 (1969), 466– 505 (Russian); English transl., Math. USSR Izv. 3 (1969), 439–476. [An70] A.N. Andrianov, Spherical functions on GLn over local fields and summation of Hecke series, Mat. Sbornik Nov. Ser. 83 (1970), no. 3, 429–451 (Russian); English transl., Math. USSR Sb. 12 (197), no. 3, 429–452. [An71] A.N. Andrianov, Dirichlet series with Euler product in the theory of Siegel modular forms of genus 2, Trudy Mat. Inst. Steklov 112 (1971), 73–94 (Russian); English transl., Proc. Steklov Inst. Math. 112 (1971), 70–93. [An74] A.N. Andrianov, Euler products corresponding to Siegel modular forms of genus 2, Uspekhi Mat. Nauk 29 (1974), no. 3, 43–110 (Russian); English transl., Russian Math. Surveys 29 (1974), no. 3, 45–116. [An87] A.N. Andrianov, Quadratic forms and Hecke operators, Grundlehren Math. Wiss. 286, Springer-Verlag, Berlin Heidelberg New York London Paris Tokyo, 1987. [An99] A.N. Andrianov, Singular Hecke–Shimura rings and Hecke operators on Siegel modular forms, Algebra i Analis 11 (1999), no. 6, 1–68; English transl., St. Petersburg Math. J. 11 (2000), no. 6, 931–987. [An03] A. Andrianov, On diagonalization of singular Frobenius operators on Siegel modular forms, Amer. J. Math. 125 (2003), 139–165. [An06] A.N. Andrianov, Zeta functions of orthogonal groups of integral positive definite quadratic forms, Uspekhi Mat. Nauk 61 (2006), no. 6, 3–44 (Russian); English transl., Russian Math. Surveys 61 (2006), no. 6. [AnK78] A.N. Andrianov and V.L. Kalinin, On the analytical properties of standard zeta functions of Siegel modular forms, Mat. Sbornik 106(148) (1978), no. 3, 323–339 (Russian); English transl., Math. USSR Sbornik 35, no. 1, 1–17. [AnZ90/95] A.N. Andrianov and V.G. Zhuravlev, Modular forms and Hecke operators, “Nauka”, Moscow, 1990 (Russian); English transl., Transl. of Math. Monographs, vol. 145, AMS, Providence Rhode Island, 1995. [AtL70] A.O.L. Atkin and J. Lehner, Hecke operators on Γ0 (m) Math. Ann. 185 (1970), 134–160. [BSD63/65] B.J. Birch and H.P.F. Swinnerton-Dyer, Notes on elliptic curves. I,II, J. Reine Angew. Math. 212 (1963), 7–25; 218 (1965), 79–108. ¨ [Bo85] S. B¨ocherer, Uber die Funktionalgleichung automorpher L-Funktionen zur Siegelschen Modulgruppe, J. Reine Angew. Math. 362 (1985), 146–168. [BSh63] Z.I. Borevich and I.R. Shafarevich, Number Theory, “Nauka”, Moscow, 1963, 1971, (1985) (Russian); English transl., Academic Press, New York, 1966. [Fr83] E. Freitag, Siegelsche Modulfunktionen, Grundlehren Math. Wiss. 254, SpringerVerlag, Berlin Heidelberg New York London Paris Tokyo, 1983.

175

176 [Ga62] [He37]

[Ki86]

[Kl90] [Ko54/55] [La67] [La69] [Ma51] [Ma55] [Ma64] [Mo17] [Ogg69] [Pe39/41]

[Ra39]

[Se40] [SC58] [Ser70] [Shi63] [Shi75] [Shi71]

[Shi89] [Si39] [Si73] [Ta63] [Tit86]

References R.C. Gunning, Lectures on modular forms, Annals of Math. Studies 48, Princeton University Press, Princeton, New Jersey, 1962. ¨ E. Hecke, Uber Modulfunktionen und die Dirichletschen Reihen mit Eulerscher Produktentwicklung. I, II, Math. Ann. 114 (1937), 1–28; 316–351; Math. Werke, Vandenhoeck & Ruprecht, G¨ottingen, 1959, 1970, pp. 644–707. Y. Kitaoka, Lectures on Siegel modular forms and representations by quadratic forms, Published for the Tata Institute of Fundamental Research, Springer-Verlag, Berlin Heidelberg New York Tokyo, 1986. H. Klingen, Introductory lectures on Siegel modular forms, Series: Cambridge Studies in Advanced Mathematics (No 20), Cambridge University Press, 1990. M. Koecher, Zur Theorie der Modulformen n-ten Grades. I, II, Math. Z. 59 (1954), no. 4, pp. 399–416; 61 (1955), no. 4, 455–466. R.P. Langlands, Euler products, Lecture Notes, Yale University, New Haven, Connecticut, 1967. R.P. Langlands, Problems in the theory of automorphic forms, Lecture Notes, Yale University, New Haven, Connecticut, 1969. H. Maass, Die Primzahlen in der Theorie der Siegelschen Modulfunktionen, Math. Ann. 124 (1951), 87–122. H. Maass, Lectures on Siegel’s modular functions, Tata Inst. Fund. Research, Bombay, 1955. H. Maass, Lectures on modular functions of one complex variable, Revised 1983, Tata Inst. Fund. Research, Bombay, 1964. L.J. Mordell, On Mr Ramanujan’s empirical expansions of modular functions, Proc. Cambridge Phil. Soc. 19 (1917), 117–124. A.P. Ogg, Modular forms and Dirichlet series, W.A. Benjamin, Inc., New York, Amsterdam, 1969. H. Petersson, Konstruktion der s¨amtlichen L¨osungen einer Riemannschen Funktionalgleichung durch Dirichletreihen mit Eulerscher Produktentwicklung. I, II, III, Math. Ann. 116 (1939), 401–412; 117 (1939), 39–64; 117 (1940/41), 277–300. R.A. Rankin, Contributions to the theory of Ramanujan’s function τ (n) and similar arithmetical functions. II, The order of the Fourier coefficients of the integral modular forms, Proc. Cambridge Phil. Soc. 35 (1939), no. 3, 357–372. A. Selberg, Bemerkungen u¨ ber eine Dirichletsche Reihe, die mit der Theorie der Modulformen nahe verbunden ist, Arch. Math. Naturvid. 43 (1940), 47–50. S`eminaire H. Cartan, Fonctions automorphes, 10e ann´ee, vol. 1, 2, Secr`etariat Math´ematique, Paris, 1958. J.-P. Serre, Cours d’arithm´etique, Presses Universitaires de France, Paris, 1970. G. Shimura, On modular correspondences for Sp(n, Z) and their congruence relations, Proc. Nat. Acad. Sci. USA 49 (1963), no. 6, 824–828. G. Shimura, On the holomorphy of certain Dirichlet series, Proc. London Math. Soc. 31 (1975), no. 1, 79–98. G. Shimura, Introduction to the arithmetic theory of automorphic functions, Publ. Math. Soc. Japon, vol. 11, Iwanami Shoten, Publishers and Princeton University Press, Princeton, 1971. G. Shimura, Yutaka Taniyama and his time, Bull. London Math. Soc. 21 (1989), 186–196. C.L. Siegel, Einf¨uhrung in die Theorie der Modulfunktionen n-ten Grades, Math. Ann. 116 (1939), 617-657; Gesam. Abhandlungen vol. II, pp. 97–137. C.L. Siegel, Topics in complex function theory, Interscience Tracts in Pure and Applied Math. 25 vol. III, Wiley-Interscience, New York London Sydney Toronto, 1973. T. Tamagawa, On the ζ -functions of a division algebra Ann. Math. II Ser. 77 (1963), no. 2, 387–405. E. C. Titchmarsh, The theory of the Riemann zeta-function, 2nd ed., Clarendon press, Oxford, 1986.

References [Wil95] [Zha74] [Zha75]

177 A. Wiles, Modular elliptic curves and Fermat’s Last Theorem, Ann. Math. II Ser. 141 (1995), no. 3, 443–551. N.A. Zharkovskaya, The Siegel operator and Hecke operators, Funkts. Anal. Prilozh. 8 (1974), no. 12, 30–38; English transl., Funct. Anal. Appl. 8 (74), 113–120. N.A. Zharkovskaya, On the connection between the eigenvalues of Hecke operators and Fourier coefficients of eigenfunctions for Siegel’s modular forms of genus n, Mat. Sb. Nov. Ser. vol 96(138) (1975) 584–593; English transl., Mat. USSR Sb. 25 (1975).

Index

A Analytic automorphisms domain H, of, 48 domain H(A), of, 53 Lie group acting as group of, 8 transitive group of, 47 Antiautomorphisms application of, 95 extension of, 72–73 second order, of, 72, 108 Associative ring, 65–67 Automorph semigroup, 169 Automorphy, factor of, 50 B Binet–Cauchy formula, 76 Block matrices, multiplication of, 13 C Cauchy’s theorem, 60 Cayley mapping, 11 Commensurability, relation of, 70 Commutative ring, 67 Complex number, nonzero, 7, 20, 85, 105, 120, 137 Congruence character, 21, 22, 24, 28–30, 35, 120, 126 subgroup, 3, 5, 20–22, 25, 28–30, 32, 92 Coprime integers, 92 Coprime numbers, 78 Cusp forms diagonalization of, 134–135 invariance of subspaces of, 120–121, 135 multiplicative relations of eigenvalues, 139 regular zeta function of, 166

D dc-rings. See Double cosets ring Dedekind zeta function, 59, 61 of the ring, 167–168 Denominators, factorization of, 115–117 Diagonalization theorem, 124–125 Diagonal matrices, 77, 88 Dirichlet character modulo, 41, 45, 125, 127, 131, 133 Dirichlet series, 96, 137, 172 cusp form, of, 143 Euler factorization of, 140–142 of F, 3 functional equation, 143–145 of Z(F, s), 4 Discrete transformation, 12 Double cosets modulo, 68, 82, 95, 98, 121, 124, 126 abstract ring of, 172–173 Double cosets ring abstract rings basic rings, 67 Hecke–Shimura rings, definition of, 66 standard bases and multiplication rules, 67–69 decomposition of, 80 general linear group of, 74 left and right rings of, 69–72 multiplication in, 77 triangular-symplectic, ring of factorization of denominators, 115–117 Frobenius elements, 107 local triangular rings, relations in, 111 spherical mappings, 112–113 triangular embeddings, 108 triangular rings, 106

179

180 E Eigenfunctions, 124–125 for all operators, 135 certain rings of linear operators, of, 4 and eigenvalues, 138 Fourier coefficients of, 137, 138, 146, 148, 174 for operator T (p), 64 of regular Hecke operators, 123, 135, 138, 141 zeta function of, 141, 143 Eigenvalues, 44 Hecke operators, of, 136, 139 matrix, of, 14 of nonzero modular form, 164 properties of, 138 Eisenstein series, 20, 34, 39, 47 for discrete subgroup, 55 and Theta series, 58–59 Elementary divisors, 76 Elementary integral operations, 76 Elementary p-divisors, 80 Elementary symmetric polynomials, 83, 104 Embeddings, extension of, 73–75 Euclidean algorithm, 13, 30, 57, 76, 79 Euclidean element, 9, 10 Euler factorization, 96, 144, 160 conditions, 164 Dirichlet series, of, 140–141 problem of, 65 radial series, of, 44 Euler formula, 42 Euler integral, 44, 59 Euler product, 2 factorization of, 4, 41 F Field Q of rational numbers, 2 Finite union left/right cosets modulo, 66 Formal power series, 102, 103 Fourier coefficients, 26, 27, 33, 35 Dirichlet series, 65 eigenfunctions and eigenvalues of modular forms, of, 138–139 f (m), 3 function, of, 64 Hecke operators, of, 126–129 modular forms, of, 34, 63, 65 multiplicative properties of, 161–164, 172 Fourier expansion, 25, 28, 30, 35, 45 convergent, 36 Fourier series, 28, 60 absolutely convergent on H, 3 Siegel operator action on, 130–134

Index Frobenius elements, 107–111 Full modular group global rings for, 86 local rings for, 90 Fundamental domains, construction of, 15 G Gamma function, 42, 61 Gaussian composition, 153 Gaussian integers, 53, 57, 59 G-invariant element, of volume on H, 9 G-invariant volume element, 52 Global rings, 75 Global zeta function, 2 H Haar measure, 43 Hadamard’s determinant theorem, 27 Hecke operators, 64 on abstract level, 119–120 acting on spaces, 4 action on modular forms for groups Fourier coefficients, 126–129 Hecke–Shimura ring, 125–126 for congruence subgroups, 120–121 diagonalization of cusp forms, 134–135 eigenfunctions, 124–125, 135–136 Petersson scalar product, 123–124 eigenvalues of, 139 invariant subspaces under Fourier coefficients and eigenvalues, 161–162 homomorphism, 121–122 problem for, 134 and Siegel operator (see Siegel operator) summation formula, 159–160 Hecke–Shimura rings, 65, 91–92, 127 definition of, 66 of double cosets, 66 homomorphism, 130–131 of l-finite pair, 67 linear representation of, 120 local subrings, 79 mapping, 133–134 q-symmetric group, of, 96 symplectic, 137, 170 triangular subgroup, 106 Hermitian matrices, 11 Hilbert space, 35 Holomorphic functions, 19 Homomorphic embedding, of group, 29 Homomorphism, 121, 133

Index I ω -images, 83 Ω-images, 100, 102 Imprimitive elements, 80, 91 Inductive hypothesis, 79, 82, 87 Integral symmetric matrix, 17, 24, 109 Integral weight cusp form of, 31, 61 Dirichlet series of modular forms of, 3 modular form of, 63 Petersson operators of, 19 Invariant polynomials, 101 Invariant subspace cusp forms, of, 135–136 under Hecke operators, 121–123 problem for, 134 Inverse isomorphisms, ring, 93 Inverse mapping, 11, 44–45 Isomorphism, of subgroup, 53 J Jacobi’s formula, 63 K K-orbit, 23 L Left coset, 64, 68 decompositions in, 77 Frobenius elements, 108 reduced representative, 79 representative, 78 Lie groups, 8 isomorphism of, 50 Linear map, 34 Linear mapping, 80, 91, 93 Lobachevsky half-spaces, transformations of, 47–48 Lobachevsky space, 49 Local Hecke–Shimura rings, 82 Local subring, 97 integral linear combinations, 80 prime numbers, 79, 90 Local triangular rings, 111–115 Local zeta functions ζ (Vp , p−s ), 2 M Mapping ring, isomorphism of, 83 Mellin integral, 3, 42, 45 Mellin transforms classical theory of, 44 of cusp forms, 44 Meromorphic function, 47, 61 Minkowski inequality, 14

181 Minkowski reduction domain, 12–13 Modular cusp form of weight k, 3 Modular forms definition of, 19–20 Fourier coefficients of, 34 Fourier expansion of, 22–25, 138, 145–148 invariant subspaces of (see Invariant subspace) multiplicative properties of, 137 nonzero (see Nonzero modular form) regular zeta function of, 141 spaces of, 30–34 zeta functions of, 41 Modular group features of, 85 global rings for, 86 Hecke–Shimura rings of, 91 local rings for, 90 q-symmetric subgroup, 95–96 subgroup K of, 92 Multiplication rules, 88 standard bases, 67–69 Multiplicative function, 63 Multiplicative homomorphism, 20 Multiplicative properties, 63 of Fourier coefficients, 161–164, 172 N Nonnegative coefficients, linear combination of, 78 Nonzero cusp forms, 35, 142 regular zeta function of, 165–166 Nonzero integral, 76 Nonzero modular form, 145 Fourier coefficients and eigenvalues of, 147, 161–164 multiplicativity of, 163–164 regular zeta function of, 165–166 Nonzero rational numbers, multiplicative group of, 70 O One-dimensional subspaces, 122 Π-Operators, 149 summation conditions in, 153 P Permutations, 99 Permutes factors, 104 Petersson operators, 19, 35, 122 of integral weights k, 19 mapping of holomorphic functions, 19 Petersson scalar product, 35, 37, 123–125 p-factors, 5 p-local subring, 90

182 Polynomial congruences modulo, 92 Polynomials, 82, 100 Q-linear homomorphism, ring of, 83 Primitive elements, 91 Principal congruence subgroup, 75, 92 Q q-integral symplectic matrix, 94 q-symmetric group, 93 local subrings, 97 Quadratic field. See Quadratic modules Quadratic modules definitions, 149 multiplication of, 150 prime and primitive in terms of, 154–157 and prime numbers, relation between, 151–153 R Radial Dirichlet series, 141, 172 cusp forms, of, 41–43, 143 and Eisenstein Series, 55–57 integral representation of, 43–44 method for obtaining, 47 properties for sums of two squares, 58 Ramanujan’s formula, 63 Rational fractions, transformations of, 99 Real matrix, 50 Riemann zeta function, 1, 34, 47 Ring epimorphism, 80 Ring homomorphism, 80 Ring monomorphism, 74 Ring of multipliers of the module, 149 Rings isomorphism, 74 S Semidefinite matrices, 45 Shimura–Taniyama conjecture, 5 Siegel modular forms, 5 Siegel modular group, 85 fundamental domains for, 11 of weight k and character χ , 20 Siegel operator, 28, 34 action on Fourier series, 130–134 Hecke operators, relations between, 67 for mapping of space, 29 Spherical mapping, 82–85, 98–99 Q-linear isomorphism, 99–102 Spinor p-polynomial, 105 Star mapping transforms, local subrings for, 108 Subspaces of modular forms Fourier coefficients of functions of, 158 invariance of, 120–121

Index Summation formulas, 103 Symmetric polynomials, 82 Symplectic case, zero divisors, 107 Symplectic divisors, 87, 95 Symplectic group, 65 dc-rings of, 75 commutative, 88 congruence subgroups, regular Hecke– Shimura rings for, 91–92 decompositions, 89 formal power series, 102, 103 full modular group, global rings for, 86 full modular group, local rings for, 90 spherical mapping, 98–99 zero-blocks, 87 Symplectic matrices, 7 of order 2n with positive multipliers, 8 Symplectic modular group, 5 T Theta series, 58 Transformations, 99, 104 Triangle, modular, 18 Triangular embeddings, 108 Triangular matrix, 10, 45, 81 Triangular-plus matrices, 83 Triangular-symplectic matrices, 106 U Unitary group, of order n, 11 Unit matrix, of order n, 7 W W -invariant polynomial, 101 Z Zero matrix, 89 Zeta functions algebraic varieties, of, 2–3 automorphic forms, of, 3–5 bilinear forms, of, 169–170 of E, 3 eigenforms, of nonzero cusp form, 165–166 nonzero modular form, 165 E p over F (p), 2 modular forms, of, 137 functional equation of, 143–145 nonzero eigenform, 145 in terms of Dirichlet series, 140–141 and symplectic Hecke–Shimura rings, 170 Zharkovskaya commutation relations, 132–134 Z-linear mapping, 95

For other titles published in this series, go to www.springer.com/series/223

Anatoli Andrianov

Introduction to Siegel Modular Forms and Dirichlet Series

ABC

Anatoli Andrianov Russian Academy of Sciences Steklov Institute of Mathematics Fontanka 27 191023 St. Petersburg Russia [email protected]

Editorial board: Sheldon Axler, San Francisco State University, San Francisco, CA, USA Vincenzo Capasso, University of Milan, Milan, Italy Carles Casacuberta, Universitat de Barcelona, Barcelona, Spain Angus MacIntyre, Queen Mary, University of London, London, UK Kenneth Ribet, University of California, Berkeley, CA, USA Claude Sabbah, Ecole Polytechnique, Palaiseau, France Endre Süli, Oxford University, Oxford, UK Wojbor Woyczynski, Case Western Reserve University, Cleveland, OH, USA

ISBN 978-0-387-78752-7 DOI 10.1007/978-0-387-78753-4

e-ISBN 978-0-387-78753-4

Library of Congress Control Number: 2008938066 Mathematics Subject Classiﬁcation (2000): 11Fxx, 11F66 This is a translation of the Dutch, Meetkunde, originally published by Epsilon–Uitgaven, 2000. c Springer Science+Business Media, LLC 2009 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identiﬁed as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed on acid-free paper springer.com

To Goro Shimura and to my granddaughter Sasha

Preface

Several years ago I was invited to an American university to give one-term graduate course on Siegel modular forms, Hecke operators, and related zeta functions. The idea to present in a concise but basically complete and self-contained form an introduction to an important and developing area based partly on my own work attracted me. I accepted the invitation and started to prepare the course. Unfortunately, the visit was not realized. But the idea of such a course continued to be alive till after a number of years this book was finally completed. I hope that this short book will serve to attract young researchers to this beautiful field, and that it will simplify and make more pleasant the initial steps. No special knowledge is presupposed for reading this book beyond standard courses in algebra and calculus (one and several variables), although some skill in working with mathematical texts would be helpful. The reader will judge whether the result was worth the effort. Dedications. The ideas of Goro Shimura exerted a deep influence on the number theory of the second half of the twentieth century in general and on the author’s formation in particular. When Andr`e Weil was signing a copy of his “Basic Number Theory” to my son, he wrote in Russian, ”To Fedor Anatolievich hoping that he will become a number theoretist”. Fedor has chosen computer science. Now I pass on the idea to Fedor’s daughter, Alexandra Fedorovna. Contents. The main objective of this book is to give a concise but basically complete and self-contained introduction to the multiplicative theory of Siegel modular forms, Hecke operators, and zeta functions, including the classical case of modular forms in one variable. Chapter 1 contains a compressed exposition of essential features of the theory of Siegel modular forms of integral weight for congruence subgroups of the symplectic modular group Spn (Z) of arbitrary genus n. Chapter 2 treats analytical properties of radial Dirichlet series attached to modular forms of genera 1 and 2. Chapter 3 is dealing with the abstract theory of Hecke–Shimura rings for symplectic and related group. Action of Hecke operators on Siegel modular forms is considered in Chapter 4. In Chapter 5, we examine applications of

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Hecke operators to a study of multiplicative properties of Fourier coefficients of modular forms and the related Euler product factorization of radial Dirichlet series attached to eigenfunctions. This leads us to Hecke zeta functions of modular forms in one variable and to spinor (or Andrianov) zeta functions of Siegel modular forms of genus two. At the end of this chapter we arrive at the proof of the analytic continuation and functional equation (under certain assumptions) of Euler products associated with modular forms of genus two. The book contains a number of exercises that usually consider some interesting points not included in the main text, partly for the reasons of space and partly because of their special character. References. I try to present the proofs in full detail whenever it is reasonable and possible. The main text contains no references. Essential references are included in the Notes at the end of the book. It should be noted that the author does not pretend to give an encyclopedic survey of modular forms or a complete bibliography, but rather to hint at certain principal points of the theory and illustrate how they have been reached. Acknowledgments. I would like to express my deep gratitude to Fedor Andrianov, the assumed co–author, who carefully read the manuscript and suggested a number of improvements. I am very grateful to Mark Spencer, of Springer New York for continued interest in this book and highly effective cooperation. Great acknowledgment must also go to Charlene Cruz Cedras, the editorial assistant of Mark Spencer, who has given me valuable assistance and support. Finally, I must acknowledge my grateful thanks to Dr. David Kramer for highly careful editing of the manuscript, that I have never encountered. St Petersburg January/April 2008

Anatoli Andrianov

Contents

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii Introduction: The Two Features of Arithmetic Zeta Functions . . . . . . . . . . .

1

1

Modular Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.1 The Symplectic Group and the Upper Half-Plane . . . . . . . . . . . . . . . . 7 1.2 Fundamental Domains for the Modular Group . . . . . . . . . . . . . . . . . . 11 1.3 Modular Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2

Dirichlet Series of Modular Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Radial Dirichlet Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Mellin Transform of Cusp Forms in One Variable . . . . . . . . . . . . . . . . 2.3 Transformations of Lobachevsky Half-Spaces . . . . . . . . . . . . . . . . . . . 2.4 Radial Series and Eisenstein Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Properties of Radial Series for Sums of Two Squares . . . . . . . . . . . . .

3

Hecke–Shimura Rings of Double Cosets . . . . . . . . . . . . . . . . . . . . . . . . . . 63 3.1 An Approach to Multiplicativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 3.2 Abstract Rings of Double Cosets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 3.3 Rings of Double Cosets of the General Linear Group . . . . . . . . . . . . . 75 3.4 Rings of Double Cosets of the Symplectic Group . . . . . . . . . . . . . . . . 85 3.5 Rings of Triangular-Symplectic Double Cosets . . . . . . . . . . . . . . . . . . 106

4

Hecke Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 4.1 Hecke Operators for Congruence Subgroups . . . . . . . . . . . . . . . . . . . . 119 4.2 Action of Hecke Operators for Γn0 (q) . . . . . . . . . . . . . . . . . . . . . . . . . . 125 4.3 Hecke Operators and Siegel Operator . . . . . . . . . . . . . . . . . . . . . . . . . . 130

5

Euler Factorization of Radial Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 5.1 Radial Series of Genus One and Zeta Functions . . . . . . . . . . . . . . . . . 138 5.2 Binary Radial Series and Gaussian Composition . . . . . . . . . . . . . . . . . 145 5.3 Zeta Functions of Eigenforms for Genus 2 . . . . . . . . . . . . . . . . . . . . . . 165

41 41 44 47 55 58

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x

Contents

Conclusion: Other Groups, Other Horizons . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179

General Notation

The letters N, P, Z, Q, R, and C are reserved for the set of positive rational integers, the set of positive rational prime numbers, the ring of rational integers, the field of rational numbers, the field of real numbers, and the field of complex numbers, respectively. K[x1 , . . . , xn ] is the ring of polynomials in x1 , . . . , xn with coefficients in K, and Am n is the set of all m × n−matrices with entries in a set A. If M is a matrix, tM always denotes the transpose of M, σ (M) for a square M is the trace of M, and M for a complex matrix M means the matrix with conjugate entries. If Y is a real symmetric matrix, then Y > 0 (resp., Y 0) means that Y is positive definite (resp., positive semidefinite). For two matrices A and B of suitable dimensions we write A[B] = tBAB.

xi

Introduction: The Two Features of Arithmetic Zeta Functions

A zeta function in arithmetic is, generally speaking, a generating function for an arithmetic problem written in the form of a Dirichlet series. A well stated zeta function must have at least two principal features: an Euler product factorization and an analytic continuation over the whole complex plane satisfying functional equations. The first reflects relations between the global arithmetic problem and its localizations, while the second provides a kind of reciprocity between the localizations. Let us illustrate this with some examples Riemann Zeta Function. The Riemann zeta function, ∞

1

∑ ns

ζ (s) =

(Re s > 1),

n=1

is the generating function for the numbers of ideals of given norm in the ring Z of rational integers. It has an Euler product factorization of the form

ζ (s) = ∏ p

1 1− s p

−1 (Re s > 1),

the product being taken over all rational prime numbers p. It was Bernhard Riemann who proved in the middle of the nineteenth century that ζ (s) has an analytic continuation over the whole complex s-plane, is holomorphic except for a simple pole of residue 1 at s = 1, and satisfies the functional equation that the function π −s/2Γ (s/2)ζ (s), where Γ is the gamma function, is invariant under the substitution s → 1 − s. He also discovered that the problem of distribution of prime numbers is closely connected with the location of complex zeros of the zeta function in the vertical strip 0 ≤ ℜs ≤ 1. At the end of the century, J. H´adamard and Ch. de la Vall´ee Poussin proved that ζ (s) has no zeros on the line ℜs = 1, which implied the famous asymptotic formula for the number π (x) of prime numbers not exceeding x, x (x → ∞). π (x) ∼ log x

1

2

Introduction

Zeta Function of Algebraic Varieties. The global zeta function of a nonsingular algebraic variety V over the field Q of rational numbers is defined by an Euler product ζ ∗ (V, s) = ∏ ζ Vp , p−s p

p−s ),

where p runs over all prime numbers such that V of local zeta functions ζ (Vp , has a “good” and, in particular, nonsingular reduction Vp modulo p, i.e., the variety over the finite field F(p) of p elements obtained by replacing equations defining V with corresponding congruences modulo p. The local zeta function is the zeta function of Vp defined by

ζ (Vp , t) = exp

∞

∑ N(pδ )t δ /δ

,

δ =1

where N(pδ ) is the number of points on Vp with coordinates in the finite field F(pδ ) of pδ elements. According to Bernhard Dwork, the local zeta functions ζ (Vp , t) of nonsingular varieties Vp are rational fractions in t. It follows that the global zeta function ζ ∗ (V, s) can be written as a Dirichlet series convergent in a right halfplane of the complex variable s. It is generally believed that the zeta function can be analytically continued over the whole s−plane as a meromorphic function and satisfies functional equations, but it is doubtful that a human being living now will see a complete proof. Nevertheless, even particular cases present considerable interest (all genuine number theory consists of particular cases; everything else is algebra). Let us consider a (projective) elliptic curve E:

y2 z = x3 + axz2 + bz3

(a, b ∈ Z).

The points on E with coordinates in Q form an abelian group, which we denote by EQ ; a theorem of L.J. Mordell tells us that the group EQ is finitely generated, i.e., is a product of a finite group by a lattice of finite rank g. A principal problem of the theory is to determine the group EQ , and, in particular, to determine the rank g. In the mid 1960’s B.J. Birch and H.P.F. Swinnerton-Dyer put forward revolutionary conjectures connecting the group EQ with the zeta function ζ ∗ (E, s) of the curve. Let us recall some details. A prime number p is said to be good if it does not divide 6(27b2 + 4a3 ). For such a prime p, the reduction Ep :

y2 z ≡ x3 + axz2 + bz3

(mod p)

of E modulo p is an elliptic curve over F(p). It is well known that the zeta function of E p over F(p) has the form

ζ (E p , t) =

1 − (1 + p − N(p))t + pt 2 . (1 − t)(1 − pt)

Introduction

3

Then we may define a zeta function of E by −1 ζ ∗ (E, s) = ∏ 1 − (1 + p − N(p))p−s + p1−2s , p

where the product is taken over all good primes. It converges for ℜs > 3/2. Then the main Birch–Swinnerton-Dyer conjecture says that ζ ∗ (E, s) has a zero of order g at s = 1. Generally, it is still open. Zeta Functions of Automorphic Forms. Despite the clear importance of zeta functions of algebraic varieties, algebraic geometry provides no means for their investigation. The only hope is to relate them to techniques coming from an analytic background, probably with zeta functions of automorphic forms, which, in contrast, usually have vast means for analytic investigation, but often lack clear arithmetic motivation. Let us consider the simplest case of a Dirichlet series of modular forms of integral weight for congruence subgroups K of the modular group SL2 (Z). Let us recall the corresponding definitions. A function F on the upper half-plane H = {x + iy ∈ C y > 0} is said to be a modular cusp form of weight k for the group K if it is holomorphic on H, equals zero at all cusps of K, and satisfies az + b ab = F(z) for each ∈ K. (cz + d)−k F cd ac + d All such functions form a finite dimensional space N = Nk (K) over the field C of complex numbers. If the group K contains the matrix 10 11 , then every F ∈ N can be presented by a Fourier series absolutely convergent on H of the form ∞

F(z) =

∑ f(m)e2π imz

m=1

with constant Fourier coefficients f(m). Let us associate with F the Dirichlet series of F defined by ∞ f(m) Z(F, s) = ∑ . s m=1 m The series converges absolutely in a right half-plane of the variable s, and can be presented there by means of a Mellin integral

Φ (s) = Φ (F, s) =

∞ 0

F(iy)ys−1 dy = (2π )−sΓ (s)Z(F, s).

Suppose that the group K satisfies −1 0 −1 0 −1 K =K q 0 q 0

4

Introduction

for a positive integer q. Then it is easy to check that for each cusp form F ∈ N, the function (F ω )(z) = q−k/2 z−k F(−1/qz) again belongs to N and satisfies F ω ω = (−1)k F. It follows that one can write the direct sum decomposition N = N+ + N− , where for F ∈ N± , F ω = ±ik F. Exercise. Prove the above assertions. If F ∈ N± , then F(i/qy) = ±(−1)k qk/2 yk F(iy)

(y > 0),

and we can write, for ℜs sufficiently large, that

Φ (s) =

q−1/2 0

∞

=

F(iy)ys−1 dy +

q−1/2

∞ q−1/2

F(iy)ys−1 dy

F(i/qy)(1/qy)s−1 (1/qy2 )dy +

= ±(−1)k qk/2−s

∞ q−1/2

∞

F(iy)yk−s−1 dy +

q−1/2 ∞

F(iy)ys−1 dy

q−1/2

F(iy)ys−1 dy.

Both of the last integrals are holomorphic for all s, and so is the function Φ (s). Moreover, the last expression implies that

Φ (k − s) = ±(−1)k qs−k/2 Φ (s), which is the functional equation for the Dirichlet series Z(F, s). Note that the simplest of the groups K satisfying the given conditions is the group

ab Γ0 (q) = ∈ SL2 (Z) c ≡ 0 (mod q) . cd The problem of Euler product factorization of the Dirichlet series corresponding to modular forms of integral weight for the groups of the type Γ0 (q) was essentially solved by Erich Hecke in 1937 and completed by A.O.L. Atkin and J. Lehner in 1970. In particular, it was found that although the Dirichlet series of a cusp form does not necessarily have an Euler product factorization, the space of cusp forms has a basis consisting of forms with Dirichlet series decomposable into Euler products, which we call the zeta functions of modular forms. Such forms can be characterized as eigenfunctions of certain rings of linear operators, the Hecke operators, acting on the spaces Nk (K). Since the nineteenth century, the main arithmetic application of modular forms had been the analytical theory of integral quadratic forms. The reason is that the generating Fourier series with coefficients equal to numbers of integral representations

Introduction

5

of positive integers by a positive definite integral quadratic form is a modular (not cusp) form. But in the middle of the twentieth century Goro Shimura and Yutaka Taniyama proposed famous conjectures relating modular forms and elliptic curves over Q. The Shimura–Taniyama conjecture includes the conjecture that the zeta function ζ ∗ (E, s) of every elliptic curve E over Q completed by appropriate p-factors for bad primes is, in fact, the zeta function of a cusp form of weight 2 for the group Γ0 (q), where q is the product of some degrees of bad primes. In 1985, G. Frey made the remarkable observation that this conjecture would imply Fermat’s last theorem. The precise relation of the two was established later by K.A. Ribet, which allowed A. Wiles in 1995 to prove the Fermat’s last theorem, one of the brightest achievements of mathematics of the twentieth century. One can hardly doubt that the relation between zeta functions of elliptic curves and zeta functions of modular forms in one variable described by the Shimura–Taniyama conjecture is only a particular case of some general links between global zeta functions of algebraic varieties and zeta functions of automorphic forms. Speaking of abelian varieties in place of elliptic curves, one can expect that modular forms in one variable should be replaced by Siegel modular forms for congruence subgroups of the symplectic modular group Γ n = Spn (Z), and this expectation is supported by some numerical evidence.

Chapter 1

Modular Forms

1.1 The Symplectic Group and the Upper Half-Plane Symplectic Matrices. A matrix M ∈ C2n 2n is said to be symplectic if it satisfies the relation 0 1n t (1.0) MJM = µ (M)J with J = Jn = −1n 0 and a nonzero scalar µ (M) called the multiplier of M, where 0 = 0n and 1n are the zero matrix and the unit matrix of order n, respectively. It is clear that the product of two symplectic matrices of the same order is again a symplectic matrix, and the multiplier of a product is the product of multipliers of factors. Lemma 1.1. Let M = CA DB , where the blocks A, B, C, and D are complex square matrices of order n, and let µ be a nonzero complex number. Then the following conditions are equivalent: (1) M is symplectic with the multiplier µ (M) = µ ; (2) tM is symplectic with the multiplier µ (tM) = µ ; (3) M is invertible, and t D −tB −1 ; µM = −tC tA

(1.1)

(4) the blocks A, B, C, and D satisfy the conditions t

AC = tCA,

BD = tDB,

and

t

AD −t CB = µ 1n

(1.2)

C tD = D tC,

and

A tD − B tC = µ 1n .

(1.3)

t

or the conditions A tB = B tA,

Proof. This is an easy but useful exercise on multiplication of block–matrices, which we leave to the reader.

A. Andrianov, Introduction to Siegel Modular Forms and Dirichlet Series, Universitext, DOI 10.1007/978-0-387-78753-4 1, c Springer Science+Business Media LLC 2009

7

8

1 Modular Forms

Exercise 1.2. Prove the lemma. In the course of our arithmetic considerations we shall be interested in discrete subgroups and subsemigroups of the general real positive symplectic group of genus n consisting of all real symplectic matrices of order 2n with positive multipliers: 2n t G = Gn = GSp+ MJM = µ (M)J, µ (M) > 0 . (1.4) n (R) = M ∈ R2n Action on Upper Half-Plane. The group G is a real Lie group acting as a group of analytic automorphisms on the n(n + 1)/2−dimensional open complex variety (1.5) H = Hn = Z = X + iY ∈ Cnn tZ = Z, Y > 0 , called the upper half-plane of genus n, with the action defined by the rule AB GM= : Z → M Z = (AZ + B)(CZ + D)−1 (Z ∈ H). CD

(1.6)

In order to verify this, we have to check that the mapping (1.6) is always defined, maps the upper half-plane into itself, and satisfies (MM ) Z = M M Z

(M, M ∈ G, Z ∈ H).

(1.7)

The relations (1.7) follow from the definition by a formal comparison of both parts, provided that both parts are defined. So it would be sufficient to prove the following two lemmas. Lemma 1.3. For every matrices M = CA DB ∈ G and Z ∈ H, the matrices J(M, Z) = CZ + D

(1.8)

are nonsingular and satisfy the rule J(MM , Z) = J(M, M Z )J(M , Z)

(M, M ∈ G, Z ∈ H).

(1.9)

Proof. The relation (1.9) formally follows from the definitions if the matrix J(M , Z) is nonsingular. As to nonsingularity, we note first that the matrix (1.8) is nonsingular for every M = CA DB ∈ G and Z = i1n . Indeed, otherwise, the matrix (Ci + D) t(Ci + D) = C tC + D tD + i(C tD − D tC) = C tC + D tD (see (1.3)) is singular, and so since it is symmetric and semidefinite, there is a nonzero real n-column T such that tT (C tC + D tD)T = 0, whence tTC tCT = 0 and tT D tDT = 0, and so tTC = tT D = 0. The last relations imply that the rank of the matrix (C, D) is less than n, which is impossible, since the matrix M is nonsingular.

1.1 The Symplectic Group and the Upper Half-Plane

9

Now note that each matrix Z = X +iY ∈ H can be written in the form Z = M i1n

with a matrix M ∈ G. It is sufficient to take M =

A1 X tA−1 1 0

tA−1 1

, if Y = A1 tA1 . Finally,

by (1.9) with Z = i1n , we get J(MM , i1n ) = J(M, M i1n )J(M , i1n ) = J(M, Z)J(M , i1n ), which implies that J(M, Z) is nonsingular. Lemma 1.4. Let M = CA DB ∈ G and Z = X +iY ∈ H then the matrix Z = X +iY = M Z is symmetric, and Y = µ (M) t(CZ + D)−1Y (CZ + D)−1 .

(1.10)

In particular, Z ∈ H. Proof. As follows easily from the relations (1.2), the matrix t

(CZ + D)M Z (CZ + D) = (Z tC + tD)(AZ + B) = Z tCAZ + tDAZ + Z tCB + tDB

is symmetric, and so the matrix Z = M Z is symmetric too. Further, by (1.2), we have

(CZ + D)(Z − Z )(CZ + D) = (Z tC + tD)(AZ + B) − (Z tA + tB)(CZ + D)

t

= Z(tCA −t AC)Z + (tDA −t BC)Z + Z(tCB −t AD) = µ (M)(Z − Z). The formula (1.10) and the lemma follow. The above formulas will allow us to find a G-invariant element of volume on H. The upper half-plane H is clearly an open subset of the n(n + 1)-dimensional real affine space, and we can consider the Euclidean element of volume on H, dZ =

∏

1≤α ≤β ≤n

dxαβ dyαβ

(Z = (xαβ + iyαβ ) ∈ H).

(1.11)

Lemma 1.5. For each matrix M = CA DB ∈ G, the element of volume (1.11) satisfies the relation dM Z = µ (M)n(n+1) | det(CZ + D)|−2n−2 dZ. + iy ) = M Z . Proof. For Z = (zαβ ) = (xαβ + iyαβ ) ∈ H, we set Z = (zγδ ) = (xγδ γδ , y with We have to compute the absolute value of the Jacobian of the variables xγδ γδ respect to the variables xαβ , yαβ . First, we shall consider the transformation of the differentials of the complex variables zαβ . For Z1 , Z2 ∈ H, since Z2 is symmetric, we get

Z2 − Z1 = (Z2 tC + tD)−1 (Z2 tA + tB) − (AZ1 + B)(CZ1 + D)−1 = µ (M)(Z2 tC + tD)−1 (Z2 − Z1 )(CZ1 + D)−1 ,

10

1 Modular Forms

where we have also used the relations (1.2). It follows that D Z = µ (M)(Z tC + tD)−1D Z(CZ + D)−1 , where D Z = (dzαβ ) and D Z = (dzγδ ) are the matrices of differentials of the variables zαβ and zγδ , respectively. Note that if ρ (U) with U ∈ GLn (C) is the transformation (vαβ ) → U(vαβ ) tU of variables vαβ = vβ α with 1 ≤ α , β ≤ n, then det ρ (U) = (detU)n+1 . This can be easily checked by ordering the variables vαβ lexicographically and replacing U by an upper triangular matrix of the form W −1UW . Let d Z and d Z be the columns with entries dzαβ (1 ≤ α , β ≤ n) and dzγδ (1 ≤ γ , ≤ δ ≤ n) arranged in a fixed order. Then the above considerations imply the relation

d Z = ρ ( µ (M) t(CZ + D)−1 )dd Z.

Taking d Z = d X + iddY , d Z = d X + iddY , and ρ ( µ (M) t(CZ + D)−1 ) = R + iS, we obtain that d X = Rdd X − SddY, d Y = Sdd X + RddY. Thus, the Jacobian equals R −S 1n −i1n R −S 1n i1n det = det 0 1n 0 1n S R S R R + iS 0 = det = µ (M)n(n+1) | det(CZ + D)|−2n−2 . 0 R − iS By combining the above lemma and formula (1.10), we get the following result. Proposition 1.6. The element of volume on H given by d ∗ Z = detY −(n+1) dZ

(Z = X + iY ∈ H),

(1.12)

where dZ = dXdY is the Euclidean element of volume (1.11), is invariant under all transformations of the group G : d ∗ M Z = d ∗ Z

(M ∈ G).

It is easy to see that two matrices M, M of G have the same action (1.6) on an open subset of H if and only if M = λ M with λ in the set R∗ of nonzero real numbers. It follows that the group of all transformations of the upper half-plane of the form (1.6) is isomorphic to the factor groups G/{R∗ 12n } S/{±12n }, where

S = Sn = Spn (R) = M ∈ Gn µ (M) = 1

(1.13)

1.2 Fundamental Domains for the Modular Group

11

is the (real) symplectic group of genus n. We have already seen in the proof of Lemma 1.3 that each matrix Z ∈ H can be written in the form Z = M iE with a matrix M ∈ S. Therefore, the upper half-plane can be identified with the homogeneous space of the symplectic group by the stabilizer U of the point iE in S. More precisely, we have the following lemma. Lemma 1.7. The map M → M i1n defines a one-to-one correspondence S/U ↔ H, which is compatible with the actions of the group S, where on the left side it acts by multiplication from the left; the stabilizer U has the form

A B n n ∈S ; U=U = −B A A B the map −B A → A + iB is an isomorphism of U onto the unitary group of order n; in particular, the group U is compact. Exercise 1.8. Prove the lemma and preceding assertions. Exercise 1.9. Show that the Cayley mapping Z → W = (Z − i1n )(Z + i1n )−1

(Z ∈ H),

is an analytic isomorphism of H onto the bounded domain W ∈ Cnn tW = W, WW < E , where the inequality is understood in the sense of Hermitian matrices. Show that the inverse mapping is given by W → Z = i(1n +W )(1n −W )−1 .

1.2 Fundamental Domains for the Modular Group Modular Group. The modular (symplectic) group or the Siegel modular group of genus n, i.e., the group of all integral symplectic matrices of order 2n with unit multiplier, (1.14) Γ = Γn = Spn (Z) = Sn Z2n 2n , is clearly a discrete subgroup of the symplectic group S. The same is true for each subgroup K of S commensurable with the group Γ, i.e., such that the intersection K ∩ Γ is of finite index both in K and Γ. Lemma 1.7 implies then that each such group K acts discretely on the upper half-plane. Automorphic forms for subgroups K of the symplectic group, which we are going to consider in this chapter, are functions on the upper half-plane having certain analytic properties and satisfying functional equations connecting its values at points of each K-orbit

12

1 Modular Forms

K Z = M Z M ∈ K

(Z ∈ H)

of K on H, so that such a function is uniquely determined by its restriction to any subset of H that meets each K-orbit. We recall that a closed subset D of a topological space X is called a fundamental domain for a discrete transformation group G acting on X if it meets each of the G-orbits G(x) = {g(x)|g ∈ G} with x ∈ X and has no distinct inner points belonging to the same orbit. It follows from the definition that the decomposition g(D) with G = g ∈ G g(x) = x, ∀x ∈ X (1.15) X= g∈G/G

holds, and its components pairwise have no common inner points. Fundamental domains do not necessarily exist. Minkowski Reduction Domain. The construction of fundamental domains for the modular symplectic group is essentially based on the Minkowski reduction theory of positive definite quadratic forms. In matrix language, the problem of reduction of positive definite quadratic forms relative to unimodular equivalence is that of construction of a fundamental domain for the unimodular group Λ = Λn = GLn (Z)

(1.16)

acting on the cone P = Pn = Y ∈ Rnn tY = Y,

Y >0

(1.17)

of real positive definite matrices of order n by ΛV :

Y → Y [V ] = tVYV.

The columns of a matrix U ∈ Λ will be denoted by u1 , . . . , un , so that U = (u1 , . . . , un ). In order to choose a special representative Y [U] of the orbit Λ(Y ) = Y [V ] V ∈ Λ of a point Y ∈ P, we determine the matrix U column by column with the help of some minimal conditions. Let Λr = Λnr = (u1 , . . . , ur ) ∈ Znr (u1 , . . . , ur , ∗, . . . , ∗) ∈ Λ be the set of all integral n × r−matrices composed of the first r columns of matrices in Λ. For a given Y ∈ P , we choose u1 ∈ Λ1 such that the value Y [u1 ] = tu1Y u1 of the quadratic form with the matrix Y on the column u1 is minimal; this can be done, since the form is positive definite. Next, we determine u2 such that (u1 , u2 ) ∈ Λ2 and the value Y [u2 ] is minimal. On replacing u2 by −u2 if necessary, one may assume

1.2 Fundamental Domains for the Modular Group

13

that tu1Y u2 ≥ 0. Proceeding in the same way, at the r−th step we choose ur such that (u1 , . . . , ur ) ∈ Λr , Y [uur ] is minimal, and tu r−1Yuur ≥ 0. Finally, when r = n, we get a unimodular matrix U = (u1 , . . . , un ) ∈ Λ and a matrix T = (tαβ ) = Y [U] ∈ Λ(Y ), which is called Minkowski reduced, or just reduced. Let us determine the conditions for a positive definite matrix to be reduced in terms of its entries. First of all, by induction on n based on the Euclidean algorithm, one can easily prove the following. Lemma 1.10. An integral n-column u belongs to Λ1 if and only if its entries are coprime. Also, as an easy exercise on multiplication of block matrices, we get the following lemma. Lemma 1.11. Two matrices U, U of Λn have the same first r columns if and only if 1 B with D ∈ Λn−r and B ∈ Zn−r U = U r r . 0 D Let U = (u1 , . . . , un ) ∈ Λn . By Lemma 1.11, the set of all r−th columns of all matrices U ∈ Λn with the first columns u1 , . . . , ur−1 coincides with the set of columns of the form Uv, where v is an integral n-column whose last n−r +1 entries vr , . . . , vn form the first column of a matrix D ∈ Λn−r+1 . By Lemma 1.10, the last condition means that the numbers vr , . . . , vn are coprime. Thus, if U = (u1 , . . . , un ) ∈ Λn and 1 ≤ r ≤ n, then (1.18) u ∈ Zn (u1 , . . . , ur−1 , u) ∈ Λnr = UVr,n , where

Vr,n = v =t (v1 , . . . , vn ) ∈ Zn gcd(vr , . . . , vn ) = 1 .

By the definition and (1.18), we conclude that a matrix T = (tαβ ) = Y [U] is reduced if and only if it satisfies the conditions Y [Uv] ≥ Y [ur ],

for all v ∈ Vr, n and 1 ≤ r ≤ n,

and ur−1Y ur ≥ 0,

t

for 1 < r ≤ n,

where U = (u1 , . . . , un ). Since Y [U] = T = (tαβ ), we have Y [ur ] = trr and = tr−1,r . Hence, the above conditions mean exactly that T belongs to the set M = Mn = {T = (tαβ ) ∈ Pn trr ≤ T [v] for all v ∈ Vr,n (1 ≤ r ≤ n);

tu r−1Y ur

tr−1,r ≥ 0 (1 < r ≤ n)}, called the Minkowski reduction domain.

(1.19)

14

1 Modular Forms

Theorem 1.12. Every orbit Λ(Y ) of the group Λ on P contains at least one and not more than finitely many points of the Minkowski domain M. If T , T are two inner points of M, and T = T [U] with U ∈ Λ, then U = ±1n ; in particular, no different inner points of M belong to the same orbit. In other words, M is a fundamental domain of Λ on P. Proof. The above consideration shows that for a given Y ∈ P, there exists U ∈ Λ such that Y [U] ∈ M, and every column of such U can be chosen in finitely many ways. Let us set M = Mn = {T ∈ Pn trr < T [v], v ∈ Vr,n , v = ±er (1 ≤ r ≤ n); tr−1,r > 0 (1 < r ≤ n)}, where e1 , . . . , en are the columns of the unit matrix 1n . It is clear that M ∈ M and ) belong to M each inner point of M is contained in M . If T = (tαβ ) and T = (tαβ = T [u ] ≥ t , and T = T [U] with U = (u1 , . . . , un ) ∈ Λ, then u1 ∈ V1, n , whence t11 1 11 and similarly, t11 ≥ t11 . It follows that t11 = t11 = T [u1 ], and so u1 = ±e1 . Then u2 ∈ V2, n , and in the same way we conclude that u2 = ±e2 . By repeating the same arguments, we see that ur = ±er for all r = 1, . . . , n. Now the conditions tr−1, r > 0,

t tr−1, r = ur−1 T ur > 0 (1 < r ≤ n)

imply that ur = er or ur = −er for r = 1, . . . , n, and T = T . The entries of reduced matrices T = (tαβ ) satisfy some useful inequalities. First of all, since trr ≤ T [er+1 ] = tr+1,r+1 , it follows that t11 ≤ t22 ≤ · · · ≤ tnn .

(1.20)

Then by tll ≤ T [er ± el ] = trr ± 2trl + tll , where 1 ≤ r < l ≤ n, we obtain |2trl | ≤ trr

if r = l.

(1.21)

Finally, Minkowski have proved a deeper inequality for reduced matrices, which we cite without proof: t11t22 · · ·tnn ≤ cn det T

(T = (tαβ ) ∈ Mn ),

(1.22)

where cn is a positive constant depending only on n. The Minkowski inequality implies that every T = (tαβ ) ∈ Mn satisfies the inequality T≥

1 nn−1 cn

diag(t11 ,t22 , . . . ,tnn ).

Indeed, let ρ1 , . . . , ρn be the eigenvalues of the matrix −1/2

T = T [diag(t11

−1/2

,t22

−1/2

, . . . ,tnn

)],

(1.23)

1.2 Fundamental Domains for the Modular Group

15

then ρ1 + · · · + ρn = n, and by (1.22),

ρ1 · · · ρn = (t11t22 · · ·tnn )−1 det T ≥ 1/cn ; it follows that ρα ≤ n and ρα ≥ 1/nn−1 cn for α = 1, . . . , n, which implies (1.23). Exercise 1.13. Show that

t11 t12 ∈ P2 0 ≤ 2t12 ≤ t11 ≤ t22 . M2 = t12 t22 Show that in the inequalities (1.22) one can take c1 = 1 and c2 = 4/3, and the values are minimal. 1 1/2 [Hint: For minimality of c2 , consider T = 1/2 1 ]. Exercise 1.14. Two binary quadratic forms f(x, y)and f (x, y) are said to be equivalent if f (x, y) = f (α x + β y, γ x + δ y) with αγ βδ ∈ Λ2 . Show that the number of classes of equivalent positive definite quadratic forms f (x, y) = ax2 + bxy + cy2 with integral coefficients a, b, c and a fixed discriminant d = b2 − 4ac is finite. Construction of Fundamental Domains. Let us return to the action of the modular group (1.14) on the upper half-plane. We consider orbits Γ Z of the modular group on H. For Z = X + iY ∈ H = Hn , we shall call the positive real number detY the height of the point Z and denote it by h(Z). By (1.10), we have AB −2 Z ∈ H, M = ∈Γ . (1.24) h(M Z ) = | det(CZ + D)| h(Z) CD Lemma 1.15. Each orbit of the group Γ on H contains points Z of maximal height. These points can be characterized by the inequalities ∗ ∗ | det(CZ + D)| ≥ 1 for every ∈ Γ. CD Proof. In view of (1.24) we have to show that | det(CZ + D)| takes a minimum on each orbit. Note that for any M = CA DB ∈ Γ and V ∈ Λ = Λn , the product t −1 t −1 t −1 V 0 V A V B M= 0 V VC V D also belongs to Γ. It follows that if (C, D) is the ”second row” of a matrix of Λ, then the same is true for (VC, V D). Replacing M in M X + iY = X + iY

16

1 Modular Forms

by the above product does not change the value | det(CZ + B)| and replaces the matrix (Y )−1 by the matrix tV (Y )−1V . Therefore, we may assume that the positive definite matrix (Y )−1 is Minkowski reduced. Let us denote by cr and dr (r = 1, . . . , n) the columns of the matrices tC and X tC + tD, respectively, and by t1 , . . . ,tn the diagonal entries of (Y )−1 . Then, by (1.10), we can write (Y )−1 = (CZ + D)Y −1 (Z tC + tD) = (CX + D)Y −1 (X tC + tD) +CY tC, whence for r = 1, . . . , n, we get tr = Y

−1

Y [cr ] [dr ] +Y [cr ] ≥ Y −1 [dr ] .

(1.25)

If for some r, the columns cr and dr are both zero, then the r−th column of the matrix (C D) is also zero, which is impossible, since M is nonsingular. Since Y > 0, the value Y [cr ] assumes a positive minimum when Z is fixed and cr is an arbitrary nonzero integral column. On the other hand, if cr = 0, then dr is the r−th column of tD and so is a nonzero integral column. Since Y −1 > 0, the value Y −1 [dr ] also assumes a positive minimum. It follows then from (1.25) that the numbers u1 , . . . , un have a positive lower bound independent of M. The relations (1.22) and (1.24) imply the inequality t1t2 · · ·tn ≤ cn (detY )−1 = cn (detY )−1 | det(CZ + D)|2 . If we now assume that a condition | det(CZ + D)| ≤ h is satisfied for an arbitrarily large number h, then it implies upper bounds for t1 ,t2 , . . . ,tn . Then from (1.25) we obtain upper bounds for entries of the columns cr and dr and hence for the entries of the matrices C and D. Therefore, the condition is satisfied only for finitely many pairs (C, D) if Z is fixed and h is a given large number. This proves the lemma. Theorem 1.16. Let D = Dn be the subset of matrices Z = X + iY ∈ Hn satisfying the following conditions: (1) | det(CZ + D)| ≥ 1 for every ( C∗ D∗ ) ∈ Γ = Γn ; reduction domain (1.19); (2) Y ∈ Mn , where Mn is the Minkowski t n (3) X ∈ Xn = X = (xαβ ) ∈ Rn X = X, |xαβ | ≤ 1/2 (1 ≤ α , β ≤ n) . Then D meets each Λ-orbit on H, and Z = M Z for two inner points Z, Z ∈ D with M ∈ Γ if and only if M = ±E, i.e., D is a fundamental domain of Γ on H. If Z = X = iY ∈ D, then Y ≥ bn E

and

σ (Y −1 ) ≤ n/bn ,

(1.26)

where bn is a positive constant depending only on n, and σ denotes the trace. The volume of D with respect to the invariant element of volume (1.12) is finite.

1.2 Fundamental Domains for the Modular Group

17

Proof. Let us consider the orbit Γ Z of a point Z ∈ H. By Lemma 1.15, the orbit contains a point Z = X +iY of maximal height, and the point satisfies the condition (1). Every transformation of the form t V SV −1 = tV Z V + S = X [V ] + S + iY [V ] Z → 0 V −1 with V ∈ Λ = Λn and an integral symmetric matrix S of order n corresponds to a matrix of Γ and does not change the height of Z . By Theorem 1.12, there is a matrix V ∈ Λ such that the matrix Y = Y [V ] belongs to M n . Also, clearly, there is S such that X [V ] + S ∈ Xn . Then Z = X + iY ∈ Γ Z

D.

Suppose now that Z = M Z for two points Z, Z ∈ D with M = CA DB ∈ Γ. Then h(Z ) = h(Z), by Lemma 1.15. It follows from (1.24) that | det(CZ + D)| = 1. On the other hand, since Z = M −1 Z , we conclude that | det(− tCZ + tA)| = 1 (see (1.1)). If C = 0, then the equations are nontrivial, and so the points Z, Z belong to the boundary of D. If C = 0, then M has the form t V SV −1 with V ∈ Λ and S = tS ∈ Znn . M= 0 V −1 So we have Z = X + iY = X[V ] + S + iY [V ],

X + iY = Z,

where

and in particular, Y = Y [V ]. Since Y and Y are both in Mn , it follows from Theorem 1.12 that Y and Y are boundary points of Mn or V = ±E. In the last case we have X = X + S, whence S = 0, unless X and X belong to the boundary of Xn . We conclude that M = ±12n , unless Z and Z are boundary points of D. Let Z = (zαβ ) = (xαβ + iyαβ ) ∈ D. the inequality | det(CZ + D)| ≥ 1 for the pair 10 0 0 (C, D) = , 00 0 1n−1 2 + y2 ≥ 1. Since |x | ≤ 1/2, it follows that implies the inequality |z11 | = x11 11 11 √ i.e., y11 ≥ 3/2. The last inequality and the inequalities (1.20) imply y211 ≥ 3/4, √ for α = 1, . . . , n. The first inequality of (1.26) follows then from that yαα ≥ 3/2 √ (1.23) with bn = 3/2nn−1 cn . The inequality implies that each eigenvalue of Y −1 is not greater than 1/bn , which proves the second inequality. Finally, by (1.12), Theorem 1.16, and (1.26) we have

v(Dn ) =

Dn

(detY )−n−1 dXdY ≤

Y ∈Mn ,Y ≥bn E

(detY )−n−1 dY,

18

1 Modular Forms

which, by (1.20), (1.21), and (1.22), can be estimated as ≤ ≤

−1 −n−1 dY bn ≤y11 ≤y22 ≤···≤ynn ; (cn y11 y22 · · · ynn ) |2yαβ |≤yαα (α =β )

y11 ,y22 ,··· ,ynn ≥bn

=c

n

∏

∞

α =1 cn

−n−1 (c−1 n y11 y22 · · · ynn )

n

α ∏ yn− αα

α =1

dy11 dy22 · · · dynn

y−α −1 dy < ∞.

Exercise 1.17. Prove that D1 is the so-called modular triangle,

1 2 2 D1 = z = x + iy ∈ H |x| ≤ , |z| = x + y ≥ 1 , 2 and one can take b1 =

√ 3 2 .

Draw the modular triangle.

Two binary quadratic forms Q(x, y) = q11 x2 + q12 xy + q22 y2 and Q (x, y) = q11 x2 + q12 xy + q22 y2 are said to be properly equivalent (over Z) if

Q (x, y) = Q(ax + by, cx + dy)

with

ab ∈ Γ1 = SL2 (Z). cd

(1.27)

Exercise 1.18. Show that any real positive definite quadratic form Q is properly equivalent to a form Q whose coefficients satisfy the inequalities |q12 | ≤ q11 ≤ q22 . Show that the cone given by the inequalities in the space of the coefficients of the form contains no distinct inner points corresponding to properly equivalent forms. [Hint: Let ω and ω be the roots belonging to H of the equations Q(x, 1) = 0 and respectively. Show that (1.27) is equivalent with the conditions ab ∈ Λ; (q12 )2 − 4q11 q22 = (q12 )2 − 4q11 q22 and ω = M −1 ω with cd

Q (x, 1) = 0,

then use Theorem 1.16 and Exercise 1.17.] Theorem 1.19. Each subgroup of the symplectic group S = Sn of the form KM = M −1 KM, where K is a subgroup of finite index in the modular group Γ = Γn , and M belongs to the general symplectic group G = Gn , has a fundamental domain D(KM ) on H = Hn , and one can take D(KM ) =

ˇ γ ∈K\Γ

(M −1 γ ) D ,

(1.28)

1.3 Modular Forms

19

where D = D(Γ) is a fundamental domain of Γ , and γ ranges over a system of representatives of different left cosets of Γ modulo the subgroup Kˇ = K ∪ (−12n )K. The invariant volume of the fundamental domain is finite. Proof. First of all, the set (1.28) is closed, as a finite union of closed subsets. Next, if (M −1 γ ) Z and (M −1 γ ) Z are two inner points of the set belonging to the same orbit of KM , i.e., (M −1 γ ) Z = (M −1 δ M) (M −1 γ ) Z

= (M −1 δ γ ) Z

with δ ∈ K, one can assume that Z and Z are inner points of D. Then, by (1.7), γ Z = (δ γ ) Z , and so Z = Z , and γ = ±δ γ , by the definition of D, which implies that γ = γ and δ = ±12n . Finally, since D meets each Γ -orbit, we have

H=

δ ∈Γ/{±12n }

whence

δ D =

(δ γ ) D ,

ˇ ˇ δ ∈K/{±1 2n } γ ∈K\Γ

H = M −1 H =

((M −1 δ M)(M −1 γ )) D .

ˇ ˇ δ ∈K/{±1 2n } γ ∈K\Γ

This proves the decomposition (1.15), whence the set D(KM ) really meets each KM -orbit. Finally, the set (1.28) is a finite union of images of the domain D under symplectic transformations and hence together with this domain has finite volume.

1.3 Modular Forms Definition of Modular Forms. Acting on the upper half-plane H = Hn , the general symplectic group G = Gn operates also on complex-valued functions F on H by Petersson operators of integral weights k: n(n+1) AB GM= : F → F|k M = µ (M)nk− 2 j(M, Z)−k F(M Z ) CD = µ (M)nk−

n(n+1) 2

det(CZ + D)−k F((AZ + B)(CZ + D)−1 ),

(1.29)

where j(M, Z) = det J(M, Z) = det(CZ + D). n(n+1)

Note that the factor µ (M)nk− 2 is introduced for aesthetic reasons, because it simplifies a number of forthcoming formulas. In fact, it can be replaced by an arbitrary quasicharacter of G or just omitted without affecting essential properties of the operators. By Lemma 1.3, the Petersson operators map holomorphic functions to holomorphic, moreover, it follows from the definition and relations (1.7), (1.9) that the operators satisfy the rules F|k rM = rn(k−n−1) F|k M

for all r ∈ R, r = 0

(1.30)

20

1 Modular Forms

and for M, M ∈ Gn F|k MM = µ (MM )nk−

n(n+1) 2

j(MM , Z)−k F((MM ) Z )

= (µ (M)µ (M ))nk− = µ (M )nk−

n(n+1) 2

n(n+1) 2

( j(M, M Z ) j(M , Z))−k F(M M Z

)

j(M , Z)−k (F|k M)(M Z ) = (F|k M)|k M .

(1.31)

Let K be a subgroup of G commensurable with the modular group Γ = Γn , χ a character of K, that is, a multiplicative homomorphism of K into the nonzero complex numbers with kernel of finite index in K, and k an integer. A complexvalued function F on H is called a (Siegel) modular form of weight k and character χ for the group K if the following conditions are satisfied: (i) F is a holomorphic function in n(n + 1)/2 complex variables on H; (ii) for every matrix M ∈ K, the function F satisfies the functional equation F|k M = χ (M)F,

(1.32)

where |k is the Petersson operator of weight k; (iii) if n = 1, then every function F|k M with M ∈ Γ1 is bounded on each subset of H1 of the form H1ε = {x + iy ∈ H1 | y ≥ ε } with ε > 0. The set Mk (K, χ ) of all modular forms of weight k and character χ for the group K is clearly a linear space over the field C. Exercise 1.20. Let k > 2 be an integer. Show that the Eisenstein series Ek (z) =

∑

c,d∈Z (c,d)=(0,0)

1 (cz + d)k

(z ∈ H1 )

converges absolutely and defines a modular form of weight k and the unit character for Γ1 . [Hint: If S is a compact subset of H1 , then |α z+ β | ≥ b(|α |+|β |) for all α , β ∈ R and z ∈ S with a positive constant b. Since there are only 4r pairs of integers (c, d) with |c| + |d| = r, it follows that Ek (z) is dominated term by term on S by the series k b−k ∑∞ r=1 4r/r .] For q ∈ N, we shall denote by Γ (q) = Γn (q) = M ∈ Γn M ≡ 12n (mod q) the principal congruence subgroup of level q of the modular group. Considering matrices of Γ modulo q, we get a homomorphism of the modular group into the finite group of symplectic matrices of order 2n over the residue class ring Z/qZ with kernel Γ (q). It follows that Γ (q) is a normal subgroup of the modular group of finite index.

1.3 Modular Forms

21

A subgroup K of the symplectic group (1.13) is called a congruence subgroup if it contains a principal congruence subgroup as a subgroup of finite index. A character of such K is said to be a congruence character if it is trivial on a principal congruence subgroup contained in K. Proposition 1.21. Let K be a congruence subgroup of Sn , χ a congruence character of K, and M a matrix of Gn with rational entries then the group KM = M −1 KM

(1.33)

is again a congruence subgroup of the symplectic group, the character M −1 KM M → χM (M ) = χ (MM M −1 )

(1.34)

of this group is a congruence character, and the image F|k M of each modular form F ∈ Mk (K, χ ) under the Petersson operator |k M is a modular form of weight k and character χM for the group KM : F|k M ∈ Mk (KM , χM ). Proof. Let q be a positive integer such that Γn (q) ⊂ K and χ is trivial on Γn (q). It can be assumed that the matrix M is integral. Let µ (M) = q then it is easy to see that the group Γn (qq ) is contained in KM and the character χM is trivial on this group. It follows from definitions and Lemma 1.3 that the function F = F|k M is holomorphic on Hn . If M ∈ KM , then MM M −1 ∈ K, and by (1.31), we have F |k M = F|k M|k M = F|k MM M −1 M = F|k MM M −1 |k M = χ (MM M −1 )F|k M = χM (M )F . Finally, if n = 1 and V ∈ Γ1 , we can write F |kV = F|k MV and present the matrix MV in the form MV = W M1 with W ∈ Γ1 = SL2 (Z) and an upper triangular matrix M1 . Then the function F |kV = F|kW M1 = (F|kW )|k M1 together with F|kW is clearly bounded on each subset of H1 of the form H1ε with ε > 0. Exercise 1.22. Let k > 2 be an integer, a, b ∈ Z, and q ∈ N. Show that the Eisenstein series 1 Ek (z; (a, b)|q) = (z ∈ H1 ) ∑ (cz + d)k c,d∈Z, (c,d)=(0,0), (c,d)≡(a,b) (mod q)

converges absolutely and defines a modular form of weight k and the unit character for Γ1 (q) satisfying Ek (z; (a, b)|q)|k M = Ek (z; (a, b)M|q) for each M ∈ Γ1 . [Hint: Use Exercise 1.20.]

22

1 Modular Forms

Fourier Expansion of Modular Forms. Theorem 1.23. Let K be a congruence subgroup of Sn , and χ a congruence character of K. Then each modular form F ∈ Mk (K, χ ) has an expansion of the form F(Z) =

πi

f (A)e q σ (AZ) ,

∑ n

(1.35)

A∈E , A≥0

with constant coefficients f (A), where En = A = (aαβ ) ∈ Znn tA = A,

a11 , a22 , . . . , ann ∈ 2Z

is the set of “even” matrices of order n, σ denotes the trace, and where q = q(K, χ ) is a positive integer such that the group K contains a subgroup of the form

1n qB n t n (1.36) T(q) = T (q) = B = B ∈ Zn . 0 1n The series (1.35) converges absolutely on H and uniformly on each subset of H of the form with ε > 0; Hε = Hnε = X + iY ∈ Hn Y ≥ ε 12n (1.37) in particular, F is bounded on each of the subsets. The coefficients f (A) satisfy the relations πi

f ( tVAV ) = (detV )k χ (M)e− q σ (AVU) f (A) for every matrix M of the group K of the form −1 V U . M = M(U, V ) = 0 tV

(∀ A ∈ En ),

(1.38)

(1.39)

The expansion (1.35) is called the Fourier expansion of F, and the numbers f (A) with A ∈ En , A ≥ 0 are the Fourier coefficients of F. Proof. The functional equations (1.32) for matrices of the subgroup T(q) ⊂ K become F(Z + qB) = F(Z) (Z = (zαβ ) ∈ H, B = tB ∈ Znn ). This means that F is periodic of period q in each of the variables zαβ = zβ α . Since F is also holomorphic, it can be expanded in a Fourier series of the form 2π i aαβ xαβ , F(Z) = ∑ f (A ; Y ) exp q 1≤α∑ A 0. Now let n ≥ 2. It was found by Max Koecher that in this case, an analogue of condition (iii) of the definition of modular forms is fulfilled automatically (Koecher’s effect). We use the same arguments. Let Λ be the group of all matrices V ∈ Λ = SLn (Z) such that the matrix M(0, V ) of the form (1.39) with U = 0 belongs to the

24

1 Modular Forms

kernel of the character χ . Since χ is a congruence character, the group Λ has finite index in Λ. By (1.38), we have f ( tVAV ) = f (A), for all A ∈ En and V ∈ Λ . Hence the expansion (1.40) can be rewritten in the form F(Z) =

∑

A∈E/Λ

f (A)η (Z, {A}),

where the sum is extended over a system of representatives for the classes {A} = { tVAV |V ∈ Λ } of the set E modulo the equivalence A ∼ tVAV with V ∈ Λ , and where πi η (Z, {A}) = ∑ e q σ (A Z) . A ∈{A}

If f (A) = 0, then the series f (A)η (Z, {A}) converges absolutely for every Z ∈ H, because it is a partial sum of an absolutely convergent series; in particular, the series

η (i1n , {A}) =

∑

π

e− q σ (Ai )

Ai ∈{A}

is convergent. Since the trace σ (Ai ) of every Ai ∈ {A} is a rational integer, it follows that the class {A} cannot contain more than a finite number of matrices Ai with σ (Ai ) < 0. On the other hand, we shall show now that the trace σ (Ai ) assumes infinitely many negative values on the class {A} of any integral symmetric matrix A of order n ≥ 2, unless A ≥ 0. If A does not satisfy A ≥ 0, then there is an integral n-column h such that thAh < 0. For r ∈ Z we set Vr = 1n + rH

with H = (t1 h, . . . ,tn h) ∈ Znn ,

where t1 , . . . ,tn are some integers. Since the rank of the matrix rH is equal to 1 or 0, it follows that detVr = 1 + σ (rH) = 1 + r(t1 h1 + · · · + tn hn ), where hα are entries of h. Since n ≥ 2, there are integers t1 , . . . ,tn not all equal to 0 such that t1 h1 + · · · + tn hn = 0. Then we have Vr ∈ Λ,

H 2 = ((t1 h1 + · · · + tn hn )hα tβ ) = 0,

and Vr = V1r .

Since the index of Λ in Λ is finite, it follows that the matrix Va = V1a belongs to Λ for some a ∈ N, and so do the matrices Vab = Vab for every integer b, hence the matrices tVab AVab belong to the class {A}, and σ tVab AVab = σ (A) + 2abσ (AH) + a2 b2 σ ( tVAV ) = σ (A) + 2abσ (AH) + a2 b2 ( thAh)(t12 + · · · + tn2 ). The last expressions is a polynomial in b of degree 2 with negative leading coefficient. Hence it takes infinitely many negative values on Z.

1.3 Modular Forms

25

The above considerations show that a coefficient f (A) in (1.30) equals zero, unless A ≥ 0, which proves the expansion (1.35). Finally, it is easy to see that the series (1.35) is majorized on each set (1.37) by a convergent series with nonnegative constant coefficients, and so converges there uniformly. Exercise 1.24. Show that if k > 2 is even, then the Fourier expansion of the Eisenstein series Ek (z) defined in Exercise 1.20 has the form ∞ (2π i)k ∞ 1 k−1 e2π inz . Ek (z) = 2 ∑ k + 2 ∑ ∑d (k − 1)! n n=1 n=1 d|n [Hint: Show first that 1

π2

∞

(2π )2t

∑ (z + d)2 = sin2 π z = (1 − t)2 = (2π i)2 ∑ dt d ,

d∈Z

d=1

where t = e2π iz . Then differentiate both parts k − 2 times.] Cusp Forms. If K is a congruence subgroup, χ is a congruence character of K, and M is a matrix of the general symplectic group (1.4) with rational entries, then by Proposition 1.21 and Theorem 1.23, the function F|k M has a Fourier expansion of the form πi (1.42) (F|k M)(Z) = ∑ fM (A)e q σ (AZ) A∈En , A≥0

with a positive integer q depending on K, χ , and M, which converges absolutely on H and uniformly on the subsets (1.37). The modular form F is called a cusp form if the coefficients fM (A) of the decomposition (1.42) satisfy the conditions fM (A) = 0 for all M ∈ G(Q) and A ∈ E with det A = 0,

(1.43)

where G(Q) = Gn (Q) = Gn ∩ Q2n 2n . The subspace of cusp forms of Mk (K, χ ) will be denoted by Nk (K, χ ). Proposition 1.25. Let K be a congruence subgroup of S = Sn and χ a congruence character of K. Then for each cusp form F ∈ Nk (K, χ ) and each matrix M ∈ Gn (Q), the modular form F|k M ∈ Mk (KM , χM ), where KM = M −1 KM and χM is the character (1.34), is a cusp form with a Fourier expansion of the form (F|k M)(Z) =

∑ n

πi

fM (A)e q σ (AZ) ,

(1.44)

A∈E , A>0

where q is a positive integer. If k ≥ 0, then the form F|k M satisfies |(F|k M)(X + iY )| ≤ c(detY )k/2

(X + iY ∈ H),

(1.45)

26

1 Modular Forms

and its Fourier coefficients satisfy | fM (A)| ≤ c (det A)k/2

for all A ∈ E

with A > 0,

(1.46)

where c and c are constants depending only on F and M. First we shall prove the following simple lemma. Lemma 1.26. Let

φ (Y ) =

∑

ϕ (A)e−ησ (AY ) ,

A∈En , A>0

where Y belongs to the cone P = Pn of positive definite matrices of order n, and η > 0, be a series with nonnegative coefficients ϕ (A) convergent for all Y ∈ P. Then for every Minkowski reduced matrix Y satisfying Y ≥ d · 1n with d > 0, the estimate

φ (Y ) ≤ d e−d

σ (Y )

holds with positive constants d and d . Proof of the lemma. If Y = (yαβ ) ∈ P and A = (aαβ ) ∈ E satisfies A > 0, then by (1.23),

σ (AY ) ≥ bσ (A diag(y11 , y22 , . . . , ynn )) = b

n

∑ aαα yαα ≥ 2bσ (Y ),

α =1

where b is a positive constant depending only on n. On the other hand, if Y > dE, then for A = (aαβ ) ∈ E with A > 0, we obtain σ (AY ) ≥ d σ (A). (Note that we have used twice the obvious inequality σ (AR) ≥ σ (BR) valid if matrices A − B and R are positive semidefinite.) It follows then from the above inequalities that 1 σ (AY ) ≥ bσ (Y ) + d σ (A), 2 hence

φ (Y ) ≤

∑

ϕ (A)e−η (bσ (Y )+ 2 d σ (A)) = e−η bσ (Y ) φ 1

A∈E, A>0

1 dE , 2

which proves the estimate. Proof of Proposition 1.25. The Fourier expansion (1.44) and the inclusion F|k M ∈ Nk (KM , χM ) follow from Proposition 1.21 and the definition of cusp forms. By Proposition 1.21 there exists a level q ∈ N such that the principal congruence subgroup Γ (q) is contained both in KM and the kernel of character χ . Let us consider the function (1.47) G = ∑ |F |k γi |, where F = F|k M. γi ∈Γ (q)\Γ

1.3 Modular Forms

27

It follows from (1.31) and (1.32) that G is independent of the choice of representatives in the cosets Γ (q)\Γ and satisfies G|k γ =

∑

|F |k γi γ | = G

for all γ ∈ Γ ,

(1.48)

γi ∈Γ (q)\Γ

because γi γ runs through a system of representatives for Γ (q)\Γ when γi does. Hence by (1.10), we see that the function H(Z) = H(X + iY ) = (detY )k/2 G(Z) = (detY )k/2

∑

|F |k γi |

(1.49)

γi ∈Γ (q)\Γ

satisfies H(γ Z ) = H(Z)

for all Z ∈ H

and γ ∈ Γ.

(1.50)

Since by Theorem 1.21, each of the functions F |k γi = F|k M γi is bounded on the sets Hε with ε > 0, it follows from (1.26) that the functions are bounded on the fundamental domain D of Γ , defined in Theorem 1.16. Furthermore, each of the these functions satisfies π |F |k γi | ≤ ∑ | fMγi (A)|e− q σ (AY ) , A∈E, A>0

and the last series converges on the cone P = Pn . If Z ∈ D, then Y is Minkowski reduced and satisfies Y ≥ bn 1n ; hence by Lemma 1.24, each of the series is dominated by a function of the form d exp(−d σ (Y )) with constants d and d depending on F and M. Therefore, since k ≥ 0, we obtain H(X + iY ) ≤ d [Γ : Γ(q)](detY )k/2 exp(−d σ (Y )) ≤δ

n

∏ yαα exp(−d yαα ), k/2

α =1

where yαα are the diagonal entries of Y , and we have also used the following consequence of the Hadamard’s determinant theorem: detY ≤ y11 y22 · · · ynn

(Y ∈ Pn ).

(1.51)

Since the last expression is clearly bounded on P, it follows that H is bounded on D. Thus by (1.50), H is bounded on H, which implies the estimate (1.45). The estimate for Fourier coefficients follows from (1.45). One can prove that the coefficients of the Fourier expansion (1.35) of an arbitrary modular form of nonnegative weight k and congruence character for a congruence subgroup of Sn satisfy | f (A)| ≤ c(det A)k where c depends only on the form.

(A ∈ En , A ≥ 0),

(1.52)

28

1 Modular Forms

The Siegel Operator. The general philosophy of modular forms emanates from the idea that consideration of arbitrary modular forms can usually be reduced to the case of cusp forms and the cases of modular forms of smaller genera. The reduction is ensured by the so-called Siegel operator and its iterations. Let F = F(Z) be a modular form of weight k for a congruence subgroup K of the symplectic group S = Sn and a congruence character χ of K. Since the Fourier series (1.25) for F converges uniformly on subsets of Hn of the form (1.27), the limit πi Z 0 (F|Φ)(Z ) = lim F f (A) lim e q σ (AZy ) , (1.53) = ∑ 0 iy y→+∞ y→+∞ A∈En , A≥O where Zy =

Z 0 0 iy

, exists for every Z ∈ Hn−1 . If A =

σ (A Z ) + iyann , hence lim e

π i σ (AZ ) y q

y→+∞

− πq yann πqi σ (A Z )

= lim e

e

y→+∞

A ∗ ∗ ann

πi e q σ (A Z ) , = 0,

Since A ≥ 0, the equality ann = 0 implies that A has the form

∑

(F|Φ)(Z ) =

f

A ∈E(n−1) , A ≥0

, then σ (AZy ) =

if ann = 0, if ann > 0. A 0 0 0

. Thus we have

πi A 0 e q σ (A Z ) , 0 0

(1.54)

for all Z ∈ H(n−1) . The last series is a partial series for the Fourier expansion of (n−1) F, and so it converges absolutely on H(n−1) and uniformly on Hε with ε > 0. If n = 1, we set (1.55) F|Φ = lim F(iy) ∈ C. y→+∞

As above, the limit exists and is equal to the constant term of the Fourier expansion of F. The linear operator (F ∈ Mk (K, χ ))

Φ : F → F|Φ

(1.56)

is called the Siegel operator. (n−1) , we shall In order to show that the function F|Φ is a modular form on H need new notation. Let n > 1. For a matrix M = CA DB with square blocks A , B ,

C , and D of order n − 1, we set − → AB M = CD where A =

A 0 0 1

,B=

B 0 0 0

,C=

− → A B = M, φ (M ) = C D

and

← − denote by S the set of all matrices to S:

(1.57)

C 0 , D = D 0 . If S is a subgroup of Gn , we 0 0 0 1 − → M = CA DB such that the matrix M belongs

1.3 Modular Forms

29

− → ← − A B 2n−2 ∈ R2n−2 M ∈ S . S = M = C D ← − Then it is clear that S is a subgroup of S(n−1) , and the map ψ given by

− → ← − S M → ψ (M ) = M ∈ S

(1.58)

← − is a homomorphic embedding of the group S into S. Lemma 1.27. Let K be a congruence subgroup of Sn with n > 1, and χ a congru← − ence character of K. Then the group K is a congruence subgroup of G(n−1) , and − given by χ the map ← → ← − −(M ) = χ (− K M → ← χ M) (1.59) ← − is a congruence character of the group K . Proof. By the assumptions, K contains a principal congruence subgroup Γ (q) = ←−− Γ n (q) of finite index belonging to kernel of χ . Then the group Γ (q) is clearly a ← − −. It principal congruence subgroup of K belonging to the kernel of the character ← χ ←−− ←−− ←−− ← − remains to prove that Γ (q) has finite index in K . Indeed, if Γ (q)Mα = Γ (q)Mβ , −→ −→ ← − where Mα , Mβ belong to K , then Γ (q)Mα = Γ(q)Mβ , since otherwise, we would −→ −→ ← − have Mα (Mβ )−1 ∈ Γ (q), and so Mα (Mβ )−1 ∈ Γ (q). Now we can prove our next result. Theorem 1.28. Let K be a congruence subgroup of Sn , and χ a congruence character of K. Then the Siegel operator Φ maps the space Mk (K, χ ) into the space ← − − Mk ( K , ← χ ): ← − − Φ : Mk (K, χ ) → Mk ( K , ← χ ), (1.60) ← − − χ ) = C. where for n = 1, we set M ( K , ← k

Proof. One can assume that n > 1. Let F ∈ Mk (K, χ ), Z ∈ Hn−1 , M = CA DB ∈ ← − S , Zλ = Z0 i0λ , and M = CA DB = ψ (M ) ∈ S, where ψ is the embedding (1.58). Then we have −1 C Z + D 0 A Z + B 0 = M Z λ , M Zλ = 0 iλ 0 1 −(M ). χ det(CZ + D) = det(C Z + D ), and χ (M) = ← λ

It follows that (F|Φ|k M )(Z ) = det(C Z + D )−k lim F(M Z λ ) λ →+∞ −k

= lim det(CZλ + D) F(M Zλ ) = (F|k M|Φ)(Z ). λ →+∞

(1.61)

30

1 Modular Forms

← − In particular, if M ∈ K , then we have −(M )F|Φ. χ F|Φ|k M = χ (M)FΦ = ← Furthermore, it follows from (1.54) and (1.61) that the function (F|Φ|k M )(Z ) is with ε > 0. holomorphic on Hn−1 , and is bounded on each subset Hn−1 ε The next lemma gives useful characterizations of cusp forms in terms of Siegel operators. Lemma 1.29. Let K be a congruence subgroup of Sn and χ a congruence character of K and let F ∈ Mk (K, χ ). Then the following three conditions are equivalent: (1) The function F is a cusp form; (2) F satisfies (F|k M)|Φ = 0

for all M ∈ Gn (Q) = Gn

where Φ is the Siegel operator; (3) F satisfies (F|k γ )|Φ = 0

Q2n 2n ,

for all γ ∈ Γn .

Proof. It follows from definition of cusp forms and formula (1.54) that condition 1 implies condition 2. Condition 3 is a particular case of condition 2. Hence it remains to prove that each function F ∈ Mk (K, χ ) satisfying condition 3 is a cusp form. It easily follows by an induction based on the Euclidean algorithm that each matrix n M ∈ Gn (Q) can be written in the form M = γ M1 , where γ ∈ Γ and M1 has the form A1 B1 M1 = 0 D1 with the zero n × n−block 0 = 0n (see Proposition 3.35(1) below). Since F|k M = F|k γ M1 = F|k γ |k M1 , the Fourier expansion (1.42) of the function F|k M can be rewritten in the form

∑ n

πi

−1 fM (A)e q σ (AZ) = (F|k M)(Z) = (det D1 )−k (F|k γ )(A1 ZD−1 1 + B1 D1 )

A∈E , A≥0

= (det D1 )−k

∑ n

A ∈E , A ≥0

π i σ (A B D−1 ) 1 1

fγ (A )e q1

π i σ (D−1 A A Z) 1 1

e q1

,

where fγ (A ) are the coefficients of the Fourier expansion (1.42) of the function F|k γ . Hence if fγ (A ) = 0 for all A ∈ En with det A = 0, then fM (A) = 0 for all A ∈ En with det A = 0. Spaces of Modular Forms. We can now prove the following important theorem. Theorem 1.30. Let K be a congruence subgroup of Sn , χ a congruence character of K, and k a nonnegative integer. Then the space Mk (K, χ ) of modular forms of weight k and character χ for the group K is finite-dimensional over the field C. The proof of the theorem is based on the following key lemma.

1.3 Modular Forms

31

Lemma 1.31. Let F(Z) =

∑

f (A)eπ iσ (AZ)

A∈E, A>0

be a cusp form of a nonnegative integral weight k and the unit character χ for the full modular group Γ = Γn . Suppose that the Fourier coefficients satisfy the conditions f (A) = 0

if

σ (A) ≤

kn , 2π bn

(1.62)

where bn is a constant satisfying the inequalities (1.26) of Theorem 1.16. Then the form F is identically equal to zero. Proof of the lemma. By Lemma 1.4 and the definition of modular forms, the function H(Z) = H(X + iY ) = (detY )k/2 |F(Z)| satisfies H(γ Z ) = H(Z) for all γ ∈ Γ and, as we have seen in the proof of Proposition 1.25, is bounded on the fundamental domain Dn of Γ defined in Theorem 1.16. Moreover, it follows from the estimates (1.51) and (1.52) that H(X + iY ) → 0, with detY → +∞ remaining in Dn . It follows from Theorem 1.16 that any subset of Dn of the form {X + iY ∈ Dn | detY ≤ c} with c > 0 is bounded and closed, and therefore is compact. Hence the function H(Z) attains its maximum µ at some point Z0 = X0 + iY0 of Dn . Since H is Γ -invariant, we conclude that µ is the maximum of H on H, that is, H(Z) ≤ H(Z0 ) = µ for all Z ∈ H. Let us set Zt = Z0 + tE, where t = u + iv is a complex parameter, and consider the function h(t) = F(Zt )e−π iλ σ (Zt ) =

∑

f (A)eπ i(σ (AZ0 )+t σ (A))−λ σ (Z0 +tE))

∑

f (A)eπ i(σ (AZ0 −λ Z0 )) eπ it(σ (A)−λ n) = h (w),

A∈E, A>0

=

A∈E, A>0

where w = eπ it and λ satisfies λ n = 1 + [kn/2π bn ], with [α ] denoting the greatest integer not exceeding α . By the assumption of the lemma, we have f (A) = 0 if σ (A) − λ n < 0, and so the expansion of the function h (w) does not contain negative powers of w. If ε > 0 is so small that Zt ∈ H for v ≥ −ε , then the expansion converges absolutely and uniformly on the half-plane v ≥ −ε , and so the function h (w) is holomorphic in the disk |w| ≤ eπε = τ . Since τ > 1, it follows, by the maximum-modulus principle, that there is a point w0 = eπ it0 satisfying |w0 | = τ and h (1) ≤ h (w0 ). Coming back to the function h, we can rewrite the last inequality in the form |F(Z0 )eπλ σ (Y0 ) | ≤ |Ft0 |eπλ σ (Y0 ) eπλ nv0 , where t0 = u0 + v0 , hence (detY0 )−k/2 H(Z0 ) ≤ (detYt0 )−k/2 H(Zt0 )eπλ nv0 .

32

1 Modular Forms

Since H(Z0 ) = µ and H(Zt0 ) ≤ µ , the last inequality implies the inequality

µ ≤ µ (detY0 )k/2 (detYt0 )−k/2 eπλ nv0 = µψ (v0 ), where ψ (v) = det(E + vY0−1 )−k/2 eπλ nv . We have ψ (0) = 1. Let us show that the derivative of ψ is positive at v = 0. We can write n

ψ (v) = eπλ nv ∏ (1 + vλ j )−k/2 , j=1

where λ1 , . . . , λn are the eigenvalues of Y0−1 , hence the value of the derivative at v = 0 is k k πλ n − (λ1 + · · · + λn ) = πλ n − σ (Y0−1 ) 2 2 kn kn kn − = π 1+ > 0, ≥ πλ n − 2bn 2π bn 2bn by (1.26), since X0 + iY0 ∈ D n . It follows that for small ε > 0, we have ψ (v0 ) = ψ (−ε ) < 1 (we recall that eπε = τ = |eπ i(u0 +iv0 ) | = e−π v0 ). Then the above inequality shows that µ = 0, and so F is identically equal to zero. Proof of Theorem 1.30. Note, first of all, that if the character χ is trivial on a principal congruence subgroup Γ (q) = Γn (q) contained in the group K, then the space Mk (K, χ ) is contained in the space M = Mk (Γ (q)) of modular forms of weight k and the unit character for the group Γ (q). Therefore, it will be sufficient to prove that each of these spaces is finite-dimensional. We recall that Γ (q) is a normal subgroup of finite index ν = [Γ : Γ (q)] in Γ . Let γ1 , . . . , γν be a system of representatives for cosets of Γ modulo Γ (q). For a function F ∈ M, let us consider the functions F1 = F|k γ1 , . . . , Fν = F|k γν .

(1.63)

By (1.31) and (1.32), the functions do not depend on the choice of the representatives γ j . Since for any γ ∈ Γ , the set γ1 γ , . . . , γν γ is again a set of representatives for the cosets, it follows that the functions F1 |k γ , . . . , Fν |k γ coincide up to a permutation with the functions (1.63). By Lemma 1.22, each of the functions (1.63) belongs to M, and by Proposition 1.25, is a cusp form if F is a cusp form. Let us now derive from Lemma 1.31 its generalization to the subspace N = Nk (Γ(q)) of cusp forms of M in the following form: if F(Z) =

∑

f (A)eπ iσ (AZ) ∈ N,

A∈E, A>0

and the Fourier coefficients f (A) satisfy f (A) = 0

if σ (A) ≤

knqν , 2π bn

(1.64)

1.3 Modular Forms

33

then F is identically equal to zero. For that we shall consider the product ν

G(Z) = ∏ Fj (Z) j=1

of the functions (1.63). By (1.29) and the above considerations, we have, for every M ∈ Γ, ν

G|kν M = j(M, Z)−kν G(M Z ) = ∏ ( j(M, Z)−k Fj ( Z )) j=1

ν

ν

j=1

j=1

= ∏ Fj |k M = ∏ Fj = G. Since G obviously satisfies the analytic conditions of the definition of cusp forms, we conclude that G is a cusp form of weight kν for the group Γ. Let f j (A) be the Fourier coefficients of the function Fj , so that Fj =

∑

πi

f j (A)e q σ (AZ) .

A∈E, A>0

Then the Fourier coefficients g(A) of G can be written in the form g(A) =

∑

f1 (A1 ) · · · fν (Aν ).

A1 +···+Aν =qA

Let A be a positive definite matrix of E satisfying σ (A) ≤ knν /2π b, where b = bn . Then the inequality 1 kν n σ (A) = (σ (A1 ) + · · · + σ (Aν )) ≤ q 2π b for positive definite even matrices A1 , . . . , Aν implies that σ (A j ) < knqν /2π b for each j = 1, . . . , ν , since the trace of any positive definite matrix is positive. If, for example, F1 = F and so f1 = f , then the condition (1.64) implies that each of the terms of the sum for g(A) has a factor of the form f1 (A1 ) = f (A1 ) with σ (A1 ) < knqν /2π b, which is zero. Then by Lemma 1.29, G = 0, and so F = 0. Now we can prove that the subspace N of cusp forms of M is finite-dimensional. Since entries of positive semidefinite matrices A = (aαβ ) satisfy the inequalities aαα ± 2aαβ + aβ β ≥ 0, it follows that the number of positive semidefinite even matrices A of order n with σ (A) ≤ 2N does not exceed the bound (N + 1)n (2N + 1)n(n−1)/2 .

(1.65)

Therefore, the number of positive definite even matrices A satisfying the inequality in (1.64) is bounded by a number of the form dn (kqν )n(n+1)/2 , where dn depends only on n. Taking dn to be integral, we see that d + 1 arbitrary functions

34

1 Modular Forms

F1 , . . . , Fd+1 of N are linearly dependent, since one can always find complex numbers c1 , . . . , cd+1 not all equal zero and such that the function F = c1 F1 + · · · + cd+1 Fd+1 satisfies the condition (1.64) and therefore is identically equal to zero. Finally, we use induction on n to prove the theorem for the entire spaces Mn = Mk (Γn (q)). Let us define for n ≥ 1 the linear map

Φ = Φn :

Mn → Mn−1 × · · · × Mn−1 ν times

by F|Φ = (F1 |Φ, . . . , Fν |Φ)

(F ∈ Mn ),

where ν is the index of Γn (q) in Γn , Φ the Siegel operator (1.56), Fj the functions (1.63), and where we set M0 = C. By Lemma 1.29, the kernel of Φn coincides with the subspace Nn of cusp forms of Mn and so is finite-dimensional. If the theorem is already proved for n − 1 ≥ 1, then the image of Φn is finite-dimensional and so is the space Mn . The image of Φ1 is obviously finite-dimensional, which proves the theorem for n = 1. One can show that M(S, χ ) = {0} if k is a negative integer, but we consider only modular forms of nonnegative integral weights and do not use this result. Exercise 1.32. Show that Mk (Γn ) = {0} if nk is odd. Exercise 1.33. Show that the spaces Mk (Γ1 ) for k = 0, 2, 4, 6, 8, 10 contain no cusp form, and there is not more than one linearly independent cusp form of weight 12 for Γ1 . [Hint: Use Lemma 1.31 for n = 1 with b1 given in Exercise 1.17.] Exercise 1.34. let k be a positive even integer. Show that the Fourier coefficients of any modular form ∞

F(z) =

∑ f(a)e2π iaz ∈ Mk (Γ1 )

a=0

satisfy

f(a) = f(0)ζ (k)−1

(2π i)k d k−1 + f (a), (k − 1)! ∑ d|n

where |f (a)| ≤ cF ak/2 ,

where ζ (s) is the Riemann zeta function. [Hint: Show first that the function F(z) − f(0)(2ζ (k))−1 Ek (z) is a cusp form. Then use Exercise 1.33 and (1.46).] Exercise 1.35. Show that the spaces Mk (Γ1 ) for k = 0, 4, 6, 8, 10 are spanned respectively by 1, E4 , E6 , E8 , and E10 , where Ek are the Eisenstein series of Exercise 1.20.

1.3 Modular Forms

35

Exercise 1.36. Show that the function ∆ (z) = ((2ζ (4))−1 E4 (z))3 − ((2ζ (6))−1 E6 (z))2 is a nonzero cusp form of the space M12 (Γ1 ) and that ∆ (z) and E12 (z) span the space. [Hint: With the help of classical formulas for ζ (4) and ζ (6) and formulas for Fourier coefficients of E4 (z) and E6 (z) show that the coefficient of e2π iz in the Fourier expansion of ∆ (z) is equal to 1728.] Petersson Scalar Product. Every space Nk (K, χ ) of cusp forms of an integral weight k and a congruence character χ for a congruence subgroup K of the symplectic group can be endowed with the structure of a Hilbert space by means of the scalar product. For two functions F and F on H = Hn , we consider the differential form on H defined by

ωk (F, F ) = F(Z)F (Z)h(Z)k d ∗ Z,

(1.66)

where h(Z) = h(X + iY ) = detY is the height of Z, d ∗ Z is the invariant element of volume (1.12), and as usual, bar means complex conjugation. It follows from Lemma 1.4 and Proposition 1.6 that for each matrix M = CA DB ∈ G = Gn , the form satisfies the relation

ωk (F, F )(M Z ) = F(M Z )F (M Z )h(M Z )k d ∗ M Z

= µ (M)nk det(CZ + D)−k F(M Z ) × det(CZ + D)−k F (M Z )h(Z)k d ∗ Z = µ (M)−nk+n(n+1) (F|k M) (Z) F |k M (Z)h(Z)k d ∗ Z = µ (M)n(n+1−k) ωk F|k M, F |k M (Z), (1.67) where |k M is the Petersson operator (1.29) of weight k. In particular, if F, F ∈ Mk (K, χ ) and M ∈ K, then by (1.32), we have

ωk (F, F )(M Z ) = ωk (χ (M)F, χ (M)F )(Z) = ωk (F, F )(Z). It follows that the integral

D(K)

ωk (F, F )(Z)

(1.68)

(1.69)

on a fundamental domain D(K) of K on H does not depend on the choice of the fundamental domain, provided that it converges absolutely. Lemma 1.37. Let K be a congruence subgroup of S = Sn , χ a congruence character of K, and k an integer. Suppose that at least one of the forms F, F ∈ Mk (K, χ ) is a cusp form. Then the integral (1.69) converges absolutely. Proof. By increasing the space Mk (K, χ ), one may assume that K is a subgroup of finite index in Γ and the character χ is trivial on K. Let us take then as D(K) a

36

1 Modular Forms

fundamental domain of the form (1.28) with M = 12n , that is, a finite union of the sets γ j D with γ j ∈ Γ, where D = Dn is the fundamental domain of Γ described in Theorem 1.16. Then, by (1.67), it is sufficient to show that each integral γ D

ωk (F, F )(Z) =

D

ωk F|k γ , F |k γ (Z) with γ ∈ Γ

converges absolutely. Assuming that F is a cusp form, by (1.42) and (1.43), we can write absolutely convergent Fourier expansions F|k γ =

∑

πi

fγ (A)e q σ (AZ) ,

A∈E,A≥0

πi

fγ (A)e q σ (AZ) ,

∑

F |k γ =

A∈E,A>.0

where E = En and q is such that Γ (q) ⊂ K. It follows that

∑

|(F|k γ )(Z)(F |k γ )(Z)| ≤

π

c(A)e− q σ (AZ) ,

A∈E,A>0

where Z = X + iY , with nonnegative coefficients c(A), and the last series converges on H. Then, by Theorem 1.16 and Lemma 1.24, we get the inequality |(F|k γ )(Z)(F |k γ )(Z)| ≤ d e−d

σ (Y )

(Z = X + iY ∈ D),

with positive constants d and d . Thus, in order to prove the lemma, it suffices to show that the integral

e−d σ (Y ) (detY )k−n−1

D

∏

1≤α ≤β ≤n

dxαβ dyαβ

with d > 0 converges. If (xαβ ) + i(yαβ ) ∈ D, then it follows from the definition of D and (1.22) that |xαβ | ≤ 1/2 for 1 ≤ α , β ≤ n and yαβ ≤ yαα /2 for α = β . In the √ proof of Theorem 1.16 we have seen that yαα ≥ 3/2 for α = 1, . . . , n. Then the inequality (1.22), if k < n, and the inequality (1.51), if k ≥ n + 1, imply that the last integral is majorized by

c

n √ yk−n−1 e−δ yαα dxαβ dyαβ αα |xαβ |≤1/2,yαα ≥ 3/2, α =1 1≤α ≤β ≤n |yαβ |≤yαα /2(α =β )

∏

∞

=c∏

α =1

∞ √

3/2

∏

k−n−1+n−α −δ yαα yαα e dyαα < ∞.

The above lemma justifies the following definition. For two modular forms F, F ∈ M(K, χ ) of an integral weight k and a congruence character χ for a congruence subgroup K ⊂ S such that at least one of the forms is a cusp form, the integral

1.3 Modular Forms

37

ˇ −1 (F, F ) = ν (G)

D(G)

ωk (F, F )(Z),

(1.70)

ˇ the where G is a congruence subgroup of Γ contained in K, Gˇ = G ∪ (−12n )G, ν (G) index of Gˇ in Γ , and D(G) a fundamental domain of G on H, is called the Petersson scalar product of these forms. Theorem 1.38. Under the above assumption, the Petersson scalar product has the following properties: (1) It converges absolutely and does not depend on the choice of fundamental domain D(G). (2) The scalar product is independent of the choice of the subgroup G of Γ contained in K. (3) It is linear in F and conjugate linear in F . (4) It satisfies (F, F ) = (F , F). (5) If F is a cusp form, then (F, F) ≥ 0, and (F, F) = 0 only if F = 0. (6) If M ∈ Gn ∩ Q2n 2n is a symplectic matrix with entries in the field Q of rational numbers and positive multiplier, then F|k M, F |k M = µ (M)n(k−n−1) (F, F ), (1.71) where functions the F|k M and F |k M are considered as elements of the space −1 Mk M KM, χM with the character χM defined by (1.34).

Proof. The first property follows from (1.68) and Lemma 1.37. In order to prove the second property, let us assume that G is another congruence subgroup contained in K. Then on replacing G by G ∩ G, one can assume that G ⊂ G. Let Γ = Gˇ γi and Gˇ = Gˇ δ j i

j

be decompositions into different left cosets. Then we have Γ=

Gˇ δ j γi ,

i, j

and the left cosets are distinct. It follows from Theorem 1.19 that one can take D(G ) =

(δ j γi ) Dn ,

i, j

hence we obtain

ν (Gˇ )−1

D(G )

ωk (F, F )(Z) = ν (Gˇ )−1 ∑ ∑ i

j

δ j γi Dn

= ν (Gˇ )−1 [Gˇ : Gˇ ] ∑ i

ωk (F, F )(Z)

γi Dn

ωk (F, F )(Z),

38

1 Modular Forms

where we have also used relations (1.68). Again by Theorem 1.19, the last expression is equal to ˇ −1 ν (G) ωk (F, F )(Z), D(G)

which proves the property (2). Properties (3), (4), and (5) follow directly from the definition. In order to prove relation (1.71), we note that since all entries of M are rational, the group M −1 KM is again a congruence subgroup of S, and so the group G(M) = G M −1 GM is again a congruence subgroup of Γ and, in particular, has finite index in Γ . It follows from Lemma 1.20, part (2), and (1.67) that −1 F|k M, F |k M = ν Gˇ (M)

D(G(M) )

ωk F|k M, F |k M (Z)

−1 = µ (M)n(k−n−1) ν Gˇ (M)

ωk (F, F ) (M Z )

−1 = µ (M)n(k−n−1) ν Gˇ (M)

ωk (F, F )(Z),

D(G(M) )

M D(G(M) )

where D Gˇ (M) is a fundamental domain for G(M) . It is clear that the set M D G(M) is a fundamental domain for the group MG(M) M −1 = MGM −1 ∩ G = G(M−1 ) . Hence, again by property (2), we can rewrite the last expression in the form −1 −1 µ (M)n(k−n−1) ν Gˇ (M) ν Gˇ (M−1 ) ν Gˇ (M−1 ) −1 ν Gˇ (M−1 ) (F, F ), = µ (M)n(k−n−1) ν Gˇ (M)

ωk (F, F D G(M −1 )

In order to prove the relation (1.71), it is sufficient to prove the equality ν Gˇ (M) = ν Gˇ (M−1 ) ,

)(Z)

(1.72)

which follows from the following more general lemma. Lemma 1.39. Let K be a congruence subgroup of Γn , and let M be a matrix of Gn with rational entries. Then the groups K(M) = K ∩ M −1 KM and K(M−1 ) = K ∩ MKM −1 are congruence subgroups of Γn with equal indices in K: K : K(M) = K : K(M−1 ) . (1.73) Proof of the lemma. By Proposition 1.21 these groups are intersections of congruence subgroups of Sn and hence themselves are congruence subgroups. Let D be a fundamental domain for the group K(M) . Since K(M−1 ) = MK(M) M −1 , it follows D as a fundamental domain for K(M−1 ) . Then Theorem that one can take the set M D 1.19 implies the following relations for the invariant volumes of the domains D and D : M D

1.3 Modular Forms

39

D) = Kˇ : Kˇ (M) v(D(K)), v(D

D ) = Kˇ : Kˇ (M−1 ) v(D(K)), v(M D

D) = v(M D D ), this relation implies both the where Kˇ = K ∪ (−12n )K. Since v(D equalities (1.72) and (1.73), We have proved the lemma and the theorem. Exercise 1.40. Show that the Eisenstein series Ek (z) defined in Exercise 1.32 is orthogonal to the space Nk (Γ1 ) of cusp forms of weight k for Γ1 . [Hint: Note that Ek (z) = (1 + (−1)k )ζ (k)

∑

j(M, z)−k ,

M∈Γ10 \Γ1

where j(M, z) is defined by (1.30) and Γ10 = a0 db ∈ Γ1 , which allows one to rewrite the scalar product of Ek (z) on a cusp form as integral over a fundamental domain for the group Γ10 , say {z = x + iy ∈ H1 | −1/2 ≤ x ≤ 1/2}.]

Chapter 2

Dirichlet Series of Modular Forms

2.1 Radial Dirichlet Series A natural way to approach zeta functions of modular forms is based on consideration of Dirichlet series constructed by means of Fourier coefficients of the forms. As was indicated in the introduction, a right zeta function must have certain analytic properties and an Euler product factorization. Therefore, the choice of appropriate Dirichlet series is motivated by a possibility of their analytic investigation plus a close relation with Euler products. These two features do not necessarily go together. Radial Series of Cusp Forms. Let us consider modular forms of the spaces Nnk (q, χ ) = Nk (Γn0 (q), χ ) of cusp forms of nonnegative integral weights k for the congruence groups

AB ∈ Γn C ≡ 0 (mod q) Γn0 (q) = CD of integral levels q with characters χ of the form AB AB = χ (det D) χ ∈ Γn0 (q) , CD CD

(2.1)

(2.2)

(2.3)

where χ is a Dirichlet character modulo q. According to Theorem 1.21 and Proposition 1.23, each such form F has the Fourier expansion F(Z) =

∑ n

f (A)eπ iσ (AZ) ,

(2.4)

A∈E ,A>0

which converges absolutely on Hn and uniformly on subsets (1.37) and has Fourier coefficients f (A) satisfying the relations f ( tVAV ) = χ (detV )(detV )k f (A)

(for all A ∈ En and V ∈ Λn )

A. Andrianov, Introduction to Siegel Modular Forms and Dirichlet Series, Universitext, DOI 10.1007/978-0-387-78753-4 2, c Springer Science+Business Media LLC 2009

(2.5) 41

42

2 Dirichlet Series of Modular Forms

and the estimate | f (A)| ≤ c(det A)k/2 ,

(2.6)

where c = cF depends only on F. If n = 1, the Fourier expansion can be written the form ∞

F(z) =

∑ f(m)e2π imz ,

m=1

and the simplest Dirichlet series associated to F is ∞

R(s) = R(s; F) =

f(m) , s m=1 m

∑

(2.7)

It converges absolutely and uniformly in right half-planes ℜs ≥ k/2 + 1 + ε with ε > 0 and uniquely determines the form F. Using the Euler formula ∞ 0

ys−1 e−α y dy = Γ(s)α −s

(α > 0, Re s > 0),

(2.8)

where Γ(s) is the gamma function, we can see that the series (2.7) can be presented in the half-planes ℜs ≥ k/2 + 1 + ε by the absolutely and uniformly convergent Mellin integral Φ(s; F) = =

∞

F(iy)ys−1 dy

0 ∞

∞

m=0

0

∑ f(m)

ys−1 e−2π my dy = (2π )−s Γ(s)R(s; F).

(2.9)

In the next section we shall see that the integral representation allows one to show that the function (2.9) has an analytic continuation over the whole s-plane as a holomorphic function and satisfies a functional equation. In Chapter 5 we shall see that the series (2.7) has close ties with Euler products. If n > 1, there is no unique natural way to introduce Dirichlet series corresponding to modular forms. The simplest generalizations of series (2.7) to genera n > 1 are given by so-called radial Dirichlet series (corresponding to the matrix “ray” A, 2A, . . . ) ∞ f (mA) (2.10) R(s) = R(s; F, A) = ∑ s m=1 m of a cusp form F with the Fourier expansion (2.4), where A is a fixed even positive definite matrix of order n. The estimate (2.6) implies that each radial series converges absolutely and uniformly in the right half-planes ℜs > nk/2 + 1 + ε

(2.11)

2.1 Radial Dirichlet Series

43

with ε > 0. For n > 1, the radial Dirichlet series does not necessarily determine the initial modular form F, since it includes only Fourier coefficients depending on matrices of the given ray A, 2A, . . . of positive definite even matrices and gives no information on Fourier coefficients with arguments belonging to other rays. Integral Representation of Radial Series. In order to derive an analytic expression of a radial series through original modular forms, we consider, for a fixed nonzero even matrix A of order n, the hyperplane in the space of real symmetric matrices of order n given by (2.12) X(A) = X = tX ∈ Rnn σ (AX) = 0 and the lattice in the hyperplane defined by X0 (A) = X(A)

Znn .

(2.13)

For every even matrix A of order n, the mapping

X → eπ iσ (A X)

(X ∈ X(A))

defines clearly a character of the compact quotient group X(A)/X0 (A), and the character is trivial if and only if A is a multiple of A, A = mA (note that the corresponding hyperplanes and lattices coincide if A = mA and have nontrivial intersections otherwise). Thus, 1 if A = mA, π iσ (A X) e [dX] = 0 otherwise, X(A)/X0 (A) where [dX] is the normalized Haar measure on X(A)/X0 (A). Integrating termwise the Fourier expansion (2.4) of the restriction of F on subsets X(A) + iY ⊂ Hn over a fundamental domain X(A)/X0 (A) of the translation group X0 (A) on X(A), we get

F(X + iY )[dX] = X(A)/X0 (A)

∑ n

f (A )e−πσ (A Y )

eπ iσ (A X) [dX]

X(A)/X0 (A)

A ∈E , A >0

=

∑n

f (mA)e−π mσ (AY ) .

m∈Q, mA∈E , mA>0

We recall that the a divisor of nonzero even matrix A is the largest number d ∈ N such that the matrix d −1 A is even. Nonzero even matrices of divisor d = 1 are called primitive. If A is even and primitive, then its rational multiple mA is even if and only if the coefficient m is integral. Since A and mA are both positive definite, m is positive. Thus, the last relation in the case of an even primitive positive definite matrix A turns into

∞

F(X + iY )[dX] = X(A)/X0 (A)

∑

m=1

f (mA)e−π mσ (AY ) .

(2.14)

44

2 Dirichlet Series of Modular Forms

There are many ways to transform the power series on √ the right to corresponding Dirichlet series. For example, if we put in (2.14) Y = v det AA−1 with v > 0, then, by using again the Euler integral (2.8), we obtain the identity ∞ √ vs−1 F(X + iv det AA−1 )[dX] dv 0 ∞

=

∑

m=1

X(A)/X0 (A) ∞

√ det Av

vs−1 e−π nm

f (mA)

0

√ dv = (π n det A)−s Γ(s)

√ = (π n det A)−s Γ(s)R(s; F, A)

∞

∑

m=1

f (mA) ms (2.15)

In Section 2.2 we briefly recall the classical theory of Mellin transforms of cusp forms in one variable. The rest of this chapter is devoted to analytic properties of radial Dirichlet series of cusp forms for genus 2. In order to illustrate the main features of the problem, we consider in detail the case of cusp forms of level q = 1 and radial series corresponding to the matrix A = 20 02 of the quadratic form x2 +y2 . Some features of the situation for other A are presented in the exercises. At present, the radial series of cusp forms for genus 2 are sufficiently investigated only in the case of level q = 1 (see Notes). The problem of Euler factorization of the radial series for genera n = 1 and n = 2 will be treated in Chapter 5. If n > 2, both the problem of analytic investigation and the problem of Euler factorization of the radial series are still essentially open.

2.2 Mellin Transform of Cusp Forms in One Variable Inversion Mapping. In this section we assume that n = 1. For a cusp form F ∈ N = N1k (q, χ ) we consider the simplest radial Dirichlet series R(s; F) defined by (2.7). As we have seen, in each of the right half-planes ℜs ≥ k/2 + 1 + ε , the series can be presented by the absolutely and uniformly convergent Mellin integral (2.9). Let us set 0 −1 1 . (2.16) Ω = Ω (q) = q 0 Since

d −c/q ab −1 , Ω Ω = −qb a cd

it follows that ΩK0 Ω−1 = K0 , where K0 = Γ10 (q), and the function F ∗ (z) = F|k Ω = q−1 z−k F(−1/qz) satisfies F ∗ |k M = F|k ΩMΩ−1 |k Ω = χ (a)F|k Ω = χ¯ (d)F ∗ , for each M =

(2.17) ab ∈ K0 , cd

2.2 Mellin Transform of Cusp Forms in One Variable

45

where χ¯ = χ −1 is the conjugate character. It is easy to see with the help of the Euclidean algorithm that among left multiples of any integral matrix of order 2 by matrices of Γ = Γ1 = SL2 (Z) there are always upper triangular matrices. In particular, for every matrix M ∈ Γ there exists a matrix M ∈ Γ such that ∗∗ . ΩM = M 0∗ Since F is a cusp form, it follows that for every M ∈ Γ the function F ∗ |k M has a Fourier expansion of the form (1.44). We conclude that F ∗ ∈ N(q, χ¯ ). Since −q 0 = (−1)k qk−2 F, F|k Ω2 = F|k 0 −q the inversion mapping F → F ∗ satisfies (F ∗ )∗ = (−1)k qk−2 F

(2.18)

and so determines an isomorphism of the space N(q, χ ) = N1k (q, χ ) onto N(q, χ¯ ) = N1k (q, χ¯ ). In particular, if χ = χ¯ , i.e., the character χ is real and takes only values ±1, the mapping is an automorphism of the space N(q, χ ). In this case the inversion mapping √ with i = −1 F → (−i)k q−k/2+1 F ∗ is also an automorphism of the space, and its square is the identity map. It follows that in the case of real χ , one can write a direct sum decomposition N(q, χ ) = N+ (q, χ ) + N− (q, χ ), where

(−i)k q−k/2 z−k F(−1/qz) = ±F

if F ∈ N± (q, χ ).

(2.19) (2.20)

Analytic Continuation and Functional Equation. Let us return to the integral (2.9). Theorem 2.1. For a nonnegative integer k, a positive integer q, and a Dirichlet character χ modulo q, let F ∈ N(q, χ ) = N1k (q, χ ) be a cusp form of weight k and character χ of the form (2.3) for the group K0 = Γ10 (q). Then the function Φ(s; F) = (2π )−s Γ(s)R(s; F), where R(s; F) is the Dirichlet series (2.7), has the following properties: (1) The function Φ(s; F) has an analytic continuation over the whole s-plane as a holomorphic function. (2) The function Φ(s; F) satisfies the functional equation Φ(k − s; F) = ik qs−k+1 Φ(s; F ∗ ), where F ∗ ∈ N(q, χ¯ ) is the function (2.17).

(2.21)

46

2 Dirichlet Series of Modular Forms

(3) If the character χ is real, and the form F belongs to one of the subspaces N± (q, χ ) of the decomposition (2.19), then the function Φ(s; F) satisfies Φ(k − s; F) = ±(−1)k qs−k/2 Φ(s; F).

(2.22)

Proof. Similarly to the case χ = 1 considered in the introduction, we can write, for ℜs sufficiently large, the identity Φ(s; F) = =

q−1/2

F(iy)ys−1 dy +

0

∞ q−1/2

∞ q−1/2

F(iy)ys−1 dy

F(i/qy)(1/qy)s−1 (1/qy2 )dy +

∞ q−1/2

F(iy)ys−1 dy,

where in the first integral we have changed the variable y → 1/qy. By substituting F(i/qy) = q(iy)k F ∗ (iy) in the first integral, we can rewrite the last relation in the form Φ(s; F) = ik q1−s

∞ q−1/2

F ∗ (iy)yk−s−1 dy +

∞ q−1/2

F(iy)ys−1 dy.

(2.23)

Since F and F ∗ are both cusp forms, the functions |F(iy)| and |F ∗ (iy)| approach exponentially zero as y → ∞ (see Lemma 1.24). It follows that the integrals in (2.23) both converge absolutely and uniformly on any compact subset of the s-plane. This proves part (1). By (2.23) with k − s in place of s and (2.18), we obtain Φ(k − s; F) = ik qs−k+1 = ik qs−k+1

∞ q−1/2 ∞

F ∗ (iy)ys−1 dy +

q−1/2

∞ q−1/2

F(iy)yk−s−1 dy

F ∗ (iy)ys−1 dy k k−s−1

∞

+ (−i) q

q−1/2

k 2−k

(−1) q

∗ ∗

k−s−1

(F ) (iy)y

dy

= ik qs−k+1 Φ(s; F ∗ ), which proves the functional equation (2.21). Finally, if, say, F ∈ N− (q, χ ), then it satisfies (−i)k q−k/2+1 F ∗ = −F, and so the functional equation turns into the equation Φ(k − s; F) = ik qs−k+1 (−1)ik qk/2−1 Φ(s; F) = −(−1)k qs−k/2 Φ(s; F). Exercise 2.2. Let F ∈ Mk (q, χ ) be an arbitrary modular form with Fourier expansion ∞

F(z) = f(0) +

∑ f(m)e2π imz .

m=1

2.3 Transformations of Lobachevsky Half-Spaces

47

Show that the Dirichlet series ∞

R(s; F) =

f(m) s m=1 m

∑

converge absolutely and uniformly in a right half-plane of the variable s; show that the functions Φ(s; F) = (2π )−s Γ(s)R(s; F) can be continued analytically to the whole s-plane as a meromorphic function having at most two simple poles at points s = 0 and s = k and satisfying the functional equation Φ(k − s; F) = ik qs−k+1 Φ(s; F ∗ ), where F ∗ ∈ N(q, χ¯ ) is the function (2.17); find the residues at the poles of Φ(s; F). [Hint: Use the integral representation Φ(s; F) =

!∞ 0

(F(it) − f(0))ys−1 dy.]

Exercise 2.3. Compute explicitly in the terms of the Riemann zeta function the Dirichlet series R(s; Ek ) for the Eisenstein series Ek (z) of even weight k > 2 defined in Exercise 1.32. [Hint: Use Exercise 1.33.]

2.3 Transformations of Lobachevsky Half-Spaces Transformation Group. The integral representation (2.15) for n = 2 shows that the radial Dirichlet series R(s; F, A) can be obtained by integration of the restriction of F to a real three-dimensional domain √ (2.24) H(A) = X(A) + iv det AA−1 ⊂ H2 . Further transformation of this representation is based on the remarkable circumstance that the real symplectic group S2 = Sp2 (R) defined by (1.13) has a rather large subgroup S(A) acting as a transitive group of analytic automorphisms of the domain H(A). By the end of this chapter the genus n is equal to 2, and the corresponding index is usually omitted. Let us consider now in detail the particular case A = 2E = 212 , where we write E in place of 12 . In this case one can take & ⎧ ⎪ x y ⎪ ⎪ X = X(2E) = x, y ∈ R , ⎪ ⎪ ⎪ y −x ⎪ ⎪ ⎪ & ⎪ ⎨ x y (2.25) X0 = X0 (2E) = x, y ∈ Z , ⎪ y −x ⎪ ⎪ ⎪ & ⎪ ⎪ ⎪ x y ⎪ ⎪ ⎪ |x| ≤ 1/2, |y| ≤ 1/2 . ⎩X/X0 = X(2E)/X0 (2E) = y −x

48

2 Dirichlet Series of Modular Forms

Since clearly, one can take [dX] = dx dy, the identity (2.15) for A = 2E turns into the identity (4π )−s Γ(s)R(s; F, 2E) & ∞ x + iv y F dx dy vs−1 dv. = |x|≤1/2, y −x + iv 0

(2.26)

|y|≤1/2

The domain of integration of this double integral is contained in the open subset

x + iv y (2.27) H = H(2E) = x, y, v ∈ R, v > 0 ⊂ H2 . y −x + iv Let us consider the set AB S = S(2E) = A, D ∈ Y, CD

B,C ∈ X,

A tD − B tC = E ,

where Y = Y (2E) =

x y x, y ∈ R . −y x

(2.28)

(2.29)

Proposition 2.4. The set S is a subgroup of the group S = Sp2 (R). For every matrix M = CA DB ∈ S, the transformation Z → M Z = (AZ + B)(CZ + D)−1

(Z ∈ H)

is a real analytic automorphism of the domain H. The action of S on H is transitive. x y y is always symmetric, it follows Proof. Since a matrix of the form −y x xy −x from (1.3) and (1.13) that S ⊂ S. Let us set

AB R= A, D ∈ Y, B,C ∈ X . CD Then S = S ∩ R. If M, M ∈ S, then M, M ∈ S and so MM ∈ S. From the obvious fact that X and Y are abelian groups under the addition of matrices, and from easily verified inclusions YY ⊂ Y,

XX ⊂ X,

XY ⊂ X,

YX ⊂ X

we conclude that MM ∈ R. Thus, MM ∈ S ∩ R = S. Further, the formula (1.1) implies that M −1 ∈ S if M ∈ S. Hence S is a subgroup of S. Now we shall show that H = M iE M ∈ S . (2.30)

2.3 Transformations of Lobachevsky Half-Spaces

Let Z =

x

y y −x

+ iv ∈ H. It is clear that the matrix ⎞ ⎛ √ x y 1 √ vE v y −x ⎠ is in S MZ = ⎝ √1 0 v

49

(2.31)

and MZ iE = Z. Thus, the left-hand side of (2.30) is contained in the right-hand side. In order to prove the reverse inclusion we note that if M = CA DB ∈ S with x1 y1 x2 y2 x3 y3 x4 y4 A= , B= , C= , D= , −y1 x1 y2 −x2 y3 −x3 −y4 x4 and M iE = X + iY , then, by (1.10) and (1.3), we have Y = t(D − iC)−1 (D + iC)−1 = (D tD +C tC)−1 −1 = x32 + x42 + y23 + y24 E.

(2.32)

Thus, it is sufficient to check that σ (X ) = 0. Since det(D + iC) = x32 + y23 + x42 + y24 , we have −1 M iE = x32 + y23 + x42 + y24 x4 − ix3 −y4 − iy3 x2 + ix1 y2 + iy1 , (2.33) × y2 − iy1 −x2 + ix1 y4 − iy3 x4 + ix3 hence

x32 + y23 + x42 + y24 σ (X ) = x2 x4 + x1 x3 + y2 y4 + y1 y3 − y2 y4 − y1 y3 − x2 x4 − x1 x3 = 0.

We have proved the relation (2.30), which implies the assertions of the proposition on the action of S on H. The above proposition shows that the domain H ⊂ H2 can be considered as a homogeneous space for the Lie group S ⊂ S2 . Next, we shall show that the pair (S, H) is naturally isomorphic to the pair (G, L), where G = SL2 (C), L is the 3-dimensional hyperbolic half-space (the Lobachevsky space) L = u = (z, v) z = x + iy ∈ C, v > 0 , and the action of G on L is given by the rule αβ Gg= : γ δ (α z + β )(γ¯z¯ + δ¯ ) + α γ¯v2 v u = (z, v) → g(u) = , , ∆(g, u) ∆(g, u)

(2.34)

50

2 Dirichlet Series of Modular Forms

where the bar means complex conjugation and where ∆(g, u) = |γ z + δ |2 + |γ |2 v2 .

(2.35)

In order to establish the relation of the pairs (S, H) and (G, L) for a real matrix of the form ⎞ ⎛ x1 y1 x2 y2 ⎜−y1 x1 y2 −x2 ⎟ AB ⎟ (2.36) M= =⎜ ⎝ x3 y3 x4 y4 ⎠ ∈ S, CD y3 −x3 −y4 x4 we set

x1 + iy1 y2 + ix2 αβ = , g(M) = y3 − ix3 x4 − iy4 γ δ

(2.37)

x y + ivE ∈ H y −x

(2.38)

u(Z) = (y + ix, v) ∈ L.

(2.39)

and for Z= we write

Proposition 2.5. In the above notation the following assertions hold: (1) g (S) = G, and the map g : S → G is an isomorphism of real Lie groups; the map u : H → L is a real analytic isomorphism. (2) The maps g and u transform the action of S on H into the action (2.34) of G on L in the sense that u (M Z ) = g(M) (u(Z))

(M ∈ S, Z ∈ H) ;

(2.40)

in particular, the action (2.34) satisfies (g, g ∈ G, u ∈ L). (gg )(u) = g(g (u)) AB (3) For all M = ∈ S and Z ∈ H, the identity CD det(CZ + D) = ∆(g(M), u(Z))

(2.41)

(2.42)

holds; in particular, the function (g, u) → ∆(g, u) is a factor of automorphy of the pair (G, L), i.e., it does not vanish on G × L and satisfies ∆(gg , u) = ∆(g, g (u))∆(g , u) (g, g ∈ G, u ∈ L).

(2.43)

Proof. The condition M ∈ S for a real matrix of the form (2.36) is equivalent to the relation A tD − B tC = E, that is, the relations x1 x4 + y1 y4 − x2 x3 − y2 y3 = 1,

−y1 x4 + x1 y4 − y2 x3 + x2 y3 = 0,

2.3 Transformations of Lobachevsky Half-Spaces

51

which can be rewritten in the form (x1 + iy1 )(x4 − iy4 ) − (y2 + ix2 )(y3 − ix3 ) = 1. The last relation is equivalent to the inclusion g(M) ∈ SL2 (C) = G. The relations g(MM ) = g(M)g(M ) for matrices M and M of the form (2.36) follow from the definitions by direct multiplication of the matrices, which we leave to the reader. The rest of part (1) is clear. In order to prove the relations (2.40), let us set A B , MMZ = M = C D where MZ is the matrix (2.31). By multiplying the matrices, we get √ x1 y1 x1 y1 = v , A = −y1 x1 −y1 x1 1 x1 x + y1 y + x2 x1 y − y1 x + y2 x2 y2 =√ , B = y2 −x2 v −y1 x + x1 y + y2 −y1 y − x1 x − x2 √ x y x y C = 3 3 = v 3 3 , y3 −x3 y3 −x3 1 x3 x + y3 y + x4 x3 y − y3 x + y4 x4 y4 √ = . D = −y4 x4 v y3 x − x3 y − y4 y3 y + x3 x + x4 For the matrix

X + iY = M Z = M MZ iE

= M iE ,

by the identities (2.32) and (2.33) with M in place of M, we obtain, respectively, −1 E Y = (x3 )2 + (y3 )2 + (x4 )2 + (y4 )2 and −1 X + iY = (x3 )2 + (y3 )2 + (x4 )2 + (y4 )2 x4 − ix3 −y4 − iy3 x2 + ix1 y2 + iy1 , × y2 − iy1 −x2 + ix1 y4 − iy3 x4 + ix3 hence

2 2 2 2 −1 x y X = (x3 ) + (y3 ) + (x4 ) + (y4 ) y −x

with x = x2 x4 + x1 x3 + y2 y4 + y1 y3 and y = −x2 y4 + x1 y3 + y2 x4 − y1 x3 . By substituting, we get 1 (x3 x + y3 y + x4 )2 + (x3 y − y3 x + y4 )2 v ∆(g, u) 1 2 2 , = v |γ | + |γ (y + ix) + δ |2 = v v

(x3 )2 + (y3 )2 + (x4 )2 + (y4 )2 = v(x32 + y23 ) +

52

2 Dirichlet Series of Modular Forms

where g = g(M) and u = u(Z) are given by (2.37) and (2.39), and y + ix = −x2 y4 + x1 y3 + y2 x4 − y1 x3 + i x2 x4 + x1 x3 + y2 y4 + y1 y3 1 = − (x1 x + y1 y + x2 )(x3 y − y3 x + y4 ) + vx1 y3 v 1 + (x1 y − y1 x + y2 )(x3 x + y3 y + x4 ) − vy1 x3 v 1 + i (x1 x + y1 y + x2 )(x3 x + y3 y + x4 ) + vx1 x3 v 1 + (x1 y − y1 x + y2 )(x3 y − y3 x + y4 ) + vy1 y3 v 1 = ((x1 + iy1 )(y + ix) + (y2 + ix2 ))((y3 + ix3 )(y − ix) + (x4 + iy4 )) v 1 + v(x1 + iy1 )(y3 + ix3 ) = (α z + β )(γ¯z¯ + δ¯ ) + α γ¯v. v The above computations imply the formula (2.40): v v x y E u (M Z ) = u(X + iY ) = u + i ∆(g, u) y −x ∆(g, u) v v (y + ix ), = g(M) (u(Z)) . = ∆(g, u) ∆(g, u) Formula (2.41) follows from (2.40) and (1.7). Direct computation shows that x + iv y x4 y4 x3 y3 + det(CZ + D) = det y3 −x3 −y4 x4 y −x + iv = (x3 x + y3 y + x4 )2 − (ivx3 )2 + (x3 y − y3 x + y4 )2 − (ivy3 )2 = ∆(g, u), as we have seen above. The relation (2.43) follows from (2.42) and (1.9). For further integration on L we shall need a G-invariant volume element. Proposition 2.6. The volume element on L given by du =

dx dy dv v3

(u = (x + iy, v) ∈ L)

(2.44)

is invariant under every transformation (2.34). Proof. It is easy to see that the group G = SL2 (C) is generated by the matrices α 0 1β 0 1 , , and (2.45) 01 −1 0 0 α −1

2.3 Transformations of Lobachevsky Half-Spaces

53

with α , β ∈ C, α = 0. For each of the matrices (2.45), the check of the assertion is an easy exercise, which we leave to the reader. Exercise 2.7. Let A be a real symmetric positive definite matrix of order 2, and sup√ pose τ ∈ GL2 (R) satisfies τ A tτ = det AE. Show that for every matrix M contained in the group t τ O −1 , with Mτ = S(A) = Mτ SMτ O τ −1 the transformation Z → M Z defines a real analytic automorphism of the domain H(A) given by (2.24) and (2.12), and the action of S(A) on H(A) is transitive. [Hint: Show first that H(A) = Mτ H .] Exercise 2.8. In the notation and assumptions of the previous exercise show that Proposition 2.5 remains true if one replaces the pair S, H by the pair S(A), H(A), and the maps g, u by the maps gτ : S(A) → G, uτ : H(A) → L defined by (M ∈ S(A)) and uτ (Z) = u Mτ−1 Z

(Z ∈ H(A)). gτ (M) = g Mτ−1 MMτ Discrete Subgroup. We turn now to integration of restrictions of modular forms to Lobachevsky subspaces of H2 . By the end of this chapter we shall restrict our consideration to the case of cusp forms for the group Γ2 = Γ20 (1) with trivial character. Each function F ∈ N2k = N2k (1, 1) satisfies the functional equations AB −k for all M = ∈ Γ2 . (2.46) det(CZ + D) F (M Z ) = F CD Let F- be the function on the Lobachevsky half-space L corresponding via (2.39) to the restriction of F on H ⊂ H2 , i.e., - = F(Z) F(u) and let g=

αβ = g(M) γ δ

if u = u(Z) with Z ∈ H, with

M=

AB ∈ S Γ2 . CD

(2.47)

(2.48)

Then the relations (2.46) and Proposition 2.6 imply the relations = ∆(g(M), u(Z))−k F(g(M)(u(Z))) ∆(g, u)−k F(g(u)) = det(CZ + D)−k F (M Z ) = F(Z) = F(u).

(2.49)

Lemma 2.9. The restriction of the map M → g(M) to the subgroup S ∩ Γ2 is an isomorphism of the subgroup onto the special linear group Λ = Λ(O) = SL2 (O)

(2.50)

of order 2 over the ring of Gaussian integers √ O = Z + iZ, i.e., the ring of integers of the imaginary quadratic field Q[i] (i = −1).

54

2 Dirichlet Series of Modular Forms

Proof. Since the map g is isomorphism of S on SL2 (C), it is sufficient to note that a matrix M of the form (2.36) contained in S belongs to the group Γ2 if and only ifall of its entries belong to Z. This is equivalent to the conditions that g(M) = αγ βδ ∈ SL2 (C) with α , β , γ , δ belonging to the ring Z + iZ = O. Analogously to the case of the symplectic modular group considered in Section 1.2, the group Λ is a discrete subgroups of G discretely acting on L. Similarly to Theorem 1.16, we have the following theorem. Theorem 2.10. The closed subset of L given by D = D(O) = (x + iy, v) ∈ L 0 ≤ x + y, x ≤ 1/2, y ≤ 1/2, 1 ≤ x2 + y2 + v2

(2.51)

is a fundamental domain of Λ on L, i.e., it meets each of the Λ -orbits and has no distinct inner points belonging to the same orbit. The fundamental domain has finite volume with respect to the invariant volume element (2.44). Proof. By analogy with the symplectic case, let us call the positive real number v2 = h(u) the height of a point u = (z, v) ∈ L. If M = CA DB ∈ S with g(M) = αγ βδ and Z ∈ H with u(Z) = (z, v), then, by (1.24), (2.42), (2.40), and (2.34), the heights of the points M Z and g(u) satisfy the relation −2

h(M Z ) = | det(CZ + D)| h(Z) =

v ∆(g, u)

2 = h(g(u)).

It follows that the values of the height of points of the orbit Λ (u) are among the values of the heights of points of the orbit Γ2 Z , where u(Z) = u. Hence by Lemma 1.15, we conclude that each orbit of Λ on L contains points u of maximal height and the points can be characterized by inequalities ∆(g, u) ≥ 1

for every g ∈ Λ .

(2.52)

Similarly to the proof of Theorem 1.16, in order to construct a fundamental domain for Λ on L, first of all, we shall choose on the Λ -orbitof a point u ∈ L a point u = α β with α , α −1 , β ∈ O (z , v ) of maximal hight. Further, the matrices g = 0 α −1 belong to Λ , and the corresponding transformations do not change the height: g(u ) = g((z , v )) = α α¯ −1 z + β α¯ −1 , v = (±z + β , v ) with β = β α¯ −1 ∈ O, since the inclusions α , α −1 ∈ O mean that α is a unit of O, i.e., α = ±1, ±i. It is clear that the sign and the number β ∈ O can be chosen such that the point u = (x + iy, v) = (±z + β , v ) satisfies the conditions 0 ≤ x + y, x ≤ 1/2, y ≤ 1/2. Since the height of u is the same as thatof u and so is again maximal 0 1 shows that for the considered orbit, the inequality (2.52) for g = −1 0

2.4 Radial Series and Eisenstein Series

55

∆(g, u) = |z|2 + v2 = x2 + y2 + v2 ≥ 1. Thus, u ∈ D. now that u = σ (u) for two points u = (z, v), u = (z , v ) ∈ D and g = Suppose α β ∈ Λ . Then by the above, the points u, u have the same height v = v, whence γ δ

∆(g, u) = 1. Similarly, ∆(g−1 , u ) = 1. If γ = 0, the equations are both nontrivial, and so points u, u belong to the boundary of D. But if γ = 0, then, as we have seen above, z = ±z + β . This implies that z = z, unless the complex numbers z = x + iy and z = x + iy belong to the boundary of the triangle 0 ≤ x + y, x ≤ 1/2, y ≤ 1/2. Finiteness of the volume is obvious. b Exercise 2.11. Let A = 2a b 2c be a positive definite even matrix. Check that the matrix 1 2c √−b ∈ SL2 (R) τ = τ (A) = √ 2c det A 0 det A √ satisfies tτ Aτ = det AE. Assuming that gcd(a, b, c) = 1, show then that the corresponding mapping gτ = gτ (A) of Exercise 2.8 maps the group Γ(A) = S(A)

Γ2

isomorphically onto the group

αβ −1 ∈ G α , δ ∈ O(A), β ∈ A(A), γ ∈ A(A) , Γ(O(A), A(A)) = γ δ where O(A) is the subring of the discriminant − det A of the ring of integers of the √ imaginary quadratic field K = Q( − det A), A(A) = ωc {a, ω }2 , and {a, ω } is the module in K of the form aZ + ω Z. [Hint: Check first that AB ˆ Γ(A) = A ∈ Y0 (A), CD

ˆ D ∈ Y0 (A), A tD − B tC = E , B ∈ X0 (A),C ∈ X0 (A),

where X0 (A) = X(A) ∩ Z22 , X(A) is the set (2.12), Y0 (A) = Y (A) ∩ Z22 , Y (A) = {Y ∈ R22 |YA tY = det Y · A, det Y ≥ 0}, and Aˆ = det A · A−1 .]

2.4 Radial Series and Eisenstein Series In this section we shall show that the integral in the representation (2.26) of the radial series R(s; F, A) for the matrix A = 2E can be interpreted as an integral convolution of the restriction of F on H considered via (2.39)√as a function on L with an Eisenstein series for the discrete subgroup Λ = SL2 (Z[ −1]) ⊂ G.

56

2 Dirichlet Series of Modular Forms

Let F be a cusp form of the space N2k = N2k (1, 1) and F- the corresponding function (2.47) on the Lobachevsky half-space L. By (2.49), the function F- satisfies the functional equations = F(u) ∆(g, u)−k F(g(u))

(u ∈ L, g ∈ Λ ) .

(2.53)

s+2 du, F(u)v

(2.54)

The identity (2.26) can be rewritten in the form (4π )−s Γ(s)R(s; F, 2E) =

P

P = {u = (x + iy, v) ∈ L |x| ≤ 1/2, |y| ≤ 1/2}

where

and du is the invariant element of volume (2.44). Note first of all that the set P can be interpreted as a fundamental domain for the group

±1 β Λ Λ0 = ∈ 0 ±1 acting on L. Let

Λ=

Λ0g j

j

be a decomposition into disjoint left cosets, and let D be a fundamental domain of the group Λ on L. Then the set P =

g j (D)

j

can also be taken as a fundamental domain of the group Λ 0 . Since the integrand in (2.54) is invariant under every transformation of Λ 0 , it follows that the integral does not change when the domain P is replaced by another sufficiently good fundamental domain of Λ0 , for example by P , provided that the integral converges absolutely. Then we obtain

s+2 du = F(u)v

P

P

s+2 du = ∑ F(u)v j

s+2 du. F(u)v

g j (D)

In each of the last integrals we make the change of variables u → g−1 j u . Then, by invariance of the volume element du and (2.53), we get P

s+2 du = ∑ F(u)v

j

=

D

D

F(u)

vs+2 du ∆(g j , u)s−k+2

∗ (u, s − k + 2)vk du, F(u)E

2.4 Radial Series and Eisenstein Series

57

where

1 . ∆(g, u)s Λ0 \Λ Λ g∈Λ We note now that two matrices αγ βδ and αγ βδ of Λ belong to the same left coset modulo Λ 0 if and only if their second rows satisfy {γ , δ } = ±{γ , δ }. Moreover, since the Euclidean algorithm holds in the ring of Gaussian integers, for every pair of relatively prime numbers γ , δ ∈ O one can find numbers α , β ∈ O such that αδ − γβ = 1. It follows that E ∗ (u, s) = vs

vs 2

E ∗ (u, s) =

∑

∑

γ ,δ ∈O γ O+δ O=O

1 s . 2 |γ z + δ | + |γ |2 v2

(2.55)

The series (2.55) is a so-called Eisenstein series for the group Λ . Lemma 2.12. The Eisenstein series (2.55) converges absolutely and uniformly on compact subsets of L in each half-plane ℜs > 2 + ε with ε > 0 and satisfies the relations for all g ∈ Λ . (2.56) E ∗ (g(u), s) = E ∗ (u, s) Proof. For u = (z, v) in a compact subset of L, we clearly have |γ z + δ |2 + |γ |2 v2 > c(|γ |2 + |δ |2 ),

(2.57)

where c is a positive constant. It follows that the series for 2c−1 v−s E ∗ (u, s) in the half-plane ℜs > 2 + ε is majorized termwise by the convergent series ∞

1

∑

(γ ,δ )=(0,0)

(|γ |2 + |δ |2 )2+ε

=

r4 (n) , 2+ε n=1 n

∑

where r4 (n) is the number of integral solutions of the equation x12 + x22 + x32 + x42 = n (for convergence, see the Jacobi formula for r4 (n)). By (2.34) and (2.43), we obtain E ∗ (g(u), s) = (v(g(u))s

∑

1

Λ0 \Λ Λ g ∈Λ −s

= v ∆(g, u) s

∑

Λ0 g ∈Λ

∆(g , g(u))s

∆(g, u)s = E ∗ (u, s). g, u)s ∆(g Λ \Λ

We summarize the results of the above considerations in the following lemma. Lemma 2.13. For every cusp form F ∈ N2k with Fourier expansion (2.4), the radial Dirichlet series ∞ f (2mE) RF (s) = ∑ ms a=1

58

2 Dirichlet Series of Modular Forms

satisfies in a right half-plane of the variable s the identity (4π )−s Γ(s)RF (s) =

D

∗ (u, s − k + 2)vk du, F(u)E

(2.58)

where F(u) is the restriction of F on H considered as a function on L, E ∗ (u, s) the Eisenstein series (2.55), du the invariant measure (2.44), and where D is a fundamental domain for the group Λ on L. b be a positive definite even matrix satisfying Exercise 2.14. Let A = 2a b 2c gcd(a, b, c) = 1. In the notation of Exercises 2.8 and 2.11, let F-A (u) = F(h−1 τ (u)) be the function on the space L corresponding via the map hτ : H(A) → L to the restriction on H(A) of a cusp form F ∈ N2k . Assuming that there exists a fundamental domain D(A) of finite volume for the group Γ(O(A), A(A)) on L, show that the radial Dirichlet series R(s; F, A) satisfies in a right half-plane of the variable s the identity 1−s 1 (2π )−s (det A) 2 Γ(s)R(s; F, A) = 2

where EA∗ (u, s) = vs

σ ∈Γ0

D(A)

F-A (u)EA∗ (u, s − k + 2)vk du,

1 , ∆( σ , u)s (O(A),A(A))\Γ(O(A),A(A))

∑

and Γ0 (O(A), A(A)) =

∗∗ ∈ Γ(O(A), A(A)) . 0∗

2.5 Properties of Radial Series for Sums of Two Squares In this section, in order to apply the integral representation (2.58), we represent the Eisenstein series E ∗ (u, s) by means of suitable theta series of a positive definite quadratic form in four variables, recall standard properties of the theta series, and finally, prove that the radial Dirichlet series R(s; F) = R(s; F, 2E) has a meromorphic analytic continuation over the whole s-plane and satisfies a functional equation with two gamma–factors. Eisenstein Series and Theta Series. Let us introduce the theta series πt (t > 0, u = (z, v) ∈ L). Θ(t, u) = ∑ exp − (|γ z + δ |2 + |γ |2 v2 ) v γ ,δ ∈O If t > η > 0 and u is in a compact subset of L, then v < b, and by (2.57), the theta series is majorized termwise by the series πη c πη c 4 ∞ 2 2 (| n2 , exp − γ | + | δ | ) = exp − ∑ ∑ b b n=−∞ γ ,δ ∈O

2.5 Properties of Radial Series for Sums of Two Squares

59

and so it converges absolutely and uniformly. If ℜs is sufficiently big, then by using the Euler integral (2.8), we get ∞ 0

t s−1 (Θ(t, u) − 1)dt = Γ(s)

v s

π

∑

1

γ ,δ ∈O, (γ ,δ )=(0,0)

(|γ z + δ |2 + |γ |2 v2 )s

.

√ Since each ideal A of the ring O = Z[ −1] of Gaussian integers is principal, A = α O, and its norm N(A) is α α¯ = |α |2 , for every nonzero pair γ , δ ∈ O with γ O + δ O = A, we have γ = αγ , δ = αδ with γ O + δ O = O, hence the last series is equal to Γ(s)

v s

π

∑ A

∑

1

γ ,δ ∈O γ O+δ O=O

N(a)s (|γ z + δ |2 + |γ |2 v2 )s

= 2π −s Γ(s)ZO (s)E ∗ (u, s),

where A ranges through all nonzero integral ideals of the ring O, ZO (s) = ∑ A

1 N(A)s

(2.59)

√ is the Dedekind zeta function of the ring Z[ −1], and E ∗ (u, s) is the Eisenstein series (2.55). We have proved the identity ∞

t s−1 (Θ(t, u) − 1)dt = 2π −s Γ(s)ZO (s)E ∗ (u, s),

0

(2.60)

valid in a right half-plane of the variable s. Lemma 2.15. (Inversion f ormula) The theta series Θ(t, u) satisfies the identity 1 1 ,u (t > 0, u ∈ L). Θ(t, u) = 2 Θ t t Proof. First we shall prove that for each symmetric positive definite matrix Q of order r, the theta series

θ (t, Q) =

∑

e−π tQ[N]

(Q[N] = tNQN)

(2.61)

N∈Zr

satisfies the inversion formula 1 θ (t, Q) = √ r θ t det Q

1 −1 ,Q . t

Indeed, the function

θ (t, X, Q) =

∑

N∈Zr

e−π tQ[N+X]

(X ∈ Rr )

(2.62)

60

2 Dirichlet Series of Modular Forms

is a periodic function of X and is equal to its Fourier series

θ (t, X, Q) =

∑ r a(M)e2π i NX , t

(2.63)

M∈Z

where 1

a(M) = 0

···

+∞

=

−∞

1 0

···

θ (t, X, Q)e−2π i MX dx1 · · · dxr t

+∞ −∞

e−π tQ[X]−2π i MX dx1 · · · dxr . t

Completing the square, we have i 1 tQ[X] + 2i tMX = tQ[X + Q−1 M] + Q−1 [M] t t

(i =

√ −1),

whence π

−1 [M]

a(M) = e− t Q

Rr

−1 [M]

e−π tQ[X+ t Q i

dX

(dX = dx1 · · · dxr ).

By Cauchy’s theorem, we can write − πt Q−1 [M]

a(M) = e

Rr

e−π tQ[X] dX

π −1 1 e− t Q [M] =√r t det Q

Rr

π −1 t 1 e− t Q [M] , e−π YY dY = √ r t det Q

where Y = SX with tSS = tQ and dY = | det S|dX, which together with (2.63) for X = 0 proves the formula (2.62). Coming back to the theta function Θ(t, u), we note that an easy straightforward calculation allows us to interpret it as a special case of the theta function (2.61), namely, (2.64) Θ(t, u) = θ (t, Q), where

⎛

⎞ [u] 0 x y 1 ⎜ 0 [u] −y x⎟ ⎟ Q= ⎜ v ⎝ x −y 1 0⎠ y x 0 1

if u = (x + iy, v) ∈ L

and [u] = x2 + y2 + v2 .

It is easy to check that det Q = 1 and ⎛ ⎛ ⎞ ⎞ 0 0 01 1 0 −x −y ⎜ ⎟ 1 ⎜ 0 1 y −x⎟ ⎟ = tUQU, U = ⎜ 0 0 1 0⎟ ∈ GL4 (Z). Q−1 = ⎜ ⎝ ⎝ ⎠ 0 −1 0 0⎠ v −x y [u] 0 −1 0 0 0 −y −x 0 [u]

2.5 Properties of Radial Series for Sums of Two Squares

61

Then by (2.64) and (2.62), we obtain 1 −1 1 ,Q Θ(t, u) = θ (t, Q) = 2 θ t t 1 1 1 1 1 1 , Q[U] = 2 θ , Q = 2Θ ,u . = 2θ t t t t t t Analytic Continuation and Functional Equation. Now we are finally able to prove the following result. Proposition 2.16. Let F be a cusp form of integral weight k for the group Γ2 . Then the function Ψ(s; F) = (2π )−2s Γ(s)Γ(s − k + 2)ZO (s − k + 2)R(s; F), where Γ(s) is the √ gamma function, ZO (s) is the Dedekind zeta function (2.59) of the ring O = Z[ −1], and R(s; F) = R(s; F, 2E) is the radial Dirichlet series corresponding to the ray of the matrix A = 2E = 2 · 12 of the sum of two squares, can be continued analytically to the whole s-plane as a meromorphic function having at most two poles at the points s = k − 2 and s = k, and satisfying the functional equation Ψ(2k − 2 − s; F) = Ψ(s; F). Proof. By (2.58), in a right half-plane of the variable s we have the identity Ψ(s; F) = π −s Γ(s − k + 2)ZO (s − k + 2)

D

∗ (u, s − k + 2)vk du, F(u)E

which, by (2.60), can be rewritten in the form ∞ 1 2−k k s−k+1 Ψ(s; F) = π t (Θ(t, u) − 1)dt du. F(u)v 2 D 0

(2.65)

In view of Lemma 2.15, we obtain 1 0

t s−k+1 (Θ(t, u) − 1)dt =

1 0

1

t s−k+1 Θ(t, u)dt −

1

t s−k+1 dt

0

1 1 t s−k−1 Θ( , u)dt − t s − k +2 0 ∞ 1 t k−s−1 Θ(t, u)dt − = s − k +2 1 ∞ 1 1 − . = t k−s−1 (Θ(t, u) − 1)dt + s−k s−k+2 1 =

62

2 Dirichlet Series of Modular Forms

Thus, the inner integral in (2.65) can be written in a right half-plane in the form ∞ 1

=

t s−k+1 (Θ(t, u) − 1)dt +

∞ 1

1 0

t s−k+1 (Θ(t, u) − 1)dt

(t s−k+1 + t k−s−1 )(Θ(t, u) − 1)dt +

1 1 − . s−k s−k+2

(2.66)

Substituting this expression in (2.65), we get ∞ k 2π k−2 Ψ(s; F) = F(u)v (t s−k+1 + t k−s−1 )(Θ(t, u) − 1)dt du D 1 1 1 k − du. (2.67) + F(u)v s−k s−k+2 D Let us take now the domain D to be the set D described in Theorem 2.10, which has a single cusp at the point v = +∞. Since F is a cusp form, the function F(u) exponentially tends to zero as u = (z, v) tends from within D to the cusp. The same is obviously true with respect to the function Θ(t, u) − 1 under the inner integral in the last formula. Thus, the integral representation (2.67) gives a meromorphic analytic continuation of Ψ(s; F) to the whole complex plane. This function is regular at all points, except possibly for simple poles at s = k and s = k − 2, when ! k D F(u)v du = 0. Furthermore, the right-hand side of (2.67) is invariant under the substitution s → 2k − 2 − s, which proves the functional equation. Exercise 2.17. Prove that the function Φ(s) = π −s Γ(s)ZO (s)E ∗ (u, s), where Γ(s) is the gamma function, ZO (s) is the zeta function (2.59), and E ∗ (u, s) is the Eisenstein series (2.55), can be continued analytically to the whole s-plane as a meromorphic function having only two simple poles at the points s = 0 and s = 2, and satisfying the functional equation Φ(2 − s) = Φ(s). [Hint: Use the integral representation (2.60) and identity (2.66) with s + k − 2 in place of s.]

Chapter 3

Hecke–Shimura Rings of Double Cosets

3.1 An Approach to Multiplicativity Numerous classical formulas for the numbers of integral representations of integers by integral quadratic forms often show remarkable multiplicative features. Thus, for example, Jacobi’s formula for the number of representations r4 (m) of an odd integer m as the sum of four of integer squares, r4 (m) = 8 ∑ d, d|m

expresses the number through the multiplicative function ς (m) = ∑d|m d with the rule of multiplication mn , ς (m)ς (n) = ∑ d ς d2 d|m,n whereas Ramanujan’s formula for the number of representations of integers as sum of 24 integral squares expresses the number as a linear combination of two multiplicative functions. The numbers of integral representations of integers by an integral positive definite quadratic form can be considered as Fourier coefficients of a modular form, the theta series of the quadratic form. In order to reveal multiplicative properties of the numbers of representations, it is reasonable to consider a more general but formulated in more invariant form problem of multiplicative properties of Fourier coefficients of modular forms. Let us go into some detail. One can argue as follows: let, for example, ∞

F(z) =

∑ f(m)e2π imz ∈ Mk = Mk (Γ )

m=0

be a modular form of integral weight k for the group Γ = Γ1 . Generally speaking, a “multiplicativity” of the function m → f(m) should at least imply that there are

A. Andrianov, Introduction to Siegel Modular Forms and Dirichlet Series, Universitext, DOI 10.1007/978-0-387-78753-4 3, c Springer Science+Business Media LLC 2009

63

64

3 Hecke–Shimura Rings of Double Cosets

regular relations between this function and functions m → f(pm) for some prime numbers p. The values f(pm) are the Fourier coefficients of the function ∞

Fp (z) =

∑

m=0

f(pm)e2π imz =

1 p−1 ∑F p b=0

z+b p

p−1

=

∑ F|k

b=0

1b , 0p

where |k M are the Petersson operators (1.29) for genus n = 1. If the linear operator F → Fp maps the space Mk into itself, then one could hope to find its eigenfunctions in Mk , i.e, functions satisfying Fp = λ (p)F. For each such function one gets relations f(pm) = λ (p)f(m) for all m = 1, 2, . . . , which provide a kind of multiplicativity. Now, the function Fp belongs to Mk if and only if Fp |k γ = Fp for all γ ∈ Γ . Since F|k γ = F if γ ∈ Γ , and |k M1 M2 = |k M1 |k M2 for every two real matrices of order two with determinant, the inclusion Fp ∈ Mk would be true if the set of positive coincides up to an order with the set left cosets Γ 10 0p , Γ 10 1p , . . . Γ 10 p−1 p Γ 10 0p γ , Γ 10 1p γ , . . . Γ 10 p−1 γ for every γ ∈ Γ . But this is not true, since, p for example, the left coset 10 0 1 0 1 p0 p0 =Γ =Γ Γ 0p −1 0 −1 0 01 01 differs from any left coset Γ 10 bp . On the other hand, we note that all the left cosets 1 b 1 01 b Γ 0 p γ = Γ 0 p 0 1 γ with γ ∈ Γ belong to the double coset Γ 10 0p Γ , and we may replace the sum Fp by the sum over all representatives of left cosets contained in that double coset. Thus, in place of the operator F → Fp , we arrive at the famous Hecke operator T (p) : F → F|T (p) = ∑ F|k M. γ ∈Γ \Γ 10 0p Γ

Since every right on an element of Γ only rearranges left Γ -cosets multiplication contained in Γ 10 0p Γ , it easily follows that T (p) maps the space Mk into itself. Since p is a prime number, one can easily check that the set

10 10 1 p−1 p0 , ,..., , 0p 1p 0 p 01 is a set of representatives of all different left Γ -classes contained in this double coset. Thus, we conclude that ∞ ∞ p0 F|T (p) = Fp + F|k = ∑ f(pm)e2π imz + pk−1 ∑ f(m)e2π ipmz 01 m=0 m=0 ∞ m e2π imz , = ∑ f(pm) + pk−1 c p m=0 where f( mp ) = 0 if p m. If F ∈ Mk is an eigenfunction for the operator T (p), F|T (p) = λ (p)F, then, by comparing Fourier coefficients, we have

3.1 An Approach to Multiplicativity

65

m k−1 = λ (p)f(m) f (pm) + p f p

(m = 0, 1, 2, . . . ),

and in particular, f (p) = λ (p)f(1). Hence we obtain m k−1 = f (p)f (m) f (1) f (pm) + p f p

(m = 0, 1, 2, . . . ),

which is not very bad kind of multiplicativity with respect to the prime number p. The above consideration points to an approach to the problem of multiplicativity of Fourier coefficients of modular forms for subgroups of modular groups as well as the problem of Euler factorization of Dirichlet series formed by the Fourier coefficients, namely the approach based on the study of linear operators on spaces of modular forms associated to certain double cosets of the corresponding subgroup. The double cosets form associative rings, the rings of double cosets or the Hecke–Shimura rings, which we consider in this chapter first in an abstract situation and then for discrete subgroups of the general linear and symplectic groups. In the next chapter we shall study the linear operators on spaces of modular forms corresponding to the rings of double cosets. Exercise 3.1. (1) Show that the subspace of cusp forms Nk (Γ ) ⊂ Mk (Γ ) is invariant with respect to all of the operators T (p). (2) Show that the function ∆ (z) ∈ N12 (Γ ) defined in Exercise 1.36 is an eigenfunction for all operators T (p). (3) (Ramanujan/Mordell) Let ∆ (z) =

∞

∑ f(m)e2π imz .

m=1

Show that the corresponding radial Dirichlet series has the Euler factorization ∞

f(m) 1 = f(1) ∏ , s −s + p11−2s m 1 − τ (p)p m=1 p∈P

∑

where τ (p) are the eigenvalues of T (p). [Hint for (3): using the multiplicative relations m 11 = τ (p)f (m) f (pm) + p f p

(m = 0, 1, 2, . . . ),

prove first that for every prime p and every m not divisible by p, we have the identity ∞

(1 − τ (p)t + p11t 2 ) ∑ f(mpδ )t δ = f(m).] δ =0

66

3 Hecke–Shimura Rings of Double Cosets

3.2 Abstract Rings of Double Cosets Definition of Hecke–Shimura rings. In this section we assign to each pair of a multiplicative semigroup with unit and its subgroup, satisfying a simple finiteness condition, an associative ring constructed of cosets of the semigroup modulo the subgroup and consider their basic properties. The simplest situation in which a ring of cosets can be naturally defined is the case of a group Σ and its normal subgroup Λ . In this case each double coset Λ gΛ of Σ modulo Λ consists of a single left coset, Λ gΛ = Λ g; hence one can define the product of double cosets just by (Λ gΛ )(Λ gΛ ) = (Λ g)(Λ g ) = (Λ gg ) = (Λ ggΛ ) and then extend the multiplication by linearity on finite formal linear combinations of the double cosets with coefficients, say, in the ring Z. As a result, we arrive at the group ring of the factor group Λ \Σ over Z. The definition of Hecke–Shimura rings of double cosets just transfers this approach to the situation in which double cosets are unions of finite numbers of left cosets. Let Σ be a multiplicative semigroup with identity element and Λ a subgroup of Σ . We shall say that the pair Λ , Σ is l(eft)-finite if each double coset Λ gΛ ⊂ Σ is a finite union of left cosets modulo Λ : ν (g)

Λ gΛ =

Λ gi

(ν (g) = #(Λ \Λ gΛ )).

(3.1)

i=1

The pair is said to be d(ouble)-finite if each of the double cosets is both a finite union of left cosets and a finite union of right cosets modulo Λ . For an l-finite pair Λ , Σ we denote by L = L(Λ , Σ ) the free Z-module consisting of all formal finite linear combinations with coefficients in Z of the symbols (Λ g) with g ∈ Σ , which are in one-to-one correspondence with the left cosets Λ g of Σ relative to the group Λ . The semigroup Σ naturally operates on L by multiplication from the right:

Σ g : τ = ∑ ai (Λ gi ) → τ g = ∑ ai (Λ gi g). i

i

Let us consider the submodule D of L consisting of all elements, that are invariant under the right multiplication by elements in Λ : (3.2) D = D(Λ , Σ ) = τ ∈ L τλ = τ for all λ ∈ Λ . If

τ = ∑ ai (Λ gi ) and τ = ∑ ai (Λ gj ) i

j

are two elements in D, then the linear combination

ττ = ∑ ai aj (Λ gi gj ) i, j

(3.3)

3.2 Abstract Rings of Double Cosets

67

is independent of the choice of the representatives gi ∈ Λ gi and gi ∈ Λ gi and again belong to D. To see this, we first note that ττ is independent of the choice of representatives gi ∈ Λ gi . Now, if we replace the representatives gj by λ j gj with λ j ∈ Λ , by definition of D, we have

∑ ai (Λ gi λ j ) = τλ j = τ = ∑ ai (Λ gi ) i

i

for each j. Hence

∑ i, j

∑

aj

=∑

aj

ai aj (Λ gi λ j gj ) =

j

j

∑ ai (Λ gi λ j )

i

∑ ai (Λ gi ) i

gj

gj = ∑ ai aj (Λ gi gj ), i, j

and so the product is independent of the choice of gj ∈ Λ gj . Finally, if λ ∈ Λ , we have (ττ )λ = ∑ ai aj (Λ gi gj λ ) = τ · (τ λ ) = ττ , i, j

whence ττ

∈ D. Since the multiplication (3.3) is clearly bilinear and associative, we see that it defines on D(Λ , Σ ) a structure of an associative ring, the Hecke–Shimura ring of the pair Λ , Σ (over Z). Basic Rings. The ring Z in the definition of the Hecke–Shimura ring of an l-finite pair Λ , Σ can be replaced by an arbitrary associative and commutative ring A with identity. It leads us to the Hecke–Shimura ring DA (Λ , Σ ) of the pair Λ , Σ over A. The choice of a basic ring A is insignificant for a majority of the properties of the Hecke–Shimura rings that we are going to consider. As a rule, we take A = Z, except for the consideration of spherical representations, when we take A to be the field Q of rational numbers, and the study of relations between Hecke operators and the Siegel operator, when it is convenient to take A = C, the field of complex numbers. If the basic ring is not indicated, it means that A = Z. Standard Bases and Multiplication Rules. Let Λ , Σ be an l-finite pair. If g1 , . . . , gν is a complete set of representatives of different left cosets modulo Λ contained in a double coset Λ gΛ ⊂ Σ , then the set g1 λ , . . . , gν λ with any λ ∈ Λ is clearly still a complete set of representatives of different left cosets modulo Λ contained in a double coset Λ gΛ . It follows that each linear combination of left cosets of the form τ (g) = τΛ (g) = ∑ (Λ gi ) (g ∈ Σ ) (3.4) gi ∈Λ \Λ gΛ

satisfies τ (g)λ = τ (g) and so belongs to the ring D = D(Λ , Σ ). By an easy induction, left to the reader, on the number of distinct left cosets that actually occur in a linear combination contained in D, one gets that each such combination is a linear combination of elements of the form τ (g) with integral coefficients.

68

3 Hecke–Shimura Rings of Double Cosets

Moreover, the elements τ (g) corresponding to different double cosets Λ gΛ are obviously linearly independent over Z. We conclude that the elements τ (g) corresponding to different double cosets Λ gΛ form a Z-basis of the ring D. The elements τ (g) will be sometimes referred as double cosets (modulo Λ ) and the ring D = D(Λ , Σ ) as the (Hecke–Shimura) ring of double cosets or just dc-ring of the pair Λ , Σ . Note that the element

τ (e) = (Λ e) = (Λ ),

(3.5)

where e is the identity of Σ , is the identity of the ring D. The following lemma describes the product of double cosets in terms of double cosets. Lemma 3.2. The product of double cosets in a dc-ring D = D(Λ , Σ ) can be computed by the following formulas: if

∑

τ (g) =

τ (g ) =

(Λ gi ) and

gi ∈Λ \Λ gΛ

∑

gj ∈Λ \Λ g Λ

(Λ gj )

are two elements of the form (3.4) in D, then

τ (g)τ (g ) =

∑

Λ hΛ ⊂Λ gΛ g Λ

c(g, g ; h)τ (h),

where h ranges over a set of representatives for the double cosets modulo Λ contained in Λ gΛ gΛ , and a coefficient c(g, g ; h) can be defined as the number of pairs i, j such that gi gj ∈ Λ h; the coefficients c(g, g ; h) can also be written in the form c(g, g ; h) = d(g, g ; h)ν (g )ν (h)−1 , where d(g, g ; h) is the number of representatives gi satisfying gi g ∈ Λ hΛ , and ν (g ), ν (h) are the numbers of left cosets in the double cosets Λ gΛ and Λ hΛ , respectively. Proof. By definition, we get

τ (g)τ (g ) =

∑

gi ∈Λ \Λ gΛ

(Λ gi )

∑

gj ∈Λ \Λ g Λ

(Λ gj ) = ∑(Λ gi gj ). i, j

Since each of the products gi gj is clearly contained in Λ gΛ gΛ , it follows that the product can be written in the form

τ (g)τ (g ) =

∑

Λ hΛ ⊂Λ gΛ g Λ

c(g, g ; h)τ (h) =

∑

Λ hΛ ⊂Λ gΛ g Λ

c(g, g ; h)

∑

(Λ hk )

hk ∈Λ \Λ hΛ

with some integral coefficients c(g, g ; h). By comparing the coefficients of (Λ h) in the two decompositions of the product into left cosets, we conclude that c(g, g ; h) is equal to the number of pairs i, j such that Λ gi gj = Λ h, i.e. gi gj ∈ Λ h. Further, it is easy to see that the number c(g, g ; h) depends only on the double cosets of g, g , and h. Thus, the sum

3.2 Abstract Rings of Double Cosets

∑

69

c(g, g ; h) = ν (h)c(g, g ; h)

Λ hk ⊂Λ \Λ hΛ

is equal to the number of pairs i, j such that gi gj ∈ Λ hΛ . Since each representative gj can obviously be taken in the form gj = g λ j with λ j ∈ Λ , we conclude that the last number equals the number d(g, g ; h) of representatives gi with gi g ∈ Λ hΛ multiplied by ν (g ). The following lemma often allows one to simplify the decomposition of double cosets into left cosets. Lemma 3.3. Let G be a group, Λ a subgroup of G, and g ∈ G. Let

Λ=

(Λ ∩ g−1Λ g)λi

(resp., Λ =

λ j (Λ ∩ gΛ g−1 ))

λ j ∈Λ /Λ ∩gΛ g−1

λi ∈Λ ∩g−1 Λ g\Λ

(3.6) be decompositions into different cosets. Then

Λ gΛ =

Λ gλi

(resp., Λ gΛ =

λi ∈Λ ∩g−1 Λ g\Λ

λ j gΛ ),

(3.7)

λ j ∈Λ /Λ ∩gΛ g−1

where the cosets are disjoint; in particular,

ν (g) = #(Λ \Λ gΛ ) = [Λ : Λ ∩ g−1Λ g] (resp., ν (g) = #(Λ gΛ /Λ ) = [Λ : Λ ∩ gΛ g−1 ]).

(3.8)

Proof. Let us consider, for example, the case of left cosets. It is clear that the right side of (3.7) is contained in the left side. Let g = λ gλ ∈ Λ gΛ . By (3.6), the element λ belongs to a left coset (Λ ∩g−1Λ g)λi , i.e. λ = αλi , where α ∈ Λ and gα g−1 ∈ Λ . Then we have g = λ gαλi = λ gα g−1 gλi ∈ Λ gλi , and so the left side of (3.7) is contained in the right side. Now, if a coset Λ gλi meets Λ gλ j , then we have λ gλi = δ gλ j with λ , δ ∈ Λ , whence g−1 δ −1 λ gλi = λ j , and so (Λ ∩ g−1Λ g)λi = (Λ ∩ g−1Λ g)λ j , because the element g−1 δ −1 λ g = λ j λi−1 is obviously contained in Λ ∩ g−1Λ g. Left and Right Rings of Double Cosets. Suppose now that Λ , Σ is a d-finite pair, i.e., such that each double coset Λ gΛ in Σ is both a finite union of left cosets and a finite union of right cosets modulo Λ . In this case, the product of double cosets of Σ modulo Λ can be defined not only by a product of right Λ -invariant linear combinations of left cosets modulo Λ , as above, but also by a similar product of left Λ -invariant linear combinations of right cosets modulo Λ . The natural question is whether we get the same product of the double cosets. The solution is not quite trivial, since a double coset can perfectly well consist of different numbers of left and right costs. Nevertheless, the answer is as simple as “yes”, at least if Σ is contained in a group.

70

3 Hecke–Shimura Rings of Double Cosets

Theorem 3.4. Let Λ , Σ be a d-finite pair contained in a group G. Then the product of any double cosets of Σ modulo Λ defined by multiplication of right Λ -invariant linear combinations of left cosets modulo Λ coincides with their product defined by a similar multiplication of left Λ -invariant linear combinations of right cosets modulo Λ . The proof of the theorem is a rather technical one; however we shall outline it, because the result has not only philosophical but quite practical value. First we have to recall definitions and simple properties of commensurability. We say that two subgroups Γ1 and Γ2 of a multiplicative group G are commensurable if their intersection is of finite index in both Γ1 and Γ2 . Then we write Γ1 ∼ Γ2 . The relation of commensurability is clearly reflexive and symmetric. It is not hard to see that the relation is also transitive. Lemma 3.5. Let G be a group, and Γ a subgroup of G. Then the commensurator of Γ in G, Γ- = Γ-G = {g ∈ G|g−1Γ g ∼ Γ }, is a group. Proof. If g ∈ Γ-, then Γ = g(g−1Γ g)g−1 ∼ gΓ g−1 , and so g−1 ∈ Γ-. Now, if g1 , g2 ∈ −1 −1 −1 Γ-, then g−1 1 Γ g1 ∼ Γ , hence g2 g1 Γ g1 g2 ∼ g2 Γ g2 ∼ Γ , and so by transitivity, −1 (g1 g2 ) Γ g1 g2 ∼ Γ . Lemma 3.6. Let G be a group, Γ a subgroup of G, and Γ- = Γ-G the commensurator of Γ in G. Then the function

ν (g) Γ- g → η (g) = ν (g−1 )

with ν (h) = [Γ : Γ ∩ h−1Γ h]

is a homomorphism of the group Γ- into the multiplicative group of nonzero rational numbers, which is trivial on the subgroup Γ . Proof. Let us denote by X the set of all subgroups of G that are commensurable with Γ . By transitivity, if Γ1 , Γ2 ∈ X , there exists a group Γ ∈ X of finite index both in Γ1 and Γ2 , for example, Γ1 ∩ Γ2 . We then set

η (Γ1 /Γ2 ) = [Γ1 : Γ ][Γ2 : Γ ]−1 .

(3.9)

It is easy to see that η is independent of the choice of Γ and satisfies the relations

η (Γ1 /Γ2 )η (Γ2 /Γ3 ) = η (Γ1 /Γ3 ),

η (g−1Γ1 g/g−1Γ2 g) = η (Γ1 /Γ2 ) (g ∈ Γ-). (3.10)

For g ∈ Γ- and Γ ∈ X , we set η (g) = η (Γ /g−1Γ g). It is readily verified that η (g) does not depend of the choice of Γ . Hence, by (3.10), we obtain

η (g1 g2 ) = η (Γ /(g1 g2 )−1Γ g1 g2 ) −1 −1 −1 = η (Γ /g−1 1 Γ g1 )η (g1 Γ g1 /(g2 g1 Γ g1 g2 )) = η (g1 )η (g2 )

3.2 Abstract Rings of Double Cosets

71

if g1 , g2 ∈ Γ-. On the other hand, if g ∈ Γ-, then

η (g) = η (Γ /Γ ∩ g−1Γ g)η (Γ ∩ g−1Γ g/g−1Γ g) = ν (g)η (gΓ g−1 ∩ Γ /Γ ) = ν (g)ν (g−1 )−1 = η (g). Proof of Theorem 3.4. It suffices to consider the products of two double cosets. By Lemmas 3.2 and 3.3, we can write

τ (g)τ (g) ´ =

∑

Λ hΛ ∈Λ gΛ g´Λ

d(g, g; ´ h)[Λ : Λ ∩ g´−1Λ g][ ´ Λ : Λ ∩ h−1Λ h]−1 τ (h),

where d(g, g; ´ h) is the number of representatives gi ∈ Λ \Λ gΛ satisfying gi g´ ∈ Λ hΛ . On the other hand, by similar reasoning for corresponding formal sums of right cosets, we get

∑

(gi Λ ) ×

=

∑

∑

g´ j ∈Λ g Λ /Λ

g-i ∈Λ gΛ /Λ

Λ hΛ ∈Λ gΛ g´Λ

(g´ j Λ )

- g; d(g, ´ h)[Λ : Λ ∩ gΛ g−1 ][Λ : Λ ∩ hΛ h−1 ]−1

∑

(hk Λ ),

hk ∈Λ hΛ /Λ

- g; where d(g, ´ h) is the number of representatives g´ j ∈ Λ g´Λ /Λ satisfying gg´ j ∈ Λ hΛ . In order to prove the equality of the two products, we have only to show that ´ Λ : Λ ∩ h−1Λ h]−1 d(g, g; ´ h)[Λ : Λ ∩ g´−1Λ g][ - g; = d(g, ´ h)[Λ : Λ ∩ gΛ g−1 ][Λ : Λ ∩ hΛ h−1 ]−1

(3.11)

if h ∈ Λ gΛ g´Λ . By Lemma 3.6 for Γ = Λ , we have ´ [Λ : Λ ∩ h−1Λ h][Λ : Λ ∩ hΛ h−1 ]−1 = η (h) = η (g)η (g) ´ Λ : Λ ∩ g´Λ g´−1 ]−1 . = [Λ : Λ ∩ g−1Λ g][Λ : Λ ∩ gΛ g−1 ]−1 [Λ : Λ ∩ g´−1Λ g][ Thus, the relation (3.11) is equivalent to - g; ´ h)[Λ : Λ ∩ g−1Λ g]. d(g, g; ´ h)[Λ : Λ ∩ g´Λ g´−1 ] = d(g,

(3.12)

By the definition and Lemma 3.3, we see that the number d(g, g; ´ h) can be written in the form d(g, g; ´ h) = #{λ ∈ Λ(g) \Λ | gλ g´ ∈ Λ hΛ }

with Λ(g) = Λ ∩ g−1Λ g.

(3.13)

For λ ∈ Λ , it is easy to see that the double coset Λ gλ g´Λ depends only on the coset Λ(g) λΛ(g´−1 ) , and the relation Λ(g) λ δ = Λ(g) λ with δ ∈ Λ(g´−1 ) holds if and only if δ ∈ λ −1Λ(g) λ . Thus we can rewrite the number (3.13) in the form d(g, g; ´ h) =

∑

[Λ(g´−1 ) : (Λ(g´−1 ) ∩ λ −1Λ(g) λ )].

λ ∈Λ(g) \Λ /Λ(g´−1 ) , gλ g∈ ´ Λ hΛ

72

3 Hecke–Shimura Rings of Double Cosets

Quite similarly, - g; d(g, ´ h) =

∑

[Λ(g) : (Λ(g) ∩ λΛ(g´−1 ) λ −1 )].

λ ∈Λ(g) \Λ /Λ(g´−1 ) , gλ g∈ ´ Λ hΛ

Using the symbol (3.9) and its properties (3.10), we obtain - g; d(g, ´ h) =

∑

λ ∈Λ(g) \Λ /Λ(g´−1 ) , gλ g∈ ´ Λ hΛ

η (Λ(g) /(Λ(g) ∩ λΛ(g´−1 ) λ −1 ))

= ∑ η (λ −1Λ(g) λ /λ −1Λ(g) λ ∩ Λ(g´−1 ) ) λ

= ∑ η (λ −1Λ(g) λ /Λ(g) )η (Λ(g) /Λ )η (Λ /Λ(g´−1 ) ) λ

× η (Λ(g´−1 ) /Λ(g´−1 ) ∩ λ −1Λ(g) λ ) = ∑ η (λ )ν (g)−1 ν (g´−1 )η (Λ(g´−1 ) /Λ(g´−1 ) ∩ λ −1Λ(g) λ ) λ

´ h), = ν (g)−1 ν (g´−1 )d(g, g; where ν is defined by (3.8). Extension of Antiautomorphisms. For a semigroup Σ and its subgroup Λ , we say that a mapping g → g∗ of Σ into itself is an antiautomorphism of the pair Λ , Σ if it is one-to-one on Σ and on Λ and satisfies (gg) ´ ∗ = g´∗ g∗

for all g, g´ ∈ Σ .

(3.14)

The main application of Theorem 3.4 is the following proposition on the extension of antiautomorphisms of d-finite pairs to the corresponding dc-rings. Proposition 3.7. Let Λ , Σ be a d-finite pair contained in a group, and let g → g∗ be an antiautomorphism of the second order of this pair. Then the linear mapping of the dc-ring D = D(Λ , Σ ) into itself defined on the double cosets (3.4) by

τ (g) → τ (g)∗ = τ (g∗ )

(g ∈ Σ )

(3.15)

is an antiautomorphism of the second order of the ring D. If, in addition, the mapping is identical on all of the double cosets, then the ring D is commutative. Proof. For the first statement it is sufficient to prove that ´ ∗ = τ (g´∗ )τ (g∗ ) (g, g´ ∈ Σ ). (τ (g)τ (g)) By Lemma 3.2, we can write the left hand side of (3.16) in the form

∑

Λ hΛ ⊂Λ gΛ g´Λ

c(g, g; ´ h)τ (h)∗ =

∑

Λ hΛ ⊂Λ gΛ g´Λ

c(g, g; ´ h)τ (h∗ ),

(3.16)

3.2 Abstract Rings of Double Cosets

73

where

∑

c(g, g; ´ h) =

1.

gi ∈Λ \Λ gΛ , g´ j ∈Λ \Λ g´Λ , gi g´ j ∈Λ h

By Theorem 3.6, we can compute the right-hand side of (3.16) by considering the double cosets τ (φ (g )) and τ (φ (g)) as sums of the right cosets modulo Λ . By similar reasoning, since the map g → g∗ is an antiautomorphism of the second order carrying Λ onto itself, we obtain

∑

g´∗j ∈Λ g´∗ Λ /Λ

where

(g´∗j Λ )

∑

g∗i ∈Λ g∗ Λ /Λ

(g∗i Λ ) =

∑

c (g´∗ , g∗ ; h∗ )

Λ h∗ Λ ⊂Λ g´∗ Λ g∗ Λ

∑

c (g´∗ , g∗ ; h∗ ) =

∑

h∗k ∈Λ h∗ Λ /Λ

(h∗k Λ ),

1.

g´∗j ∈Λ g´∗ Λ /Λ ; g∗i ∈Λ g∗ Λ /Λ ; g´∗j g∗i ∈h∗ Λ

Furthermore, the conditions g´∗j ∈ Λ g´∗Λ /Λ , g∗i ∈ Λ g∗Λ /Λ and Λ h∗Λ ⊂ Λ g´∗Λ g∗Λ , h∗k ∈ Λ h∗Λ /Λ , g´∗j g∗i ∈ h∗Λ are equivalent to the conditions g´ j ∈ Λ \Λ g´Λ , gi ∈ Λ \Λ gΛ

and Λ hΛ ⊂ Λ gΛ g´Λ , hk ∈ Λ \Λ hΛ , gi g´ j ∈ Λ h,

´ h), which proves (3.16). respectively. It follows that c (g´∗ , g∗ ; h∗ ) = c(g, g; As to the second statement, the assumption implies that τ ∗ = τ for all τ ∈ D; hence we have ´ τ τ´ = (τ τ´ )∗ = τ´ ∗ τ ∗ = ττ

for all τ , τ´ ∈ D.

Extension of Embeddings. Next we consider embeddings of Hecke–Shimura rings based on inclusions of corresponding pairs. The embeddings turn out to be useful when initial rings seem to be too narrow and we are looking for suitable extensions. Suppose that Λ , Σ and Λ0 , Σ0 are two l-finite pairs contained in the same group and satisfying the following conditions:

Λ0 ⊂ Λ ,

Σ ⊂ Λ Σ0 ,

and

Λ ∩ Σ0 Σ0−1 ⊂ Λ0 ,

(3.17)

where Σ0−1 = {g−1 |g ∈ Σ0 }. Since Σ ⊂ Λ Σ0 , each linear combination of left cosets τ ∈ L(Λ , Σ ) can be written in the form

τ = ∑ ai (Λ gi )

with gi ∈ Σ0 .

i

Let us consider then the formal linear combination of left cosets

ι (τ ) = ∑ ai (Λ0 gi ). i

74

3 Hecke–Shimura Rings of Double Cosets

The element ι (τ ) is independent of the choice of representatives gi ∈ Σ0 in Λ gi , since if gi ∈ Λ gi ∩ Σ0 are other representatives, the last condition of (3.17) implies −1 that λi = gi g−1 i ∈ Λ ∩ Σ 0 Σ 0 ⊂ Λ0 , whence

ι (τ ) = ∑ ai (Λ0 gi ) = ∑ ai (Λ0 λi gi ) = ∑ ai (Λ0 gi ) = ι (τ ). i

i

i

The condition Λ0 ⊂ Λ implies that the map ι carries different left cosets modulo Λ to different left cosets modulo Λ0 . Thus we obtain an additive embedding of the modules of left cosets L(Λ , Σ ) → L(Λ0 , Σ0 ). Proposition 3.8. Let Λ , Σ and Λ0 , Σ0 be two l-finite pairs contained in a group and satisfying conditions (3.17). Then the restriction of the map ι to the Hecke– Shimura ring D(Λ , Σ ) ⊂ L(Λ , Σ ) is a ring monomorphism of D(Λ , Σ ) into the ring D(Λ0 , Σ0 ): ι : D(Λ , Σ ) → D(Λ0 , Σ0 ). (3.18) If, moreover,

Σ0 ⊂ Σ

and

νΛ (g) = νΛ0 (g) for all g ∈ Σ0 ,

(3.19)

where ν stands for the indices (3.8), then the mapping (3.18) is an isomorphism of the rings. Proof. The first assertion follows directly from definitions and the inclusion Λ0 ⊂ Λ . As to the second, it suffices to show that ι (τΛ (g)) = τΛ0 (g) for all g ∈ Σ0 and double cosets (3.4) if the conditions (3.19) are fulfilled. Let λ1 , . . . , λν with ν = νΛ0 (g) be a set of representatives of the left cosets of Λ0 modulo Λ0 ∩ g−1Λ0 g. By Lemma 3.3, we have ν

τΛ0 (g) = ∑ (Λ0 gλi ). i=1

On the other hand, the elements gλ1 , . . . , gλν belong to Λ gΛ and are contained in different left cosets modulo Λ , since a relation gλi = δ λ j with δ ∈ Λ would imply that δ = (gλi λ j−1 )g−1 ⊂ Λ ∩ Σ0 Σ0−1 ⊂ Λ0 . By (3.19) and (3.7), we obtain ν

τΛ (g) = ∑ (Λ gλi ), i=1

and so ι (τΛ (g)) = τΛ0 (g) . Exercise 3.9. Let Λ , Σ and Λ0 , Σ0 be two l-finite pairs contained in a group and satisfying conditions (3.17). Let g → g∗ be an antiautomorphism of the second order of the both pairs. Show then that

ι (τ )∗ = ι (τ ∗ ) for all τ ∈ D(Λ , Σ ).

3.3 Rings of Double Cosets of the General Linear Group

75

Exercise 3.10. Let Λ , Σ be an l-finite pair. Show that the index map D(Λ , Σ ) ∑ ai τΛ (gi ) → ν (∑ ai τ (gi )) = ∑ ai ν (gi ), i

i

i

where ν (gi ) are the indices (3.8), is a ring homomorphism into Z.

3.3 Rings of Double Cosets of the General Linear Group Although our main objective is the study of dc-rings of the symplectic group, we shall start our consideration of concrete rings with the case of the general linear group because this case is easier, it coincides with the symplectic case for matrices of order two, and, as we shall see later, in some aspects the symplectic case can be reduced to the general linear case. Global Rings. Our basic pair here will be the pair Λ = Λn = GLn (Z) = g ∈ Znn det g = ±1 , Σ = Σn = g ∈ Znn det g = 0 with n = 1, 2, . . . . Lemma 3.11. The pair Λn , Σn is d-finite for every n. Proof. By considering the entries of matrices in Λ modulo a positive integer q, we get a homomorphism of Λ into the finite group GLn (Z/qZ). The kernel of this homomorphism is the principal congruence subgroup of level q of Λ , Λ (q) = Λn (q) = λ ∈ Λ λ ≡ 1n (mod q) , (3.20) which is therefore a normal subgroup of finite index in Λ . If g ∈ Λ , then clearly the groups gΛ (d)g−1 and g−1Λ (d)g, where d = | det g|, are both contained in Λ . It follows that the groups Λ ∩ g−1Λ g and Λ ∩ gΛ g−1 both contain the group Λ (d), and so have finite indices in Λ . The lemma follows then from Lemma 3.3. The lemma allows us to define the Hecke–Shimura ring (3.1), H = Hn = D(Λn , Σn ),

(3.21)

of the pair Λn , Σn . As we have seen in the previous section, elements of the form (3.4), which, for the group Λ = Λn , will be denoted by t(g) = τΛ (g)

(g ∈ Σn ),

(3.22)

corresponding bijectively to the double cosets Λ gΛ contained in Σ , can be taken as a (free) basis of the ring H (over Z). The following theorem gives a convenient parametrization of the double cosets.

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3 Hecke–Shimura Rings of Double Cosets

Theorem 3.12. (The theorem on elementary divisors). Each of the double cosets Λ gΛ with Λ = Λn and g ∈ Σ = Σn contains a unique representative of the form ed(g) = diag(d1 , . . . , dn ) with d1 , . . . , dn ∈ N and d1 |d2 | · · · |dn .

(3.23)

First we cite a simple lemma, which like Lemma 1.10 is an easy consequence of the Euclidean algorithm. Lemma 3.13. Let l be a nonzero integral n-row (resp.,n-column) and d the greatest common divisor of the entries of l. Then there exists a matrix λ ∈ Λn such that lλ = (d, 0, . . . , 0) (resp., λ l = t(d, 0, . . . , 0)) . Proof of Theorem 3.12. First we note that elementary integral operations on rows of matrices g ∈ Σ such as a permutation of rows and addition to one row of another row multiplied by an integer (resp., similar operations on columns) can be derived by multiplication of the matrix from the left (resp., from the right) by a matrix of Λ and so does not change the coset Λ gΛ . In order to prove that a representative of the form (3.23) exists, we apply induction to n. If n = 1, there is nothing to prove. Suppose that n > 1, and the theorem has already been proved for matrices of order n − 1. Denote by δ = δ (g) the minimum of the greatest common divisor of entries of rows of a matrix g ∈ Σn . By rearranging rows and applying Lemma 3.13, we may replace g by a matrix g = (gi j ) ∈ Λ gΛ with δ (g ) = δ (g) and the first row (g11 , . . . , g1n ) = (δ , 0, . . . , 0). Now we apply induction to δ . If δ = 1, then by adding to other rows suitable multiples of the first row, we can replace g by a matrix g = (gi j ) in the same double coset with g21 = 0, . . . , gn1 = 0, and the theorem is true by the induction assumption on n. But if δ > 1, then by similar reasoning, we can replace g by a the matrix g = (gi j ) in the same double coset with 1 ≤ g21 ≤ δ , . . . , 1 ≤ gn1 ≤ δ . If the greatest common divisor of entries of each row of the matrix g is equal to δ , then g21 = · · · = gn1 = δ , all entries of g are divisible by δ , and we clearly can replace g by a matrix of the form

δ 0 0 δ g1

with g1 ∈ Σ(n−1) , which is again contained in Λ gΛ , and we can use the induction assumption on n. Otherwise, δ (g ) < δ , and we may apply the induction assumption on δ . This proves the existence of a representative of the form (3.23). If D = diag(d1 , . . . , dn ) and D = diag(d1 , . . . , dn ) are two matrices of the form (3.23) satisfying D = λ Dλ with λ , λ ∈ Λ , then the Binet–Cauchy formula implies that each r-minor of D with 1 ≤ r ≤ n is divisible by d1 · · · dr ; in particular, d1 · · · dr |d1 · · · dr . Similarly, d1 · · · dr |d1 · · · dr . Hence D = D . The matrix ed(g) = diag(d1 , . . . , dn ) is called the matrix of elementary divisors of g, and the numbers dr = dr (g) are the elementary divisors of g. The Binet–Cauchy formula implies that the product d1 · · · dr is equal to the greatest common divisor of the minors of order r of g; in particular d1 · · · dn = | det g|.

(3.24)

It follows that the additive structure of the ring H is very simple: it is just a lattice spanned by the generators

3.3 Rings of Double Cosets of the General Linear Group

77

t[d1 , . . . , dn ] = t(diag(d1 , . . . , dn )) with d1 , . . . , dn ∈ N and d1 |d2 | · · · |dn , (3.25) where t(g) are the double cosets (3.22). Let us now turn to the multiplicative structure. First of all, we have the following theorem. Theorem 3.14. The dc-ring Hn = D(Λn , Σn ) is commutative for every n. Proof. The mapping g → tg, where tg is the transpose of g, is clearly an antiautomorphism of order two of the pair Λn , Σn that does not change diagonal matrices. Theorem 3.12 and Proposition 3.7 complete the proof. As to explicit rules of multiplication, we have the following basic theorem. Theorem 3.15. The double cosets (3.25) satisfy the rules t[d1 , . . . , dn ]t[d1 , . . . , dn ] = t[d1 d1 , . . . , dn dn ] if gcd(dn /d1 , dn /d1 ) = 1.

(3.26)

In particular, [d]t[d1 , . . . , dn ] = t[d1 , . . . , dn ][d] = t[dd1 , . . . , ddn ],

(3.27)

[d] = [d]n = t(d · 1n ) = (Λ d · 1n ).

(3.28)

where Proof. Since the double coset (3.28) consists of a single left coset, the relation (3.27) follows directly from the definition of multiplication in dc-rings. It follows from (3.27) that it is sufficient to prove the general relations (3.26) only when d1 = d1 = 1 and gcd(dn , dn ) = 1, or under the more general assumption that the numbers d1 · · · dn and d1 · · · dn are coprime. In order to prove it, we shall use the following lemma. Lemma 3.16. The elements

∑

t(a) = t n (a) =

t[d1 , . . . , dn ] ∈ Hn

(3.29)

1≤d1 |···|dn , d1 ···dn =a

with a ∈ N have decompositions into left cosets of the form t(a) =

∑

(Λ g)

(3.30)

g∈Λ \Σ , | det g|=a

and satisfy the relations t(a)t(a ) = t(aa)

if gcd(a, a ) = 1.

(3.31)

Proof of the lemma. The decompositions (3.30) follow from (3.24) and Theorem 3.12. Let g1 , . . . , gν and g1 , . . . , gµ be systems of representatives for the left cosets modulo Λ of matrices in Σ with determinants ±a and ±a , respectively. We have to show that the product of these systems g1 g1 , . . . , dν dµ is a system of representatives for

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3 Hecke–Shimura Rings of Double Cosets

the left cosets modulo Λ of matrices in Σ with determinant ±aa , provided that a and a are coprime. First, all of the products have determinant ±aa . Further, the products belong to different left cosets modulo Λ : if λ gi gj = gk gl with λ ∈ Λ , then −1 has determinant ±1 and is integral, since its prodthe matrix γ = g−1 k λ gi = gl (g j ) ucts by coprime numbers det gk = ±a and det gj = ±a are both integral, and hence γ ∈ Λ and γ gj = gl , so gj = gl and λ gi = gk , whence gi = gk . Finally, if h ∈ Σ and det h = ±aa , then by Theorem 3.12, it follows easily that h can be written in the form h = h1 h2 with h1 , h2 ∈ Σ satisfying det h1 = ±a and det h2 = ±a . We have h2 = δ gj and h1 δ = γ gi with δ , γ ∈ Λ ; hence h = h1 h2 = γ gi gj ∈ Λ gi gj . We can now return to the proof of the theorem. Let g = ed(g) and g = ed(g ) be two matrices of elementary divisors satisfying det g = a and det g = a . Since clearly gg ∈ Λ gΛ gΛ , it follows from Lemma 3.2 that the product of corresponding double cosets (3.22) has the form t(g)t(g ) = t(gg ) + s(g, g ), where s(g, g ) is a linear combination of double cosets with nonnegative coefficients. Hence, we obtain t(a)t(a ) =

∑

g=ed(g), det g=a

=

∑

g=ed(g), det g=a, g =ed(g ), det g =a

t(g)

∑

t(g )

∑

s(g, g ).

g =ed(g ), det g =a

t(gg ) +

g=ed(g), det g=a, g =ed(g ), det g =a

Since the numbers a and a are coprime, it follows easily that the first sum on the right is equal to t(aa ), which equals t(a)t(a ) by Lemma 3.14. Hence, the second sum on the right is zero, and so each of the sums s(g, g ) is zero too. Exercise 3.17. For g, g ∈ Σn satisfying gcd(dn (g)/d1 (g), dn (g )/d1 (g )) = 1, prove that the elementary divisors satisfy the relations di (gg ) = di (g)di (g ) (i = 1, 2, . . . , n). The following useful lemma gives a convenient description of representatives for the left cosets of Σn modulo Λn . A matrix g = (gi j ) ∈ Σn will be called reduced if its entries satisfy ⎧ ⎪g11 > 0, ⎨ (3.32) if 1 ≤ i < j ≤ n, 0 ≤ gi j < g j j ⎪ ⎩ if i > j. gi j = 0 Lemma 3.18. Each left coset Λn g ⊂ Σn contains its unique reduced representative. Proof. We shall apply induction to n. If n = 1 there is nothing to prove. Let n > 1 and suppose that the lemma has already been proved for n − 1. By Lemma 3.13 applied

3.3 Rings of Double Cosets of the General Linear Group

79

to the first column of g, we conclude that the coset Λ g contains a representative g d1 ∗ n−1 of the form 0 g1 with d1 > 0 and g1 ∈ Σ . By the inductive hypothesis, we can

write g1 = λ g0 , where λ ∈ Λn−1 and g0 is reduced. It follows from the Euclidean such that the matrix algorithm that there exists a row l ∈ Zn−1 1 1 0 1 l g ∈ Λ g 0 1n−1 0 λ −1 is reduced. The uniqueness follows easily by contradiction. Exercise 3.19. Show that for every l ∈ N, the element t(l) = t 2 (l) of H2 has the following decomposition into left cosets ab Λ2 . t(l) = ∑ 0d a,b∈N, ad=l, 0≤b 0. An element of H p is called primitive (resp., imprimitive) if it is a linear combination of primitive (resp., imprimitive) elements. It is clear that each element t ∈ H p has a unique decomposition t = t pr + t im , where t pr is primitive and t im is imprimitive. It follows from (3.27) that the set J of all imprimitive elements of H p is the principal ideal generated by the element [p] = [p]n , J = [p]H p . satisfying Lemma 3.21. The linear mapping φ : Hnp → Hn−1 p φ t[pδ1 , . . . , pδn ] =

t[pδ2 , . . . , pδn ] 0

if 0 = δ1 ≤ δ2 ≤ · · · ≤ δn , if 0 < δ1 ≤ δ2 ≤ · · · ≤ δn ,

is a ring epimorphism with kernel J . Proof. It is obvious that this mapping is an epimorphism of Z-modules. It follows from (3.27) that the kernel φ equals J . Thus, we have only to check that it is a ring homomorphism. For this, it suffices to verify the relations

3.3 Rings of Double Cosets of the General Linear Group

φ t[1, pδ 2 , . . . , pδ n ]t[1, pδ 2 , . . . , pδ n ] = t[pδ 2 , . . . , pδ n ]t[pδ 2 , . . . , pδ n ],

81

(3.39)

where 0 ≤ δ2 ≤ · · · ≤ δn and 0 ≤ δ2 ≤ · · · ≤ δn . Let us set 1 0 1 0 , g = diag(1, pδ 2 , . . . , pδ n ) = . g = diag(1, pδ 2 , . . . , pδ n ) = 0 g0 0 g0 By Lemma 3.2 and the definition of φ , we obtain

φ (t(g)t(g )) = ∑ c(g, g ; h)t(h0 ), h

where h ranges over the matrices of the form 10 h00 , with h0 = diag(pγ2 , . . . , pγn ), 0 ≤ γ2 ≤ · · · ≤ γn , and γ2 + · · · + γn = δ2 + · · · + δn + δ2 + · · · + δn . Similarly, in the we have the relation ring Hn−1 p t(g0 )t(g0 ) = ∑ c(g0 , g0 ; h0 )t(h0 ), h0

where h0 ranges over the matrices indicated above. Thus, in order to prove (3.39), we have only to show that c(g, g ; h) = c(g0 , g0 ; h0 ).

(3.40)

Since the coefficient c(g, g ; h) depends only on the double cosets of matrices g, g , h h0 0 modulo Λ , we may replace h by h = 0 1 . By Lemmas 3.2 and 3.18, the coefficient c(g, g ; h) is equal to the number of pairs d, d of reduced matrices satisfying the conditions d ∈ Λ gΛ , d ∈ Λ gΛ , and dd = λ h . Since λ = dd (h )−1 , the matrix λ is an upper triangular matrix; in particular, it can be written in the form λ = λ00 λ∗ with λ0 ∈ Λn−1 and λnn = ±1. We can write d = d00 dvnn , nn d v d = 00 d , where d0 and d0 are reduced matrices of order n − 1, dnn > 0, and nn

all of the entries of the columns v and v are nonnegative integers smaller than , respectively. Then the relation dd = λ h implies that d d = λ dnn and dnn nn nn nn = 1, and hence and d0 d0 = λ0 h0 . By the first of the relations we have d = d nn nn v = v = 0. Thus, d = d00 01 , d = d00 01 , where d0 and d0 are reduced matrices of

order n − 1 satisfying d0 d0 = λ0 h0 with λ0 ∈ Λn−1 . It is clear that d0 ∈ Λn−1 g0 Λn−1 and d0 ∈ Λn−1 g0 Λn−1 . Conversely, arbitrary matrices d and d satisfying these conditions are reduced, belong to the double cosets Λ gΛ and Λ gΛ , respectively, and satisfy dd = λ h with λ ∈ Λ . This proves the equality (3.40). Theorem 3.22. For every n = 1, 2, . . . and every prime number p, the local ring Hnp is generated by the elements 1n−i 0 n n (1 ≤ i ≤ n); (3.41) πi (p) = πi (p) = t(Di ) with Di = Di = 0 p · 1i

82

3 Hecke–Shimura Rings of Double Cosets

the elements π1 (p), . . . , πn (p) are algebraically independent over Z, where Z is identified with the isomorphic subring Z [1]n ⊂ Hnp . Proof. For the first assertion it suffices to show that each element t of the form (3.38) is a polynomial in π1 (p), . . . , πn (p) with coefficients in Z. We shall use induction on n and N = δ1 + · · · + δn . For n = 1 the assertion is clear, since t[pδ ] = π11 (p)δ . Let us assume that n > 1 and the assertion has already been proved for n − 1. If N = 1, then t = π1 (p). Let N > 1 and the assertion is true for N < N. If δ1 ≥ 1, by (3.27) we have t = πn (p)δ1 t , where t = t[1, pδ2 −δ1 , . . . , pδ2 −δ1 ] with N = N − nδ1 < N, and hence t is an integral polynomial in elements (3.41). But if δ1 = 0, then the element t is primitive, and so by the inductive assumption on n, the image t = t[pδ2 , . . . , pδn ] = φ (t) of t under the mapping of Lemma 3.21 has the form t = n−1 (p)), where P(π1n−1 (p), . . . , πn−1 P(x1 , . . . , xn−1 ) =

∑

i

n−1 ai1 ,...,in−1 x1i1 · · · xn−1 ,

i1 +2i2 +···+(n−1)in−1 =N n−1 (p)in−1 since each double coset (h) entering into the product π1n−1 (p)i1 · · · πn−1 i +2i +···+(n−1)i δ +···+ δ N n n−1 = p 2 must satisfy | det h| = p 1 2 = p . By the definition of n (p)) bethe mapping φ , we conclude that the element t1 = t − P(π1n (p), . . . , πn−1 longs to the kernel of φ , and hence, by Lemma 3.21, is divisible by [p] = πnn (p) in the ring H p : t1 = πnn (p)t1 . By the above, t1 is a linear combination of ele ments t[pδ1 , . . . , pδn ] with δ1 + · · · + δn = N, then t1 is a linear combination of elements t[pδ1 , . . . , pδn ] with δ1 + · · · + δn = N − n. The first assertion follows. Now by induction on n we shall prove that the elements (3.41) are algebraically independent over Z. It is clear if n = 1, since the elements π11 (p)δ = t1 (pδ ) with δ = 0, 1, 2, . . . correspond to distinct double cosets modulo Λ1 = {±1} and so are linearly independent. Let us assume now that n > 1 and that the assertion has already been proved for n − 1. If π1n (p), . . . , πnn (p) are algebraically dependent and P(x1 , . . . , xn ) is a polynomial of minimal total degree such that P(π1n (p), . . . , πnn (p)) = 0, then, by applying the homomorphism φ of Lemma 3.21, we have

φ (P(π1n (p), . . . , πnn (p)) = P(φ (π1n (p)), . . . , φ (πnn (p))) = P(π1n−1 (p), . . . , πnn−1 (p), 0) = 0. By the inductive hypothesis, we obtain that P(x1 , . . . , xn ) = xn P (x1 , . . . , xn ), where P is a polynomial of smaller degree. By (3.27), the element πnn (p) = [p]n is not a zero divisor in Hn . Thus, P (π1n (p), . . . , πnn (p)) = 0, a contradiction. Spherical Mapping. By Theorem 3.22, every element of a local Hecke–Shimura ring of the general linear group can be written as a polynomial in a finite number of generators. Sometimes one needs to find these polynomials explicitly. However, a direct calculation proves to be rather complicated. To simplify the calculation, it turns out to be convenient to introduce certain mappings of the local Hecke–Shimura rings into rings of symmetric polynomials. Since the polynomials that appear do

3.3 Rings of Double Cosets of the General Linear Group

83

not necessarily have integral coefficients, it will be reasonable to consider Hecke– Shimura rings over the field Q of rational numbers in place of the ring Z. We shall denote the rings of double cosets over Q by the same symbol as before, but with an upper tilde, for example, - Λ , Σ ) = DQ (Λ , Σ ) D(

-p = H - n = DQ (Λn , Σn ). H p p

or

A matrix g = (gi j ) ∈ Σn is called (upper) triangular if gi j = 0 for all i > j if, moreover, all diagonal entries gii are positive, the matrix is called triangular-plus. By Lemma 3.18, each left coset Λn g ⊂ Σn contains a triangular-plus representative, which is not unique, in contrast to the reduced representative. On the other hand, a product of triangular-plus matrices is again triangular-plus, which is generally not true for reduced matrices. We fix an order n ∈ N and a prime number p. Diagonal entries of a triangular-plus representative contained in a left coset Λ g ⊂ Σ p = Σnp are nonnegative powers of p, say, pδ1 , pδ2 , . . . , pδn . It is clear that such a diagonal depends only on the left coset. Let x1 , x2 , . . . , xn be algebraically independent commuting variables. We set n

ω pn ((Λ g)) = ∏(xi p−i )δi ,

(3.42)

i=1

-p = H - n with a j ∈ Q we define the and for a given element t = ∑ j a j (Λ g j ) ∈ H p polynomial ω pn (t) = (ω pn (t))(x1 , . . . , xn ) = ∑ a j ω pn ((Λ g j )). j

Theorem 3.23. (1) The mapping - n → Q[x1 , . . . , xn ] ω = ω pn : H p

(3.43)

- n into the ring of polynomials in is a Q-linear homomorphism of the ring H p x1 , . . . , xn over Q. (2) The ω -images of the generators (3.41) are given by ω (πin (p)) = p−i(i+1)/2 si (x1 , . . . , xn ) (1 ≤ i ≤ n), where si (x1 , . . . , xn ) =

∑

1≤α1 0, is a ring epimorphism with kernel I. Exercise 3.39. Prove the lemma. From this lemma, similarly to the proof of Theorem 3.22, using induction on n, we obtain the following structural theorem. Theorem 3.40. For every n = 1, 2, . . . and every prime number p, the ring Lnp is generated over the subring Z Z 1 n by the element T (p) = T n (p) = T (1, . . . , 1, p, . . . , p) n

(3.64)

n

and the elements Ti (p2 ) = Tin (p2 ) = T (1, . . . , 1, p, . . . , p, p2 , . . . , p2 , p, . . . , p) n−i

i

n−i

(3.65)

i

for i = 1, . . . , n; the elements T (p), T1 (p2 ), . . . , Tn (p2 ) are algebraically independent over Z. Exercise 3.41. Provide details of the proof of the theorem. Regular Hecke–Shimura Rings for Congruence Subgroups. We shall see here that the basic structure theorems established above for Hecke–Shimura rings of the full modular group remain essentially true in the case of congruence subgroups.

92

3 Hecke–Shimura Rings of Double Cosets

We recall that a subgroup K of the modular group Γ = Γn is called a congruence subgroup if it contains a principal congruence subgroup Γ (q) = Γn (q) defined above (1.33). It is easy to check by induction on n that the natural map modulo q, t (3.66) Γn → Spn (Z/qZ) = M ∈ (Z/qZ)2n 2n MJn M ≡ Jn (mod q) , is an epimorphism with kernel Γn (q), whence Γn (q) is a normal subgroup of the modular group with the factor group isomorphic to Spn (Z/qZ). In particular, we have the following equality for the index of Γn (q):

ν (Γn (q)) = [Γn : Γn (q)] = #(Spn (Z/qZ)).

(3.67)

Lemma 3.42. If q and q are coprime integers, then Γn (q)Γn (q ) = Γn . Proof. The relation (3.67) shows that the index ν (Γn (q)) is the number of solutions of a system of polynomial congruences modulo q. Then the Chinese remainder theorem implies that the index is multiplicative in q, i.e., ν (Γn (qq )) = ν (Γn (q))ν (Γn (q )), provided that q and q are coprime. On the other hand, if q and q are coprime, then clearly Γn (q) ∩ Γn (q ) = Γn (qq ), which together with the multiplicativity of indexes implies that the embedding Γn (q) ⊂ Γn defines an isomorphism of the factor groups Γn (q)/Γn (qq ) and Γn /Γn (q ). Thus, for every γ ∈ Γn there is γ ∈ Γn (q ) such that γγ ∈ Γ(q). Then γ ∈ Γn (q)(γ )−1 ⊂ Γn (q)Γn (q ). Let us turn now directly to Hecke–Shimura rings of congruence subgroups. Let K = K n be a congruence subgroup of Γ = Γn . It is easy to see that for every M ∈ ∆ = ∆n , the intersection M −1 KM ∩ K is again a congruence subgroup of Γ . Hence, it has finite index in K. By Lemma 3.3, the double coset KMK is a finite union of left cosets modulo K. Similarly, KMK is a finite union of right cosets modulo K. Thus, the pair K, ∆ is d-finite, and we can define the Hecke–Shimura ring D(K, ∆ ) of the pair. However, this ring is generally too big to have good properties and is usually replaced by appropriate subrings corresponding to certain subsemigroups of ∆ . First of all, for positive integers n and q, we define the subsemigroup

1n 0 n (mod q) gcd(µ (M), q) = 1 ∆ (q) = ∆ (q) = M ∈ ∆ M ≡ 0 µ (M) · 1n (3.68) of ∆ = ∆n and define the dc-ring L(q) = Ln (q) = D(Γn (q), ∆ n (q))

(3.69)

as the (regular) Hecke–Shimura ring of the principal congruence subgroup Γn (q). We shall restrict ourselves to consideration of congruence subgroups K of Γ containing a principal congruence subgroup Γ (q) that satisfies the condition K ∆ (q) = ∆ (q)K.

(3.70)

3.4 Rings of Double Cosets of the Symplectic Group

93

Such group K will be called a q-symmetric group of genus n. For a q-symmetric group of genus n, we set

∆ (K) = ∆ (K, q) = K ∆ (q)K = K ∆ (q) = ∆ (q)K.

(3.71)

It is clearly a subsemigroup of ∆ , and the pair K, ∆ (K) is d-finite. The Hecke– Shimura ring (3.72) L(K) = D(K, ∆ (K)) will be called the (regular) Hecke–Shimura ring of the group K. By TK (M) = τK (M)

(M ∈ ∆ )

(3.73)

we shall denote the basic elements of the ring L(K) of the form (3.4) corresponding to double cosets KMK ⊂ ∆ (K) and refer to these elements as double classes. The following important theorem discloses relations between regular Hecke–Shimura rings for various q-symmetric groups of given genus. Theorem 3.43. Let K and K be two q-symmetric groups of a given genus n. Suppose that K ⊂ K. Then the following assertions are true: (1) ∆ (K) = K ∆ (K ) = ∆ (K )K. (2) Γ ∩ ∆ (K) = K. (3) If M, M ∈ ∆ (K ) and M ∈ KMK, then M ∈ Γ (q)MK . (4) Let M ∈ ∆ (K ) and let TK (M) and TK (M) be two elements of the form (3.73) then a decomposition TK (M) = ∑i (KMi ) with Mi ∈ ∆ (K ) implies the decomposition TK (M) = ∑i (K Mi ) and vice versa. (5) The linear mappings ι : L(K) → L(K ) and ι : L(K ) → L(K) defined on the double classes by

ι (TK (M)) = TK (M ) if M ∈ ∆ (K) and M ∈ KMK ∩ ∆ (K ) and

ι (TK (M )) = TK (M ) if M ∈ ∆ (K ),

respectively, are mutually inverse isomorphisms of the rings. Proof. By (3.71), we get ∆ (K) = K ∆ (q) = KK ∆ (q) = K ∆ (K ). Similarly, ∆ (K) = ∆ (K )K. This proves part (1). If γ ∈ Γ ∩ ∆ (K), then γ = δ M, where δ ∈ K and M ∈ ∆ (K). Hence M = δ −1 γ ∈ Γ ∩ ∆ (q) = Γ (q). Thus, γ ∈ KΓ (q) ⊂ K, and part (2) is proved. Let M = γ M γ with γ , γ ∈ K. Let us choose an integer q prime to q such that the matrix q M −1 is integral. By Lemma 3.42, the matrix γ can be written in the form γ = γ1 γ2 , where γ1 ∈ Γ (q) and γ2 ∈ Γ (q ). Then we can write M = γ1 M γ3 , where γ3 = M −1 γ2 M γ . Since γ2 ∈ Γ (q ), it follows that the matrix γ3 is integral, and so γ3 ∈ Γ . On the other hand, γ3 = M −1 γ1−1 M . If M = M0 δ and M = M0 δ , where M0 , M0 ∈ ∆ (q) and δ , δ ∈ K , then γ3 ≡ δ −1 M0−1 M0 δ ≡ δ −1 δ (mod q). Hence γ3 ∈ K . This proves part (3).

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3 Hecke–Shimura Rings of Double Cosets

In order to prove part (4), we note that all of the left cosets K Mi are distinct and their union contains the double coset K MK . On the other hand, it follows from part (3) that all of the matrices Mi are contained in the double coset K MK , which proves the direct assertion of part (4). The inverse assertion follows from the direct one and part (1). Finally, it follows from parts (1), (2), and (4) that the pairs K, ∆ (K) and K , ∆ (K ) satisfy the conditions (3.17) and (3.19). Thus, by Proposition 3.8, the corresponding mapping (3.18) is an isomorphism. It follows from (4) that this mapping coincides with the mapping ι defined in part (5). The mapping ι is just the inverse of ι . The above theorem shows that the Hecke–Shimura rings L(K) of various q-symmetric groups K are all naturally isomorphic to each other and to the ring L(q). Thus, any of the groups K can be taken to study the ring. Moreover, the isomorphisms not only establish one-to-one correspondences between the basic elements (3.73) of the rings, but also between the left cosets entering in their decompositions. The last circumstance plays an essential part in the theory of representations of these rings by Hecke operators. An important example of q-symmetric groups apart from the group Γ (q) = Γ n (q) is presented by the group Γ0 (q) = Γn0 (q) defined by (2.2). Lemma 3.44. Let

∆0 (q) = ∆n0 (q)

AB = M= ∈ ∆ gcd(det M, q) = 1, C ≡ 0 (mod q) . CD

(3.74)

Then Proposition 3.35 remains true for the pair Γ0 (q), ∆0 (q) in place of Γ , ∆ , i.e., each left coset of ∆0 (q) modulo Γ0 (q) contains a “triangular” representative of the form (3.57). Moreover,

∆0 (q) = Γ0 (q)∆ (q) = ∆ (q)Γ0 (q);

(3.75)

in particular, the group Γ0 (q) is q-symmetric, and ∆ (Γ0 (q)) = ∆0 (q). Proof. By Proposition 3.35, there is γ ∈ Γ such that the matrix γ M has the “triangular” form Then obviously, γ ∈ Γ0 (q). This proves the first asser (3.57). tion. Let M = CA DB ∈ ∆0 (q). In order to prove that M ∈ Γ0 (q)∆ (q), by the first part, we can assume that C = 0. Thus by (1.3), we get the relations A tD = µ (M)1n and A tB = B tA; hence the matrices A and D are invertible modulo q, and the matrix A−1 B is symmetric modulo q. It follows that the matrix −1 −1 A A −A−1 BD−1 0 1n −µ −1 A−1 B = , γ= 0 1n 0 µ D−1 0 µ D−1 where µ = µ (M), is a rational q-integral symplectic matrix with multiplier 1. Since the mapping (3.66) is epimorphic, there exists a matrix γ ∈ Γ such that γ ≡ γ

3.4 Rings of Double Cosets of the Symplectic Group

95

(mod q). Then it follows from the definitions that γ ∈ Γ0 and γ M ∈ ∆ (q). Hence, Γ0 (q) ⊂∈ Γ0 (q)∆ (q). The inverse inclusion is obvious. The second of the decompositions (3.75) follows from the first one by application of the antiautomorphism M → µ (M)M −1 of the semigroup ∆0 (q). For positive integers n and q, we semigroup define the ∆ q = ∆n q = M ∈ ∆n gcd(q, det M) = 1

(3.76)

and the Hecke–Shimura ring L{q} = Ln q = D(Γn , ∆n q ).

(3.77)

Since ∆ q ⊂ ∆ , the ring L q can be naturally considered as a subring of the ring L. Theorem 3.45. Let K be a q-symmetric subgroup of the modular group Γ = Γn . Then the Z-linear mapping ιK : L(K) → L q

(3.78) defined on the elements of the form (3.73) by

ιK (TK (M)) = T (d1 , . . . , dn , e1 , . . . , en )

(M ∈ ∆ (K)),

where di = di (M) and e j = e j (M) are the symplectic divisors of M, is a ring isomorphism. In particular, the ring L(K) is commutative. Proof. It follows from Theorem 3.43 and Lemma 3.44 that each of the rings L(K) is isomorphic to the ring L0 (q) = Ln0 (q) = D(Γn0 (q), ∆n0 (q)),

(3.79)

and in addition, the isomorphism links elements (3.73) of the rings corresponding to matrices from the same double cosets modulo Γ , and hence, with the same symplectic divisors. Therefore, it suffices to prove the theorem only for the group Γ0 (q). Note that the group Γ is also q-symmetric and

∆ (Γ , q) = Γ ∆ (q) = ∆ (q)Γ = ∆ q .

(3.80)

Thus, Theorem 3.43 can be applied to the groups K = Γ and K = Γ0 (q). Since the matrix sd(M ) of symplectic divisors of a matrix M ∈ ∆0 (q) again belongs to ∆0 (q), it follows from part (3) of this theorem that sd(M ) ∈ Γ0 (q)M Γ0 (q). Hence in part (5) of the theorem we can take M = M = sd(M ), which completes the proof. The above theorem implies that for every q-symmetric subgroup K of Γ , the double class TK (M) of a matrix M ∈ ∆ (K) depends only on the symplectic divisors d1 , . . . , dn , e1 , . . . , en of M, which can be arbitrary positive integers satisfying the conditions (3.47) and gcd(di ei , q) = 1. This justifies the notation

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3 Hecke–Shimura Rings of Double Cosets

TK (M) = TK (sd(M)) = TK (d1 , . . . , dn , e1 , . . . , en ) (M ∈ ∆ (K))

(3.81)

and TK (m) =

∑

TK (M)

∑

TK (d1 , . . . , dn , e1 , . . . , en ),

M∈K\∆ (K)/K, µ (M)=m

=

(3.82)

d1 e1 =···=dn en =m, d1 |d2 |···|dn |en |en−1 |···|e1

where m is prime to q. From Lemma 3.32, Theorem 3.31, and Theorem 3.45 we obtain the following theorem. Theorem 3.46. The elements (3.81) of the Hecke–Shimura ring L(K) of a qsymmetric group K ⊂ Γ = Γn satisfy the relations TK (d1 , . . . , dn , e1 , . . . , en )TK (d1 , . . . , dn , e1 , . . . , en ) = TK (d1 d1 , . . . , dn dn , e1 e1 , . . . , en en ), (3.83) provided that gcd(e1 /d1 , e1 /d1 ) = 1; in particular, d K TK (d1 , . . . , dn , e1 , . . . , en ) = TK (d1 , . . . , dn , e1 , . . . , en ) d K = TK (dd1 , . . . , ddn , de1 , . . . , den ),

(3.84)

where d K = T (d · 12n ) = (Γn d · 12n ).

(3.85)

The elements (3.82) satisfy the relations TK (m)TK (m ) = TK (mm ) if gcd(m, m ) = gcd(mm , q) = 1.

(3.86)

Exercise 3.47. Show that the elements (3.82) for n = 1 satisfy the relations TK (m)TK (m ) =

∑ d · d K TK (mm /d 2 )

if gcd(mm , q) = 1,

d|m,m

where d K are the elements (3.85). Use the relations to show that the formal Dirichlet series with coefficients TK (m) for m prime to q has a formal Euler factorization of the form −1 TK (m) = ∏ TK (1) − TK (p)p−s + p K p1−2s . s m p∈P, pq m∈N, gcd(m,q)=1

∑

[Hint: For K = Γ10 (q), show first that the linear map L10 (q) → H1 = D(Λ2 , Σ2 ) transforming elements TK (D) to tΛ (D) is an embedding of the rings and then use Exercise 3.19.]

3.4 Rings of Double Cosets of the Symplectic Group

97

In the same way as was done for the full modular group, consideration of global Hecke–Shimura rings L(K) for q-symmetric subgroups K of Γ reduces to the study of local subrings. For a prime number p not dividing q, let us introduce the subsemigroup of ∆ (K) given by (3.87) ∆ p (K) = M ∈ ∆ (K) µ (M)|p∞ and the local subring L p (K) of L(K) defined by L p (K) = D(K, ∆ p (K)).

(3.88)

Similarly to the case of full modular group, using corresponding results and Theorem 3.45, we get the following result. Theorem 3.48. The ring L(K) of each q-symmetric group K of genus n is generated by the local subrings L p (K) for all prime numbers p not dividing q; each of the local subrings L p (K) is naturally isomorphic to the local subring L p = Lnp of the ring L = Ln and generated over Z Z 1 K by the n + 1 algebraically independent elements (3.89) TK (p) = TK (1, . . . , 1, p, . . . , p) n

n

and Ti, K (p2 ) = TK (1, . . . , 1, p, . . . , p, p2 , . . . , p2 , p, . . . , p) (i = 1, . . . , n). n−i

i

n−i

(3.90)

i

The following lemma describes explicit decompositions of certain generators in terms of left cosets, which will be needed below. Lemma 3.49. The following decompositions hold for the group K = Γn0 (q) and each prime number p not dividing q: n .B pD K , (3.91) TK (p) = ∑ ∑ 0 D i=0 D∈Λ \Λ D (p)Λ i

B∈B(D)/mod D

where Λ = Λn = GLn (Z), B(D) are the sets (3.58), residue classes taken modulo the relation (3.59), Di (p) = Dni (p) = diag(1, . . . , 1, p, . . . , p), n−i

and we set

. = tD−1 ; D

mod D are (3.92)

i

(3.93)

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3 Hecke–Shimura Rings of Double Cosets

in the same notation, Tn−1,K (p2 ) =

2 .B p D K ∑ 0 D D∈Λ \Λ Dn−1 (p)Λ , B 2 .B p D K + ∑ 0 D D∈Λ \Λ pD1 (p)Λ , B 2 .B p D K , + ∑ 0 D D∈Λ \Λ D (p)Λ , B

(3.94)

n

where the matrix B in each of the three sums on the right ranges over all representa2 tives of residue classes B(D)/mod D such that the rank of the matrix p 0D. DB over the field F p = Z/pZ of residues modulo p is equal to 1; finally, Tn,K (p2 ) = p K = K(p · 12n ) . (3.95) Proof. Without loss of generality it suffices to consider the case K = Γ = Γn . It follows from Theorem 3.28 that the set ∆ (p) = {M ∈ ∆n |µ (M) = p} consists of the single double coset modulo Γ of the diagonal matrix generating the element (3.89). By (1.2), an integral matrix of the form (3.57) belongs to ∆ (p) if . and B ∈ B(D). According to Theorem 3.12, the matrix pD . and only if A = pD with an integral matrix D is integral if and only if D belongs to one of the double cosets Λ Di (p)Λ for i = 0, 1, . . . , n. Thus, the decomposition (3.91) follows from Proposition 3.35(2). Further, it follows from Theorem 3.28 or formula (3.54) that the set ∆ (p2 ) = {M ∈ ∆n |µ (M) = p2 } is the union of the double cosets modulo Γ of the matrices generating elements (3.90), and the double coset corresponding to an element Ti (p2 ) consists of all matrices in ∆ (p2 ) whose rank over the field F p is equal to n − i. It is easy to see that the rank over F p of a matrix in ∆ (p2 ) of the form (3.57) can be equal to 1 only if D ∈ Λ Di (p)D j (p)Λ , where 0 ≤ j ≤ i ≤ n and j + n − i ≤ 1, which implies that (i, j) = (n, 0), (n − 1, 0), or (n, 1). Then the decomposition (3.94) follows from Proposition 3.35(2). The decomposition (3.95) is obvious. Exercise 3.50. Show that for K = Γn0 (q) the decompositions of the elements (3.90) with i = 1, . . . , n into left cosets can be written in the form 2 .B p D 2 K Ti (p ) = , ∑ ∑ 0 D α +β ≤n,α ≥i D∈Λ \Λ D (p)D (p)Λ , B α

β

where the matrix B in each of the inner sums ranges overall representatives of 2D . B p residue classes B(D)/mod D such that the rank of the matrix 0 D over the field F p is equal to n − i. Spherical Mapping. Calculations in local Hecke–Shimura rings of the symplectic group can be considerably simplified if one uses a realization of the rings given

3.4 Rings of Double Cosets of the Symplectic Group

99

by spherical polynomials. Since these polynomials have, generally speaking, not integral but rational coefficients, it will be convenient to extend the rings of double cosets under consideration to the corresponding rings over the field Q of rational numbers, i.e., the rings L(K) = DQ (K, ∆(K)),

L-p (K) = DQ (K, ∆ p (K)), and

L-p = L-np = DQ (Γn , ∆np ).

It follows from Theorem 3.48 that each of the local rings L-p (K), for a q−symmetric group K and a prime p not dividing q, is naturally isomorphic to the ring L-p . Hence, one can restrict oneself to consideration of spherical mappings of the rings L-p . Let M ∈ ∆ p = ∆np . According to Lemma 3.35 and (1.3), the left coset Γ M, where Γ = Γn , contains a representative of the form δ .B p D (3.96) M = , where pδ = µ (M). 0 D It is clear that the left coset Λ D modulo Λ = Λn depends only on the left coset Γ M. Let x0 , x1 , . . . , xn be independent (commuting) variables. We assign to the left coset (Γ M) the monomial Ω((Γ M)) = x0δ ω ((Λ D)), where ω ((Λ D)) is the monomial in the variables x1 , . . . , xn assigned to the left coset (Λ D) by (3.42), and to every element T = ∑ j a j (Γ M j ) ∈ L-p = L-np with a j ∈ Q we assign the polynomial Ω(T ) = (Ωnp (T ))(x0 , x1 , . . . , xn ) = ∑ a j Ω((Γ M j )) ∈ Q[x0 , x1 , . . . , xn ].

(3.97)

j

We shall call the map Ω = Ωnp : L-np → Q[x0 , x1 , . . . , xn ] the spherical mapping of the ring L-np . Theorem 3.51. The spherical mapping Ωnp is a Q-linear isomorphism of the ring L-np onto the subring Q[x0 , x1 , . . . , xn ]W of Q[x0 , x1 , . . . , xn ] consisting of all polynomials invariant under the group W = W n of transformations of rational fractions in x0 , x1 , . . . , xn generated by all permutations in variables x1 , . . . , xn and by the transformations ρ1 , . . . , ρn of the variables x0 , x1 , . . . , xn defined by

ρi (x0 ) = x0 xi ,

ρi (xi ) = xi−1 , and ρi (x j ) = x j if j = 0, i.

(3.98)

Proof. In order to prove the theorem, it is sufficient, by Theorem 3.40, to show that the Ω-images of the generators (3.64)–(3.65) are algebraically independent and generate the ring Q[x0 , x1 , . . . , xn ]W . Here we shall restrict ourselves to small genera n = 1 and n = 2, since these are the only cases we use below. The general proof is based on similar ideas but demands more computations and space. For n = 1 it follows from the decompositions of Lemma 3.49 that Ω(T (p)) = x0 + x0 x1 and Ω(T1 (p2 )) = Ω( p ) = p−1 x02 x1 . These polynomials are clearly algebraically independent. Thus, it is sufficient to prove that

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3 Hecke–Shimura Rings of Double Cosets

Q[x0 , x1 ]W = Q[x0 + x0 x1 , x02 x1 ]. The substitution x0 = z1 , x0 x1 = z2 transforms the ring Q[x0 + x0 x1 , x02 x1 ] into Q[z1 + z2 , z1 z2 ] and the ring Q[x0 , x1 ]W into the ring Q[z1 , z2 ]S of all symmetric polynomials in z1 , z2 over Q (note that if a polynomial P(x0 , x1 ) = ∑i, j≥0 ci j x0i x1j is W -invariant, i.e., P(x0 x1 , x1−1 ) = P(x0 , x1 ), then ci j = 0 implies that i ≥ j, and so the image P(z1 , z2 /z1 ) of P(x0 , x1 ) is a polynomial in z1 , z2 ). Hence, the theorem on symmetric polynomials proves the assertion for n = 1. Standard computations based on the definitions and Proposition 3.35 show that the following formulas hold for n = 2:

Ω(T1 (p

2

Ω(T (p)) = x0 (1 + x1 )(1 + x2 ), 2 )) = x0 ω (t[1, p]) + x02 p3 ω (t[p, p2 ]) + x02 (p2 − 1)ω (t[p, p]) 1 p2 − 1 2 x x1 x2 + x02 (x1 + x2 )(1 + x1 x2 ), = p3 0 p

(3.99)

(3.100)

where t[d, d ] are the elements (3.25) (note that the number of symmetric matrices of order 2 with rank 1 over the field F p is p2 − 1), and Ω(T2 (p2 )) = Ω( p ) =

1 2 x x1 x2 . p3 0

(3.101)

According to these formulas, it suffices to show that the polynomials t = x0 (1 + x1 )(1 + x2 ),

u = x02 x1 x2 ,

and v = x02 (x1 + x2 )(1 + x1 x2 )

are algebraically independent and generate the ring Q[x0 , x1 , x2 ]W . Suppose that these polynomials are algebraically dependent, and let G(y, y1 , y2 ) be the polynomial of minimal degree such that G(t, u, v) = 0. Let us expand G in powers of y1 : G = ∑i gi (y, y2 )yi1 . If g0 = 0, then G is divisible by y1 , and y−1 1 G is a polynomial of smaller degree with the same property as G. Hence, g0 = 0. By the assumption, we have ∑i gi (t, v)ui = 0. Since it is an identity in x0 , x1 , x2 , we can set here x2 = 0. The substitution transfers the polynomials t, v into x0 (1 + x1 ), x02 x1 , respectively, and u turns to zero. Hence, we get the identity g0 (x0 (1 + x1 ), x02 x1 ) = 0, a contradiction, because the polynomials x0 + x0 x1 and x02 x1 are algebraically independent. Now we shall prove by induction on m that each W −invariant polynomial F(x0 , x1 , x2 ) of degree m relative to x0 is a polynomial in t, u, v. If m = 0, then F = F(x1 , x2 ) is a symmetric polynomial satisfying F(x1−1 , x2 ) = F(x1 , x2 ), and therefore it must be equal to a constant. Suppose that the assertion has already been proved for the polynomials whose degree in x0 is less than m with m ≥ 1. Let F(x0 , x1 , x2 ) =

∑

0≤i≤m

x0i fi (x1 , x2 )

3.4 Rings of Double Cosets of the Symplectic Group

101

be W −invariant polynomial of degree m relative to x0 . Since the variables x0 , x1 , x2 are algebraically independent, it follows from the definition of the group W that each of the polynomials x0i fi (x1 , x2 ) is also W −invariant. Thus, we may assume that F = x0m f (x1 , x2 ). Since F is W 2 −invariant, it is also W 1 −invariant, where W 1 acts only on x0 , x1 . In particular, it is true for the polynomial F(x0 , x1 , 0). Since we have already proved the assertion for n = 1, we can write F(x0 , x1 , 0) = P(x0 (1+x1 ), x02 x1 ), where P(y1 , y2 ) = ∑i, j ai j yi1 y2j is a polynomial over Q. Since F is of the form x0m f (x1 , x2 ), and the variables x0 , x1 are algebraically independent, it follows that ai j = 0 if i+2 j = m. Thus, the polynomial F (x0 , x1 , x2 ) = F(x0 , x1 , x2 )−P(t, v) can be written in the form F (x0 , x1 , x2 ) = x0m f (x1 , x2 ) −

∑

ai j t i v j = x0m G(x1 , x2 ),

i+2 j=m

where G is a polynomial. Since the polynomials t, v turn at x2 = 0 into x0 (1 + x1 ) and x02 x1 , respectively, it follows that the polynomial F is identically zero at x2 = 0. Thus, it is divisible by x2 . Since F is symmetric in x1 , x2 , it is divisible by x1 x2 . Hence F (x0 , x1 , x2 ) = x0m x1 x2 G (x1 , x2 ), where G is a polynomial. Since F is W -invariant, it is invariant under the transformation ρ1 , i.e., it satisfies F (x0 x1 , x1−1 , x2 ) = F (x0 , x1 , x2 ). It follows that G satisfies x1m−2 G (x1−1 , x2 ) = G (x1 , x2 ). If m = 0 or m = 1, than G is not a polynomial in x1 . Thus, m ≥ 2, and we can write F(x0 , x1 , x2 ) = F (x0 , x1 , x2 ) − P(t, v) = x02 x1 x2 F (x0 , x1 , x2 ) − P(t, v), where F is a W −invariant polynomial of degree m − 2 in x0 . The theorem will allow us to reduce calculations in local Hecke–Shimura rings of the symplectic group to calculations with invariant polynomials. In order to illustrate this approach, we shall consider generating formal power series for the sums T (pδ ) ∈ Lnp of all double cosets of multiplier pδ . It follows from (3.54) and Proposition 3.35 that δ .B p D Γ , (3.102) T (pδ ) = ∑ 0 D 0≤δ ≤···≤δ ≤δ , 1

n

D∈Λ \Λ diag(pδ1 ,...,pδn )Λ , B∈B(D)/mod D

where Λ = Λn and Γ = Γn . It is easy to check that the number of elements of the set B(D)/mod D for a matrix D ∈ Σn with elementary divisors d1 , . . . , dn depends only on the elementary divisors and is equal to d1n d2n−1 · · · dn . Therefore, by the definition of Ω, we obtain the following formula for Ω-images of the elements T (pδ ): Ω(T (pδ )) = x0δ

∑

0≤δ1 ≤···≤δn ≤δ

pnδ1 +(n−1)δ2 +···+bn ω (t(pδ1 , . . . , pδn )),

(3.103)

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3 Hecke–Shimura Rings of Double Cosets

where ω is defined by (3.42) and (3.43). Hence, the Ω-image of the generating formal power series of genus n with coefficients T (pδ ), ∞

Z p (v) =

∑ T (pδ )vδ ,

(3.104)

δ =0

can be written in the form Ω(Z p )(v) =

∞

∑ Ω(T (pδ ))vδ

δ =0

∑

=

pnδ1 +(n−1)δ2 +···+bn ω (t(pδ1 , . . . , pδn ))(x0 v)δ .

(3.105)

0≤δ1 ≤···≤δn ≤δ

For the genus n = 1, we get Ω(Z p )(v) =

∑

pδ1 (x1 p−1 )δ1 (x0 v)δ

0≤δ1 ≤δ

= (1 − x0 v)−1 (1 − x0 x1 v)−1 = (1 − Ω(T (p))v + pΩ((p))v2 )−1 , (3.106) where (p) = (p)1 = T1 (p2 ) is the double coset of the form (3.53) (formulas for Ω(T (p)) and Ω(T1 (p2 )) follow from Proposition 3.35). Hence, since the spherical mapping is monomorphic, we obtain summation formulas for the series Z p (v). Proposition 3.52. For every prime number p, the following identity holds in the ring of formal power series over L1p : ∞

∑ T (pδ )vδ = (1 − T (p)v + p p v2 )−1 .

(3.107)

δ =0

Similarly, for genus n = 2 we obtain Ω(Z p )(v) =

∑

p2δ1 +δ2 ω (t(pδ1 , pδ2 ))(x0 v)δ .

0≤δ1 ≤δ2 ≤δ

Since by (3.27) and Theorem 3.23,

ω (t(pδ1 , pδ2 )) = ω ([pδ1 ]t(1, pδ2 −δ1 )) = (p−3 x1 x2 )δ1 ω (t(1, pδ2 −δ1 ), where [d] = [d]2 are elements (3.28), the sum for Ω(Z p )(v) after the replacement of δ2 and δ by δ1 + α and δ2 + β , respectively, can be rewritten in the form Ω(Z p )(v) =

∑

p2δ1 +δ1 +α (p−3 x1 x2 )δ1 ω (t(1, pα ))(x0 v)δ1 +α +β

α ,β ,δ1 ≥0

= (1 − x0 v)−1 (1 − x0 x1 x2 v)−1

∑ ω (t(1, pα ))(px0 v)α .

α ≥0

3.4 Rings of Double Cosets of the Symplectic Group

103

In order to compute the last sum, we note that with the help of Proposition 3.35 it can be easily seen that any set of the form α α −β

p 0 1 a p b α β (a mod p ), (0 < β < α , b mod p , p b) , 0 pα 0 1 0 pβ can be taken as a set of of the left cosets modulo Λ = Λ2 contained representatives in the double coset Λ 10 p0α Λ . Hence, by the definition of ω , we see that

∑ ω (t(1, pα ))(px0 v)α

α ≥0

=

∑

α

−2 α

β

−1 γ

−1 α

p (x2 p ) + (x1 p ) +

α ≥0

=

∑

∑

β

β −1

(p − p

−1 α −β

)(x1 p )

−2 β

(x2 p )

(px0 v)α

0 1 is an element of the form (3.19), and Ω(T) = Ωnp (T)(x0 , x1 , . . . , xn ) is the spherical polynomial (3.140), then Ωn−1 p (Ψ(T(M)))(x0 , x1 , . . . , xn−1 ) = Ωnp (T(M))(χ (p)pk−n x0 , x1 , . . . , xn−1 , χ (p)pn−k ),

(4.32)

where we assume that the spherical mapping is extended to complexifications of Hecke–Shimura rings by linearity. Proof. All assertions of thetheorem, δ except for the formula (4.32), have already . i Bi n D p , then by formula (4.30) and the definibeen proved. If T (M) = ∑i Γ0 0 D i tion of spherical mapping (see (3.140) and (3.43)), we obtain (k−n) δ ) (χ (p)p−k )di x0δ ω pn−1 (Λn−1 Di )(x1 , . . . , xn−1 ) Ωn−1 p (Ψ(T(M))) = ∑( χ (p)p i

Di ∗ = ∑(x0 χ (p)pk−n )δ ω pn Λn (x1 , . . . , xn−1 , χ (p)pn−k ), di 0 p i

which proves the formula (4.32). ´p Let us look now at the restriction of the mapping Ψ to the complexification L L of the isomorphic image (3.123) in p of the local symplectic Hecke–Shimura ring L p (Γ0 (q)). Proposition 4.20. In the notation and under the assumption of Theorem 4.19, the ´ np into the ring mapping Ψ = Ψnk,χ with n > 1 transforms the complexification L L´ n−1 p ; the mapping ´ np → L ´ n−1 (4.33) Ψ = Ψnk,χ : L p is epimorphic, except for the case n = k and χ (p) = −1, in which case the image ´ n−1 ´ np ) is a proper subring of L Ψ(L p . Proof. It follows from Theorem 3.51 that the extended spherical mapping Ω = ´ np → C[x0 , x1 , . . . , xn ] is an isomorphism of the ring L ´ np onto the subring Ωnp : L n C[x0 , x1 , . . . , xn ]W of W = W -invariant polynomials. Relations (4.32) imply that n−1 −invariant ´n the spherical polynomial Ωn−1 p (Ψ(T)) of an element T ∈ L p is W ´ np ) ⊂ L´ n−1 if Ωnp (T) is W n -invariant. Thus, Ψ(L p .

134

4 Hecke Operators

´ np ) is generated over C by Ψ-images of a system of generators of The ring Ψ(L n ´ the ring L p or the ring Lnp . The relations (4.32) allow one to find spherical polynomials corresponding to the images if spherical polynomials for the generators are known. Computation of spherical polynomials corresponding to generators for an arbitrary genus n > 1 is not hard, but it requires a lot of space. In order to illustrate the computations, we restrict ourselves here to the simplest case n = 2. It was shown in the course of the proof of Theorem 3.51 that the sets {x02 x1 , x0 (1 + x1 )} and {x02 x1 x2 , x0 (1 + x1 )(1 + x2 ), x02 (x1 + x2 )(1 + x1 x2 )} consist of spherical polynomials corresponding to certain systems of generators (over Q) of the rings L1p ´ 2p ) is generated and L2p , respectively. It follows from (4.32) that the image Ψ(L over C by the elements whose spherical polynomials are equal to the polynomials χ (p)pk−2 x02 x1 , χ (p)pk−2 x0 (1 + x1 )(1 + χ (p)p2−k ), and χ (p)2 p2(k−2) x02 (x1 + χ (p)p2−k )(1 + χ (p)p2−k x1 ). If 1 + χ (p)p2−k = 0, then clearly these elements gen´ 1p . But if 1 + χ (p)p2−k = 0, i.e., k = 2 and χ (p) = χ (p) = −1, erate the whole ring L then the image is generated by the elements with spherical polynomials x02 x1 and 2 ´ 1p . x02 (1 − x1 )2 = x0 (1 + x1 ) − 4x02 x1 and so cannot be equal to L Diagonalization of Hecke Operators for the Full Modular Group. The Zharkovskaya relations allow one to reduce certain problems relating to Hecke operators on modular forms of a given genus to similar problems for cusp forms and modular forms of smaller genera. In particular, the problem of simultaneous diagonalization of Hecke operators on invariant spaces of modular forms is a problem of this kind. Theorem 4.7 solves the problem of diagonalization for cusp forms. Here we shall consider the problem for invariant subspaces of the space Mnk = Mk (Γn ) = Mk (Γn , 1)

(4.34)

of modular forms of integral nonnegative weight k for the full modular group Γn with the trivial character, but first we shall prove a useful lemma on cusp forms contained in these spaces. Lemma 4.21. Let F=

∑ n

f (A)eπ iσ (AZ) ∈ Mnk

(4.35)

A∈E , A≥0

be a modular form of weight k for the full modular group Γn then the following three conditions are equivalent: (a) F is a cusp form. (b) F|Φ = 0, where Φ is the Siegel operator (1.56). (c) The Fourier coefficients of F satisfy the rule f (A) = 0 if det A = 0. Proof. Condition (a) implies (b), by Lemma 1.29. According to formula (1.54), condition (b) implies (c). Let M ∈ Gn (Q) be a symplectic matrix of genus n with rational entries and positive multiplier µ = µ (M). It follows from Proposition 3.35(1) B n A that there is a matrix γ ∈ Γ such that M = γ M with triangular M = 0 D . Then, if the condition (c) is fulfilled, we have

4.3 Hecke Operators and Siegel Operator

135

F|k M = F|k γ M = F|k γ |k M = F|k M = µ nk−

n(n+1) 2

(det D )−k

−1 )

∑ n

f (A)eπ iσ (A(A Z+B )(D )

∑ n

f (A)eπ iσ (AB (D ) eπ iσ ((D )

A∈E , A>0

=µ

n(n+1) nk− 2

(det D )−k

−1

−1 AA Z)

.

A∈E , A>0

Each of the matrices (D )−1 AA = µ −1 tA AA is symmetric and positive definite together with A, whence F|k M|Φ = 0, by the definition of Φ. Thus, by Lemma 1.29, F is a cusp form. Theorem 4.22. Every subspace V ⊂ Mnk invariant with respect to all Hecke operators |k T = |k,1 T with T ∈ Ln = L(Γn ) has a basis of common eigenfunctions for all of the operators. Proof. The same arguments that were used to prove Theorem 4.7 show that the theorem is true for invariant subspaces of the space Nnk = Nk (Γn ) of cusp forms. Now let V ⊂ Mnk be an arbitrary invariant subspace. We set V1 = V ∩ Nnk and denote by V2 = V⊥ 1 = { f ∈ V|(F, G) = 0 for all G ∈ V1 } the orthogonal complement of V1 in V with respect to the Petersson scalar product (1.70). It follows in a standard way from the properties of scalar product listed in Theorem 1.38 that the space V is the orthogonal direct sum of the subspaces V1 and V2 : V = V1 ⊕ V2 .

(4.36)

By Proposition 4.4 and our assumption, the subspace V1 is the intersection of two invariant subspaces and so is also invariant with respect to all of the Hecke operators. By Lemma 4.6 with q = 1, each of the Hecke operators |k T with T ∈ Ln satisfies (F|k T, F ) = (F, F |k T )

(4.37)

if one of the forms F, F ∈ Mnk is a cusp form. It follows that the subspace V2 is again invariant under all of the Hecke operators, and it suffices to prove that this subspace has a basis of common eigenfunctions for all of the operators. Since the subspace V2 does not contain nonzero cusp forms, it follows from Lemma 4.21 that the Siegel operator Φ maps this subspace isomorphically onto its image V = V2 |Φ in Mn−1 k . Since the mapping (4.33) with χ = 1 is epimorphic for every prime p, it follows is invariant with respect to all of from Theorem 4.19 that the subspace V ⊂ Mn−1 k for all prime p and hence, by Theorem 3.37, the Hecke operators |k T with T ∈ Ln−1 p n−1 with all T ∈ L . Assuming that the theorem has already been proved for invariant subspaces of Mn−1 k , we see that there is a basis F1 , . . . , Fd of the space V consisting −1 of common eigenfunctions. The inverse images F1 = F1 |Φ , . . . , Fd = Fd |Φ−1 of the functions F1 , . . . , Fd in V2 clearly form a basis of V2 , and each of the functions is a common eigenfunctions for all of the Hecke operators on Mnk . In order to see the latter, it is sufficient to consider only the operators corresponding to elements of a local Hecke–Shimura ring Lnp . For T ∈ Lnp , by (4.31), we obtain

136

4 Hecke Operators

(Fi |k T )|Φ = Fi |Ψ(T) = λ (Ψ(T))Fi = (λ (Ψ(T))Fi )|Φ, where |k = |k,1 , T is the image of T in Lnp , and λ (Ψ(T)) is a scalar. Hence Fi |k T − λ (Ψ(T))Fi |Φ = 0, and so Fi |k T − λ (Ψ(T))Fi = 0. In order to complete the induction, it remains to prove the theorem for n = 1. If we again present an invariant subspace V of M1k in the form (4.35) of the direct sum of two invariant subspaces, then since M0k = C, the dimension of V2 is 0 or 1. If dim V2 = 0, then V ⊂ N1k and the assertion has already been proved. But if dim V2 = 1, then every function of V2 is a common eigenfunction. Exercise 4.23. Show that all eigenvalues of all Hecke operators on Mnk are real numbers. Exercise 4.24. Let p be a prime number and F ∈ Mnk a nonzero common eigenfunction of all Hecke operators corresponding to elements T of Lnp with the eigenvalues λ (T ) = λF (T ). Let Q p,F (v) = ∑ (−1)n λ (qni (p))vi 0≤i≤2n

be the spinor p-polynomial of F resulting from the replacement of coefficients of the spinor p-polynomial (3.112) with eigenvalues of corresponding Hecke operators acting on the eigenfunction F, and let α0 (p), α1 (p), . . . , αn (p) be parameters of the linear extension on L-np of the homomorphism T → λ (T ) defined in Proposition 3.55. Prove the following formulas: (1) Q p,F (v) = (1 − α0 (p)v) ∏nr=1 ∏1≤i1 1 variables, since generally, there is no multiplication of the classes, except for some isolated cases such as Gauss composition of binary forms. A possible way out of the situation is to consider suitable generating series for values of the functions instead of the values themselves. Traditionally, starting from Euler, multiplicativity of arithmetic sequences is customarily expressed in the form of an Euler product factorization of the generating Dirichlet series. It turns out that in the situation of modular forms, suitable Dirichlet series constructed by Fourier coefficients of eigenfunctions of Hecke operators can be expressed through Dirichlet series formed by the corresponding eigenvalues. The latter, according to the multiplicative properties of the relevant Hecke–Shimura rings, may have an Euler product factorization. Such relations between Dirichlet series constructed by Fourier coefficients of the eigenfunctions and the Euler product formed with the help of the corresponding eigenvalues prove to be useful in both directions: on the one hand, they establish the multiplicativity of the Fourier coefficients, and on the other hand, the relations allow one to express the Euler products, called zeta functions of modular forms, in terms of underlying modular forms and so to relate their analytic properties with those of the forms, as has been noted in the introduction. A. Andrianov, Introduction to Siegel Modular Forms and Dirichlet Series, Universitext, DOI 10.1007/978-0-387-78753-4 5, c Springer Science+Business Media LLC 2009

137

138

5 Euler Factorization of Radial Series

5.1 Radial Series of Genus One and Zeta Functions In this section,

F = F(z) ∈ Mk (q, χ ) = Mk (Γ0 (q), χ )

(5.1)

is a modular form in one complex variable z belonging to the upper half-plane H = H1 = {z = x + iy ∈ C|y > 0} of a positive integral weight k and a character χ of the form (2.3) for the group Γ0 (q) = Γ10 (q), where q ∈ N and χ is a Dirichlet character modulo q. We shall write the Fourier expansion (1.35) of F in the form F(z) =

∑

f (2a)eπ i2az =

∞

∑ f(a)e2π iaz

(5.2)

a=0

2a∈E1 , a≥0

with Fourier coefficients f(a) = f (2a) indexed by all nonnegative integers. Fourier Coefficients of Eigenfunctions and Eigenvalues. It turns out that for modular forms in one variable, there are close relations between individual Fourier coefficients of the eigenfunctions and the corresponding eigenvalues. Proposition 5.1. Let F be a modular form (5.1) with Fourier expansion (5.2). Suppose that F is an eigenfunction for the Hecke operator |T (m) = |k,χ T (m), where T (m) ∈ L0 (q) = L10 (q) with m prime to q is an element of the form (3.82) for K = Γ0 (q): (5.3) F|T (m) = λ (m)F. Then the Fourier coefficients f(a) of F and the eigenvalue λ (m) = λF (m) are linked by the relations ma for all a ≥ 0. (5.4) λ (m)f(a) = ∑ χ (d)d k−1 f 2 d d|m, a Proof. By Lemma 4.12, Proposition 3.61, and Lemma 4.14, the Fourier coefficients (f | T (m))(a) of the function F|T (m) can be written in the form (f |T (m))(a) = (f |T(m))(a) = =

∑

f|

∑

d,d1 ∈N,dd1 =m

Π1+ (d)Π1− (d1 ) (a)

((f |Π1+ (d))|Π1− (d1 ))(a)

d, d1 ∈N,dd1 =m

=

∑

χ (d1 )d1k−1 (f |Π1+ (d))(a/d1 )

∑

χ (d1 )d1k−1 f(da/d1 ) =

dd1 =m, d|a

=

dd1 =m, d|a

∑

χ (d)d k−1 f(ma/d 2 ).

d|m, a

On the other hand, by (5.3), this Fourier coefficient is equal to λ (m)f(a). The relation (5.4) follows.

5.1 Radial Series of Genus One and Zeta Functions

139

First of all, the relations (5.4) are useful for studying properties of the eigenvalues. Theorem 5.2. The eigenvalues λ (m) = λF (m) of the Hecke operators |T (m) = |k,χ T (m) corresponding to a nonzero common eigenfunction F ∈ Mk (q, χ ) of the operators with gcd(m, q) = 1 have the following properties: (1) For every m and m prime to q, the eigenvalues satisfy the multiplicative relations mm . (5.5) λ (m)λ (m ) = ∑ χ (d)d k−1 λ d2 d|m, m (2) If F is a cusp form, then the eigenvalues satisfy the inequalities |λ (m)| ≤ cmk/2 ,

(5.6)

where the constants c = cF depend only on F. Proof. Let f(0), f(1), . . . be the Fourier coefficients of F. Since k > 0, the form F is not a constant, and there are positive integers a with f(a) = 0. Let κ = κ (F) be the smallest of such integers, and let δ be the greatest divisor of κ that is coprime to q. Since the numbers δ and κ /δ are coprime, it follows from (5.4) with m = δ and a = κ /δ that f(κ ) = f(δ · κ /δ ) = χ (δ )λ (δ )f(κ /δ ). Thus f(κ /δ ) = 0, and so δ = 1, which means that

κ (F)|q∞ , in particular, κ (F) = 1 if q = 1.

(5.7)

Hence by (5.4) with a = κ , we obtain the relations

λ (m) = f(mκ )/f(κ ) for all m prime to q.

(5.8)

By (5.8) and (5.4), we get

λ (m)λ (m ) = λ (m)f(m κ )/f(κ ) =

∑

d|m, m κ

χ (d)d

k−1

mm · κ f d2

/f(κ ) =

∑

d|m, m

χ (d)d

k−1

λ

mm d2

,

since common divisors of m and m κ must divide m . The estimates (5.6) follow from (5.8) and (1.46). It follows from the estimate (1.52) of the Fourier coefficients and relations (5.8) that, for, the eigenvalues of an arbitrary eigenfunction satisfy the inequalities |λ (m)| ≤ cF mk

(m ∈ N, gcd(m, q) = 1).

(5.9)

Exercise 5.3. Prove the relations (5.5), using the relations of Exercise 3.47 and relations (4.26).

140

5 Euler Factorization of Radial Series

Exercise 5.4. In the notation of Theorem 5.2, show that if f(0) = 0, then

λ (m) = ∑ χ (d)d k−1

(m ∈ N, gcd(m, q) = 1).

d|m

Exercise 5.5. In the notation of Theorem 5.2, prove that the eigenvalues λ (m) = λF (m) corresponding to the Eisenstein series F = Ek (z) ∈ Mk (see Exercise 4.9) are given by the formula

λ (m) = ∑ d k−1

(m ∈ N);

d|m

use the formula to show that the Fourier coefficients f(a) of Ek satisfy the relations

f(a) = f(1) ∑ d k−1

(a ∈ N)

d|a

(compare Exercise 1.24). Dirichlet Series and Zeta Functions of Modular Forms. Here we reformulate the relations (5.4) and (5.5) in the terms of generating Dirichlet series. Let us look first at a formal consequence of relations (5.5) in terms of Dirichlet series. Lemma 5.6. Let q ∈ N then the relations (5.5) for a nonzero function

λ : {m ∈ N| gcd(m, q) = 1} → C are equivalent to the following formal Euler factorization of the Dirichlet series with coefficients λ (m): −1 λ (m) λ (p) χ (p)pk−1 1 − = + . ∑ ∏ ms ps p2s p∈P, pq m∈N, gcd(m,q)=1

(5.10)

Proof. The relations (5.5) are equivalent to the relations

λ (m)λ (m ) = λ (mm ) if gcd(mm , q) = 1 and gcd(m, m ) = 1

(5.11)

together with the relations

λ (p)λ (pδ ) = λ (pδ +1 ) + χ (p)pk−1 λ (pδ −1 ) for each p ∈ P, p q and δ ≥ 1. (5.12) The relations (5.11) are obviously equivalent to the formal factorization ∞ λ (m) λ (pδ ) = ∏ ∑ ∑ δs , ms p∈P, pq δ =0 p m∈N, gcd(m, q)=1 whereas the recursion relations (5.12) can be written as the identities for formal power series of the form

5.1 Radial Series of Genus One and Zeta Functions

141

(1 − λ (p)t + χ (p)pk−1t 2 )

∞

∑ λ (pδ )t δ

=1

δ =0

or, by substituting t = p−s , as the summation formulas −1 λ (pδ ) λ (p) χ (p)pk−1 . ∑ δ s = 1 − ps + p2s δ =0 p ∞

Exercise 5.7. Under the assumptions of Lemma 5.6 prove the following formal Euler factorization of the formal Rankin Dirichlet series:

λ (m2 ) ms m∈N, gcd(m,q)=1 −1 χ (p)pk−1 λ (p2 ) − χ (p)pk−1 χ (p2 )p2k−2 1+ 1− = ∏ + . ps ps p2s p∈P, pq

∑

[Hint: First sum the formal power series ∑δ ≥0 λ (p2δ )t δ .] Let F ∈ Mk (q, χ ) be a nonzero common eigenfunction for all regular Hecke operators, i.e., operators |T = |k,χ T with T ∈ L0 (q) and in particular, F|T (m) = λ (m)F for all m prime to q. Then we call the Dirichlet series

ζr (s, F) =

λ (m) ms m∈N, gcd(m,q)=1

∑

(5.13)

the regular zeta function of the eigenfunction F. If q = 1, we omit the adjective “regular” and the lower index r and speak just about the zeta function ζ (s, F) of F. From Lemma 5.6 and the estimate (5.6), by standard calculus, we have the following result. Proposition 5.8. Let a nonzero cusp form F ∈ Nk (q, χ ) be a common eigenfunction for all the regular Hecke operators. Then the regular zeta function ζr (s, F) of F converges absolutely and uniformly in each right half-plane ℜs ≥ 2k + 1 + ε with ε > 0 and has the factorization into an absolutely and uniformly convergent Euler product of the form

ζr (s, F) =

∏

1−

p∈P, pq

λ (p) χ (p)pk−1 + ps p2s

−1 .

(5.14)

For an arbitrary nonzero eigenform F ∈ Mk (q, χ ), by Lemma 5.6 and (5.9), the factorization (5.14) holds in every half-plane ℜs ≥ k + 1 + ε with ε > 0. Exercise 5.9. Show that the zeta function of the Eisenstein series Ek (z) has the form

ζ (s, Ek ) = ζ (s)ζ (s − k + 1), where ζ (s) is the Riemann zeta function.

142

5 Euler Factorization of Radial Series

In Section 2.1 we associated with each cusp form of the space Nk (q, χ ) = N1k (q, χ ) the radial Dirichlet series of the form (2.7) constructed by means of Fourier coefficients of F, which, according to Theorem 2.1, have nice analytic properties. On the other hand, by Theorem 4.7, the space Nk (q, χ ) is spanned by eigenfunctions for all of the regular Hecke operators, and according to Proposition 5.8, the regular zeta function of every such eigenfunction is a Dirichlet series with Euler product factorization. A natural question arises whether radial Dirichlet series and Dirichlet series with Euler product associated with an eigenform are related to each other, or in other words, whether the Dirichlet series constructed by Fourier coefficients of an eigenform have an Euler product factorization. The following theorem gives a partly positive answer. Theorem 5.10. Let a nonzero cusp form F ∈ Nk (q, χ ) with Fourier coefficients f(a) be a common eigenfunction for all regular Hecke operators, in particular, F|k,χ T (m) = λ (m)F for every m prime to q. Then the identity −1 ∞ f(a) f(a) λ (p) χ (p)pk−1 1 − = + (5.15) ∑ s ∑ as ∏ ps p2s a=1 a a|q∞ p∈P, pq is valid in the half-plane ℜs > 2k + 1, where the series and product are absolutely convergent. In particular, if q = 1, then −1 f(a) λ (p) χ (p)pk−1 ∑ s = f(1) ∏ 1 − ps + p2s a=1 a p∈P ∞

(ℜs >

k + 1). 2

(5.16)

Proof. By Proposition 5.1, and we have ∞

f(a) f(am) = ∑ as = ∞ ∑ (am)s a=1 a|q , gcd(m, q)=1

f(a) ∑∞ as a|q

λ (m) , ms m, gcd(m,q)=1

∑

whence, by Proposition 5.8, we obtain the identity (5.15). For an arbitrary eigenform of all regular Hecke operators on the space Mk (q, χ ), for the same reasons and by the corresponding estimates, the identity (5.15) holds in the half-plane ℜs > k + 1. We shall use the notation Zq (s, F) =

∑∞

a|q

f(a) as

(5.17)

for the q-part of the Dirichlet series Z(s, F) of a modular form F ∈ Mk (q, χ ) whose coefficients are the Fourier coefficients of F indexed only by positive integers dividing a power of q. In this and the above notation, the identity (5.15) takes the form Z(s, F) = Zq (s, F)ζr (s, F) (resp., Z(s, F) = f(1)ζ (s, F) if q = 1).

(5.18)

5.1 Radial Series of Genus One and Zeta Functions

143

Exercise 5.11. Let F1 , . . . , Fh be a basis of an invariant subspace of the space Mk (q, χ ) under all regular Hecke operators, in particular, Fi |k,χ T (m) =

∑

λi j (m)Fj

(i = 1, . . . , h; gcd(m, q) = 1).

1≤ j≤h

Let f(a) with a = 0, 1, . . . be the h-column with i-entry for i = 1, . . . , h equal to the Fourier coefficient fi (a) of Fi , and let Λ(m) = λi j (m) . Prove the following relations: ma (a = 0, 1, . . . ; gcd(m, q) = 1); Λ(m)f(a) = ∑ χ (d)d k−1 f d2 d|m,a mm (gcd(mm , q) = 1); Λ(m)Λ(m ) = ∑ χ (d)d k−1 Λ 2 d d|m,m −1 ∞ f(a) −s −s k−1−2s 1h ∑ a f(a) = ∏ 1h − p Λ(p) + χ (p)p ∑ as . a=1 a|q∞ p∈P, pq Exercise 5.12. Let F ∈ Mk be a nonzero eigenform with F|k T (m) = λ (m)F for all m ∈ N. Prove that if the eigenvalues satisfy the Ramanujan–Petersson inequality k−1 |λ (p)| ≤ 2p 2 for each prime number p, then for arbitrary m ∈ N, the eigenvalues satisfy k−1 |λ (m)| ≤ τ (m)m 2 , where τ (m) is the number of positive divisors of m. [Hint: use Exercise 4.23.] Functional Equation for the Zeta Function. According to Theorem 2.1, the radial Dirichlet series Z(s, F) of a cusp form F ∈ Mk (q, χ ) has an analytic continuation over the whole s-plane such that the function Φ(s, F) = (2π )−s Γ(s)Z(s, F), where Γ(s) is the gamma function, is holomorphic everywhere and satisfies the functional equation 0 −1 ∈ Mk (q, χ ). (5.19) Φ(k − s, F) = ik qs−k+1 Φ(s, F ∗ ), with F ∗ = F|k 0 q The following proposition complements properties of the Dirichlet series entering into the functional equation. Proposition 5.13. If a modular form F ∈ Mk (q, χ ) is an eigenfunction for all the Hecke operators |k,χ T (m) with m prime to q, and λ (m) are the corresponding eigenvalues, then the form F ∗ ∈ Mk (q, χ ) is an eigenfunction for all Hecke operators |k,χ T (m) with m prime to q, and the corresponding eigenvalues are equal to χ (m)λ (m); the regular zeta function of the eigenfunction F ∗ has properties similar

144

5 Euler Factorization of Radial Series

to those of the regular zeta function of F listed in Theorem 5.10; in particular, it has an Euler factorization of the form

χ (m)λ (m) ms m∈N, gcd(m,q)=1 −1 χ (p)λ (p) χ (p)pk−1 1− = ∏ + ps p2s p∈P, pq

ζr (s, F ∗ ) =

∑

Proof. To prove the first part it is sufficient to show that 0 −1 0 −1 |k,χ T (p) = χ (p)F|k,χ T (p)|k F|k 0 q 0 q

(5.20)

(5.21)

p not dividing q. By Lemma 3.49, the for every F ∈ Mk (q, χ ) and primenumber matrix smap, 0, 0, 1. and matrices 10 bp with b = 0, 1, . . . , p − 1 form a system of representatives of left cosets modulo K = Γ10 (q) contained in the double coset T (p). If p b, then there is b ∈ {1, . . . , p − 1} such that 1 + qbb ≡ 0 (mod p). Hence for b = 1, 2, . . . , p − 1, we get the relation 1b 0 −1 0 −1 1b p −b , = −qb (1 + qbb )/p 0 p q 0 q 0 0p where the first factor on the right belongs to the group K. Hence by (4.13), (4.14), and (1.31), we obtain 0 −1 0 −1 p0 |k,χ T (p) = χ (p)F|k F|k 0 q 0 q 01 p−1 0 −1 10 0 −1 1b + F|k + ∑ F|k 0 q 0p 0 q 0p b=1 10 0 −1 p0 0 −1 + F|k = χ (p)F|k 0p q 0 01 q 0 p−1 1 b 0 −1 p −b + ∑ F|k −qb (1 + qbb )/p 0 p q 0 b=1 10 p0 + F|k,χ = χ (p) F|k,χ 0p 01 p−1 1b 0 −1 + ∑ F|k,χ |k , 0 p q 0 b=1 which proves the relation (5.21). The Euler factorization (5.20) follows from Proposition 5.8 with F ∗ in place of F. It follows from the proposition and Theorem 5.10 that the function Φ(s, F) and Φ(s, F ∗ ) in the functional equation (5.19), for a nonzero eigenfunction of all regular

5.2 Binary Radial Series and Gaussian Composition

145

Hecke operators on the space Nk (q, χ ), in the half-plane ℜs > 2k + 1, can be written in the form Φ(s, F) = (2π )−s Γ(s)Zq (s, F)ζr (s, F) and

Φ(s, F ∗ ) = (2π )−s Γ(s)Zq (s, F ∗ )ζr (s, F ∗ ),

where ζr (s, F) and ζr (s, F ∗ ) are the regular zeta functions of the eigenforms F and F ∗ , and Zq (s, F), Zq (s, F ∗ ) are the corresponding series (5.17). Thus, the functional equation can be considered as a functional equation for the zeta functions. In particular, since for q = 1 clearly F ∗ = F and Z1 (s, F) = Z1 (s, F ∗ ) = f(1) = 0, we get the following theorem. Theorem 5.14. The zeta function ζ (s, F) of a nonzero eigenform F ∈ Nk has a meromorphic continuation to the whole s-plane; the function Φ(s, F) = (2π )−s Γ(s)ζ (s, F) is holomorphic and satisfies the functional equation Φ(k − s, F) = ik Φ(s, F).

(5.22)

Exercise 5.15. Show that the zeta function ζ (s, Ek ) of the Eisenstein series Ek with even k has a meromorphic continuation over the whole s-plane, and that the function Φ(s, Ek ) = (2π )−s Γ(s)ζ (s, Ek ) is analytic except for two simples poles at s = 0 and s = k and satisfies the functional equation Φ(k − s, F) = ik Φ(s, F). [Hint: Use Exercise 2.2, or Exercise 5.9 and properties of the Riemann zeta function.] In framework of the theory of old and new forms initiated by A.O.L. Atkin and J. Lehner, which we do not touch on in this book, it turns out that under certain conditions, the q-partial Dirichlet series Zq (s, F) is a finite Euler product extending only over prime divisors of q.

5.2 Binary Radial Series and Gaussian Composition In this section we fix a nonzero modular form F = F(Z) ∈ M2k (q, χ ) = Mk (Γ20 (q), χ )

(5.23)

of positive integral weight k and character χ modulo q ∈ N for the group Γ20 (q) defined on the upper half-plane H2 of genus 2. The Fourier expansion (1.35) of such an F takes the form F(Z) =

∑

A∈E2 ,

A≥0

f (A)eπ iσ (AZ)

(5.24)

146

5 Euler Factorization of Radial Series

with Fourier coefficients indexed by matrices A of integral nonnegative semidefinite binary quadratic forms 1 2a b x X= . (5.25) a(x, y) = tXAX = ax2 + bxy + cy2 ⇔ A = b 2c y 2 Depending on circumstances, we shall use the language of even matrices or of integral quadratic forms with similarly defined basic notation. So we say that a number d ∈ N divides the form or matrix (5.25) if it divides the coefficients a, b, c, and the greatest divisor e = e(a) = e(A) = gcd(a, b, c) (5.26) is the divisor of a nonzero form or matrix. If the divisor is equal to 1, the form and matrix are called primitive. The number det a = det A is the determinant of the form, and the number δ = δ (a) = δ (A) = − det a = b2 − 4ac (5.27) is the discriminant of the form and the matrix. If A and A are even matrices and A = tλ Aλ with λ ∈ Λ2+ = SL2 (Z), then matrices A and A and the corresponding quadratic forms are called (properly) equivalent, and we write then A ∼ A

and

a ∼ a.

Equivalent even matrices or integral forms make up a (proper) class. We denote by {A}+ and {a}+ the proper classes of A and a, respectively. All even matrices and integral forms from a fixed class have the same signature, divisor, and discriminant. According to (4.16), the Fourier coefficients f (A) of every modular form for Γ20 (q) depend only on the proper class of the matrix A. Fourier Coefficients of Eigenfunctions and Eigenvalues. As in the case of modular forms in one variable, Fourier coefficients of the eigenfunctions of genus 2 are closely related to the corresponding eigenvalues, but the relations are not so simple. Proposition 5.16. Let F be a modular form (5.23) with Fourier expansion (5.24). Suppose that F is an eigenfunction for the Hecke operator |T (m) = |k,χ T (m), where T (m) ∈ L0 (q) = L20 (q) with m prime to q is an element of the form (3.82) for K = Γ20 (q): (5.28) F|T (m) = λ (m)F. Then the Fourier coefficients f (A) of F and the eigenvalue λ (m) = λF (m) are linked by the relations

∑

λ (m) f (A) =

χ (d1 )χ (d22 )d1k−2 d22k−3

d,d1 ,d2 ∈N; A∈d2 E2 , dd1 d2 =m

×

∑

D∈Λ+ \Λ+ diag(1,d1 )Λ+ ; d2−1 A[ tD]∈d1 E2

where Λ+ = Λ+2 .

f

d A[ tD] , d1 d2

(5.29)

5.2 Binary Radial Series and Gaussian Composition

147

Proof. For the Fourier coefficient of the function F|T (m) indexed by a matrix A ∈ E2 , by formula (3.132) of Proposition 3.61, we obtain the formula

λ (m) f (A) = ( f |T (m))(A) =

∑

( f |Π+ (d)Π(d1 )Π− (d2 ))(A),

d,d1 ,d2 ∈N, dd1 d2 =m

where Π± = Π2± . By formula (4.24) of Lemma 4.14, we get ( f |Π+ (d)Π(d1 )Π− (d2 ))(A) = ( f |Π+ (d)|Π(d1 ))|Π− (d2 ) (A) χ (d2 )2 d22k−3 ( f |Π+ (d)|Π(d1 ))(d2−1 A) if A ∈ d2 E2 , = 0 if A ∈ / d2 E2 . Further, by formulas (4.27) and (4.25) of Lemma 4.14, we conclude that ( f |Π+ (d))|Π(d1 ) (A ) = χ (d1 )d1k−2 ( f |Π(d)) d1−1 A [ tD] ∑ D∈Λ+ \Λ+ diag(1, d1 )Λ+ ; A [ tD]∈d1 E2

= χ (d1 )d1k−2

∑

f

D∈Λ+ \Λ+ diag(1, d1 )Λ+ ; A [ tD]∈d1 E2

d t A [ D] . d1

These formulas imply the formulas (4.29). Although formulas (5.29) look rather complicated, we shall need only certain particular cases. Corollary 5.17. Let F ∈ M2k (q, χ ) be a nonzero modular form with Fourier expansion (5.24) satisfying F|T (m) = λ (m)F for all of the Hecke operators |T (m) = |k,χ T (m) with m prime to q. Then for every primitive matrix A ∈ E2 and integer ν coprime to q, the Fourier coefficients of F and the eigenvalues λ (m) = λF (m) are linked by the relation λ (m) f (ν A) = f dd1−1 ν A[ tD] . ∑ χ (d1 )d1k−2 ∑ d,d1 ∈N, dd1 =m

D∈Λ+ \Λ+ diag(1, d1 )Λ+ , A[ tD]∈d1 E2

(5.30) Proof. With ν A in place of A, where A and ν satisfy the stated conditions, the formula (5.29) turns into (5.30), since for primitive A, the index d2 must be equal to 1, and the conditions ν A[ tD] ∈ d1 E2 and A[ tD] ∈ d1 E2 are obviously equivalent. Let us look at some attractive formal implementations of these last relations. Note that for every matrix D entering the inner sum on the right, the matrix d1−1 A[ tD] is even and has the same (positive) determinant as the matrix A. Let us suppose for a moment that the discriminant of a primitive nonsingular matrix A is a so-called single-class discriminant, i.e., such that each even primitive matrix A of order 2 with

148

5 Euler Factorization of Radial Series

the same discriminant is (properly) equivalent to A. Then, since Fourier coefficients depend only on the classes of indexing matrices, the last formula takes the shape

λ (m) f (ν A) =

∑

d,d1 ∈N, dd1 =m

f (d ν A)χ (d1 )d1k−2 ρ (d1 , A),

ρ (d, A) = # D ∈ Λ+ \Λ+ diag(1, d)Λ+ A[ tD] ∈ dE2 .

where

(5.31)

Now if we divide both parts of the last relation by ms = d s d1s , considered just a formal quasicharacter of the semigroup N, and then sum up these relations over all m prime to q, then as a result we get the following identity for formal Dirichlet series: f (ν A)

λ (m) ms m∈N, gcd(m, q)=1

∑

=

∑

d,d1 ∈N, gcd(dd1 , q)=1

=

∑

d∈N, gcd(d, q)=1

f (d ν A) χ (d1 )d1k−2 ρ (d1 , A) ds d1s

f (d ν A) ds d

χ (d1 )ρ (d1 , A) . d1s−k+2 1 ∈N, gcd(d1 , q)=1

∑

(5.32)

Therefore, by dividing both sides by ν s and summing up over all ν ∈ N with ν |q∞ , since n = d ν ranges over all of N, we obtain for every primitive matrix A of singleclass discriminant the relation

∑

ν ∈N, ν |q∞

f (ν A) λ (m) ∑ ν s m∈N, gcd(m, ms q)=1 ∞

=

∑

n=1

f (nA) χ (d)ρ (d, A) . ∑ ns d∈N, gcd(d, d s−k+2 q)=1

(5.33)

It follows from Theorem 3.46 and Proposition 3.53 that the Dirichlet series on the left with coefficients λ (m) has an Euler product factorization. Thus, an identity of the form (5.33), on the one hand, expresses this Euler product by means of a radial Dirichlet series on the right and so opens a way to derive analytic properties of the product from those of the radial series, and on the other hand, it reveals multiplicative properties of Fourier coefficients of the eigenfunction in the form of a (partial) Euler product factorization of radial series (we shall see later that the second Dirichlet series on the right also has an Euler product factorization). Note that in order to reveal multiplicativity of the Fourier coefficients, one needs similar identities for a variety of radial series, whereas in order to study analytic properties of the Euler product, it suffices to take such an identity only for a single nonzero radial series. Both of these problems will be considered below. Exercise 5.18. In the notation and under assumptions of Proposition 5.16, if, moreover, f (02 ) = 0, prove that

5.2 Binary Radial Series and Gaussian Composition

λ (m) =

∑

d,d1 ,d2 ∈N; dd1 d2 =m

149

χ (d1 )χ (d22 )d1k−2 d22k−3 #{Λ+ \Λ+ diag(1, d1 )Λ+ }.

Π -Operators and Multiplication of Quadratic Modules. Along with the simple operations of multiplication and division of even matrices or corresponding quadratic forms by positive integers, the above formulas for the action of Hecke operators on Fourier coefficients of modular forms include more complicated operators |Π(d) of the form (4.27). It turns out that these operators can also be interpreted with the help of a multiplication of quadratic forms, namely the multiplication in the sense of Gaussian composition. We shall treat the Gaussian composition of quadratic forms using the equivalent language of multiplication of quadratic modules. First we have to recall some common definitions related to modules in a quadratic number field. For details see any textbook on algebraic numbers. Let K be a quadratic extension of the field Q of rational numbers. An additive subgroup A of K finitely generated over Z is called a module in K. Two modules A and A are said to be similar or belong to one class {A } = {A} if A = α A with a nonzero α of K; in this case one writes A ∼ A. The set of all modules in K is a disjoint union of the classes of similar modules. A module A in K is said to be full if QA = K. Each full module in K has a Z−basis of two elements. A full module in K that contains 1 and is a ring is called an order in K. Every order in K is contained in the maximal order O = O(K), the ring of all (algebraic) integers of the field K over Q. Let O be an order and ω1 , ω2 a basis of O . Then the number δ (O ) = det(tr(ωi ω j )), where tr means the trace from K to Q, is independent of the choice of the basis and called the discriminant of O . The discriminant of the maximal order O is called the discriminant of the field K. If A is a module in K, then the ring O(A) = α ∈ K α A ⊂ A is called the ring of multipliers of the module. Similar modules have equal rings of multipliers. The ring of multipliers of a full module is an order in K. For each full module, there is a similar module contained in its ring of multipliers. For a full module A with O(A) = O , let α1 , α2 and ω1 , ω2 be bases of A and O , respectively. Then the absolute value of the determinant of the transition matrix from the first basis to the second, N(A) = | det(ai j )|,

where αi = ∑ ai j ω j ,

is independent of the choice of bases, and is called the norm of A. If A ⊂ O , then N(A) = [O : A] (the index of A in O ). Let A and A be two full modules in K. Then the set AA is again a full module in K, which is called the product of A and A . The norm of the product is equal to the product of norms: N(AA ) = N(A)N(A ).

150

5 Euler Factorization of Radial Series

For a full module A in K we denote by A the module consisting of the conjugates α¯ over Q of the elements α ∈ A. The conjugate module A is again a full module with the same ring of multipliers as that of A, and the following formula holds: AA = N(M)O(A). All full modules in K with a fixed ring of multipliers O form a commutative group under multiplication of modules. The quotient group of this group by the subgroup of modules similar to O is a finite group H(O ) called the class group of O . The order h(O ) = #H(O ) of the class group is the class number of O . Full modules A in K contained in its ring of multipliers O(A) = O are called regular ideals of the ring O . Each regular ideal of an order is uniquely (up to order) decomposable into a product of regular prime ideals. √ Every quadratic extension of Q has the form K = Q( d0 ), where d0 = 0, 1 is a square-free integer. As a basis of the maximal order O of such K one can take the numbers 1 and ω , where √ (1 + d0 )/2 if d0 ≡ 1 (mod 4), ω= √ d0 if d0 ≡ 2 or 3 (mod 4), and the discriminant of the field is equal to if d0 ≡ 1 (mod 4), d0 d = δ (K) = δ (O) = 4d0 if d0 ≡ 2 or 3 (mod 4). √ In both cases one can write K = Q( d). Any order O of K has the form

Ol = Z + l ω Z, where l is the index [O : O ]. The discriminant of Ol is equal to dl 2 . For α , β , . . . ∈ K we shall denote by {α , β , . . . } the full module in K generated over Z by α , β , . . . : {α , β , . . . } = Zα + Zβ + · · · . The following lemma is often useful in computations with quadratic modules. / Q, and let aγ 2 + bγ + c = 0, Lemma 5.19. Let K be a quadratic field, γ ∈ K, γ ∈ where a, b, and c are the rational integers satisfying gcd(a, b, c) = 1 and a > 0. Then the module A = {1, γ } satisfies the conditions

O(A) = {1, aγ },

N(A) = 1/a,

δ (O(A)) = b2 − 4ac.

Proof. A number α = r + r γ ∈ K with r, r ∈ Q satisfies α A ⊂ A if and only if α = r + r γ ∈ A and αγ = −r c/a + (r − r b/a)γ ∈ A, which is equivalent to the inclusions r, r , r c/a, r b/a ∈ Z. Since the numbers a, b, c are coprime, these inclusions are equivalent to the inclusions r ∈ Z, r ∈ aZ, whence O(A) = {1, aγ }. For the basis 1, γ of A and the basis 1, aγ of O(A), we have N(A) = | det(diag(1, 1/a))| = 1/a. The formula for the discriminant follows from the definition.

5.2 Binary Radial Series and Gaussian Composition

151

The relation between quadratic modules and prime numbers is described by the following proposition. √ Proposition 5.20. Let K = Q( d) be the quadratic field with discriminant d, and Ol the order of K with discriminant dl 2 . Suppose that p is a rational prime number not dividing l. Then a full module A in K satisfying the conditions

O(A) = Ol ,

A ⊂ Ol ,

N(A) = p,

(5.34)

exists if and only if the congruence x2 ≡ d (mod 4p) is solvable. If the congruence is solvable and p does not divide d, then there are precisely two different modules P and P satisfying (5.34), and P = P. If the congruence has a solution and p divides d, then there is exactly one module P satisfying (5.34), and P = P. Finally, if the congruence has no solutions, then there is the unique complete module P satisfying

O(P) = Ol ,

P ⊂ Ol ,

N(P) = p2 ,

(5.35)

and this module is P = pOl . All of the listed modules A = P are regular prime ideals of the ring Ol . Proof. Suppose that the module A satisfies conditions (5.34). Then the index of A in Ol is p, and so the least positive integer contained in A equals p. Hence, the / Q. module A has a basis of the form p, pγ , A = {p, pγ }, where γ ∈ K and γ ∈ The number γ satisfies the equation aγ 2 + bγ + c = 0, where a, b, and c are rational integers with gcd(a, b, c) = 1 and a > 0. Then, by Lemma 5.19, we have p = N(A) = N(pOl )N({1, γ }) = p2 /a, whence a = p. On the other hand, b2 − 4ac = δ (O(A)) = dl 2 , and so b2 ≡ dl 2 (mod 4p). Since d ≡ 0 or 1 (mod 4) and p l, we conclude that the congruence x2 ≡ d (mod 4p) has a solution. Conversely, if this congruence has a solution, then there are rational integers b, c such that b2 − 4ac = d. It follows from the definition of d that gcd(p, b, c) = 1, and so gcd(p, bl, cl 2 ) = 1. Let γ be a root of the equation px2 + blx + cl 2 = 0. Then the module A = {p, pγ } satisfies (5.34). Let Ai = {p, pγi }, where i = 1, 2, be two modules satisfying (5.34). As has been shown, each of the numbers γi satisfies a relation pγi2 + bi γi + ci = 0, where bi , ci are rational integers, gcd(p, bi , ci ) = 1, and b2i − 4pci = dl 2 . In particular, b2i ≡ dl 2 (mod 4p), whence b1 ≡ b2 (mod 2p) or b1 ≡ −b2 (mod 2p). In the first case we obtain b2 = b1 + 2t p and √ √ √ A2 = {p, (−b2 ± l d)/2} = {p, (−b1 ± l d)/2 − t p} = {p, (−b1 ± l d)/2}. The last module is equal to A1 or A1 . Similarly, in the second case, b2 = −b1 + 2t p and √ √ A2 = {p, (b1 ± l d)/2 − t p} = {p, (−b1 ∓ l d)/2} = A1 or A1 . If p|d, then b1 ≡ −b1 (mod 2p); hence A1 = A1 . Let us assume now that the congruence x2 ≡ d (mod 4p) has no solutions. If A is a complete module satisfying (5.35), then the least positive integer contained in A

152

5 Euler Factorization of Radial Series

is equal to pδ with δ = 1 or δ = 2, and A = {pδ , pδ γ }. If aγ 2 + bγ + c = 0, where a, b, c ∈ Z, a > 0, and gcd(a, b, c) = 1, then by Lemma 5.19, we conclude that p2 = N(A)N(pδ )N({1, γ }) = p2δ /a, O(A) = {1, aγ }, and δ (O(A)) = b2 − 4ac. If δ = 2, then a = p2 and dl 2 = b2 − 4ac ≡ b2 (mod 4p2 ), which, since p l, contradicts the assumption. Thus δ = 1, a = 1, O(A) = Ol = {1, γ }, and A = p{1, γ } = pOl . On the other hand, the module pOl clearly satisfies (5.35). Every ideal A ⊂ Ol with prime norm is clearly a prime ideal. Let N(A) = p2 . / A, then the module A = αOl + A is complete and If α ∈ O(A) = Ol and α ∈ N(A) = p or 1, since A = A. By the above, the equality N(A ) = p in this case is impossible; hence N(A ) = 1, A = Ol , and α is invertible modulo A. Thus, the quotient Ol /A is a field, and the ideal A is prime. In view the above proposition it will be convenient to define the sign ε p (δ ) of δ modulo a prime p by ⎧ ⎪ if the congruence x2 ≡ d (mod 4p) is solvable and p d, ⎨1 ε p (d) = 0 if the congruence x2 ≡ d (mod 4p) is solvable and p | d, ⎪ ⎩ −1 if the congruence x2 ≡ d (mod 4p) has no solutions. (5.36) We conclude the survey part of this subsection with a description of the correspondence between modules in quadratic fields of negative discriminant and integral positive definite binary quadratic forms. For the quadratic forms, their matrices, and related notion, we shall use the notation √ (3.25)–(3.27) and the definitions of the beginning of this section. Let K = Q( d) be an imaginary quadratic field with discriminant d < 0, and let A be a full module in K with the ring of multipliers of the form O(A) = Ol . With every Z basis α , β of A ordered by the condition ℑ(β /α ) > 0 we associate the binary quadratic form a = a(A) =

1 (α x + β y)(α x + β y) = ax2 + bxy + cy2 . N(A)

The form a is clearly positive definite. It follows easily from Lemma 5.1 that the form is integral and primitive with discriminant δ (a) = b2 − 4ac = dl 2 equal to the discriminant of the ring of multipliers O(A). Conversely, if a(X) = ax2 +bxy+cy2 is a positive definite integral primitive quadratic form of discriminant b2 − 4ac = dl 2 , then the module √ & −b + l d A = A(a) = a, 2 is a full module in K with ring of multipliers Ol contained in the ring. It is not difficult to check that the indicated correspondences define a bijection between the set of all classes of equivalent full modules in K with the ring of multipliers Ol and the set of all classes of properly equivalent positive definite integral primitive binary quadratic forms of discriminant dl 2 . Let a and a be two integral primitive positive definite binary quadratic forms of the same negative discriminant δ , and let A and A be the matrices of the forms. Since √ the modules A = A(a) and A = A(a ) are full modules in the field K = Q( δ )

5.2 Binary Radial Series and Gaussian Composition

153

with the same ring of multipliers of discriminant δ = dl 2 , the same is true for their product AA . We define the Gaussian composition a × a of the forms a and a as the form corresponding to the product AA , a × a = a(AA ), and denote by A × A the matrix of a × a . For any full module A in K with the same ring of multipliers, we denote by A × A the matrix of the quadratic form a × a(A), A × A = A × A(A),

(5.37)

where A(A) is the matrix of the form a(A). The proper equivalence classes of the products depend only on proper classes of the quadratic forms or their matrices and the equivalence class of A. Thus, the above formulas define the products of corresponding classes. Since the set of equivalence classes of full modules with the ring of multipliers of discriminant δ forms an abelian group under multiplication of the classes, the set of classes with respect to proper equivalence of integral primitive positive definite binary quadratic forms of discriminant δ and the set of proper classes of their matrices can also be endowed with a group structure. We shall denote by H(δ ) all three of these naturally isomorphic class groups and by h(δ ) = #{H(δ )} the class number. We can now return to the action of Π-operators. First we shall examine the summation conditions in the formula (4.27) defining Π-operators. By Lemma 3.3, one can take Λ+ \Λ+ diag(1, d)Λ+ = diag(1, d)U U ∈ Λd \Λ+ , where Λd = Λ+

αβ ∈ Λ+ β ≡ 0 diag(1, d) Λ+ diag(1, d) = γ δ −1

(mod d) .

It is easy to check that two matrices of Λ+ with first rows (u1 , u2 ) and (u1 , u2 ) belong to the same left coset modulo Λd if and only if (u1 , u2 ) ∼ = (u1 , u2 ) (mod d) ⇐⇒ u1 u2 − u2 u1 ≡ 0

(mod d).

(5.38)

Thus, we have proved the following lemma. Lemma 5.21. One can take Λ+ \Λ+ diag(1, d)Λ+

u1 u2 1 , (u1 , u2 ) ∈ P (Z/dZ) , = diag(1, d)U U = v1 v2

(5.39)

where P1 (Z/dZ) is the “projective line modulo d”, i.e., a set of representatives for classes of all coprime pairs of integers (u1 , u2 ) modulo the equivalence (5.38).

154

5 Euler Factorization of Radial Series

Exercise 5.22. Prove that the cardinal numbers κ (d) = #(P1 (Z/dZ)) satisfy the following rules: κ (dd ) = κ (d)κ (d ) if gcd(d, d ) = 1, and κ (pδ ) = pδ −1 (p + 1) if p is a prime number. Lemma 5.21 will allow us to rewrite formula (4.27) for the action of Hecke operators |k,χ Π(d) on Fourier coefficients of functions of Fε in terms of points of the projective line modulo d. Let a = a(u) with u = (u1 , u2 ) be the quadratic form (5.25) with the matrix A, and b(u, v) = uA tv = b(v, u) the corresponding bilinear form. Then for D of the form u 10 u1 u2 = , D= dv v1 v2 0d we have

A[ tD] = DA tD =

2a(u) db(u,v) uA tu duA tv = . db(v,u) 2d 2 a(v) dvA tu d 2 vA tv

By Lemma 5.21, in this notation we can rewrite formula (4.27) in the form ( f |k,χ Π(d))(A) = χ (d)d k−2 ( f |P(d))(A), where ( f |P(d))(A) =

∑

u∈P1 (Z/dZ), a(u)≡0 (mod d)

2a(u)/d b(u,v) f , b(v,u) 2da(v)

(5.40)

(5.41)

and where for each u ∈ P1 (Z/dZ), by v is denoted a 2-row such that ( uv ) ∈ Λ+ . Since the definition of operators |P(d) is independent of the choice of k and χ , according to (5.40) we can express the operators |P(d) on Fourier coefficients of functions of Fε through the operators |k,χ Π(d), for arbitrary k and χ satisfying (−1)k χ (−1) = ε (−1) and χ (d) = 0, by the formula |P(d) = χ (d)d 2−k |k,χ Π(d).

(5.42)

The next key lemma interprets values of the images ( f |P(p))(A) for prime p and primitive A in terms of multiplication of quadratic modules. b Lemma 5.23. Let A = 2a b 2c be the matrix of an integral primitive quadratic form a of negative discriminant δ = b2√ − 4ac = dl 2 , where d is the discriminant of the imaginary quadratic field K = Q( δ ). Then for each function f on E2 depending only on classes of even matrices relative to proper equivalence and each prime number p not dividing the number l, we have ⎧ ⎪ ⎨ f (A × P) + f (A × P) if ε p (d) = 1, ( f |P(p))(A) = f (A × P) if ε p (d) = 0, ⎪ ⎩ 0 if ε p (d) = −1,

5.2 Binary Radial Series and Gaussian Composition

155

where ε p (d) is the sign (5.36), P a regular prime ideal of norm p of the order Ol ⊂ K with discriminant dl 2 , P the conjugate ideal, and where × is the multiplication (5.37). Proof. Let us consider the congruence a(x1 , x2 ) = ax12 + bx1 x2 + cx22 ≡ 0

(mod p).

(5.43)

It is easy to check in the framework of the elementary theory of quadratic congruences that for each prime p not dividing l, the number of different points of the projective line P1 (Z/pZ) modulo p, i.e., the number of inequivalent coprime pairs of integers with respect to the equivalence (5.38) modulo p, satisfying this congruence is equal to 1 + ε p (d). Let ε p (d) = 1, and let u = (u1 , u2 ) and u = (u1 , u2 ) be two inequivalent modulo p primitive solutions of (5.43). Since the discriminant δ = b2 − 4ac is not divisible by p, we may assume that a(u) = pa1 , a(u ) = pa2 , where a1 and a2 are not divisible by p. We can find integers v1 , v2 , v1 , v2 such that u1 u2 u u , U2 = 1 2 ∈ Λ+ . U1 = v1 v2 v1 v2 We set

2pa1 b1 A1 = U1 A U1 = , b1 c1 2pa2 b2 , A2 = U2 A tU2 = b2 c2

t

2a1 b1 , = b1 2pc1 2a2 b2 A2 = . b2 2pc2 A1

Then by (5.41), we get (P(p) f )(A) = f (A1 ) + f (A2 ), and it is sufficient to prove that with the appropriate choice of regular prime ideal P of norm p in Ol , one has A1 ∼ A × P and

A2 ∼ A × P,

(5.44)

where the symbol ∼ stands for proper equivalence. Let us consider the modules √ & √ & −b2 + δ −b1 + δ , A = p, A = p, 2 2 of the field K. It follows from Lemma 5.19 that the ring of multipliers of each of these modules is Ol , that they are both contained in Ol , and that they have the norm N(A) = N(A ) = p. Let us show that A + A = Ol . To prove this it is obviously sufficient to check that the number (b2 − b1 )/2 is not divisible by p. We set t 1 t2 −1 ∈ Λ+ . U2U1 = T = t3 t4

156

5 Euler Factorization of Radial Series

Since the solutions (u1 , u2 ) and (u1 , u2 ) are not equivalent modulo p, we have t2 = u1 u2 − u2 u1 ≡ 0 (mod p). It follows from the relation TA1 tT = A2 that pa1t12 + b1t1t2 + c1t22 = pa2 and 2pa1t1t3 + b1 (t1t4 + t2t3 ) + 2c1t2t4 = b2 . From the first equality we obtain the congruence b1t1 + c1t2 ≡ 0 (mod p), and it follows from the second that b1 (t1t4 + t2t3 ) + 2c1t2t4 − b1 b2 − b1 ≡ = t2 (b1t3 + c1t4 ) (mod p). 2 2 If (b2 − b1 )/2 were divisible by p, we would get the system of congruences b1t1 + c1t2 ≡ b1t3 + c1t4 ≡ 0 (mod p), which would imply that b1 ≡ c1 ≡ 0 (mod p), a contradiction, because δ = b21 −4pa1 c1 is not divisible by p. It follows from what we have proved and Proposition 5.20 that one can set A = P and A = P, where P, P are the unique pair of regular conjugate distinct prime ideals in Ol with norm p. In order to prove (5.43), let us denote by a1 , a2 , a1 , and a2 the quadratic forms with matrices A1 , A2 , A1 , and A2 , respectively, and √ by A1 , A2 , A1 , and A2 the correspond2 ing modules in K. If we set γ1 = (−b1 + δ )/2, then clearly γ1 = −pa1 c1 − b1 γ , and since a1 and p are coprime, we obtain A1 P = {a1 , γ1 }{p, γ1 } = {pa1 , pγ1 , a1 γ1 , −pa1 c1 − b1 γ1 } = {pa1 , γ1 } = A1 . Hence A1 P = A1 PP = A1 N(P) ∼ A1 , and A1 ∼ A1 × P ∼ A × P. Similarly, A2 ∼ A × P. This proves (5.44) and the first formula of the lemma. Now let ε p (d) = 0 and let u = (u1 , u2 ) be the unique solution of (5.43) modulo equivalence (5.38). We can find integers v1 , v2 such that U1 = ( uv11 uv22 ) ∈ Λ+ and set 2pa1 b1 2a1 b1 , A1 = . A1 = U1 A tU1 = b1 c1 b1 2pc1 Then the right-hand side of (5.41) is equal to f (A1 ), and it is sufficient to prove that A1 ∼ A × P,

(5.45)

where P is the unique regular prime ideal of norm p in Ol . By our assumption, the discriminant δ = dl 2 = b2 − 4ac = b21 − 4pa1 c1 is divisible by p and p l. It follows that b1 ≡ 0 (mod p) and dl 2 ≡ −4pa1 c1 (mod p2 ). If p = 2, since d is not divisible by p2 , the last congruence implies that a1 c1 is not divisible by p. If p = 2, then d = 4d0 with d0 ≡ 2, 3 (mod 4). Let us set b1 = 2b0 . Then δ = 4d0 l 2 = 4(b20 − 2a1 c1 ), whence 2a1 c1 = b20 − d0 l 2 ≡ b20 − d0

(mod 4).

The number b20 − d0 is not divisible by 4, since b20 ≡ 0, 1 (mod 4) and d0 ≡ 3, 3 (mod 4). Hence, a1 c1 is not divisible by p also for p = 2. Further, since the quadratic form a1 with matrix A1 together with a is primitive, and p a1 , it follows that the form a1 with matrix A1 is also primitive. Let us consider the module A = {p, γ },

5.2 Binary Radial Series and Gaussian Composition

157

√ where γ = (−b1 + δ )/2 satisfies γ 2 + b1 γ + pa1 c1 = 0. By Lemma 5.19 and Proposition 5.20, the module A = P is the unique regular prime ideal of Ol of norm p. Let us denote by a1 and a1 the quadratic forms with matrices A1 and A1 , respectively, and by A1 and A1 the corresponding modules in K. Since γ 2 = −b1 γ − pa1 c1 and p a1 , we have A1 P = {a1 , γ }{p, γ } = {pa1 , pγ , a1 γ , −pa1 c1 − b1 γ1 } = {pa1 , γ } = A1 . Hence, since P = P, we have A1 P = A1 PP ∼ A1 , and the corresponding matrices satisfy (5.45). Finally, if ε p (d) = −1, then, as was noted at the beginning of the proof, the congruence (5.43) has no solutions on the projective line modulo p, which proves the last formula of the lemma. Exercise 5.24. In the notation of Lemma 5.23, assuming that ε p (d) = 0 and p|l, prove the formula ( f |P(p))(A) = f (A × Ol/p ). Along with operators corresponding to elements Π(p) we shall also need operators associated with elements Ψ(p) ∈ L 2p defined by formula (3.145). The action of these operators can be naturally reduced to computation of simple trigonometric sums, which will be done in the next lemma. Lemma 5.25. In the notation and under the assumptions of the previous lemma and Lemma 3.68, the following formulas hold ⎧ ⎪ if ε p (d) = 1, ⎨p − 1 π iσ (AB)/p (5.46) e = S p (A) = ε p (d) = 0, −1 if ∑ ⎪ ⎩ B= tB∈Z22 /pZ22 , −(p + 1) if ε p (d) = −1, r p (B)=1

Proof. The set of matrices of the form

2 α0 γβ γβ , α , γ = 1, . . . , p − 1, β = 0, 1, . . . , p − 1 0 0 γβ γ can clearly be taken as a summation set in the sum S p (A). Thus, since A = we obtain 2 S p (A) = ∑ e2π iaα /p + ∑ e2π iγ (aβ +bβ +c)/p . α

2a

b b 2c

,

γ ,β

The sum on α is obviously equal to p − 1 if p|a and −1 if p a; similarly, the sum on γ is p − 1 if p|aβ 2 + bβ + c and −1 if p aβ 2 + bβ + c. Hence if, for example, ε p (d) = −1, then the second of the alternatives is realized in all cases, and we have S p (A) = −1 + p(−1) = −(p + 1). The other cases are similar, and we leave their consideration to the reader as a useful exercise on quadratic congruences. The above lemmas together with Lemma 3.71 will allow us to discover the action of the operators |P(d) with composite d on Fourier coefficients of functions of Fε in terms of characters of class groups.

158

5 Euler Factorization of Radial Series

2 Theorem 5.26. Let δ be a negative integer written in the form √ δ = dl , where d is the discriminant of the imaginary quadratic field K = Q( δ ) and l is a positive integer. Let Ol be the order in K of the discriminant δ , A1 , . . . , Ah with h = h(δ ) a full system of representatives of classes of properly equivalent even positive definite primitive matrices of order two and the discriminant δ , and let ψ be a character of the class group H(δ ). Then for every function f on E2 whose values are Fourier coefficients of a function of Fε and positive integer d prime to l, the following formulas hold for the action of operators (5.41) on averaging

ψ (A1 ) f (A1 ) + · · · + ψ (Ah ) f (Ah ) of f over the group H(δ ) with character ψ : h

∑ ψ (Ai ) f (Ai )

|P(d)

i=1

= λ (P(d); ψ , δ )

h

= ∑ ψ (Ai )( f |P(d))(Ai ) i=1

h

∑ ψ (Ai ) f (Ai )

,

(5.47)

i=1

where the numbers λ (P(d); ψ , δ ) satisfy the formal identity λ (P(d); ψ , δ ) ψ (P) −1 1 1 − 2s ∏ 1 − = ∏ , ∑ ds p N(P)s P| p p∈P, p l d∈N, gcd(d,l)=1

(5.48)

where P in the inner product on the right runs through all regular prime ideals of the ring Ol dividing the principal ideal pOl . Proof. Since the function f depends only on proper classes of even matrices, by Lemma 5.23 in the notation of the lemma, for a prime p not dividing l, we obtain h

∑ ψ (Ai ) f (Ai )

i=1

h

|P(p) = ∑ ψ (Ai )( f |P(p))(Ai ) i=1

⎧ h ⎪ ⎨∑i=1 ψ (Ai ) f (Ai × P) + f (Ai × P) if ε p (d) = 1, = ∑hi=1 ψ (Ai ) f (Ai × P) if ε p (d) = 0, ⎪ ⎩ 0 if ε p (d) = −1,

which clearly implies the formula (5.47) with d = p and ⎧ ⎪ ⎨ψ (P) + ψ (P) if ε p (d) = 1, λ (P(p); ψ , δ ) = ψ (P) = ψ (P) if ε p (d) = 0, ⎪ ⎩ 0 if ε p (d) = −1.

(5.49)

In order to reduce the action of operators |P(d) for composite d to the cases of prime divisors of d, we shall use the relation (5.42) between operators |P(d) and

5.2 Binary Radial Series and Gaussian Composition

159

|k Π = |k,1 Π(d) with χ = 1 (the unit character modulo 1) and relations for elements Π(d) obtained in Section 3.5. First of all, by Lemma 3.62, we have |P(d)|P(d1 ) = d k−2 d1k−2 |Π(d)|Π(d1 ) = (dd1 )k−2 |Π(dd1 ) = |P(dd1 ) if gcd(d, d1 ) = 1.

(5.50)

Then, by Lemma 3.71, for each prime p we get the following summation formula for the formal power series with operators |P(pν ) as coefficients:

∑ |P(pν )vν = ∑ |k Π(pν )(p2−k v)ν

ν ≥0

ν ≥0

= (1 − |k p2 p 2 (p2−k v)2 ) −1 × 1 − |k Π(p)p2−k v + (|k p p 2 + |k pΨ(p))(p2−k v)2 −1 = (1 − |k p 2 p6−2k v2 ) 1 − |P(p)v + (|k p 2 + |k Ψ(p))p5−2k v2 . (5.51) Let us consider now the action of operators |k p 2 and |k Ψ(pi ) on the average. By formula (4.26) for n = 2 and χ = 1, we have h

∑ ψ (Ai ) f (Ai )

|k p 2 = p2k−6

i=1

h

∑ ψ (Ai ) f (Ai )

.

(5.52)

i=1

By formula (4.22) of Lemma 4.13 for χ = 1 applied to the double cosets entering in the decomposition (3.45) of the element Ψ(p), for each A ∈ E2 , one can write ( f |k Ψ(p))(A) = p2k−6 S p (A) f (A), where S p (A) is the trigonometric sum (5.46). Hence, by Lemma 5.25, we derive the formula h

∑ ψ (Ai ) f (Ai )

|k Ψ(p) = p2k−6 λ (Ψ(p); ψ , δ )

i=1

where

⎧ ⎪ ⎨(p − 1) λ (Ψ(p); ψ , δ ) = −1 ⎪ ⎩ −(p + 1)

h

∑ ψ (Ai ) f (Ai )

,

(5.53)

i=1

if ε p (d) = 1, if ε p (d) = 0, if ε p (d) = −1.

It follows from the relations (5.50) and (5.51) that each of the operators |P(d) with d = ∏i pνi i , where pi are different prime divisors of d, is a polynomial in operators |P(pi ), |k pi 2 , and |k Ψ(pi ) that commute with each other. Then the relations (5.47) with some coefficients λ (P(d); ψ , δ ) follow for every d prime to l from the formulas for primes proved above. In order to compute the coefficients, we note first that by (5.50), we have

λ (P(dd1 ); ψ , δ ) = λ (P(d); ψ , δ )λ (P(d1 ); ψ , δ )

if gcd(d, d1 ) = 1,

(5.54)

160

5 Euler Factorization of Radial Series

which implies the formal Euler product factorization λ (P(d); ψ , δ ) 1 ν = ∏ ∑ ∑ λ (P(p ); ψ , δ ) pν s . ds p∈P, pl ν ≥0 d∈N, gcd(d,l)=1

(5.55)

It follows from the summation formula (5.51) and formulas (5.47) with d = p, (5.52), and (5.53) that each of the p-factors of the last Euler product can be summed in the form 1 1 ν λ (P(p ); ψ , δ ) = 1 − ∑ pν s p2s ν ≥0 −1 λ (P(p); ψ , δ ) (1 + λ (Ψ(p); ψ , δ ))p−1 × 1− + . ps p2s By substituting here the values for λ (P(p); ψ , δ ) and λ (|k Ψ(p); ψ , δ ) computed above, we obtain that the fraction on the right is equal to −1 (ψ (P) + ψ (P)) (1 + p − 1)p−1 1 1− + 1 − 2s p ps p2s −1 ψ (P) ψ (P) 1 1− 1 − = 1 − 2s p N(P)s N(P)s if ε p (d) = 1, where P and P is a pair of conjugate regular prime ideals of Ol of norm p, 1−

1 p2s

1−

ψ (P) (1 − 1)p−1 + ps p2s

−1

ψ (P) −1 1 1− = 1 − 2s p N(P)s

if ε p (d) = 0, where P is a regular prime ideal of Ol of norm p, and 1−

1 p2s

−1 (1 − (p + 1))p−1 1 ψ (P) −1 1+ 1 − = 1 − p2s p2s N(P)s

if ε p (d) = −1, where P = pOl is a regular prime ideal of Ol of norm p2 (with ψ (P) = 1). Hence, according to Proposition 5.20, we conclude that in all three cases, the p-factor of the factorization (5.55) can be written in the form 1 1 ψ (P) −1 ∑ λ (P(p ); ψ , δ ) pν s = 1 − p2s ∏ 1 − N(P)s , ν ≥0 P|p ν

(5.56)

where P ranges over all regular prime ideals of the ring Ol dividing the principal ideal pOl . The formal Euler factorization (5.48) follows from (5.55) and (5.56). Exercise 5.27. Let ρ (d, A) be the number (5.31) of solutions of the congruence a(u) ≡ 0 (mod d), where a is the quadratic form with matrix A, on the projective

5.2 Binary Radial Series and Gaussian Composition

161

line modulo d. In the notation and under the assumptions of Theorem 5.26, prove the formal identity −1 ρ (d, A) 1 1 1 − 2s ∏ 1 − = ∏ . ∑ ds p N(P)s P| p p∈P, p l d∈N, gcd(d,l)=1 Multiplicative Properties of Fourier Coefficients. We return to a nonzero modular form F of weight k > 0 and character χ modulo q for the group Γ20 (q) with Fourier expansion (5.24) and assume that F is an eigenfunction for Hecke operators |T (m) = |k,χ T (m) corresponding to all elements T (m) ∈ L0 (q) = L20 (q) of the form (3.82), where K = Γ20 (q), with eigenvalues λ (m) = λF (m): F|T (m) = λ (m)F

for all m prime to q.

(5.57)

Then by Corollary 5.17, for every primitive matrix A ∈ E2 and positive integer ν coprime to q, the Fourier coefficients of F and the eigenvalues are linked by the relation (5.30), which we shall rewrite in the form

λ (m) f (ν A) =

∑

d,d1 ∈N, dd1 =m

χ (d1 )d1k−2 (( f |Π+ (ν d))|P(d1 ))(A),

(5.58)

where |Π+ = |k,χ Π+ are the operators (4.25) and the operators P are defined by 2 (5.41). Let, as in Theorem 5.26, δ be a negative integer written in the form √ δ = dl , where d is the discriminant of the imaginary quadratic field K = Q( δ ) and l is a positive integer. Let Ol be the order in K of discriminant δ , A1 , . . . , Ah with h = h(δ ) a full system of representatives of classes of properly equivalent even positive definite primitive matrices of order two with discriminant δ , and let ψ be a character of the class group H(δ ). Let us multiply both sides of the relation (5.58) with A = Ai by ψ (Ai ) and sum these relations over i = 1, . . . , h. We get the relation h

λ (m) ∑ ψ (Ai ) f (ν Ai ) = i=1

∑

d, d1 ∈N, dd1 =m

h

χ (d1 )d1k−2 ∑ ψ (Ai )(( f |Π+ (d ν ))|P(d1 ))(Ai ). i=1

If m is also coprime to l, then, since values of the functions f |Π+ (ν d) together with values of f can be considered as Fourier coefficients of functions of Fε , we can apply relations (5.47) of Theorem 5.26 to the inner sums on the right of the last formula, which leads us the relations h

λ (m) ∑ ψ (Ai ) f (ν Ai ) i=1

=

χ (d1 )d1k−2 λ (P(d1 ); ψ , δ ) ∑ ψ (Ai )( f |Π+ (d ν )))(Ai )

∑

χ (d1 )d1k−2 λ (P(d1 ); ψ , δ ) ∑ ψ (Ai ) f (d ν Ai ).

d, d1 ∈N, dd1 =m

=

h

∑

d, d1 ∈N, dd1 =m

i=1 h

i=1

162

5 Euler Factorization of Radial Series

Hence, dividing both parts by ms = d s d1s and summing over all m ∈ N with gcd(m, lq) = 1, we come to the following generalization of the relations (5.32), obtained under the assumption that h(δ ) = 1, to the case of arbitrary class number: h

∑ ψ (Ai ) f (ν Ai )

i=1

∑

=

d1 ∈N, gcd(d1

λ (m) ms m∈N, gcd(m, lq)=1

∑

χ (d1 )d1k−2 λ (P(d1 ); ψ , δ ) h f (d ν Ai ) , ∑ ψ (Ai ) ∑ s d ds 1 i=1 , lq)=1 d∈N, gcd(d, lq)=1 (5.59)

where ν is an arbitrary positive integer prime to all integers m ∈ N satisfying gcd(m, lq) = 1, in other words, ν satisfies ν |(lq)∞ , i.e., divides a power of lq. Ultimately, we come to the following theorem on multiplicative properties of Fourier coefficients of eigenfunctions. Theorem 5.28. Let F be a nonzero modular form of weight k ∈ N and character χ modulo q for the group Γ20 (q). Suppose that F is an eigenfunction satisfying (5.57). Let√δ be a negative integer, δ = dl 2 , where d is the discriminant of the field K = Q( δ ) and l ∈ N, A1 , . . . , Ah with h = h(δ ) a full system of representatives of classes of properly equivalent even positive definite primitive matrices of order two with discriminant δ , and let ψ be a character of the group H(δ ) of the classes of matrices with respect to the Gaussian composition (5.37). Then for each positive integer ν dividing a power of lq, the following formal identity holds: h

∑ ψ (Ai ) f (ν Ai ) ∏

i=1

=

∏

P lq

p∈P, p lq

−s Q−1 p, F (p )

ψ (P)χ (N(P)) 1− N(P)s−k+2

−1

h

∑ ψ (Ai )

i=1

∑

d∈N, gcd(d, lq)=1

f (d ν Ai ) , ds

(5.60)

where Q p, F (v) =1 − λ (p)v + (λ (p)2 − λ (p2 ) − χ (p2 )p2k−4 )v2 − χ (p2 )p2k−3 λ (p)v3 + χ (p4 )p4k−6 v4 ,

(5.61)

and P on the right ranges over all regular prime ideals of the ring Ol not dividing the principal ideal lqOl . Proof. By Theorem 3.46, we have

λ (mm ) = λ (m)λ (m )

if

gcd(m, m ) = gcd(mm , q) = 1.

By Proposition 3.53, for each prime number p not dividing q, we get the formal identity

5.2 Binary Radial Series and Gaussian Composition ∞

163

∞

∑ λ (pβ )vβ = ∑ λ (T (pβ ))vβ

β =0

β =0

= (1 − p λ ( p )v2 )(1 − λ (q1 (p))v + λ (q2 (p))v2 − λ (q3 (p))v3 + λ (q4 (p))v4 )−1 , 2

where λ (q1 (p)) = λ (T (p)) = λ (p),

λ (q2 (p)) = λ (T (p)2 ) − λ (T (p2 )) − p2 λ ( p ) = λ (p)2 − λ (p2 ) − p2 λ ( p ), λ (q3 (p)) = p3 λ ( p )λ (p), and λ (q4 (p)) = p6 λ ( p )2 . Since by (4.26), we can write λ ( p ) = λ ( p ) = χ (p2 )p2k−6 , it follows that ∞

∑ λ (pβ )vβ = (1 − χ (p2 )p2k−4 v2 )Q−1 p, F (v),

β =0

where Q p, F (v) is the polynomial (5.61). Thus, we obtain the formal Euler product factorization ∞ λ (m) λ (pβ ) = ∏ ∑ s βs m p∈P, p lq β =0 p m∈N, gcd(m, lq)=1

∑

=

∏

−s (1 − χ (p2 )p−2(s−k+2) )Q−1 p, F (p ).

(5.62)

p∈P, p lq

On the other hand, by (5.48) we can write

χ (d)d k−2 λ (P(d); ψ , δ ) χ (d) = λ (P(d); ψ , δ ) s−k+2 ∑ s d d d∈N, gcd(d,l)=1 d∈N, gcd(d,l)=1

∑

=

∏

p∈P, p l

1−

χ (p2 ) p2(s−k+2)

∏

P| p

χ (N(P))ψ (P) 1− N(P)s−k+2

−1 .

(5.63)

Let us return now to the identity (5.59). On replacing the Dirichlet series on both sides of this identity by their product factorizations obtained above and dividing both sides by ∏ p∈P, p lq (1 − χ (p2 )p−2(s−k+2) ), we obtain to the relation (5.58). Let us reformulate the theorem in the terms of the multiplicativity of functions of integral argument. For an integer r, we say that a nonzero function

ϕ : N r = {m ∈ N| gcd(m, r) = 1} → C is r −multiplicative if it satisfies the following two conditions: (1) ϕ (m m ) = ϕ (m)ϕ (m ) if m and m are coprime (and prime to r); (2) for each prime number p not dividing r, the formal power series ∑ν ≥0 ϕ (pν )vν is formally equal to a rational fraction Np (v)D p (v)−1 , where Np (v) and D p (v) are polynomials with Np (0) = D p (0) = 1.

164

5 Euler Factorization of Radial Series

Alternatively, these conditions are equivalent to the condition that the following formal Euler factorization holds

ϕ (m) = ∏ Np (p−s )D p (p−s )−1 , s m p∈P, p r m∈N, gcd(m, r)=1

∑

where Np (v) and D p (v) are polynomials with Np (0) = D p (0) = 1. The function m → λ (m), where λ (m) are eigenvalues of a nonzero modular form F ∈ M2k (q, χ ) satisfying (5.57), gives an example of q -multiplicative functions. Other examples are given by the following corollary. Corollary 5.29. In the notation of Theorem 5.28, each of the functions d → f (d ν A), where A is a positive definite even primitive matrix of order two with discriminant δ = dl 2 and ν divides a power of lq, is a linear combination with constant coefficients of at most h = h(δ ) lq -multiplicative functions. Proof. It follows from Theorem 5.28 that each of the functions d →

∑

ψ (Ai ) f (d ν Ai )

1≤i≤h

is proportional to an lq -multiplicative function. If, say, A ∼ A j , then 1 f (d ν A) = f (d ν A j ) = ∑ ψ (A j ) ∑ ψ (Ai ) f (d ν Ai ) , h ψ 1≤i≤h by the orthogonality relations for the characters. Exercise 5.30. In the notation of Theorem 5.28 prove the following generalization of the relations (5.33): h

∑ ψ (Ai )

i=1

=

∑

ν ∈N, ν | (lq)∞

f (ν Ai )

∏

p∈P, p lq

−s Q−1 p, F (p )

∞ ψ (P)χ (N(P)) −1 h f (n Ai ) 1 − ψ (Ai ) ∑ . ∑ ∏ s−k+2 ds N(P) i=1 n=1 P lq

Exercise 5.31. Prove that for each prime number p not dividing q, the polynomial (5.61) has a factorization of the form Q p, F (v) = 1− α0 (p)v 1− α0 (p)α1 (p)v 1− α0 (p)α2 (p)v 1− α0 (p)α1 (p)α2 (p)v , where α0 (p)2 α1 (p)α2 (p) = χ (p2 )p2k−3 . [Hint: Use Proposition 3.55.]

5.3 Zeta Functions of Eigenforms for Genus 2

165

5.3 Zeta Functions of Eigenforms for Genus 2 According to Theorem 4.7, each invariant subspace of cusp forms for the group Γ20 (q) has a basis of eigenfunctions for all regular Hecke operators. The Zharkovskaya commutation relations (Theorem 4.19) allow one to prove that all regular Hecke operators can be simultaneously diagonalized on certain invariant spaces of modular forms consisting not only of cusp forms, but, for example, on the whole space M2k of all modular forms of weight k for the full modular group Γ2 (Theorem 4.22). This justifies a look at the eigenforms. Regular Zeta Functions of Eigenforms. As in Section 5.2, we fix a nonzero modular form F = F(Z) ∈ M2k (q, χ ) = Mk (Γ2o (q), χ ) of positive integral weight k and character χ modulo q ∈ N for the group Γ20 (q) and suppose that F is an eigenfunction for all Hecke operators |T (m) = |k,χ T (m), where T (m) ∈ L0 (q) = L20 (q) are the elements of the form (3.82), with the eigenvalues λ (m), (gcd(m, q) = 1). F|T (m) = λ (m)F It follows from Theorem 3.46 and Proposition 3.53 that one can write the formal identities ∞ λ (m) λ (pβ ) = ∏ ∑ s βs m p∈P, p q β =0 p m∈N, gcd(m, q)=1 χ (p2 ) −s 1 − 2(s−k+2) = ∏ ∏ Q−1 p, F (p ), p p∈P, p q p∈P, p q

∑

(5.64)

where Q p, F (v) =1 − λ (p)v + (λ (p)2 − λ (p2 ) − χ (p2 )p2k−4 )v2 − χ (p2 )p2k−3 λ (p)v3 + χ (p4 )p4k−6 v4 . We call the infinite product

ζr (s, F) =

∏

p∈P, p q

−s Q−1 p, F (p )

(5.65)

the regular zeta function of the eigenform F; if q = 1, we omit the adjective “regular” as well as the lower subscript r and get the zeta function ζ (s, F) of F. Lemma 5.32. The regular zeta function ζr (s, F) of a nonzero cusp form F ∈ N2k (q, χ ) converges absolutely and uniformly in each right half-plane ℜs > k +1+ ε with ε > 0.

166

5 Euler Factorization of Radial Series

Proof. Let f (A) be a fixed nonzero Fourier coefficient of F. By relation (2.29) of Proposition 5.16 and estimate (1.46) of Proposition 1.25, for each m prime to q, we obtain the inequalities c |λ (m)| ≤ | f (A)| d, d =

∑

1 , d2 ∈N, dd1 d2 =m

c(det A)k/2 | f (A)| d, d

∑

d1k−2 d22k−3 κ (d1 ) det

1 , d2 ∈N, dd1 d2 =m

k/2 d A[diag(1, d1 )] d1 d2

d k d1k−2 d2k−3 κ (d1 ) < c τ3 (m)mk ,

where c is a constant, κ (d1 ) = #(Λ+ \Λ+ diag(1, d1 )Λ+ ) ≤ d12 is the number of points on the projective line modulo d1 , and τ3 (m) is the number of factorizations of m into products of three positive integers. Hence using the elementary estimate τ3 (m) < c(ε )mε with arbitrary ε > 0, it follows that |λ (m)| < c mk+ε

(gcd(m, q) = 1, ε > 0),

(5.66)

where c = c (ε ) is a constant depending only on ε . It follows from the last estimate that the Dirichlet series on the left in (5.54) converges absolutely and uniformly on each of the indicated right half-planes. On the other hand, by (5.54), we can write the product (5.65) in the form −1 ∞ χ (p2 ) λ (pβ ) ζr (s, F) = ∏ 1 − p2(s−k+2) ∏ ∑ βs p∈P, p q p∈P, p q β =0 p ∞ ∞ χ (p2α ) λ (pβ ) = ∏ ∑ 2α (s−k+2) ∏ ∑ βs , p∈P, p q α =0 p p∈P, p q β =0 p where absolute and uniform convergence of both products in the half-planes of the stated form follows from estimates (5.66) by a well-known classical test. For example, |λ (pβ )| |λ (m)| 1 1 < ∑ < c ∑ ℜs−k−ε < c ∑ 1+ε −ε ℜs β s| m m m |p m>1 m>1 m>1 p, β ≥1

∑

if ℜs > k + 1 + ε with ε > ε , and the last series with positive constant terms is convergent. It follows from the lemma that the regular zeta function ζr (s, F) of a cusp form F ∈ N2k (q, χ ) is a holomorphic function of the complex variable s in each of the right half-planes ℜs > k + 1 + ε with ε > 0. The question of analytic continuation of the zeta functions of cusp forms of level q = 1 will be considered in the next subsection.

5.3 Zeta Functions of Eigenforms for Genus 2

167

Exercise 5.33. Prove that the regular zeta function of an arbitrary eigenform F ∈ M2k (q, χ ) converges absolutely and uniformly in each right half-plane ℜs > 2k + 1 + ε with ε > 0. [Hint: Use the estimate (1.52).] Exercise 5.34. Let F ∈ M2k (q, χ ) be an eigenfunction for all regular Hecke operators. Prove that if k = 2 or the character χ is trivial, then the image F|Φ ∈ M1k (q, χ ) of F under the Siegel operator (1.56) is again an eigenfunction for all regular Hecke operators, and if F|Φ = 0, the regular zeta functions of F and F|Φ are linked by the relation ζr (s, F) = ζr (s, F|Φ)ζr (s − k + 2, χ ; F|Φ), where

ζr (s, χ ; F|Φ) =

∏

p∈P, p q

−s Q−1 p, F|Φ ( χ (p)p ).

Analytic Continuation and Functional Equation. Here we shall prove that under certain conditions, the zeta function of a cusp eigenform of level one has a meromorphic continuation over the whole s-plane and satisfies a functional equation with two gamma factors. eigenform for all Hecke operators with Theorem 5.35. Let F ∈ N2k be a cusp Fourier coefficients f (A). Suppose that f 20 02 = 0. Then the zeta function ζ (s, F) of F can be continued to the whole s-plane as a meromorphic function. More precisely, the function Ψ(s, F) = (2π )−2s Γ(s)Γ(s − k + 2)ζ (s, F),

(5.67)

where Γ(s) is the gamma function, has a continuation to the whole s-plane as a meromorphic function with at most two simple poles at the points s = k − 2 and s = k and satisfies the functional equation Ψ(2k − 2 − s, F) = Ψ(s, F).

(5.68)

Proof. Let us use Theorem 5.28 √of the discriminant δ = d = −4 of the √ in the case Gaussian number field K = Q( −1) = Q( −4) with class number h = h(−4) = 1, the representative A1 = 2 · 12 of proper classes of discriminant −4, and the unit characters χ and ψ . Then the identity (5.60) takes the form ∞

f (A1 )ζ (s, F) = ZO (s − k + 2) ∑

d=1

where ZO (s) = ∏ P

1 1− N(P)s

−1

f (dA1 ) , ds

=∑ A

(5.69)

1 , N(A)s

with P √ and A ranging over all prime ideals and nonzero integral ideals of the ring O = Q[ −1] of the Gaussian integers, respectively, is the Dedekind zeta function

168

5 Euler Factorization of Radial Series

of the ring. It follows from Lemma 5.32 and estimate (1.46) that both sides of (5.69) define functions holomorphic in the half-plane ℜs > k + 1. Since f (A1 ) = 0, in this half-plane we can write the functional identity Ψ(s, F) = (2π )−2s Γ(s)Γ(s − k + 2)ζ (s, F) ∞

= f (A1 )−1 (2π )−2s Γ(s)Γ(s − k + 2)ZO (s − k + 2) ∑

d=1

f (dA1 ) . ds

According to Proposition 2.16, the function on the right has a meromorphic continuation to the whole s-plane with at most two simple poles at the points s = k − 2 and s = k and satisfies the functional equation (5.68). Exercise 5.36. Let F ∈ M2k be an eigenfunction for all Hecke operators. Suppose that F is not a cusp form, i.e., F = f |Φ = 0. Prove that in this case, the function (5.67) is defined and holomorphic in the half-plane ℜs > 2k + 1, has an analytic continuation to the whole s-plane as a holomorphic function if F ∈ M1k is a cusp form, and as a meromorphic function with four simple poles at the points s = 0, k − 2, k, 2k − 2, if F is an Eisenstein series. In all cases it satisfies the functional equation (5.68).

Conclusion: Other Groups, Other Horizons

Although the theory of zeta functions of Siegel modular forms is far from being complete, and many fundamental questions (such as analogues of the Shimura– Taniyama conjecture for abelian varieties) have not even been touched, one must keep in mind that the symplectic group is but one example from a big family of arithmetically significant groups. The most natural situation, when it is possible to approach both analytic properties and factorization into an Euler product of arithmetic zeta functions, is the case in which zeta functions can be associated with representations of suitable arithmetic discrete subgroups of Lie groups on related function spaces of automorphic forms. The typical example of this kind is provided by “zeta functions of bilinear forms.” Let, for example, ⎛ ⎛ ⎞ ⎛ ⎞⎞ x1 y1 ⎜ ⎜ .. ⎟ ⎜ .. ⎟⎟ t q = q(X, Y ) = ∑ qi j xi y j = XQY ⎝X = ⎝ . ⎠ , Y = ⎝ . ⎠⎠ i, j=1,...,m

xm

ym

be a bilinear form of order m with integral nonsingular matrix Q = (qi j ). With the form q we associate the automorph semigroup of q, A(q) = D ∈ Zm m q(DX, DY ) = µ q(X, Y ) with µ = µ (D) > 0 , and the group of units of q, E(q) = {D ∈ A(q)| µ (D) = 1} . The pair (E(q), A(q)) is quite often left-finite in the sense of Section 3.2, and we may define the Hecke–Shimura ring D = D(E(q), A(q)) of the pair (over Z), which is also called the automorph class ring of q and is denoted by H(q). Generally speaking, one has little to say about the ring H(q), and it should be replaced by the more complicated construction of a matrix Hecke–Shimura ring. Each bilinear form

169

170

Conclusion: Other Groups, Other Horizons

is the sum of uniquely defined symmetric and skew-symmetric forms, i.e., the forms q satisfying q(X, Y ) = q(Y, X) or q(X, Y ) = −q(Y, X), respectively, and corresponding groups and semigroups of automorphs are intersections of those for the components. In the case of either a pure symmetric or skewsymmetric form q, each double coset of the semigroup A(q) modulo the group E(q) is a finite union of left cosets. In such a case, a theory can be developed that would resemble the Hecke–Shimura theory outlined above for the most interesting skewsymmetric case of the bilinear form with matrix Q = Jn of order 2n defined in (0.1) (with the integral symplectic group Γn = Spn (Z) as the group of units and Siegel modular forms as representation spaces for symplectic Hecke–Shimura rings). In the symmetric case, when the form q is symmetric and the group of units E(q) coincides with the proper integral orthogonal group of the quadratic form q(X, X), the theory of orthogonal matrix Hecke–Shimura rings is similar in many respects to the theory of symplectic rings, including the formal Euler factorization of generating Dirichlet series. In the orthogonal case, Hecke–Shimura rings operate on spaces of harmonic polynomials related to basic quadratic forms, and one can define zeta functions corresponding to these representations. It was revealed recently that in the case of positive definite quadratic forms in two and four variables, the relevant (orthogonal) zeta functions coincide with zeta functions corresponding to representations of symplectic Hecke–Shimura rings on spaces of theta series of genus 1 and 2 with harmonic coefficients, respectively. The coincidence is even more striking because Hecke–Shimura rings of quite different groups of units are involved: finite orthogonal groups and infinite symplectic groups. Other examples of relations between zeta functions of different arithmetic groups are provided by various lifting of automorphic forms and related zeta functions to similar groups of higher orders such as Saito–Kurokawa and Ikeda lifts, and various liftings of zeta functions of the general linear group to their symmetric degrees. There is no doubt that further progress in number theory will be closely connected with the investigation of relations among zeta functions of representations of Hecke–Shimura rings of various arithmetic discrete subgroups of Lie groups on automorphic functions.

Notes

Introduction For the Riemann zeta function, see, e.g., [Ti86]. On zeta functions of algebraic varieties one can read [Shi71, Chapter 7]. For the Birch–Swinnerton-Dyer conjectures see [BSD63/65]. For modular forms in one variable and corresponding Dirichlet series see [Ogg69]. The Hecke theory of the Euler product factorization of Dirichlet series of modular forms is stated in [He37]. The original Atkin–Lehner theory was set forth in [AtL70]. For the history of the Shimura–Taniyama conjecture see [Shi89]. For the proof of Fermat’s last theorem and related questions see [Wil95].

Chapter 1 The general references on this chapter, where one can find all omitted details and much more, are [An87, Chapters 1 and 2] and [AnZ90/95]. The later book also treats modular forms of half-integer weight. For a classical introduction to Siegel modular forms and Dirichlet series see [Kl90]. More details on the theory of Siegel modular forms and functions can be found in [Fr83], [Ma55], and [Si39]. For the treatment of automorphic forms on Lie groups see [SC58]. For modular forms in one variable see the books [Ogg69] and [Ma64]; for elementary presentations see [Ser70] and [Ga62]; relations to algebraic geometry are considered in [Fr83] and [Shi71]. Relations of modular forms to integral quadratic forms are considered in [Ki86] (numbers of representations of quadratic forms by quadratic forms) and in [An87] (multiplicative properties of the representations). Section 1.2. For details on the reduction theory of positive definite quadratic forms see, e.g., [Ca78, Chapter 12]. For the construction of the fundamental domain of the modular group see also [Si73].

171

172

Notes

Section 1.3. The notation (1.29) is a variation of the notation introduced by H. Petersson. The Koecher effect was discovered in [Ko54/55]. For the rather complicated proof of the estimate (1.52) see, e.g., [An87, Theorem 2.3.4]. The scalar product of modular forms in one variable was introduced by H. Petersson in [Pe39/41]; the consideration of the general case is based on a similar idea and was initiated by H. Maass in [Ma51].

Chapter 2 Section 2.1. Radial Dirichlet series (2.10) are not the only kind of Dirichlet series constructed with the help of Fourier coefficients of common eigenfunctions for regular Hecke operators that have an Euler product factorization and good analytic properties. Another kind of such Dirichlet series is given by series of the form

∑

M∈SLn (Z)\{M∈Znn | det M>0}

ψ (det M) f (MA tM) , (det M)s

where n ≥ 1, f are Fourier coefficients of a common eigenfunction for regular Hecke operators on the space Mk (Γn0 (q), χ ) with integral or half-integral k, ψ is a Dirichlet character, and A is a fixed even positive definite matrix of order n. For n = 1 these series were considered in [Ra39], [Se40], and [Shi75]; Euler product factorizations for an arbitrary n and integral k were discussed in [An87, §4.3.3], whereas both integral and half-integral k were considered in [AnZ90/95, §3.3]; analytic properties of corresponding Euler products (the standard zeta functions of eigenforms F) of level q = 1 with ψ = 1 were considered in [AnK78] with certain restrictions, and in [Bo85] without restrictions. Section 2.3. Our exposition of the theory of radial Dirichlet series corresponding to the ray mA1 = 2m · 12 (m = 1, 2, . . . ) of the matrix of a quadratic form x12 + x22 was outlined first in [An71, §3]; the general case of radial series corresponding to rays of matrices of arbitrary positive definite integral primitive binary quadratic forms was considered in detail in [An74, §§3.3–3.7].

On Chapter 3 Section 3.1. The multiplicative properties of the Fourier coefficients of the modular form ∆ (z) ∈ N12 (Γ ) cited in Exercise 3.1(3) were observed (!) and conjectured by Ramanujan and a little later proved by Mordell in [Mo17]. Mordell’s proof actually contains the idea of Hecke operators for the particular case of the space N12 (Γ ), an idea that had to wait twenty more years to be reborn in the case of more general spaces of modular forms by Hecke in [He37]. Section 3.2. Our definition of an abstract ring of double cosets by means of multiplication of right-invariant linear combinations of left cosets is equivalent to the

Notes

173

definition given by Shimura in [Shi63] with the help of the direct multiplication of double cosets, but in some respects it turns out to be more convenient. Section 3.3. The rings of double cosets of the general linear group are discussed here and in [An87, §3.2] in the spirit of the fundamental paper [Ta63] of Tamagawa. For the explicit formulas for spherical functions on GLn see [An70]. Section 3.4. The symplectic case is described as in [An87, §3.3]. The structure of Hecke–Shimura rings for Γn was indicated first in [Sa63] and [Shi63]. I cannot judge who was the first. The passage to congruence subgroups goes back to Hecke’s ideas in [He37]. Our presentation uses common sense and an analogy with the case of the general linear group rather than any historical reminiscences. For the theory of singular Hecke–Shimura rings see [An99]. The theory of spherical mappings in the form of a general theory of zonal spherical functions on reductive algebraic groups over p-adic fields is due to Satake; see [Sa63]. We use an elementary approach based on explicit formulas. The series (3.107) and (3.109) were summed up in [Shi63]; similar series for arbitrary n were computed in [An69] and [An70]. Section 3.5. The embedding of symplectic Hecke–Shimura rings into rings of triangular-symplectic double cosets allows one to split their elements into elementary components naturally related to the general linear group. This gives impetus to the theory of factorization of standard symplectic polynomials in triangular extensions, presented in detail in [An87, §§3.4–3.5]. Here we use only the elementary aspects of the theory.

Chapter 4 General references are [An87, §§4.1–4.2] or [AnZ90/95]. Section 4.1. Hecke operators for the group Γ1 = SL2 (Z) and some congruence subgroups were introduced in [He37]. Hecke operators on Siegel modular forms were first introduced by Sugawara in [Su37] and [Su38] and after the war were studied by Maass in [Ma51]. The existence of a basis of common eigenfunctions for Hecke operators acting on cusp forms in one variable was proved by Petersson in [Pe39/41]. For Siegel modular forms, Petersson’s ideas were developed by Maass in [Ma51]. For diagonalization of singular Hecke operators see [AtL70] in the case of genus n = 1, and [An99], [An03] in the general case. Section 4.3. Zharkovskaya commutation relations for the group Γn and the unit character are due to Natasha Zharkovskaya [Zha74], who generalized the Maass relation for Γ2 obtained in [Ma51]. Our consideration follows the same idea.

Chapter 5 Section 5.1. Relations of Fourier coefficients of eigenforms with eigenvalues can be found in [He37].

174

Notes

Section 5.2. For more details on the relations of binary quadratic forms and modules in quadratic number fields see, e.g., [An87, Appendix 3], or the excellent book on algebraic numbers [BSh63, Chapter 2]. Section 5.3. It is proved in [An74, Theorem 3.1.1] without any restrictions that the zeta function ζ (s, F) of an arbitrary eigenfunction for all Hecke operators F ∈ M2k of integral weight k ≥ 0 can be continued to the whole s-plane as a meromorphic function; the function Ψ(s, F) = (2π )−2s Γ(s)Γ(s − k + 2)ζ (s, F), where Γ is the gamma function, is meromorphic on the s-plane with the only possible poles at s = 0, k − 2, k, 2k − 2 and satisfies the functional equation Ψ(2k − 2 − s, F) = (−1)k Ψ(s, F). This does not contradict the functional equation (5.67), because the assumptions of Theorem 5.35 imply that k there must be even. The relation between the eigenvalues of Hecke operators and Fourier coefficients of eigenfunctions for Siegel modular forms of genus n discovered by Zharkovskaya in [Zha75] can possibly provide the first step to approach the still open problem of analytic properties of similar zeta functions of cusp form for genera n > 2.

Conclusion On relations of zeta functions of orthogonal and symplectic groups see [An06]. For a general outlook on problems of Euler products attached to automorphic forms see Langlands’ lectures [La67] and [La69].

References

[An69]

A.N. Andrianov, Rationality theorems for Hecke series and zeta-functions of the groups GLn and Spn over local fields, Izv. Akad. Nauk Ser. Mat. 33 (1969), 466– 505 (Russian); English transl., Math. USSR Izv. 3 (1969), 439–476. [An70] A.N. Andrianov, Spherical functions on GLn over local fields and summation of Hecke series, Mat. Sbornik Nov. Ser. 83 (1970), no. 3, 429–451 (Russian); English transl., Math. USSR Sb. 12 (197), no. 3, 429–452. [An71] A.N. Andrianov, Dirichlet series with Euler product in the theory of Siegel modular forms of genus 2, Trudy Mat. Inst. Steklov 112 (1971), 73–94 (Russian); English transl., Proc. Steklov Inst. Math. 112 (1971), 70–93. [An74] A.N. Andrianov, Euler products corresponding to Siegel modular forms of genus 2, Uspekhi Mat. Nauk 29 (1974), no. 3, 43–110 (Russian); English transl., Russian Math. Surveys 29 (1974), no. 3, 45–116. [An87] A.N. Andrianov, Quadratic forms and Hecke operators, Grundlehren Math. Wiss. 286, Springer-Verlag, Berlin Heidelberg New York London Paris Tokyo, 1987. [An99] A.N. Andrianov, Singular Hecke–Shimura rings and Hecke operators on Siegel modular forms, Algebra i Analis 11 (1999), no. 6, 1–68; English transl., St. Petersburg Math. J. 11 (2000), no. 6, 931–987. [An03] A. Andrianov, On diagonalization of singular Frobenius operators on Siegel modular forms, Amer. J. Math. 125 (2003), 139–165. [An06] A.N. Andrianov, Zeta functions of orthogonal groups of integral positive definite quadratic forms, Uspekhi Mat. Nauk 61 (2006), no. 6, 3–44 (Russian); English transl., Russian Math. Surveys 61 (2006), no. 6. [AnK78] A.N. Andrianov and V.L. Kalinin, On the analytical properties of standard zeta functions of Siegel modular forms, Mat. Sbornik 106(148) (1978), no. 3, 323–339 (Russian); English transl., Math. USSR Sbornik 35, no. 1, 1–17. [AnZ90/95] A.N. Andrianov and V.G. Zhuravlev, Modular forms and Hecke operators, “Nauka”, Moscow, 1990 (Russian); English transl., Transl. of Math. Monographs, vol. 145, AMS, Providence Rhode Island, 1995. [AtL70] A.O.L. Atkin and J. Lehner, Hecke operators on Γ0 (m) Math. Ann. 185 (1970), 134–160. [BSD63/65] B.J. Birch and H.P.F. Swinnerton-Dyer, Notes on elliptic curves. I,II, J. Reine Angew. Math. 212 (1963), 7–25; 218 (1965), 79–108. ¨ [Bo85] S. B¨ocherer, Uber die Funktionalgleichung automorpher L-Funktionen zur Siegelschen Modulgruppe, J. Reine Angew. Math. 362 (1985), 146–168. [BSh63] Z.I. Borevich and I.R. Shafarevich, Number Theory, “Nauka”, Moscow, 1963, 1971, (1985) (Russian); English transl., Academic Press, New York, 1966. [Fr83] E. Freitag, Siegelsche Modulfunktionen, Grundlehren Math. Wiss. 254, SpringerVerlag, Berlin Heidelberg New York London Paris Tokyo, 1983.

175

176 [Ga62] [He37]

[Ki86]

[Kl90] [Ko54/55] [La67] [La69] [Ma51] [Ma55] [Ma64] [Mo17] [Ogg69] [Pe39/41]

[Ra39]

[Se40] [SC58] [Ser70] [Shi63] [Shi75] [Shi71]

[Shi89] [Si39] [Si73] [Ta63] [Tit86]

References R.C. Gunning, Lectures on modular forms, Annals of Math. Studies 48, Princeton University Press, Princeton, New Jersey, 1962. ¨ E. Hecke, Uber Modulfunktionen und die Dirichletschen Reihen mit Eulerscher Produktentwicklung. I, II, Math. Ann. 114 (1937), 1–28; 316–351; Math. Werke, Vandenhoeck & Ruprecht, G¨ottingen, 1959, 1970, pp. 644–707. Y. Kitaoka, Lectures on Siegel modular forms and representations by quadratic forms, Published for the Tata Institute of Fundamental Research, Springer-Verlag, Berlin Heidelberg New York Tokyo, 1986. H. Klingen, Introductory lectures on Siegel modular forms, Series: Cambridge Studies in Advanced Mathematics (No 20), Cambridge University Press, 1990. M. Koecher, Zur Theorie der Modulformen n-ten Grades. I, II, Math. Z. 59 (1954), no. 4, pp. 399–416; 61 (1955), no. 4, 455–466. R.P. Langlands, Euler products, Lecture Notes, Yale University, New Haven, Connecticut, 1967. R.P. Langlands, Problems in the theory of automorphic forms, Lecture Notes, Yale University, New Haven, Connecticut, 1969. H. Maass, Die Primzahlen in der Theorie der Siegelschen Modulfunktionen, Math. Ann. 124 (1951), 87–122. H. Maass, Lectures on Siegel’s modular functions, Tata Inst. Fund. Research, Bombay, 1955. H. Maass, Lectures on modular functions of one complex variable, Revised 1983, Tata Inst. Fund. Research, Bombay, 1964. L.J. Mordell, On Mr Ramanujan’s empirical expansions of modular functions, Proc. Cambridge Phil. Soc. 19 (1917), 117–124. A.P. Ogg, Modular forms and Dirichlet series, W.A. Benjamin, Inc., New York, Amsterdam, 1969. H. Petersson, Konstruktion der s¨amtlichen L¨osungen einer Riemannschen Funktionalgleichung durch Dirichletreihen mit Eulerscher Produktentwicklung. I, II, III, Math. Ann. 116 (1939), 401–412; 117 (1939), 39–64; 117 (1940/41), 277–300. R.A. Rankin, Contributions to the theory of Ramanujan’s function τ (n) and similar arithmetical functions. II, The order of the Fourier coefficients of the integral modular forms, Proc. Cambridge Phil. Soc. 35 (1939), no. 3, 357–372. A. Selberg, Bemerkungen u¨ ber eine Dirichletsche Reihe, die mit der Theorie der Modulformen nahe verbunden ist, Arch. Math. Naturvid. 43 (1940), 47–50. S`eminaire H. Cartan, Fonctions automorphes, 10e ann´ee, vol. 1, 2, Secr`etariat Math´ematique, Paris, 1958. J.-P. Serre, Cours d’arithm´etique, Presses Universitaires de France, Paris, 1970. G. Shimura, On modular correspondences for Sp(n, Z) and their congruence relations, Proc. Nat. Acad. Sci. USA 49 (1963), no. 6, 824–828. G. Shimura, On the holomorphy of certain Dirichlet series, Proc. London Math. Soc. 31 (1975), no. 1, 79–98. G. Shimura, Introduction to the arithmetic theory of automorphic functions, Publ. Math. Soc. Japon, vol. 11, Iwanami Shoten, Publishers and Princeton University Press, Princeton, 1971. G. Shimura, Yutaka Taniyama and his time, Bull. London Math. Soc. 21 (1989), 186–196. C.L. Siegel, Einf¨uhrung in die Theorie der Modulfunktionen n-ten Grades, Math. Ann. 116 (1939), 617-657; Gesam. Abhandlungen vol. II, pp. 97–137. C.L. Siegel, Topics in complex function theory, Interscience Tracts in Pure and Applied Math. 25 vol. III, Wiley-Interscience, New York London Sydney Toronto, 1973. T. Tamagawa, On the ζ -functions of a division algebra Ann. Math. II Ser. 77 (1963), no. 2, 387–405. E. C. Titchmarsh, The theory of the Riemann zeta-function, 2nd ed., Clarendon press, Oxford, 1986.

References [Wil95] [Zha74] [Zha75]

177 A. Wiles, Modular elliptic curves and Fermat’s Last Theorem, Ann. Math. II Ser. 141 (1995), no. 3, 443–551. N.A. Zharkovskaya, The Siegel operator and Hecke operators, Funkts. Anal. Prilozh. 8 (1974), no. 12, 30–38; English transl., Funct. Anal. Appl. 8 (74), 113–120. N.A. Zharkovskaya, On the connection between the eigenvalues of Hecke operators and Fourier coefficients of eigenfunctions for Siegel’s modular forms of genus n, Mat. Sb. Nov. Ser. vol 96(138) (1975) 584–593; English transl., Mat. USSR Sb. 25 (1975).

Index

A Analytic automorphisms domain H, of, 48 domain H(A), of, 53 Lie group acting as group of, 8 transitive group of, 47 Antiautomorphisms application of, 95 extension of, 72–73 second order, of, 72, 108 Associative ring, 65–67 Automorph semigroup, 169 Automorphy, factor of, 50 B Binet–Cauchy formula, 76 Block matrices, multiplication of, 13 C Cauchy’s theorem, 60 Cayley mapping, 11 Commensurability, relation of, 70 Commutative ring, 67 Complex number, nonzero, 7, 20, 85, 105, 120, 137 Congruence character, 21, 22, 24, 28–30, 35, 120, 126 subgroup, 3, 5, 20–22, 25, 28–30, 32, 92 Coprime integers, 92 Coprime numbers, 78 Cusp forms diagonalization of, 134–135 invariance of subspaces of, 120–121, 135 multiplicative relations of eigenvalues, 139 regular zeta function of, 166

D dc-rings. See Double cosets ring Dedekind zeta function, 59, 61 of the ring, 167–168 Denominators, factorization of, 115–117 Diagonalization theorem, 124–125 Diagonal matrices, 77, 88 Dirichlet character modulo, 41, 45, 125, 127, 131, 133 Dirichlet series, 96, 137, 172 cusp form, of, 143 Euler factorization of, 140–142 of F, 3 functional equation, 143–145 of Z(F, s), 4 Discrete transformation, 12 Double cosets modulo, 68, 82, 95, 98, 121, 124, 126 abstract ring of, 172–173 Double cosets ring abstract rings basic rings, 67 Hecke–Shimura rings, definition of, 66 standard bases and multiplication rules, 67–69 decomposition of, 80 general linear group of, 74 left and right rings of, 69–72 multiplication in, 77 triangular-symplectic, ring of factorization of denominators, 115–117 Frobenius elements, 107 local triangular rings, relations in, 111 spherical mappings, 112–113 triangular embeddings, 108 triangular rings, 106

179

180 E Eigenfunctions, 124–125 for all operators, 135 certain rings of linear operators, of, 4 and eigenvalues, 138 Fourier coefficients of, 137, 138, 146, 148, 174 for operator T (p), 64 of regular Hecke operators, 123, 135, 138, 141 zeta function of, 141, 143 Eigenvalues, 44 Hecke operators, of, 136, 139 matrix, of, 14 of nonzero modular form, 164 properties of, 138 Eisenstein series, 20, 34, 39, 47 for discrete subgroup, 55 and Theta series, 58–59 Elementary divisors, 76 Elementary integral operations, 76 Elementary p-divisors, 80 Elementary symmetric polynomials, 83, 104 Embeddings, extension of, 73–75 Euclidean algorithm, 13, 30, 57, 76, 79 Euclidean element, 9, 10 Euler factorization, 96, 144, 160 conditions, 164 Dirichlet series, of, 140–141 problem of, 65 radial series, of, 44 Euler formula, 42 Euler integral, 44, 59 Euler product, 2 factorization of, 4, 41 F Field Q of rational numbers, 2 Finite union left/right cosets modulo, 66 Formal power series, 102, 103 Fourier coefficients, 26, 27, 33, 35 Dirichlet series, 65 eigenfunctions and eigenvalues of modular forms, of, 138–139 f (m), 3 function, of, 64 Hecke operators, of, 126–129 modular forms, of, 34, 63, 65 multiplicative properties of, 161–164, 172 Fourier expansion, 25, 28, 30, 35, 45 convergent, 36 Fourier series, 28, 60 absolutely convergent on H, 3 Siegel operator action on, 130–134

Index Frobenius elements, 107–111 Full modular group global rings for, 86 local rings for, 90 Fundamental domains, construction of, 15 G Gamma function, 42, 61 Gaussian composition, 153 Gaussian integers, 53, 57, 59 G-invariant element, of volume on H, 9 G-invariant volume element, 52 Global rings, 75 Global zeta function, 2 H Haar measure, 43 Hadamard’s determinant theorem, 27 Hecke operators, 64 on abstract level, 119–120 acting on spaces, 4 action on modular forms for groups Fourier coefficients, 126–129 Hecke–Shimura ring, 125–126 for congruence subgroups, 120–121 diagonalization of cusp forms, 134–135 eigenfunctions, 124–125, 135–136 Petersson scalar product, 123–124 eigenvalues of, 139 invariant subspaces under Fourier coefficients and eigenvalues, 161–162 homomorphism, 121–122 problem for, 134 and Siegel operator (see Siegel operator) summation formula, 159–160 Hecke–Shimura rings, 65, 91–92, 127 definition of, 66 of double cosets, 66 homomorphism, 130–131 of l-finite pair, 67 linear representation of, 120 local subrings, 79 mapping, 133–134 q-symmetric group, of, 96 symplectic, 137, 170 triangular subgroup, 106 Hermitian matrices, 11 Hilbert space, 35 Holomorphic functions, 19 Homomorphic embedding, of group, 29 Homomorphism, 121, 133

Index I ω -images, 83 Ω-images, 100, 102 Imprimitive elements, 80, 91 Inductive hypothesis, 79, 82, 87 Integral symmetric matrix, 17, 24, 109 Integral weight cusp form of, 31, 61 Dirichlet series of modular forms of, 3 modular form of, 63 Petersson operators of, 19 Invariant polynomials, 101 Invariant subspace cusp forms, of, 135–136 under Hecke operators, 121–123 problem for, 134 Inverse isomorphisms, ring, 93 Inverse mapping, 11, 44–45 Isomorphism, of subgroup, 53 J Jacobi’s formula, 63 K K-orbit, 23 L Left coset, 64, 68 decompositions in, 77 Frobenius elements, 108 reduced representative, 79 representative, 78 Lie groups, 8 isomorphism of, 50 Linear map, 34 Linear mapping, 80, 91, 93 Lobachevsky half-spaces, transformations of, 47–48 Lobachevsky space, 49 Local Hecke–Shimura rings, 82 Local subring, 97 integral linear combinations, 80 prime numbers, 79, 90 Local triangular rings, 111–115 Local zeta functions ζ (Vp , p−s ), 2 M Mapping ring, isomorphism of, 83 Mellin integral, 3, 42, 45 Mellin transforms classical theory of, 44 of cusp forms, 44 Meromorphic function, 47, 61 Minkowski inequality, 14

181 Minkowski reduction domain, 12–13 Modular cusp form of weight k, 3 Modular forms definition of, 19–20 Fourier coefficients of, 34 Fourier expansion of, 22–25, 138, 145–148 invariant subspaces of (see Invariant subspace) multiplicative properties of, 137 nonzero (see Nonzero modular form) regular zeta function of, 141 spaces of, 30–34 zeta functions of, 41 Modular group features of, 85 global rings for, 86 Hecke–Shimura rings of, 91 local rings for, 90 q-symmetric subgroup, 95–96 subgroup K of, 92 Multiplication rules, 88 standard bases, 67–69 Multiplicative function, 63 Multiplicative homomorphism, 20 Multiplicative properties, 63 of Fourier coefficients, 161–164, 172 N Nonnegative coefficients, linear combination of, 78 Nonzero cusp forms, 35, 142 regular zeta function of, 165–166 Nonzero integral, 76 Nonzero modular form, 145 Fourier coefficients and eigenvalues of, 147, 161–164 multiplicativity of, 163–164 regular zeta function of, 165–166 Nonzero rational numbers, multiplicative group of, 70 O One-dimensional subspaces, 122 Π-Operators, 149 summation conditions in, 153 P Permutations, 99 Permutes factors, 104 Petersson operators, 19, 35, 122 of integral weights k, 19 mapping of holomorphic functions, 19 Petersson scalar product, 35, 37, 123–125 p-factors, 5 p-local subring, 90

182 Polynomial congruences modulo, 92 Polynomials, 82, 100 Q-linear homomorphism, ring of, 83 Primitive elements, 91 Principal congruence subgroup, 75, 92 Q q-integral symplectic matrix, 94 q-symmetric group, 93 local subrings, 97 Quadratic field. See Quadratic modules Quadratic modules definitions, 149 multiplication of, 150 prime and primitive in terms of, 154–157 and prime numbers, relation between, 151–153 R Radial Dirichlet series, 141, 172 cusp forms, of, 41–43, 143 and Eisenstein Series, 55–57 integral representation of, 43–44 method for obtaining, 47 properties for sums of two squares, 58 Ramanujan’s formula, 63 Rational fractions, transformations of, 99 Real matrix, 50 Riemann zeta function, 1, 34, 47 Ring epimorphism, 80 Ring homomorphism, 80 Ring monomorphism, 74 Ring of multipliers of the module, 149 Rings isomorphism, 74 S Semidefinite matrices, 45 Shimura–Taniyama conjecture, 5 Siegel modular forms, 5 Siegel modular group, 85 fundamental domains for, 11 of weight k and character χ , 20 Siegel operator, 28, 34 action on Fourier series, 130–134 Hecke operators, relations between, 67 for mapping of space, 29 Spherical mapping, 82–85, 98–99 Q-linear isomorphism, 99–102 Spinor p-polynomial, 105 Star mapping transforms, local subrings for, 108 Subspaces of modular forms Fourier coefficients of functions of, 158 invariance of, 120–121

Index Summation formulas, 103 Symmetric polynomials, 82 Symplectic case, zero divisors, 107 Symplectic divisors, 87, 95 Symplectic group, 65 dc-rings of, 75 commutative, 88 congruence subgroups, regular Hecke– Shimura rings for, 91–92 decompositions, 89 formal power series, 102, 103 full modular group, global rings for, 86 full modular group, local rings for, 90 spherical mapping, 98–99 zero-blocks, 87 Symplectic matrices, 7 of order 2n with positive multipliers, 8 Symplectic modular group, 5 T Theta series, 58 Transformations, 99, 104 Triangle, modular, 18 Triangular embeddings, 108 Triangular matrix, 10, 45, 81 Triangular-plus matrices, 83 Triangular-symplectic matrices, 106 U Unitary group, of order n, 11 Unit matrix, of order n, 7 W W -invariant polynomial, 101 Z Zero matrix, 89 Zeta functions algebraic varieties, of, 2–3 automorphic forms, of, 3–5 bilinear forms, of, 169–170 of E, 3 eigenforms, of nonzero cusp form, 165–166 nonzero modular form, 165 E p over F (p), 2 modular forms, of, 137 functional equation of, 143–145 nonzero eigenform, 145 in terms of Dirichlet series, 140–141 and symplectic Hecke–Shimura rings, 170 Zharkovskaya commutation relations, 132–134 Z-linear mapping, 95

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