Introduction to the Spectral Theory of Automorphic Forms

246 Pages·1995·2.108 MB·English

by  Iwaniec H.

Introduction to the
Spectral Theory
of
Automorphic Forms
Henryk Iwaniec
Revista Matema(cid:19)tica Iberoamericana
1995
El proyecto que hace ya diez an~os puso en marcha la Revista
Matem(cid:19)atica Iberoamericana, inclu(cid:19)(cid:16)a la publicacio(cid:19)n espora(cid:19)dica de
monograf(cid:19)(cid:16)as sobre temas de gran inter(cid:19)es y actualidad en a(cid:19)reas cuya
actividad hiciera aconsejable una recapitulacio(cid:19)n llevada a cabo por uno
de sus artistas importantes.
Pretendemos que estas monograf(cid:19)(cid:16)as de la Biblioteca de la Re-
vista Matem(cid:19)atica Iberoamericana puedan servir de gu(cid:19)(cid:16)a a aquellos
que no siendo especialistas deseen explorar territorios de matema(cid:19)ticas
en parte consolidados, pero vivos y con mucho por descubrir y entender.
Es nuestro propo(cid:19)sito ofrecer verdaderas expediciones desde el confor-
table hogar de las matema(cid:19)ticas que todos compartimos hasta la terra
incognita en los con(cid:12)nes del (cid:12)rmamento lejano, donde las ideas estan
en continua ebullicio(cid:19)n.
Para iniciar esta andadura hemos tenido la gran fortuna de poder
contar con la presente monograf(cid:19)(cid:16)a que sobre formas modulares y su
teor(cid:19)(cid:16)a espectral ha escrito el profesor Henryk Iwaniec. Como directores
delaRevista Matem(cid:19)atica Iberoamericanaqueremosagradecerleel
entusiasmo que desde un principio mostro(cid:19) en este empen~o, y el cuidado
exquisito que ha puesto en su elaboracio(cid:19)n.
Ha sido una suerte poder contar con el magn(cid:19)(cid:16)(cid:12)co trabajo de com-
posicio(cid:19)n y maquetacio(cid:19)n de Domingo Pestana. Su dedicacio(cid:19)n y buen
hacer han sido una ayuda inestimable.
Antonio Co(cid:19)rdoba Jos(cid:19)e L. Ferna(cid:19)ndez
CONTENTS
Preface xiii
Introduction 1
Chapter 0
Harmonic analysis on the euclidean plane 3
Chapter 1
Harmonic analysis on the hyperbolic plane 7
1.1. The upper half plane 7
1.2. H as a homogeneous space 12
1.3. The geodesic polar coordinates 16
1.4. Bruhat decomposition 18
1.5. The classi(cid:12)cation of motions 18
1.6. The Laplace operator 20
1.7. Eigenfunctions of (cid:1) 21
1.8. The invariant integral operators 28
1.9. The Green function on H 35
Chapter 2
Fuchsian groups 39
2.1. De(cid:12)nitions 39
2.2. Fundamental domains 41
2.3. Basic examples 44
2.4. The double coset decomposition 48
2.5. Kloosterman sums 51
2.6. Basic estimates 53
IX
X Contents
Chapter 3
Automorphic forms 57
3.1. Introduction 57
3.2. The Eisenstein series 61
3.3. Cusp forms 63
3.4. Fourier expansion of the Eisenstein series 65
Chapter 4
The spectral theorem. Discrete part 69
4.1. The automorphic Laplacian 69
4.2. Invariant integral operators on C((cid:0)nH) 70
4.3. Spectral resolution of (cid:1) in C((cid:0)nH) 75
Chapter 5
The automorphic Green function 77
5.1. Introduction 77
5.2. The Fourier expansion 78
5.3. An estimate for the automorphic Green function 81
5.4. Evaluation of some integrals 83
Chapter 6
Analytic continuation of the Eisenstein series 87
6.1. The Fredholm equation for the Eisenstein series 87
6.2. The analytic continuation of Ea(z;s) 90
6.3. The functional equations 93
6.4. Poles and residues of the Eisenstein series 95
Chapter 7
The spectral theorem. Continuous part 103
7.1. The Eisenstein transform 104
7.2. Bessel’s inequality 107
7.3. Spectral decomposition of E((cid:0)nH) 110
7.4. Spectral expansion of automorphic kernels 113
Contents XI
Chapter 8
Estimates for the Fourier coe(cid:14)cients of Maass forms 117
8.1. Introduction 117
8.2. The Rankin-Selberg convolution 119
8.3. Bounds for linear forms 121
8.4. Spectral mean-value estimates 123
8.5. The case of congruence groups 126
Chapter 9
Spectral theory of Kloosterman sums 133
9.1. Introduction 133
9.2. Analytic continuation of Zs(m;n) 134
9.3. Bruggeman-Kuznetsov formula 138
9.4. Bruggeman-Kuznetsov formula reversed 141
9.5. Petersson’s formulas 144
Chapter 10
The trace formula 149
10.1. Introduction 149
10.2. Computing the spectral trace 154
10.3. Computing the trace for parabolic classes 157
10.4. Computing the trace for the identity motion 161
10.5. Computing the trace for hyperbolic classes 161
10.6. Computing the trace for elliptic classes 163
10.7. Trace formulas 166
10.8. The Selberg zeta-function 168
10.9. Asymptotic law for the length of closed geodesics 170
Chapter 11
The distribution of eigenvalues 173
11.1. Weyl’s law 173
11.2. The residual spectrum and the scattering matrix 179
11.3. Small eigenvalues 181
11.4. Density theorems 185
XII Contents
Chapter 12
Hyperbolic lattice-point problems 189
Chapter 13
Spectral bounds for cusp forms 195
13.1. Introduction 195
13.2. Standard bounds 196
13.3. Applying the Hecke operator 198
13.4. Constructing an ampli(cid:12)er 200
13.5. The unique ergodicity conjecture 202
Appendix A
Classical analysis 205
A.1. Self-adjoint operators 205
A.2. Matrix analysis 208
A.3. The Hilbert-Schmidt integral operators 209
A.4. The Fredholm integral operators 210
A.5. Green function of a di(cid:11)erential equation 215
Appendix B
Special functions 219
B.1. The gamma function 219
B.2. The hypergeometric functions 221
B.3. The Legendre functions 223
B.4. The Bessel functions 224
B.5. Inversion formulas 228
References 233
Subject Index 239
Notation Index 245
Preface
IwascaptivatedbyagroupofenthusiasticSpanishmathematicians
whose desire for cultivating modern number theory I enjoyed recently
during two memorable events, the (cid:12)rst at the summer school in San-
tander, 1992, and the second while visiting the Universidad Auto(cid:19)noma
in Madrid in June 1993. These notes are an expanded version of a series
(cid:3)
of eleven lectures I delivered in Madrid . They are more than a survey
of favorite topics since proofs are given for all important results. How-
ever, there is a lot of basic material which should have been included
for completeness, but was not because of time and space limitations.
Instead, to make a comprehensive exposition we focus on issues closely
related to the the spectral aspects of automorphic forms (as opposed
to the arithmetical aspects to which I intend to return on another oc-
casion).
Primarily, the lectures are addressed to advanced graduate stu-
dents. I hope the student will get inspiration for his own adventures in
the (cid:12)eld. This is a goal which Professor Antonio Co(cid:19)rdoba has a vision
of pursuing throughout the new volumes to be published by the Revista
Matema(cid:19)tica Iberoamericana. I am pleased to contribute to part of his
plan.
Many people helped me prepare these notes for publication. In
particular I am grateful to Fernando Chamizo, Jos(cid:19)e Luis Ferna(cid:19)ndez,
Charles Mozzochi, Antonio Sa(cid:19)nchez-Calle and Nigel Pitt for reading
and correcting an early draft. I also acknowledge the substantial work
in the technical preparation of this text by Domingo Pestana and Mar(cid:19)(cid:16)a
Victoria Melia(cid:19)n without which this project would not exist.
New Brunswick, October 1994 Henryk Iwaniec
(cid:3)
The author would like to thank the participants and the Mathematics Department
for their warm hospitality and support.
XIII
Introduction
The concept of an automorphic function is a natural generalization
of that of a periodic function. Furthermore an automorphic form is a
generalization of the exponential function
2(cid:25)iz
e(z) = e :
To de(cid:12)ne an automorphic function in an abstract setting one needs
a group (cid:0) acting discontinuously on a locally compact space X; the
functions on X which are invariant under the group action are called
automorphic functions (the name was given by F. Klein in 1890). A
typical case is the homogeneous space X = G=K of a Lie group G
where K is a closed subgroup. In this case the di(cid:11)erential calculus is
available since X is a riemannian manifold. The automorphic functions
which are eigenfunctions of all invariant di(cid:11)erential operators (these
include the Laplace operator) are called automorphic forms. The main
goal of harmonic analysis on the quotient space (cid:0)nX is to decompose
every automorphic function satisfying suitable growth conditions into
automorphic forms. In these lectures we shall present the basic theory
for Fuchsian groups acting on the hyperbolic plane.
When the group (cid:0) is arithmetic, there are interesting consequences
for number theory. What makes a group arithmetic is the existence of
a large family (commutative algebra) of certain invariant, self-adjoint
1
2 Introduction
operators, the Hecke operators. We shall get into this territory only
brie(cid:13)yinSections8.4and13.3todemonstrateitstremendouspotential.
Many important topics rest beyond the scope of these lectures; for
instance, the theory of automorphic L-functions is missed entirely.
A few traditional applications are included without straining for
the best results. For more recent applications the reader is advised to
see the original sources (see the surveys [Iw 1, 2] and the book [Sa 3]).
There is no dearth of books on spectral aspects of automorphic
functions, but none covers and treats in detail as much as the expansive
volumes by Dennis Hejhal [He1]. I recommend them to anyone who is
concerned with doing reliable research. In these books one also (cid:12)nds a
very comprehensive bibliography. Those who wish to learn about the
theory of automorphic forms on other symmetric spaces in addition to
the hyperbolic plane should read Audrey Terras [Te]. A broad survey
with emphasis on new developments is given by A. B. Venkov [Ve].
Chapter 0
Harmonic analysis
on the euclidean plane
We begin by presenting the familiar case of the euclidean plane
R2 = (x;y) : x;y R :
2
(cid:8) (cid:9)
The group G = R2 acts on itself as translations, and it makes R2 a
homogeneous space. The euclidean plane carries the metric
ds2 = dx2 +dy2
of curvature K = 0, and the Laplace-Beltrami operator associated with
this metric is given by
@2 @2
D = + :
@x2 @y2
Clearly the exponential functions
’(x;y) = e(ux+vy); (u;v) R2;
2
are eigenfunctions of D;
(D+(cid:21))’ = 0; (cid:21) = (cid:21)(’) = 4(cid:25)2(u2 +v2):
3