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Spectral Methods of Automorphic Forms SECOND EDITION Spectral Methods of Automorphic Forms SECOND EDITION Henryk Iwaniec Graduate Studies in Mathematics Volume 53 American Mathematical Society Revista Matemática Iberoamericana Spectral Methods of Automorphic Forms Spectral Methods of Automorphic Forms Second Edition Henryk Iwaniec Graduate Studies in Mathematics Volume 53 American Mathematical Society Providence, Rhode Island Revista Matemática Iberoamericana Editorial Board Walter Craig Nikolai Ivanov Steven G. Krantz David Saltman (Chair) 2000 Mathematics Subject Classification. Primary 11F12, 11F30, 11F72. Library of Congress Cataloging-in-Publication Data Iwaniec, Henryk. Spectral methods of automorphic forms / Henryk Iwaniec.—2nd ed. p. cm. — (Graduate studies in mathematics, ISSN 1065-7339 ; v. 53) First ed. published in Revista matem´atica iberoamericana in 1995. Includes bibliographical references and index. ISBN 0-8218-3160-7 (acid-free paper) 1. Automorphic functions. 2. Automorphic forms. 3. Spectral theory (Mathematics) I. Title. II. Series. QA353.A9 I88 2002 511.33—dc21 2002027749 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294, USA. Requests can also be made by e-mail to [email protected]. c 2002 by Henryk Iwaniec. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America. ∞ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at http://www.ams.org/ 10 9 8 7 6 5 4 3 2 17 16 15 14 13 12 Contents Preface ix Preface to the AMS Edition xi 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 15 1.4. Group decompositions 16 1.5. The classification of motions 17 1.6. The Laplace operator 19 1.7. Eigenfunctions of Δ 19 1.8. The invariant integral operators 26 1.9. The Green function on H 32 Chapter 2. Fuchsian Groups 37 2.1. Definitions 37 2.2. Fundamental domains 38 2.3. Basic examples 41 2.4. The double coset decomposition 45 v vi Contents 2.5. Kloosterman sums 47 2.6. Basic estimates 49 Chapter 3. Automorphic Forms 53 3.1. Introduction 53 3.2. The Eisenstein series 56 3.3. Cusp forms 57 3.4. Fourier expansion of the Eisenstein series 59 Chapter 4. The Spectral Theorem. Discrete Part 63 4.1. The automorphic Laplacian 63 4.2. Invariant integral operators on C(Γ\H)64 4.3. Spectral resolution of Δ in C(Γ\H)68 Chapter 5. The Automorphic Green Function 71 5.1. Introduction 71 5.2. The Fourier expansion 72 5.3. An estimate for the automorphic Green function 75 5.4. Evaluation of some integrals 76 Chapter 6. Analytic Continuation of the Eisenstein Series 81 6.1. The Fredholm equation for the Eisenstein series 81 6.2. The analytic continuation of Ea(z,s)84 6.3. The functional equations 86 6.4. Poles and residues of the Eisenstein series 88 Chapter 7. The Spectral Theorem. Continuous Part 95 7.1. The Eisenstein transform 96 7.2. Bessel’s inequality 98 7.3. Spectral decomposition of E(Γ\H) 101 7.4. Spectral expansion of automorphic kernels 104 Chapter 8. Estimates for the Fourier Coefficients of Maass Forms 107 8.1. Introduction 107 8.2. The Rankin–Selberg L-function 109 8.3. Bounds for linear forms 110 8.4. Spectral mean-value estimates 113 Contents vii 8.5. The case of congruence groups 116 Chapter 9. Spectral Theory of Kloosterman Sums 121 9.1. Introduction 121 9.2. Analytic continuation of Zs(m, n) 122 9.3. Bruggeman–Kuznetsov formula 125 9.4. Kloosterman sums formula 128 9.5. Petersson’s formulas 131 Chapter 10. The Trace Formula 135 10.1. Introduction 135 10.2. Computing the spectral trace 139 10.3. Computing the trace for parabolic classes 142 10.4. Computing the trace for the identity motion 145 10.5. Computing the trace for hyperbolic classes 146 10.6. Computing the trace for elliptic classes 147 10.7. Trace formulas 150 10.8. The Selberg zeta-function 152 10.9. Asymptotic law for the length of closed geodesics 154 Chapter 11. The Distribution of Eigenvalues 157 11.1. Weyl’s law 157 11.2. The residual spectrum and the scattering matrix 162 11.3. Small eigenvalues 164 11.4. Density theorems 168 Chapter 12. Hyperbolic Lattice-Point Problems 171 Chapter 13. Spectral Bounds for Cusp Forms 177 13.1. Introduction 177 13.2. Standard bounds 178 13.3. Applying the Hecke operator 180 13.4. Constructing an amplifier 181 13.5. The ergodicity conjecture 183 Appendix A. Classical Analysis 185 A.1. Self-adjoint operators 185 viii Contents A.2. Matrix analysis 187 A.3. The Hilbert–Schmidt integral operators 189 A.4. The Fredholm integral equations 190 A.5. Green function of a differential equation 194 Appendix B. Special Functions 197 B.1. The gamma function 197 B.2. The hypergeometric functions 199 B.3. The Legendre functions 200 B.4. The Bessel functions 202 B.5. Inversion formulas 205 References 209 Subject Index 215 Notation Index 219 Preface I was captivated by a group of enthusiastic Spanish mathematicians whose desire for cultivating modern number theory I enjoyed recently during two memorable events, the first at the summer school in Santander, 1992, and the second while visiting the Universidad Aut´onoma in Madrid in June 1993. These notes are an expanded version of a series of eleven lectures I delivered in Madrid1. They are more than a survey of favorite topics, since proofs are given for all important results. However, 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 occasion). Primarily, the lectures are addressed to advanced graduate students. I hope the student will get inspiration for his own adventures in the field. This is a goal which Professor Antonio C´ordoba has a vision of pursuing throughout the new volumes to be published by the Revista Matem´atica 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´e Luis Fern´andez, Charles Mozzochi, Antonio S´anchez-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´ıa Victoria Meli´an, without which this book would not exist. New Brunswick, October 1994 Henryk Iwaniec 1I would like to thank the participants and the Mathematics Department for their warm hospitality and support. ix Preface to the AMS Edition Since the initial publication by the Revista Matem´atica Iberoamericana I have used this book in my graduate courses at Rutgers several times. I have heard from my colleagues that they also had used it in their teaching. On several occasions I have been told that the book would be a good source of ideas, results and references for graduate students, if it were easier to obtain. The American Mathematical Society suggested publishing an edition, which would increase its availability. Sergei Gelfand should be given special credit for his persistence during the past five years to convince me of the need for a second publication. I am also very indebted to the editors of the Revista for the original edition and its distribution and for the release of the copyright to me, so this revised edition could be realized. I eliminated all the misprints and mistakes which were kindly called to my attention by several people. I also changed some words and slightly expanded some arguments for better clarity. In a few places I mentioned recent results. I did not try to include all of the best achievements, since this had not been my intention in the first edition either. Such a task would require several volumes, and that could discourage beginners. I hope that a subject as great as the analytic theory of automorphic forms will eventually be presented more substantially, perhaps by several authors. Until that happens, I am recommending that newcomers read the recent survey articles and new books which I have listed at the end of the regular references. New Brunswick, June 2002 Henryk Iwaniec xi 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 e(z)=e2πiz. To define an automorphic function in an abstract setting, one needs a group Γ 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.
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