The spectrum of hyperbolic surfaces 9783319276649, 9783319276663, 9782759805648, 2759805646, 3319276646

Table of contents :
Preface.- Introduction.- Arithmetic Hyperbolic Surfaces.- Spectral Decomposition.- Maass Forms.- The Trace Formula.- Multiplicity of lambda1 and the Selberg Conjecture.- L-Functions and the Selberg Conjecture.- Jacquet-Langlands Correspondence.- Arithmetic Quantum Unique Ergodicity.- Appendices.- References.- Index of notation.- Index.- Index of names.

Citation preview

Universitext

Nicolas Bergeron

The Spectrum of Hyperbolic Surfaces Translated by Farrell Brumley

Universitext

Universitext Series Editors Sheldon Axler San Francisco State University Vincenzo Capasso Università degli Studi di Milano Carles Casacuberta Universitat de Barcelona Angus MacIntyre Queen Mary, University of London Kenneth Ribet University of California, Berkeley Claude Sabbah CNRS, École Polytechnique, Paris Endre Süli University of Oxford Wojbor A. Woyczy´nski Case Western Reserve University Cleveland, OH

Universitext is a series of textbooks that presents material from a wide variety of mathematical disciplines at master’s level and beyond. The books, often well classtested by their author, may have an informal, personal even experimental approach to their subject matter. Some of the most successful and established books in the series have evolved through several editions, always following the evolution of teaching curricula, to very polished texts. Thus as research topics trickle down into graduate-level teaching, first textbooks written for new, cutting-edge courses may make their way into Universitext. More information about this series at http://www.springer.com/series/223

Nicolas Bergeron

The Spectrum of Hyperbolic Surfaces

123

Nicolas Bergeron IMJ-PRG Universite Pierre et Marie Curie Paris France Translated by Farrell Brumley

Translation from the French language edition: Le spectre des surfaces hyperboliques by Nicolas Bergeron Copyright © 2011 EDP Sciences, CNRS Editions, France. http://www.edpsciences.org/ http://www.cnrseditions.fr/ ISSN 0172-5939 Universitext ISBN 978-3-319-27664-9 DOI 10.1007/978-3-319-27666-3

ISSN 2191-6675

(electronic)

ISBN 978-3-319-27666-3

(eBook)

Library of Congress Control Number: 2015960755 Mathematics Subject Classification: 11F72, 11M36, 35P15, 35P20, 35R01, 58J50, 58J51 Springer Cham Heidelberg New York Dordrecht London © Springer International Publishing Switzerland 2016 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. Printed on acid-free paper Springer International Publishing AG Switzerland is part of Springer Science+Business Media (www.springer.com)

Preface

The Laplacian, like few other objects, is nearly ubiquitous in mathematics. The aim of this book is to provide an introduction to the study of the spectrum of the Laplacian on hyperbolic surfaces. These are the Riemannian surfaces of constant negative curvature 1. In the more familiar context of the Euclidean circle the Laplacian is given simply by the second derivative, and the analysis of its spectrum is the subject of classical Fourier theory. Besides its intrinsic interest, Fourier analysis has long been one of the principal tools in analytic number theory. Indeed, one of the proofs of the analytic continuation of the Riemann zeta function is based on the Poisson summation formula. In the first chapter of this book, in order to get things rolling, we will recall this and other basic Fourier analytic facts. In 1956 the Norwegian mathematician Atle Selberg proposed a vast generalization of the Poisson summation formula, now referred to as the Selberg trace formula. This formula, which we prove in Chap. 5, is reminiscent of the so-called explicit formulae in the analytic theory of L-functions. In the powerful analogy that the Selberg trace formula evokes, the closed geodesics on a compact hyperbolic surface S play the role of prime numbers. The trace formula establishes a relation between these closed geodesics and the Laplacian spectrum of S. These results were, upon their publication, immediately recognized as a new perspective on the Riemann hypothesis. While the latter remains an enigma, the trace formula has, since the late 1960s, played an increasingly central role in the ambitious program of Robert Langlands. The latter aims to link number theory with harmonic analysis on locally symmetric spaces, chief among which are the hyperbolic surfaces. There are several textbooks already dedicated to the extension of the classical Fourier theory to hyperbolic surfaces; we mention in particular the now classic text of Iwaniec [63] (geared towards applications to analytic number theory) as well as the book by Buser [24], which has a more geometric outlook. In French, the closest

v

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Preface

work to the present one is probably the book by Kowalski [71] where a large part of the basic theory is presented.1

Why Another Book? For well over 30 years now the Langlands program has made enormous advances, and spectral theory and number theory alike have harvested its fruits. But it has become difficult to penetrate such a larger and larger mathematical landscape. We wanted to write a book on the Langlands program in a more classical, and hopefully more accessible, language. This inclination has naturally led us to explore in more detail compact arithmetic hyperbolic surfaces. In particular, we present a proof of the first striking result in the Langlands program: the Jacquet-Langlands correspondence. The proof we give – due to Bolte and Johansson – should be more readily comprehensible to a reader unfamiliar with the language of adeles. With any luck, this expository simplification will encourage the reader to dive headlong into the original work of Jacquet and Langlands. An added motivation for an updating of the literature was given by three other recent results: the lower bound on the Laplacian eigenvalues of arithmetic hyperbolic surfaces by the method Luo-Rudnick-Sarnak, the lovely proof of the existence of cusp forms for congruence subgroups of SL.2; Z/ due to Lindenstrauss and Venkatesh, and finally Lindenstrauss’ proof, by purely ergodic theoretic arguments, of the arithmetic quantum unique ergodicity conjecture of Rudnick and Sarnak in the compact setting. We describe these in detail in the introduction, and we later give complete proofs of the first two results. The last chapter contains an introduction to the work of Lindenstrauss on the quantum unique ergodicity conjecture. Via this last chapter in particular, we hope to lead the reader to the heart of current research.

To Whom Is This Book Addressed? We presuppose a basic knowledge of differential geometry and functional analysis. Our desire has been to make the entire text comprehensible to an ambitious first year graduate student – or possibly a colleague whose speciality lies elsewhere but whose curiosity is piqued by the subject. Each chapter has its own level of difficulty, however. In the first chapters, for example, we develop the spectral theory of the Laplacian on hyperbolic surfaces using only the basic algebraic and analytic tools of a first year graduate course (integration, Fourier analysis, Hilbert spaces). By contrast, the three last chapters, being more directly plugged into current research, would more naturally find their place in an advanced graduate level course.

1

Strictly speaking, the closest book in French to the present one is, well,. . .

Preface

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Nevertheless, with an eye on highlighting the most illustrative cases, we have tried to simplify certain proofs; we then give references for the stronger statements. Finally, to the reader eager to learn more on the subject than is presented here, we could not do better than to recommend the beautiful article of Sarnak [112], entitled “Spectra of hyperbolic surfaces”, from which this book has borrowed its own title as well as a large part of its structural organization.

Acknowledgments We extend an immense thank you to Nalini Anantharaman who proofread the text in depth and to Farrell Brumley who translated it into English. This book would have contained many more errors without their numerous corrections. I’m also grateful to a second anonymous reader whose constructive remarks allowed me to simplify certain proofs and to correct several mistakes. A big thank you to Aurélien Alvarez, Jean-Pierre Otal, Gabriel Rivière, Claude Sabbah and Emmanuel Schenck for their proofreading and the numerous corrections they pointed out. Finally, we are deeply indebted to Sébastien Gouëzel for granting us permission to reproduce a part of his notes on Host’s proof of Rudolph’s theorem presented in § 9.4 and, of course, Valentin Blomer and Farrell Brumley who were willing to write Appendix C on elementary estimates on hyper-Kloosterman sums and who explained to me how to simplify the proof of Theorem 7.38. Paris, France

Nicolas Bergeron

Contents

1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.1 Spectral Analysis on the Torus .. . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.2 The Hyperbolic Plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.3 Hyperbolic Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.3.1 Fundamental Domains .. . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.3.2 Hyperbolic Laplacian . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.4 Description of the Main Results . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.4.1 Examples of Hyperbolic Surfaces .. . . . . .. . . . . . . . . . . . . . . . . . . . 1.4.2 The Spectral Theorem . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.4.3 Maaß Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.4.4 The Selberg Trace Formula .. . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.4.5 The Selberg Eigenvalue Conjecture .. . . .. . . . . . . . . . . . . . . . . . . . 1.4.6 The Jacquet-Langlands Correspondence.. . . . . . . . . . . . . . . . . . . 1.4.7 Arithmetic Quantum Unique Ergodicity .. . . . . . . . . . . . . . . . . . . 1.5 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.6 Remarks and References . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

1 1 4 11 12 14 16 16 18 19 20 22 23 24 24 25 25

2 Arithmetic Hyperbolic Surfaces .. . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.1 The Space of Lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.2 Quaternion Algebras and Arithmetic Groups . . . .. . . . . . . . . . . . . . . . . . . . 2.2.1 An Exceptional Isomorphism . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.2.2 A Subset of the Space of Lattices . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.2.3 The Cocompactness Criterion . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.2.4 Commensurability Classes . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.3 Arithmetic Hyperbolic Surfaces . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.3.1 Eliminating Torsion .. . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.3.2 The Modular Surface and Its Covers .. . .. . . . . . . . . . . . . . . . . . . . 2.4 Commentary and References.. . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

31 31 35 37 40 40 42 43 44 45 49 50

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3 Spectral Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.1 The Laplacian .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.2 Eigenfunctions of the Laplacian on H . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.2.1 Radial Functions . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.3 Invariant Integral Operators on H. . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.4 The Selberg Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.5 A Family of Examples: The Heat Kernel .. . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.6 The Laplacian on  nH. . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.7 Integral Operators on  nH . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.7.1 The Heat Kernel .. . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.7.2 The Non-compact Case . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.8 Review of Functional Analysis . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.9 Proof of the Spectral Theorem . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.10 The Minimax Principle .. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.10.1 Small Eigenvalues I: Geometric Existence Criterion .. . . . . . 3.10.2 Small Eigenvalues II: The Selberg Conjecture . . . . . . . . . . . . . 3.11 Commentary and References.. . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.12 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

53 53 56 58 60 64 67 70 73 75 76 81 83 89 91 93 94 96

4 Maaß Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.1 Eisenstein Series for SL.2; Z/ . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.1.1 Eisenstein Series and the Riemann Zeta Function I. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.2 Eisenstein Series and the Spectrum of the Laplacian . . . . . . . . . . . . . . . . 4.2.1 Mellin Inversion Formula, Phragmén-Lindelöf Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.2.2 The Space of Incomplete Eisenstein Series . . . . . . . . . . . . . . . . . 4.2.3 Regularity of E.z; s/ on the Vertical Line Re.s/ D 1=2 .. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.2.4 Eisenstein Series and the Riemann Zeta Function II .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.2.5 The Eisenstein Transform . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.2.6 The Spectral Theorem for the Modular Surface .. . . . . . . . . . . 4.3 Existence of Cusp Forms .. . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.3.1 Automorphic Wave Equation .. . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.3.2 Construction of Cusp Forms. . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.4 Hyperbolic Periods of Eisenstein Series . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.4.1 Primitive Geodesics on the Modular Surface .. . . . . . . . . . . . . . 4.4.2 Hyperbolic Fourier Series of E.z; s/ . . . .. . . . . . . . . . . . . . . . . . . . 4.5 Explicit Construction of Maaß Forms . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.6 Commentary and References.. . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

99 99

116 118 121 123 123 125 127 127 135 138 145 149

5 The Trace Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.1 The Selberg Trace Formula I: General Framework . . . . . . . . . . . . . . . . . . 5.2 The Selberg Trace Formula II: The Case of Compact Surfaces . . . . . 5.2.1 The Pretrace Formula .. . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

153 153 157 157

103 106 107 110 113

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5.2.2 The Geometric Side of the Trace Formula .. . . . . . . . . . . . . . . . . 5.2.3 Contribution of Hyperbolic Elements .. .. . . . . . . . . . . . . . . . . . . . 5.2.4 The Trace Formula .. . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . The Selberg Trace Formula III: The Case of SL.2; Z/.. . . . . . . . . . . . . . 5.3.1 The Kernel Kcont . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.3.2 The Kernel K  Kcont . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.3.3 The Spectral Side of the Trace Formula . . . . . . . . . . . . . . . . . . . . 5.3.4 The Geometric Side of the Trace Formula: Parabolic Term Contribution . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.3.5 The Trace Formula .. . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Applications.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.4.1 The Weyl Law .. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.4.2 The Prime Geodesic Theorem .. . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Comments and References . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

159 160 162 165 167 167 171

6 Multiplicity of 1 and the Selberg Conjecture .. . . . . . .. . . . . . . . . . . . . . . . . . . . 6.1 Point Counting in Arithmetic Lattices . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.2 Multiplicity of the First Eigenvalue .. . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.3 Representation Theory of PSL.2; Z=pZ/ . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.3.1 Review of the Representation Theory of Finite Groups .. . . 6.3.2 Proof of Theorem 6.8 . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.4 Lower Bound on the First Non-zero Eigenvalue .. . . . . . . . . . . . . . . . . . . . 6.5 Comments and References . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

193 193 196 205 205 207 210 211

7 L-Functions and the Selberg Conjecture . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 7.1 The L-Function Attached to a Maaß Form . . . . . . .. . . . . . . . . . . . . . . . . . . . 7.2 Hecke Operators and Applications.. . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 7.2.1 Hecke Operators . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 7.2.2 Atkin-Lehner Theory . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 7.2.3 Multiplicative Properties of Fourier Coefficients .. . . . . . . . . . 7.3 Dirichlet Characters and Twisted Maaß Forms. . .. . . . . . . . . . . . . . . . . . . . 7.3.1 Dirichlet Characters .. . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 7.3.2 Maaß Forms Twisted by a Character .. . .. . . . . . . . . . . . . . . . . . . . 7.3.3 Twisted Eisenstein Series . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 7.4 Rankin-Selberg L-Functions . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 7.5 The Luo-Rudnick-Sarnak Theorem .. . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 7.6 Bounds on Fourier Coefficients . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 7.7 Comments and References . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 7.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

213 214 217 217 221 227 230 230 232 234 239 253 260 262 265

8 Jacquet-Langlands Correspondence .. . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 8.1 Arithmetic of Quaternion Algebras .. . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 8.1.1 Orders in Quaternion Algebras .. . . . . . . . .. . . . . . . . . . . . . . . . . . . . 8.1.2 Orders in Quadratic Extensions of Q . . .. . . . . . . . . . . . . . . . . . . . 8.1.3 Strong Approximation Theorems . . . . . . .. . . . . . . . . . . . . . . . . . . .

267 267 267 270 272

5.3

5.4

5.5 5.6

175 179 181 181 183 188 191

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8.2 8.3 8.4 8.5

Optimal Embeddings of Quadratic Fields . . . . . . . .. . . . . . . . . . . . . . . . . . . . The Trace Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Jacquet-Langlands Correspondence and Applications .. . . . . . . . . . . . . . Commentary and References.. . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

273 282 286 292

9 Arithmetic Quantum Unique Ergodicity .. . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 9.1 Quantization of the Geodesic Flow . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 9.1.1 The Classical System: The Geodesic Flow . . . . . . . . . . . . . . . . . 9.1.2 Quantum Systems . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 9.1.3 Quantum Mechanics on the Poincaré Upper Half-Plane .. . 9.2 Microlocal Lift .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 9.2.1 The Microlocal Lift . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 9.2.2 Quantum Ergodicity . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 9.3 First Links with Ergodic Theory .. . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 9.4 Multiplication by 2 and 3 on the Circle. . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 9.4.1 The Circle as a Foliated Space . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 9.4.2 Conditional Measures .. . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 9.4.3 Recurrence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 9.4.4 Invariant Measures .. . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 9.4.5 Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 9.5 Hecke Operators and Lindenstrauss’s Theorem ... . . . . . . . . . . . . . . . . . . . 9.5.1 The Tree of PGL.2; Qp / . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 9.5.2 Hecke Operators . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 9.5.3 Lindenstrauss’s Theorem . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 9.6 Use of Hecke Operators . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 9.6.1 Local Contributions .. . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 9.6.2 Intersections of Hecke Translates . . . . . . .. . . . . . . . . . . . . . . . . . . . 9.6.3 Strongly Positive Entropy . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 9.6.4 Tp -Recurrence .. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 9.7 Commentary and References.. . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 9.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

295 295 296 299 300 302 304 308 310 312 312 313 315 317 319 322 323 325 326 327 328 333 337 339 339 342

Appendices A Three Coordinate Systems for H . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 343 B The Gamma Function and Bessel Functions . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 345 C Elementary Bounds on Hyper-Kloosterman Sums by Valentin Blomer and Farrell Brumley . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 349 References .. .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 355 Index of notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 363

Contents

xiii

Index . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 365 Index of names . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 369

Chapter 1

Introduction

1.1 Spectral Analysis on the Torus Before entering into the heart of the subject, we begin with the motivating and familiar example of spectral analysis on the torus. Let .Rn ; h ; i/ be standard ndimensional Euclidean space, where Rn D fx D .x1 ; : : : ; xn / j x1 ; : : : ; xn 2 Rg and hx; yi D x1 y1 C    C xn yn : The torus Tn is the quotient of Rn by the group of translations Zn . One thinks of the latter as a discrete (torsion-free) subgroup of the real group G D Rn acting on Euclidean space by translations. We write q W Rn ! Tn for the quotient map. Let  D @2 =@x21      @2 =@x2n be the Laplacian1 on Rn . A C1 - function f on the torus Tn lifts to a Zn -periodic C1 -function f ı q on Rn . By abuse of notation we identify these two functions. The Laplacian commutes with the action of G on Euclidean space. In particular, since f is Zn -periodic, the function f is also Zn periodic and defines a function on the torus. The Laplacian  descends in this way to a differential operator on Tn , which we continue to denote by ; this is the Laplacian on the torus. It is clear that the exponential functions2 ' .x/ D e.hx; i/;

 2 Rn ;

We consider the negative of the usual Laplacian in order for its spectrum to lie in RC . One sometimes calls this operator the “geometric Laplacian”.

1

2

Throughout this book we shall use the notation e.z/ D e2iz .

© Springer International Publishing Switzerland 2016 N. Bergeron, The Spectrum of Hyperbolic Surfaces, Universitext, DOI 10.1007/978-3-319-27666-3_1

1

2

1 Introduction

are  eigenfunctions;  D ./ D 4 2 jj2 :

' D ' ;

Here j  j denotes the norm associated to h ; i. Let S.Rn / be the space of Schwartz class functions. Recall that a function is in the Schwartz class if it and all of its derivatives decay rapidly. Let f 2 S.Rn /. The classical Fourier inversion identities (see [106, Chap. 9]) b f ./ D

Z Z

f .x/ D

Rn

Rn

f .x/e.hx; i/ dx (1.1) b f ./' .x/d

can be interpreted as the “spectral resolution of the Laplacian ”. Formally, we can write b f ./ D . f ; ' /, but the functions ' do not belong to L2 .Rn /, so this expression does not literally make sense. Moreover, for f 2 L2 .Rn / the Fourier integral does not have a well-defined meaning in general and the Fourier inversion formula is only obtained by analytic continuation. The functions ' for  2 Zn descend to C1 -functions of L2 norm 1 on the torus Tn . They form a Hilbert orthonormal basis for L2 .Tn /. Since they are also Laplacian eigenfunctions, we say that they form a complete orthonormal system of solutions to the spectral problem ' D ': In this way, we recover Fourier theory (see [145, Chap. IV]). We remark that analytic number theory profits enormously from harmonic analysis on the torus since it often exploits properties of periodic functions, notably via the Poisson summation formula (see [145, p. 94]). Poisson’s formula Let f 2 S.Rn /. Then, X

f .m/ D

m2Zn

X

b f .m/:

(1.2)

m2Zn

Proof Consider the function defined by F.x/ D

X

f .x C m/

(1.3)

m2Zn

for x 2 R. From the fact that f 2 S.Rn /, it follows that this series converges absolutely and uniformly on compact sets. The function F is therefore C1 , and being Zn -periodic it comes from a function on the torus Tn .

1.1 Spectral Analysis on the Torus

3

We can thus expand F with respect to the basis .' /2Zn : X

F.x/ D

c ' .x/:

(1.4)

2Zn

According to Dirichlet’s theorem [145, p. 86] this series again converges absolutely and uniformly. We now compute the coefficients c . By definition we have Z c D Z

Tn

F.x/e.hx; i/ dx X

D

x2Œ0;1n m2Zn

D D

XZ

f .x C m/e.hx; i/ dx f .x C m/e.hx; i/ dx

m2Zn

x2Œ0;1n

m2Zn

x2mCŒ0;1n

XZ

Z

D x2Rn

f .x/e.hx; i/ dx

f .x/e.hx; i/ dx D b f ./:

Here all interchanging of sums and integrals is justified by absolute convergence. Setting x D 0 in (1.4) gives the desired formula. t u The analytic continuation of the Riemann zeta function is a classical application of the Poisson summation formula (in the case n D 1 above) to number theory; see Exercise 1.16. This is the precise proof that Riemann gave in the sole paper [102] that the famous geometer and analyst dedicated to arithmetic. Let us then consider the case n D 1; we first recall the following classical lemma. 2

Lemma 1.1 Let y > 0 and fy .x/ D eyx . Then for every  1 1 2 b f y ./ D p e =y D p fy1 ./: y y Proof Put f D f1 . Changing variables, we have 1    b f y ./ D p b f p : y y It suffices then to treat the case y D 1. We must show that b f D f . Now one has b f ./ D

Z

2

ex e.x/ dx: R

4

1 Introduction

Differentiating under the integral (a valid operation since f is Schwartz class), we obtain (via an integration by parts): b f 0 ./ D 2b f ./: This first order linear differential equation admits f as a solution as well. We deduce that there exists  2 C such that b f D f . Applying once again the Fourier transform we obtain (recall that f is even): b f D b f D b f D 2 f : Thus  D ˙1 and since b f .0/ > 0,  D 1.

t u

The modern theory of automorphic forms in concerned with the spectral theory of quotients of more general (non-abelian) groups. In this book we concentrate exclusively on the case of the group G D SL.2; R/ and the hyperbolic plane associated with it. This basic example is absolutely central to the entire automorphic theory. It has the advantage of being explicit yet sufficiently rich to serve as a model case; in fact we shall see that many important questions in this setting remain stubbornly open.

1.2 The Hyperbolic Plane As a model for the hyperbolic plane we shall use the Poincaré model of the upper half-plane H D fz D x C iy 2 C j y > 0g: This is endowed with the complex structure coming from C. To gain our footing we proceed in the spirit of Euclid, who put forward a notion of geometry based on axioms – although we shall end up excluding the famous fifth axiom. We begin by defining hyperbolic lines in H as • the vertical half-lines orthogonal to the real axis R, and • the half-circles perpendicular to R. See Fig. 1.1. Given any two distinct points in H there is exactly one hyperbolic line passing through them. On the other hand, given any point there is in general an infinite number of lines passing through it which are parallel to – in the sense of not intersecting with – a given line. A more modern point of view consists in viewing H as a Riemannian manifold. This point of view is not necessary for a good understanding of the rest of the text but it would be a pity not to review it, at least rapidly. The reader is encouraged to see [6, 24, 67, 92] for more detailed presentations.

1.2 The Hyperbolic Plane

5

Fig. 1.1 Hyperbolic lines

y

0

x

One can in fact endow the half-plane H with the Riemannian metric ds2 D

dx2 C dy2 I y2

(1.5)

one calls this the hyperbolic metric. This defines a distance on H, by taking the lim inf of the hyperbolic lengths Z `. / D

Z

1

ds D 0



p x0 .t/2 C y0 .t/2 dt y.t/

of piecewise C1 curves joining z to w, .t/ D .x.t/; y.t// with t 2 Œ0; 1. The Riemannian metric allows us to define the angle3 between two tangent vectors at a point z 2 H. As ds2 is proportional to the Euclidean metric, we find that the hyperbolic angles are the same as the Euclidean angles. The presence of a large group of isometries will allow us to calculate explicitly the distance function on H. For that we begin by recalling a few well known properties of homographic transformations z 7!

az C b ; cz C d

a; b; c; d 2 C; ad  bc ¤ 0

(1.6)

of the Riemann sphere b C D C [ f1g. We can always assume that ad  bc D 1 so that the homography determines a matrix   ab 2 SL.2; C/ cd

Let .M; g/ be a Riemannian manifold and x a point in M. Given two tangent vectors v; w 2 Tx .M/ of norm 1, we have gx .v; w/ D cos where is the angle between v and w.

3

6

1 Introduction

    up to sign. In particular, the matrices I D 1 1 and I D 1 1 both define the identity transformation. A homography g sends a Euclidean circle to another one – as long as Euclidean lines in b C are deemed circles (these are the circles passing through infinity). Of course, the centers are not in preserved since g is not a Euclidean isometry,   general apart from the case g D ˙ 10 1 representing a translation. Let G D SL.2; R/ be the group of real 2  2 matrices of determinant 1. Given an element g D . c d / 2 G we write j.g; z/ D cz C d:

(1.7)

The function j satisfies the cocycle condition: j.gh; z/ D j.g; hz/j.h; z/:

(1.8)

Taking into account the formula jj.g; z/j2 Im.gz/ D Im.z/

(1.9)

the Riemann sphere b C is the union of three G-orbits: the upper half-plane H, the R D R [ f1g. Moreover, we lower half-plane H and the boundary separating them b have Im.gz/1 jd.gz/j D Im.z/1 jdzj; which implies that the metric ds on H is invariant under the action of G. The homographies are therefore the isometries of the hyperbolic plane. The action of the group G on H is, moreover, transitive, meaning that for any given pair z; w 2 H, there exists g 2 G such that gz D w. To see this, it suffices to take z D i and w D x C iy with y > 0. Then the matrix  1=2 1=2  y xy gD 0 y1=2 does the job. The group G preserves the orientation of H and the only homography acting trivially on H is the one associated to (plus or minus) the identity matrix. Besides these isometries there is also the orthogonal symmetry about the imaginary axis, z 7! z, which reverses orientation. One can show that the symmetry z 7! z along with all real homographies generate the whole group of isometries of H; we will not, however, have need of this fact in the rest of the text. For the proof one can refer to [6, §7.2], for example.

1.2 The Hyperbolic Plane

7

If z D i and w D iy, y > 0, one sees immediately that ˇZ ˇ `. / > ˇˇ

y 1

ˇ dt ˇˇ D j log yj tˇ

for every curve adjoining z to w. This lower bound is realized by the vertical segment linking z to w, and so the hyperbolic distance .z; w/ between z and w is given by

.i; iy/ D j log yj. Now let z and w be arbitrary. Since the action of G on H is transitive there exists g0 2 G such that g0 z D i. We can replace g0 by k. /g0 for every k. / D

  cos  sin 2 SO.2; R/  SL.2; R/ sin cos

for such a k fixes i. As Re.k.=2/g0 w/ and Re.g0 w/ have opposite sign, by continuity there exists such that Re.k. /g0 w/ D 0. We have thus found an element g 2 G such that gz D i and gw D iy for some y > 0, allowing us to calculate the distance .z; w/ D .i; iy/. Explicitly we have

.z; w/ D log

jz  wj C jz  wj : jz  wj  jz  wj

(1.10)

Equivalently, one can write cosh .z; w/ D 1 C 2u.z; w/;

(1.11)

where u.z; w/ D

jz  wj2 I 4 Im z Im w

(1.12)

the latter function is often more practical to use than the true distance function

.z; w/. By using the above method, we can show that the geodesics of H – the curves locally realizing the distance between two points – are precisely the hyperbolic lines introduced at the beginning of this section. In particular, the space H is complete (in the Riemannian sense of the word, or as a metric space relative to ). Finally, we isolate a few properties of these hyperbolic lines, see [24]. 1. Distances are globally minimized by geodesics. 2. If  is a geodesic and p …  , there exists a unique geodesic passing through p and perpendicular to  . 3. If  and  are two geodesics at distance > 0 apart, there exists a unique geodesic perpendicular to  and .

8

1 Introduction

The set C.i; / of points at hyperbolic distance from i coincides with the orbit of the point ie under the action of the group SO.2; R/. Since k. / D

  i  1  1 1 1 e 1 i 0 ; 1 i 0 ei 2 i i

the set C.i; / is the image by a homography of b C of a Euclidean circle centered at 0, and is thus again a Euclidean circle. We note that the orbit is traced out twice: k./ is the identity transformation. The set C.i; / is therefore the Euclidean circle of center i cosh. / and radius sinh. /. More generally, since the action of G on H is transitive, every hyperbolic circle – the locus of points at fixed hyperbolic distance from a given point – is represented by a Euclidean circle in H (but of course with a different center). Having at our disposal a notion of a line, we can now speak about hyperbolic triangles – and, more generally, polygons: these are closed arcwise geodesic Jordan curves. One defines the sides and internal angles of hyperbolic polygons in the usual way. There are several cute and surprising relations in the hyperbolic plane. For example, an exercise in hyperbolic trigonometry shows (see for example [6, §7.7]) that in each hyperbolic triangle one has sin ˛ sin ˇ sin  D D ; sinh a sinh b sinh c

(1.13)

sin ˛ sin ˇ cosh c D cos ˛ cos ˇ C cos ;

(1.14)

cosh c D cosh a cosh b  sinh a sinh b cos ;

(1.15)

as well as

where ˛; ˇ;  are the interior angles opposite the sides of length a; b; c, respectively. The relation (1.14) implies, contrary to the Euclidean case, that the length of a given side depends only on the interior angles of the triangle. Even more counter-intuitive phenomena appear when we consider the notion of area. As for any Riemannian metric, one can define a measure starting from the hyperbolic metric. By using formulae valid for an arbitrary Riemannian metric (see for example [41, p. 140]), one finds that the hyperbolic measure is d .z/ D

dx dy : y2

(1.16)

According to the general theory, this measure is invariant under isometric motions and hence invariant under the action of the group G. The following theorem reduces easily to the case of a triangle. We refer the reader to [92, Lem. 1.4.4] for a proof.

1.2 The Hyperbolic Plane

9

Theorem 1.2 The area of an n-sided hyperbolic polygon P with internal angles 1 ; : : : ; n is given by the formula: area. P/ D .n  2/  . 1 C    C n /: There exist other models for hyperbolic geometry. For example, the Poincaré disk is obtained from the homography z 7!

zi ; zCi

z2H

sending the half-plane H biholomorphically onto the open unit disk D, D D fw D x C iy 2 C j x2 C y2 < 1g: This map transports the hyperbolic metric on H to the Riemannian metric ds2 D

4.dx2 C dy2 / .1  .x2 C y2 //2

(1.17)

and the corresponding geodesics are the half-circles orthogonal to the boundary unit circle. Although we will not often use the unit disk model, it nevertheless has its advantages. Being more symmetric,4 it sometimes lends itself more easily to a qualitative understanding of hyperbolic geometry. The very accessible article by Ghys [44] adopts this point of view and gives a beautiful proof of Theorem 1.2 and also provides a rich bibliography that one can consult for more details. That article begins in particular with a quote of Poincaré which highlights a certain phenomenon of concentration, beautifully illustrating a fundamental difference between Euclidean and hyperbolic geometry. On one hand, the hyperbolic disk of radius centered at i has hyperbolic area 4.sinh. =2//2 and perimeter 2 sinh . On the other, since the Euclidean center of this same disk is i cosh with radius sinh , the Euclidean area is .sinh /2 while its perimeter is again 2 sinh . Thus, despite having the same perimeter, the Euclidean area is much larger than the hyperbolic area as gets large (and the ratio of the two areas tends to 1 as tends toward 0). The bulk of the hyperbolic area is therefore concentrated close to the edge of the disk. We conclude this section on hyperbolic geometry with a classification of the isometries of H. We group them into three subsets according to the way in which they displace points. We begin by remarking that conjugate isometries act in the same way; the classification should therefore be invariant under conjugation. Given

4

We in fact made indirect use of the unit disk model to show that the hyperbolic circles are Euclidean circles.

10

1 Introduction

a homography g 2 PSL.2; R/, we let Œg D fhgh1 j h 2 PSL.2; R/g denote its conjugacy class. The identity transformation forms a class by itself; there is not much more we can say about it. Each   ab ¤ ˙I gD cd has one or two fixed points in b C. There are precisely three possibilities: 1. g has one fixed point in b R, 2. g has two distinct fixed points in b R, 3. g fixes a unique point in H and fixes the conjugate of this point in H. The matrix (and the corresponding transformation) is said to be parabolic, hyperbolic, or elliptic, respectively, and each classification obviously applies to their entire conjugacy class. Each conjugacy class contains a representative which acts on H in one of the following three ways: 1. z 7! z C v, with v 2 R (translation, fixed point 1), points 2. z ! 7 pz, with p 2 RC (homothety,  0 and 1),  fixed  sin (rotation, fixed point i). 3. z ! 7 k. /z, with 2 R, k. / D cos sin cos A parabolic transformation is always of finite order, and moves points along horocycles. The geodesic line between the two points fixed by a hyperbolic transformation g is preserved by g. Of the two fixed points, one attracts, the other repels. The homothety factor p is called the norm of g. For every point z on this geodesic, j log pj is the hyperbolic distance between z and gz. An elliptic transformation can be of finite or infinite order and moves points along circles centered at its fixed point in H. The trace – or more its absolute value, since the transformation g  precisely  determines the matrix ac db up to sign – is invariant under conjugation and the above classes (other than the identity transformation) are characterized in the following way: 1. g is parabolic if and only if ja C dj D 2, 2. g is hyperbolic if and only if ja C dj > 2, 3. g is elliptic if and only if ja C dj < 2. The isometries of H do not in general commute, but it is clear that if gh D hg, then the fixed points of g are those of h. The converse is also true. More precisely, two non-trivial isometries of H commute if and only if they have the same set of fixed points. The centralizer fh j gh D hgg of a parabolic (resp. hyperbolic, elliptic)

1.3 Hyperbolic Surfaces

11

isometry g is just the subgroup consisting of the identity transformation along with all the parabolic (resp. hyperbolic, elliptic) isometries fixing the same points as g.

1.3 Hyperbolic Surfaces A Fuchsian group is a discrete subgroup   SL.2; R/. We are tacitly taking the topology to be the usual one on SL.2; R/  R4 . Fuchsian groups shall play the role of Zn – a discrete subgroup of Rn – in the non-abelian version of the Fourier theory that we have in mind. One considers this time the quotient space  nH, endowed with the quotient topology rendering the projection H !  nH continuous. One can show (see [92, Cor. 1.5.3]) that the Fuchsian groups of SL.2; R/ are precisely the subgroups of SL.2; R/ – viewed as the isometry group of H – acting properly discontinuously on H, in other words no orbit  z, z 2 H, has an accumulation point in H. Let us fix a Fuchsian group   SL.2; R/. Since  acts properly discontinuously on H – a Hausdorff space – the quotient  nH is also Hausdorff (see for example [92, Lem. 1.7.2]). We write  for the image of  in PSL.2; R/. If moreover  acts freely on H, that is to say that for every z 2 H the stabilizer z WD f 2  j  z D zg is contained in f˙Ig, then the projection p W H !  nH is a covering map. (See for example [37, 91] for an introduction to covering theory.) One can use p to transport the C1 , Riemannian or even hyperbolic structures. The quotient S D  nH is naturally a hyperbolic surface; its universal cover is H and its fundamental group is isomorphic to  . We shall speak throughout this text of functions defined on the hyperbolic surfaces S D  nH, but this point of view is not strictly necessary: one could more simply think of such a function f as being defined on H while being periodic with respect to the action of the group  : f . z/ D f .z/ for every  2  . The latter approach has the advantage of generalizing to all Fuchsian groups. Finally, we shall say that a Fuchsian group   G is of the first kind if it is finitely generated and if for every z 2 H, the compactification of the real axis b Rb C is contained in the closure (for the topology on b C) of the orbit  z.

12

1 Introduction

1.3.1 Fundamental Domains One can visualize a Fuchsian group   G with the help of a fundamental domain. An open D  H is a fundamental domain for  if: 1. the sets S  D, for  2  , are pairwise disjoint; and 2. H D  2  D. Every Fuchsian group   G admits a fundamental domain which is of course far from unique. Nevertheless, all of the “classical” fundamental domains – those which we are going to construct – have a boundary of measure zero and therefore have the same area Z jDj D d .z/: D

If   G is of the first kind, we can choose for a fundamental domain a convex polygon (in the sense of hyperbolic lines). Indeed, let w 2 H be a point fixed by no element  2  other than ˙I, then the set D.w/ D fz 2 H j .z; w/ < .z;  w/ for every  2 ;  ¤ ˙Ig is a fundamental domain for  called a Dirichlet domain. One can show that D.w/ is a polygon (connected and convex) with an even number of sides. (We adopt the convention that if a side contains the fixed point of an elliptic element of order 2 of  , then this point is considered as a vertex and the hyperbolic line on which it lies counts for two sides.) The sides of D.w/ can be grouped into pairs in such a way that the isometries identifying each pair generate the group  . From these properties of D.w/, one can show – it is a rather delicate theorem of Siegel [123, pp. 75–77], see also [92, Th. 1.9.1] – that a Fuchsian group is of the first kind if and only if it admits a a fundamental domain of finite area. Fuchsian groups of the first kind break further into two classes according to whether the closure D in H of a fundamental domain D is compact or not. In the former case, we shall say that  is a cocompact group. For  of the first kind, if  nH is non-compact this fact can be read off from the fundamental domain by the presence of a finite number of cusps x. These are points where D – or rather its closure in H [ b R – meets b R. Modifying D if necessary we can always assume that the two sides of D adjacent to a cusp are identified by an element of  fixing the cusp. Lemma 1.3 Let  be a Fuchsian group of the first kind, x a cusp and x D f 2  j  x D xg its stabilizer in  .

1.3 Hyperbolic Surfaces

13

1. Every element  2 x is parabolic. 2. The stabilizer x is isomorphic to Z or Z=2Z  Z according to whether I …  or I 2  , respectively; the image  x of x in PSL.2; R/ is always isomorphic to Z. 3. Let  be a generator of  x . There exists a matrix x 2 SL.2; R/ such that

x 1 D x

and x1  x D

  11 I 01

(1.18)

moreover every matrix satisfying these two properties is of the form   1 t

D ˙ x 01

with t 2 R:

Proof Changing D if necessary, we can in fact assume that the cusps of D are pairwise nonequivalent. Let x be a cusp of D and  the element of  identifying the two sides of D adjacent to x (and fixing x). To see that  is parabolic, we consider the two hyperbolic lines d0 and d1 bounding D in a neighborhood of x and such that .d0 / D d1 . Since  fixes the point x 2 b R, it is a parabolic or hyperbolic transformation. Arguing by contradiction we assume that it is hyperbolic and consider the position of its second fixed point y2b R relative to the two lines. Now the hyperbolic line d adjoining x to y cannot lie strictly between d0 and d1 , for otherwise (as one sees by considering the germ of D along d at x) the intersection of the fundamental domain D with .D/ would be non-empty. Replacing  by  1 if necessary, we can assume that x repels and y attracts, so that the sequence of geodesics dn D  n .d0 / tends towards d. The side of D abutted by S d0 is then sent by  n to a side of Dn D  n .D/ abutted by dn . The tiling H D  2 .D/ is therefore not locally finite in a neighborhood of d. Since  acts properly discontinuously on H, we have found a contradiction. The above argument in fact shows that the stabilizer x contains only parabolic elements. As such, it is contained in a one-parameter subgroup consisting entirely of parabolic elements. In fact there exists in SL.2; R/ such that 1 D x and thus

1  x 

  ˇ 1t ˇ ˇt2R : 01

As 1  x is discrete, it must therefore be of the form   1 hZ : 0 1

14

1 Introduction

We easily conclude the argument by using the formula     1   2   a 0 1 t 1a t a 0 D : 01 0 a 0 1 0 a1

t u

Conversely, translating by  if necessary, the cusps are precisely the fixed points of parabolic elements of  . We deduce the following proposition. Proposition 1.4 A Fuchsian group of the first kind is cocompact if and only if it contains no parabolic element. Now a given fundamental domain D is not canonically associated with  , so for a rigorous definition of a cusp of  we take a point of b R which is fixed by a parabolic element in  . The group  induces an equivalent relation on the set of cusps and each equivalence class has a cusp of D as a representative. In particular, it follows from Siegel’s theorem mentioned earlier that there are only a finite number of cusps modulo  (again, the latter is always assumed to be of the first kind). We saw that if a Fuchsian group  acts freely on H, the quotient S D  nH is a hyperbolic Riemann surface. The surface S is obtained by gluing a fundamental domain of  along identified sides. The Riemannian measure on the quotient has a concrete interpretation as the restriction of to D. This measure is in particular of finite area if  is of the first kind. Finally, the surface S is compact if  is a cocompact group. A general Fuchsian group  does not necessarily act freely on H. If z 2 H and  z ¤ fIg then all of the elements of z are elliptic. Since  is discrete and the stabilizer of z in G is compact, the group z is necessarily finite. If  is of the first kind, it admits a fundamental domain with a finite number of sides. We easily deduce from this that  contains only a finite number of elliptic elements. The cover H !  nH therefore has ramification points precisely at the fixed points of elliptic elements in  . Figure 1.2, illustrated by Frank Loray for the collective work [34], represents a quotient with two ramification points and one cusp.

1.3.2 Hyperbolic Laplacian The Laplacian admits a natural generalization to any Riemannian manifold. Using the general formulae for a Riemannian metric (see for example [41, p. 183]), one finds that in the coordinates z D x C iy of the Poincaré upper half-plane the hyperbolic Laplacian is the differential operator  D y2 acting on C1 functions on H.



@2 @2 C @x2 @y2

 (1.19)

1.3 Hyperbolic Surfaces

15

∞ ∞

π

P

δ1 δ1 S1

δ2

δ2 S1

S

S2

S2

Fig. 1.2 A quotient with one cusp and two ramification points

According to the general theory, the Laplacian is an object intrinsic to the Riemannian metric, invariant under the action by isometries, and so in particular by G. In other words, if Lg f .z/ D f .g1 z/ then Lg  D Lg ;

for g 2 G:

(1.20)

In polar coordinates .r; / of the hyperbolic plane (see Appendix A), one has D

1 @2 @2 1 @   : 2 2 @r tanh r @r .2 sinh r/ @ 2

(1.21)

Property (1.20) is fundamental: one can show that it, in a sense, characterizes the Laplacian. Indeed, the differential operators which commute with all isometries form a polynomial algebra in the Laplacian  (see [56]). Finally, we note that, as in the case of the torus, it follows from (1.20) that if f is a C1 function on the hyperbolic surface S D  nH, viewed as a C1  -periodic function on H, the function f is again  -periodic on H. This function therefore comes from a function on S. The operator defined in this way on S is the Laplacian associated with the Riemannian metric on S; we shall continue to denote it by .

16

1 Introduction

1.4 Description of the Main Results 1.4.1 Examples of Hyperbolic Surfaces There are several ways to construct hyperbolic surfaces. One can begin by drawing a convex hyperbolic polygon D  H of finite area having an even number of sides. The polygon D will not in general be the fundamental domain of a Fuchsian group of the first kind  ; for that D must satisfy certain conditions which were completely explicated by Poincaré. We shall not enter into the details here. We only mention that it is easy to show using this construction (see [34]) that every surface of genus g at least 2 can be endowed with a hyperbolic structure and that the set of such structures is a real manifold of dimension 6g  6. There are therefore a great many hyperbolic surfaces (up to isometry). From the number theoretic viewpoint, of all the hyperbolic surfaces of finite area, the most interesting are the so-called arithmetic hyperbolic surfaces; we shall construct them in Chap. 2. Among other results we give a proof of the following important theorem. Theorem 1.5 Given two positive integers a and b, the group p p ˇ x0 C x1 a x2 C x3 a ˇ x0 ; x1 ; x2 ; x3 2 Z; p p ˇ 2 b.x2  x3 a/ x0  x1 a x0  ax21  bx22 C abx23 D 1

 a;b D

is Fuchsian of the first kind. It is cocompact if and only if .0; 0; 0/ is the unique integral solution to the Diophantine equation x2  ay2  bz2 D 0. The most important example corresponds to the choice a D b D 1, where one obtain a finite index subgroup of the modular group  .1/ D SL.2; Z/: The associated surface X.1/ D  .1/nH is called the modular surface. It is not strictly speaking a hyperbolic surface: the group  .1/ contains elliptic elements. Let us determine the cusps of  .1/. The fixed points of parabolic elements are necessarily in Q [ f1g since if az C b Dz cz C d

1.4 Description of the Main Results

17

has a unique solution, it is either rational or 1. The point 1 is indeed a cusp as is every rational number since they are  -equivalent:  1 D a=c runs through all of Q. We take 1 as a representative of the unique equivalence class of cusps for  .1/. Its stabilizer is the subgroup of integer translations (up to sign)   ˇ 1n ˇ n 2 Z :  .1/1 D ˙ ˇ 01 In the notation of Lemma 1.3, a suitable choice of 1 is I. In this case there exists an explicit construction of a fundamental domain. Starting from the fundamental domain D1 D fz 2 H j jxj < 1=2g for  .1/1 we consider the subset D D fz 2 D1 j Im.z/ > Im. z/ for all  2  .1/;  …  .1/1 g: For a proof that D is indeed a fundamental domain, see for example [118, Chap. VII, Th. 1]. This method of construction generalizes to all Fuchsian groups with at least one cusp. The general construction – different from that of Dirichlet – is due to Ford [39]. The polygon D is a hyperbolic triangle with angles =3, =3 and 0, see Fig. 1.3.5 Its area is therefore equal to =3. Thus, although the modular surface X.1/ is noncompact, one has area.X.1// D =3:

(1.22)

p One can moreover show that i and  D .1 C i 3/=2 are representatives of the two classes of elliptic fixed points of  (of order 2 and 3, respectively). Topologically the modular surface is therefore a sphere minus a point (the cusp) with two conical points of respective angles  and 2=3. See Fig. 1.2. The modular group is naturally accompanied by the congruence subgroups of SL.2; Z/  .N/ D f 2 SL.2; Z/ j   I .mod N/g: They are of finite index. The associated quotients X.N/ D  .N/nH (N > 1) are finite covers of X.1/ called congruence covers. For N > 2, we shall see that the

5

Figure 1.3 was realized by McMullen, see http://www.math.harvard.edu/~ctm/gallery/index.html.

18

1 Introduction

Fig. 1.3 Tesselation of H by translates of D

surfaces X.N/ are truly hyperbolic surfaces whose genus grows roughly like N 3 as N grows to infinity. Just like the modular group, the groups a;b also contain congruence subgroups a;b .N/, each of finite index. Congruence subgroups are the heroes of this text.

1.4.2 The Spectral Theorem In Chap. 3, which is completely independent of Chap. 2, we shall be interested in the spectral decomposition of the Laplacian on compact hyperbolic surfaces. More precisely, the goal of this chapter is to provide the proof of the following theorem. Theorem 1.6 (Spectral theorem) Let S be a compact hyperbolic surface. The spectral problem ' D ' admits a complete orthonormal system of C1 eigenfunctions '0 ; '1 ; : : : in L2 .S/ with corresponding eigenvalues 0 D 0 < 1 6 2 6 : : : ;

n ! 1 as n ! 1:

This result is rather clearly the basis of the entire theory. The proof that we propose is far from original; in a sense it is intermediate between those given in

1.4 Description of the Main Results

19

the books of Buser [24] and Iwaniec [63], from which we once again borrow many ideas. The spectral theorem allows one to formulate the conjecture – due to Selberg [116], see also Vignéras [134] – that will occupy us for a large part of this book: Conjecture 1.7 For every N > 1, 1 .a;b .N/nH/ > 1=4: At the end of Chap. 3 we explain the overall geometric tenor of such lower bounds; what we find will be quite surprising. The explanation of the precise bound 1=4 shall come later!

1.4.3 Maaß Forms In the case of the non-compact surfaces X.N/, one can formulate the spectral problem in the following way. We are looking for non-zero solutions  W H ! C to the spectral problem  D  . z/ D .z/;  2  .N/

Z

(1.23)

j.z/j2 d .z/ < C1: X.N/

Contrary to the case of the torus, the solutions to the problem (1.23) are not at all explicit. The same can be said of the associated eigenvalues. In the case of a compact surface, however, we do have an existence theorem. This follows essentially from the spectral theorem, and so it is not constructive. In the case of a non-compact but finite area surface, Theorem 1.6 is no longer applicable. In Chap. 4 we begin by establishing its analog for the modular surface X.1/; a new feature here is the presence of a continuous spectrum. The eigenvalues appearing in the latter do not literally correspond to solutions to (1.23). We already encountered an example of a continuous spectrum when we interpreted Fourier inversion as the spectral resolution of the Laplacian on Rn . One cannot deduce from this spectral theorem the existence of non-constant solutions to the problem (1.23). It could very well happen that the entire spectrum is continuous. Nevertheless, a recent idea of Lindenstrauss and Venkatesh [83] allows one to deduce rather easily from the spectral theorem the existence of an infinite number of solutions of (1.23) for the modular surface X.1/. This theorem, proved for the first time by Selberg using more involved techniques, brings to the fore the arithmetic nature of the modular surface.

20

1 Introduction

A solution to problem (1.23) is called a Maaß form in honor of Hans Maaß who was the first to construct explicit solutions to problem (1.23). By all appearances analytic objects, Maaß forms are every bit as arithmetic as their cousins the holomorphic modular forms.6 On the other hand, at present we know of no single explicit Maaß form for X.1/. It is one of the fundamental problems in the theory; see [80] for some recent results. We conclude Chap. 4 with a result – a special case of a more general theorem of Maaß – which sheds light on the arithmetic nature of certain Maaß forms. This result can be thought of as providing a construction of explicit solutions to problem (1.23) for the surface X.8/: Theorem 1.8 For every integer k > 1, there exists a solution to the spectral problem (1.23) for N D 8 and D

2 k 1  p C : 4 log.1 C 2/

1.4.4 The Selberg Trace Formula To prove his general theorem on the existence of Maaß forms mentioned earlier, Selberg developed an analog of the Poisson formula for the group SL.2; R/: the famous Selberg trace formula. This is the main result of Chap. 5. We shall give the proof in the case of compact surfaces and then in the case of the modular surface. We deduce from it two applications, one of which is the prime geodesic theorem. Theorem 1.9 For a positive integer N let N .x/ be the number of prime closed geodesics of norm7 at most x on X.N/. Then N .x/ 

x log.x/

as x goes to infinity. A closed geodesic on X.N/ is the projection of a hyperbolic line d  H which is preserved by a non-trivial element  2  .N/. Such an element is necessarily

6 There are quite a few works dedicated to the holomorphic theory [70, 92, 118]. Although many ideas in this text were originally developed in the context of holomorphic modular forms and are simply extended here to Maaß forms, we wanted to focus our attention on analytic questions and have therefore not recalled – even briefly – this beautiful theory. 7

The norm of a closed geodesic is the exponential of its length. The prime closed geodesics are the closed geodesics which are traced out just once.

1.4 Description of the Main Results

21

hyperbolic of axis d. Hence there exists 2 G such that 

t 0 0 t1

 1 

ˇ ˇ  t 2 R : ˇ

As  .N/ 1 is discrete, it must be of the form ˇ  m t0 0 ˇ m 2 Z : ˇ 0 t0m The subgroup of  .N/ made up of hyperbolic elements preserving d is therefore isomorphic to Z and generated by an element of norm t02 . Note that 2j log t0 j is equal to the (hyperbolic) length of the projection of d on X.N/. Such an element is called primitive. In particular,  is primitive if it cannot be written as a non-trivial power – different from ˙1 – of a hyperbolic element of  .N/. Two conjugate hyperbolic elements in  .N/ have axes projecting onto the same closed geodesic in X.N/. The multiset of norms of prime closed geodesics in X.N/ is therefore equal to the multiset of norms N.0 /, where f0 g runs through the set of conjugacy classes of primitive hyperbolic elements in  .N/. The norms N.0 / have the same asymptotic behavior as the prime numbers,8 namely Z N .x/ D jff0 g j N.0 / 6 xgj 

x 2

dt : log t

Many mathematicians have had a hand in establishing a good error term for the asymptotic. Among them we mention just a few: Huber [59], Hejhal [53, 54], Venkov [131] and Kuznetsov [72] (sometimes for difference groups). Analogously to the Riemann hypothesis we expect that Z N .x/ D

2

x

dt C O" .x1=2C" /; log t

(1.24)

as x ! C1. The role of the Riemann zeta function is in this context played by the Selberg zeta function ZN .s/ D

Y C1 Y 0 kD0

8

1 1  N.0 /sk

We sketch a proof of the prime number theorem in Chap. 4. We refer the reader to the book of Titchmarsh [129] for a complete proof of this theorem, its link with the Riemann zeta function and the Riemann hypothesis.

22

1 Introduction

which converges absolutely in the half-plane Re.s/ > 1; see Selberg [116] or Iwaniec [63]. It is well-known that the analogous asymptotic behavior to (1.24) for prime numbers follows from the Riemann hypothesis on the zeros of the Riemann zeta function. This feature does not continue to hold here (see [62] for a detailed discussion of the problems one encounters). We shall nevertheless show that the analog of the Riemann hypothesis for the Selberg zeta function would imply Z N .x/ D

x 2

dt C O.x3=4 = log x/: log t

(1.25)

1.4.5 The Selberg Eigenvalue Conjecture The Riemann hypothesis for the Selberg zeta function is in fact equivalent to the non-existence of small non-zero Laplacian eigenvalues. It is better known in the following form, and it is this statement which motivates a large part of this work. Conjecture 1.10 (Selberg) For every N > 1, 1 .X.N// >

1 : 4

Roelcke [105] has verified the conjecture for N D 1; see Theorem 3.38. Selberg stated his conjecture in [116], motivated by questions on the cancellation in certain arithmetic exponential sums introduced earlier, and in another context, by Hendrik Kloosterman. Selberg showed that the absence of small eigenvalues gives rise to non-trivial cancellation in these sums. These questions are at the heart of most arithmetic applications of the spectral theory of hyperbolic surfaces. In fact, every result in the direction of the Selberg conjecture is useful and is not simply to be viewed as abstract evidence in support of the conjecture. One can find several illustrations of this phenomenon in Sarnak’s article [112]. Selberg himself showed [116] the following approximation to his conjecture. Theorem 1.11 (Selberg) For every N > 1, 1 .X.N// >

3 : 16

We shall see in Chap. 6 the following theorem. While it gives a weaker lower bound than that of Selberg, it has the conceptual advantage of being essentially geometric.

1.4 Description of the Main Results

23

Theorem 1.12 For every sufficiently large prime number p, 1 .X. p// >

5 : 36

The proof of the above theorem is based on an idea of Kazhdan. It was subsequently developed by Sarnak and Xue [113] in the case of arithmetic compact surfaces and then by Gamburd [42] for the covers X. p/ of the modular surface. Given a positive integer N, we put  0 .N/ D

 ˇ ab ˇ 2 SL.2; Z/ ˇ Njc : cd

In light of the link between arithmetic and Maaß forms, we shall come back to the Selberg conjecture in Chap. 7 via the study of L-functions associated to Maaß forms. We show: Theorem 1.13 For every square free integer N > 1, we have 1 .0 .N/nH/ >

5 : 36

The proof, due to Luo, Rudnick and Sarnak [84], is analytic. We restrict ourselves to congruence subgroups of the above form (including the square free assumption on N) to simplify the exposition, but the argument of [84] is general enough to apply to any of the congruence surfaces Y.N/. It can moreover be adapted to the setting of automorphic representations of GL.n/. This method is at the heart of the more recent proof by Kim and Sarnak [68] of the best known bound towards the Selberg  7 2 eigenvalue conjecture, namely 1 > 14  64 .

1.4.6 The Jacquet-Langlands Correspondence Chapter 8 is dedicated to the proof of the surprising correspondence – due to Jacquet and Langlands [65] – between the spectra of different arithmetic surfaces. We shall present the proof in a special case and in classical language, following the work of Bolte and Johansson [12]. Let a and b be positive integers such that the quotient a;b nH is compact. Consider a discrete subgroup Qa;b  G containing a;b and maximal for this property. The Jacquet-Langlands correspondence [65] implies, as a special case, the following theorem. Theorem 1.14 The set of Laplacian eigenvalues in L2 .Qa;b nH/ is contained in the set of eigenvalues associated with Maaß forms for a group 0 .N/ for some positive square free integer N.

24

1 Introduction

In particular, this allows us to prove the following approximations to the Selberg eigenvalue conjecture. Theorem 1.15 We have 5 1 .Qa;b nH/ > : 36 By working with adelic groups one can generalize Theorem 1.14 to an arbitrary congruence subgroup of a;b , see [43]. This allows one to transport the theorem of Kim and Sarnak to the groups a;b .N/.

1.4.7 Arithmetic Quantum Unique Ergodicity The behavior of Maaß forms  as  ! C1 is especially interesting, for one begins to broach questions in mathematical physics. We conclude this book with a description of the recent breakthroughs concerning this problem in the case of arithmetic surfaces.

1.5 Notation Throughout this text we shall use the following notation: given a set X and two functions f and g on X, we write f .x/ D O.g.x// for x 2 Y  X if there exists a constant c > 0 such that, for every x 2 Y, one has jf .x/j 6 cg.x/. If the set Y is not specified, it is taken to be X by default. If the constant c depends on auxiliary parameters, for example " and , we indicate this as a subscript: f .x/ D O"; .g.x//. If x ! x0 , the notation f .x/  g.x/ means that the limit lim

x!x0

f .x/ g.x/

is well-defined and equal to 1. Given two integers n and m the notation njm means that n divides m. We write .n; m/ for the greatest common divisor of n and m. We adopt the usual notation Mn .K/ for the space of n  n matrices over a field K and In (or simply I if n D 2) for the identity matrix in Mn .K/. The symbols GL.n; K/ and SL.n; K/ have their standard meaning. Finally, we denote by jAj the cardinality of a set A.

1.7 Exercises

25

1.6 Remarks and References Most of the references are grouped at the end of each chapter in a section entitled “Remarks and References”. When a result is stated without proof in the body of the text, the references where one can find the proof are given in the same section. One will also find there more specialized commentary and general statements of theorems which are stated and proved in special cases in the body of the text. We note finally that certain proofs – sometimes entire paragraphs – are typeset in small letters. These passages are not strictly necessary for the rest of the text and are more difficult in nature.

1.7 Exercises Exercise 1.16 (Poisson summation formula and the Riemann zeta function) The zeta function .s/ is defined on Re.s/ > 1 by .s/ D

1 X

ns D

Y

.1  ps /1 :

p

nD1

Here the product runs over all prime numbers and the equality is equivalent to the uniqueness of the prime factorization of an integer. 1. By applying the Poisson summation formula to fy show that for every y > 0 X n2Z

1 X n2 =y 2 en y D p e : y n2Z

2. Show that for Re.s/ > 1, Z

C1 0

X n2Z

2y

en

 dy D .s/ WD  s=2  .s=2/ .s/; ys=2 y

where  denotes the Euler Gamma function: Z  .s/ D

C1 0

ex xs

dx : x

(1.26)

The latter is first defined on Re.s/ > 0 but admits a meromorphic continuation to all of C. See Appendix B for a brief summary of these basic properties.

26

1 Introduction

3. Deduce that the function s 7! .s/ admits a meromorphic continuation to all of C with two simple poles at s D 0 and s D 1 and that it satisfies the functional equation .s/ D .1  s/:

(1.27)

This last exercise is non-trivial, but it is worthwhile to try solving it without help. In case of difficulty, the reader can consult the proof of Theorem 4.2, which contains the answer. Exercise 1.17 (The hyperbolic disk) Let D D fz 2 C j jzj < 1g be the open unit disk in C. We endow the disc D with the hyperbolic Riemannian metric ds D

jdzj : 1  jzj2

As in the case of the half-plane, this defines a distance disthyp , that we again call hyperbolic. 1. If ˛ 2 R and a 2 D, show that the transformation f˛;a W z 7! exp.i˛/

za 1  az

preserves the disk D. 2. Show that the set of transformations f˛;a forms a group acting transitively on D. 3. Check that jdf˛;a .z/j jdzj D I 1  jf˛;a .z/j2 1  jzj2 in other words, check that the hyperbolic metric is invariant under the group of transformations f˛;a . Exercise 1.18 (Geodesics in the disk model) 1. Let  W Œ0; 1 ! D be a piecewise C1 curve joining 0 2 D to a point r 2 Œ0; 1Œ D. Show that the radial projection j j W Œ0; 1 ! Œ0; 1Œ D is again a piecewise C1 curve joining 0 to r and that its hyperbolic length is no greater than that of  . 2. Deduce from the first question that the unique curve minimizing the  hyperbolic  length between 0 and r is the ray Œ0; r of length Argth.r/ D 12 log 1Cr . 1r 3. Recall that a homography z 2 C [ f1g 7!

az C b 2 C [ f1g; cz C d

ad  bc ¤ 0;

1.7 Exercises

27

sends a circle to a circle in the Riemann sphere C [ f1g and preserves the intersection angle between two circles. Deduce that the geodesics of D are precisely the arcs of circles orthogonal to the unit circle or the diameters. Exercise 1.19 (Hyperbolic distance and cross-ratio) Recall that the cross-ratio of four distinct point x, y, z, t in C [ f1g is defined as Œx W y W z W t D

zxty zytx

and that for any homography f , we have Œ f .x/ W f .y/ W f .z/ W f .t/ D Œx W y W z W t: 1. Deduce from the previous exercise that disthyp .0; r/ D

1 j logŒ1 W 1 W 0 W rj: 2

2. More generally, show that if z0 and z1 are two distinct points in D, there exists a unique geodesic passing through z0 and z1 , and that if u and v are the endpoints of this geodesic on the unit circle we have disthyp .z0 ; z1 / D

1 j logŒu W v W z0 W z1 j: 2

Exercise 1.20 (From the disk to the half-plane) Show that the transformation 1 sends the disk D biholomorphically to the half-plane H, that it transports z 7! zi the hyperbolic metric on D to the hyperbolic metric on H, and that it conjugates (in the group of all homographies) the group of transformations f˛;a to the group PSL.2; R/. Exercise 1.21 Show that the group generated by the symmetry z 7! z and all of the real homographies is the group of all isometries of H. Exercise 1.22 Show that the geodesics in H are precisely the hyperbolic lines introduced at the beginning of § 1.2. Exercise 1.23 1. Check directly, with the help of (1.9), that the measure d is invariant under the action of the group G. 2. Show that such an invariant measure is unique up to scalar multiples. Exercise 1.24 (Area of hyperbolic triangles) By a triangle in H or D we understand three distinct points joined by three geodesic segments. Since the two models H and D of the hyperbolic plane are conformal with the Euclidean metric jdzj, the hyperbolic angles are the same as the Euclidean angles.

28

1 Introduction

1. Show that every ideal triangle in H or D (those having all three vertices on R [ f1g or on the unit circle) has area . 2. Consider a triangle T.˛/ having one angle equal to ˛ 2 Œ0;  and whose two other vertices are at infinity (lying in R [ f1g or on the unit circle). Show that area.T.˛ C ˇ// D area.T.˛// C area.T.ˇ//  : 3. Let F.˛/ D  area.T.˛//. Show that there exists a constant c such that F.˛/ D c˛ (˛ 2 Œ0; ). Deduce from the value of F./ that for every ˛ 2 Œ0;  one has area.T.˛// D   ˛: 4. Now consider a triangle T.˛; ˇ;  / whose three vertices are properly in the hyperbolic plane H or D. By extending the sides to the boundary at infinity (R [ f1g or the unit circle), one obtains an ideal hexagon. Each vertex of the triangle determines two isometric copies of T.˛/, T.ˇ/, T. /, respectively. Deduce that the area of the hexagon is 2Œ.  ˛/ C .  ˇ/ C .   /  2 area.T.˛; ˇ;  //: By cutting up this hexagon into ideal triangles, deduce finally that area.T.˛; ˇ;  // D   .˛ C ˇ C  /: Exercise 1.25 (Area of hyperbolic polygons) If P is a polygon in H with n sides and angles ˛1 ; : : : ; ˛n we put A. P/ D .n  2/ 

n X

˛i :

iD1

1. Show that A is invariant under isometries and that if we cut up P into two polygons P1 and P2 along a geodesic, we have A. P/ D A. P1 / C A. P2 /: 2. Show that A. P/ goes to 0 as the diameter of P goes to 0. 3. By mimicking the construction of the Lebesgue measure in the plane, show that A defines a measure in the hyperbolic plane which is invariant under isometries. 4. From the uniqueness of such a measure show that there is a constant c > 0 such that   n X area. P/ D c ..n  2/  ˛i / : iD1

1.7 Exercises

29

Exercise 1.26 (Discrete subgroups of PSL.2; R/) Show that a subgroup   PSL.2; R/ is discrete if and only if n ! I (n 2  ) implies n D I for sufficiently large n. Exercise 1.27 (Proper discontinuous actions) Let  be a subgroup of PSL.2; R/. Show that the following statements are equivalent. 1. The group  acts properly discontinuously on H. 2. For every compact set K  H, the set f 2  j .K/ \ K ¤ ¿g is finite. 3. Every point x in H admits a neighborhood V such that if .V/ \ V ¤ ¿ then .x/ D x. Exercise 1.28 (Fuchsian cyclic groups) 1. Show that every cyclic group generated by a hyperbolic or parabolic transformation is Fuchsian. 2. Show that a cyclic group generated by an elliptic element is Fuchsian if and only if it is finite. Exercise 1.29 Show that if  is a Fuchsian group then it acts properly discontinuously on H. Exercise 1.30 Show that the polygon D in Fig. 1.3 is a fundamental domain for the group p PSL.2; Z/, that i is the unique elliptic fixed point of order 2 and that  D .1 C i 3/=2 and  C 1 are the only elliptic fixed points of order 3.

Chapter 2

Arithmetic Hyperbolic Surfaces

As we recalled in the introduction, there are several ways to construct Fuchsian groups of the first kind. Of all of these groups, the most important from the number theoretic viewpoint are the arithmetic groups. The general definition of these groups is a bit technical so we content ourselves for the moment with describing a family of examples: the arithmetic groups coming from a quaternion algebra over Q. Before describing them, we begin by proving several general results concerning the space of lattices.

2.1 The Space of Lattices A lattice in Rn is a discrete subgroup isomorphic to Zn . In this section we study the set Ln of lattices in Rn . We endow Rn with the Euclidean norm j  j for which the canonical basis .e1 ; : : : ; en / is orthonormal. We define two fundamental invariants of a lattice L 2 Ln : 1. its height H D H.L/ D minv2Lf0g fjvjg, and 2. its volume V D V.L/, i.e., the Euclidean volume of the paralleletope subtended by a basis of L. The group GL.n; R/ acts transitively (on the left) on the set Ln . Denote by L0 D Zn the standard lattice in Rn generated by the canonical basis. A lattice L generated by n (linearly independent) vectors v1 ; : : : ; vn is the image of L0 under the action of the invertible matrix whose columns are the vi . The stabilizer of the Z-module L0 in GL.n; R/ is the subgroup GL.n; Z/ consisting of matrices that are invertible in Mn .Z/. Note that the determinant of such a matrix is necessarily invertible in Z and is therefore equal to ˙1. Finally, the set Ln is naturally identified with the quotient GL.n; R/=GL.n; Z/. This identification induces a topology on Ln : the quotient topology. By definition, a sequence Lm of lattices in Rn converges, in the quotient © Springer International Publishing Switzerland 2016 N. Bergeron, The Spectrum of Hyperbolic Surfaces, Universitext, DOI 10.1007/978-3-319-27666-3_2

31

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2 Arithmetic Hyperbolic Surfaces

topology, towards a lattice L of Rn if and only if there exists a basis . f1m ; : : : ; fnm / of Lm which converges towards a basis . f1 ; : : : ; fn / of L. The lattice L0 corresponds to the identity class in GL.n; R/=GL.n; Z/; it will serve as a convenient base point for the space Ln . Recall that the determinant of n vectors v1 ; : : : ; vn relative to the canonical basis is equal to the Euclidean volume of the paralleletope they subtend. Thus, if L D g.L0 / with g 2 GL.n; R/, we have V.L/ D j det gj. The set Ln is not compact: the volume can explode or the height can tend towards 0. The following theorem tells us that these are the only two ways that a family of lattices can degenerate. Theorem 2.1 (Hermite-Mahler criterion) A subset M  Ln is relatively compact if and only if there exist constants " > 0 and C > 0 such that (

H>" V 1; N WD fg 2 GL.n; R/ j g  In is strictly upper triangularg and; Nt WD fn 2 N j jnij j 6 t for 1 6 i < j 6 ng with t > 0: According to the Iwasawa decomposition of GL.n; R/ – which boils down to the orthonormalization procedure of Schmidt – the map .k; a; n/ 7! kan is a homeomorphism of K  A  N onto GL.n; R/. We denote by Ss;t the Siegel domain Ss;t D KAs Nt

2.1 The Space of Lattices

33

and  D SL.n; Z/: One knows that N is a closed subgroup of GL.n; R/, homeomorphic to Rm (m D n.n  1/=2) by the map W n 7! .nij /16i 2= 3, t > 1=2. Proof Let g be in GL.n; R/ and L D g.L0 /. We shall proceed by induction on n. We begin by remarking that given a linear subspace V of Euclidean space Rn , the quotient space Rn =V can be identified with the orthogonal of V in Rn and, as such, is itself naturally a Euclidean vector space. One can then define, by induction on n, the notion of an admissible family of vectors in L. This is a family . f1 ; : : : ; fn / of vectors in L such that • f1 is a vector in L  f0g of minimal norm, • the images fP2 ; : : : ; fPn of f2 ; : : : ; fn in the lattice LP WD L=Zf1 of the Euclidean space P and Rn =Rf1 form an admissible family of L, • the vectors fi are of minimal norm among the vectors of L whose image in LP is fPi . (When n D 1, we pay no attention to the last two conditions.) It is a standard fact that the Z-module L admits an admissible family . f1 ; : : : ; fn / and that this family is in fact a Z-basis. Multiplying g on the right by an element of  , if necessary, we can assume that for every i D 1; : : : ; n we have gei D fi . We show by induction on n that if g 2 GL.n; R/ sends the canonical basis of Rn to an admissible family of vectors of the lattice g.L0 / then g 2 S2=p3;1=2 . Write g D kan. As .k1 f1 ; : : : ; k1 fn / is an admissible basis of k1 L, we can assume that k D In . One then has g D an, so that f1 D a11 e1 f2 D a22 e2 C a11 n12 e1 ::: fi D aii ei C ai1;i1 ni1;i ei1 C    C a11 n1i e1 : Denote by Rn1 the subspace of Rn orthogonal to Rf1 D Re1 . This is the Euclidean subspace of Rn generated by e2 ; : : : ; en . The linear transformation g W Rn ! Rn defines on the quotient by Rf1 a linear transformationg W Rn1 ! Rn1 which, by the definition of an admissible family, sends the canonical basis .e2 ; : : : ; en / to an admissible family of the quotient lattice L=Zf1 obtained by the orthogonal projection of L onto Rn1 . By the induction hypothesis, we therefore

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2 Arithmetic Hyperbolic Surfaces

have jnij j 6 1=2 for 2 6 i < j 6 n and p aii 6 2= 3 aiC1;iC1

for 2 6 i 6 n  1:

It remains to show p jn1j j 6 1=2 for 2 6 j 6 n and a11 6 2= 3a22 : The first inequality is a consequence of j fj j2 6 jfj C pf1 j2 ;

8p 2 Z:

By subtracting the left-hand side from the expanded right-hand side and factoring out by a211 (which is non-zero!), we deduce that for all p 2 Z, p2 C 2n1;j p > 0: This forces n1j to be less than 1=2 in absolute value. The second inequality comes from jf1 j2 6 jf2 j2 , as this, when written out, is 2 a11 6 a222 C a211 n212 6 a222 C 1=4a211. t u p Proof of Theorem 2.1 Fix s > 2= 3 and t > 1=2. It is clear that a subset M of R is relatively compact if and only if there exists a compact subset S of Ss;t such that M  S  L0 WD fgL0 j g 2 Sg. We first show the forward direction of the equivalence stated in Theorem 2.1. Let us fix 0 < r < R such that for every g D kan in S and for every i D 1; : : : ; n we have r 6 aii 6 R. Then j det gj D

n Y

aii 6 Rd

iD1

and min jgvj > r;

v2Zn f0g

for if v D

P` iD1

mi ei 2 Zn with m` ¤ 0, then jgvj > jhke` ; gvij D jhe` ; anvij D a`` jm` j > r:

2.2 Quaternion Algebras and Arithmetic Groups

35

We now show the converse direction. Let S D fg 2 Ss;t j gL0 2 Mg. It follows from Lemma 2.2 that M is contained in S  L0 . But for every g D kan in S we have • a11 D jge1 j > ", • Q aii 6 saiC1;iC1 for i D 1; : : : ; n  1, n • iD1 aii 6 C. We deduce that there exists R > r > 0 such that, for every g D kan in S and for all t u i D 1; : : : ; n, we have r 6 aii 6 R. Thus S is compact, as is M.

2.2 Quaternion Algebras and Arithmetic Groups Let F be an arbitrary field and let a and b be two non-zero elements of F. The corresponding quaternion algebra over F is the ring Da;b .F/ D fx0 C x1 i C x2 j C x3 k j x0 ; : : : ; x3 2 Fg; where • addition is defined in the obvious way, so as to form a vector space of dimension 4 over F; • the map x 7! x C 0i C 0j C 0k is an (injective) ring homomorphism from F into Da;b .F/ whose image – again written F – is contained in the center of Da;b .F/. In other words, x˛ D ˛x;

for all x 2 F; ˛ 2 Da;b .F/I

• multiplication is determined by the relations i2 D a; j2 D b; ij D k D ji: The reduced norm of ˛ D x0 C x1 i C x2 j C x3 k 2 Da;b .F/ is Nred .˛/ D x20  ax21  bx22 C abx23 :

(2.1)

The conjugate of ˛ is ˛ D x0  x1 i  x2 j  x3 k; so that Nred .˛/ D ˛˛ D ˛˛. One defines the trace of ˛ by tr.˛/ D ˛ C ˛ D 2x0 :

(2.2)

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2 Arithmetic Hyperbolic Surfaces

The terminology naturally comes from the famous example D1;1 .R/ of Hamilton’s quaternions. On the other hand the map D1;1 .R/ ! M2 .R/, which on basis vectors is defined by 1 7! I2 ;

  1 0 i 7! ; 0 1

  01 j 7! ; 10



 0 1 k 7 ! ; 1 0

extends in a unique way to a ring isomorphism. This will be justified at the beginning of the proof of Theorem 2.3. One calls a quaternion algebra D a division algebra if every non-zero element ˛ 2 D admits an inverse. This happens if and only if Nred .˛/ ¤ 0 for all ˛ ¤ 0, in which case ˛ 1 D ˛=Nred .˛/. Henceforth we fix two positive integers a and b. We can then naturally speak of the subring O WD Da;b .Z/ in Da;b .Q/. Our goal is to prove the following theorem. Theorem 2.3 1. There exists a group isomorphism from Da;b .R/1 WD SL.1; Da;b .R// D fg 2 Da;b .R/ j Nred .g/ D 1g to G D SL.2; R/. 2. The image a;b in G of the group O1 D SL.1; Da;b .Z// via this isomorphism is a Fuchsian group of the first kind. 3. The following statements are equivalent: a. a;b is cocompact in G; b. .0; 0; 0/ is the unique solution in integers of the Diophantine equation x2  ay2  bz2 D 0; c. Da;b .Q/ is a division algebra. 4. Two subgroups a;b and a0 ;b0 are commensurable in G if and only if the quadratic forms x2  ay2  bz2 and x2  a0 y2  b0 z2 are similar over Q. Two subgroups  and  in G are said to be commensurable in G if, conjugating  in G if necessary, the group  \  is of finite index in both  and . Recall furthermore that two quadratic forms over a field of characteristics ¤ 2 are said to be equivalent if a change of basis over the field transforms one into the other and are said to be similar if they are equivalent up to a non-zero scalar multiple (in the base field). According to Property 4 of Theorem 2.3, there exist infinitely many commensurability classes of discrete cocompact subgroups in G. Indeed, if p and q are two distinct prime numbers congruent to 1 modulo 4, the groups p;p and q;q are not commensurable, for otherwise there would exist M 2 GL.3; Q/ and  2 Q such that 0 D  tMM, with  D diagf1; p; pg, 0 D diagf1; q; qg. We can therefore assume that det M D ˙q=p (and  D 1). Let p˛ n be the least common multiple of

2.2 Quaternion Algebras and Arithmetic Groups

37

the denominators of the coefficients of M; we have p˛ nM D .mij / with mij 2 Z, and ˛ > 1. The first equation obtained in identifying the coefficients of 0 and  tMM is .np˛ /2 D m211  pm221  pm231 : Thus m11  0 .mod p/ and m221 C m231  0 .mod p/. Since 1 is not a square modulo p, we necessarily have m21  m31  0 .mod p/. Likewise, all of the coefficients of p˛ nM are divisible by p. Therefore p˛1 nM 2 M3 .Z/, in contradiction with the choice of p˛ n. Proof of Theorem 2.3 The linear map ˚ W Da;b .R/ ! M2 .R/ defined on basis vectors by ˚.1/ D I2 ; ˚.i/ D

p     p  a 0 a 0 01 p ; ˚.j/ D p ; ˚.k/ D 0  a b a 0 b0

is bijective. Moreover, ˚ preserves multiplication, making it a ring isomorphism. Finally, if ˛ D x0 C x1 i C x2 j C x3 k 2 Da;b .R/, we have p p p p det.˚.˛// D .x0 C x1 a/.x0  x1 a/  .x2 C x3 a/.bx2  bx3 a/ D x20  ax21  bx22 C abx23 D Nred .˛/: Thus ˚.Da;b .R/1 / D SL.2; R/, the first claim of the theorem is proved, and  a;b D

p p ˇ x0 C x1 a x2 C x3 a ˇ x0 ; x1 ; x2 ; x3 2 Z; p p : ˇ 2 b.x2  x3 a/ x0  x1 a x0  ax21  bx22 C abx23 D 1

The proofs of the other claims will be taken up in the following subsections.

t u

2.2.1 An Exceptional Isomorphism From now on we denote by Da;b the real quaternion algebra Da;b .R/, remembering that this algebra is defined over Q (and even over Z) and that we can therefore speak of its rational (or integral) points. Let P D f˛ 2 Da;b j tr.˛/ D 0g be the set of pure quaternions in Da;b . This is a subspace of Da;b defined over Q and isomorphic to R3 . The reduced norm restricts to P as the quadratic form q D ax21  bx22 C abx23 . Lemma 2.4 There is a rational isomorphism '

Da;b .Q/ =Q ! SO.q; Q/

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which induces an isomorphism on the real groups1 D1a;b =f˙1g ! SO0 .q/: Proof It suffices to construct the first isomorphism. For that we consider the action of the group Da;b .Q/ on Da;b .Q/ by interior automorphisms. For ˛ 2 Da;b .Q/ and ˇ 2 Da;b .Q/ we write S˛ .ˇ/ D ˛ˇ˛ 1 : We verify immediately the following properties: • S˛ preserves the reduced norm, • .S˛ /jQ D IdQ , and • S˛ .P/ D P. Let s˛ D .S˛ /jP . Then s W Da;b .Q/ ! O.q/ is a homomorphism whose kernel is Q \ Da;b .Q/ D Q . Before continuing we make a few preliminary calculations. For ˇ 2 P such that q.ˇ/ ¤ 0, the reflection ˇ associated with ˇ is given by the formula ˇ .x/ D x  2

q.x; ˇ/ ˇ q.ˇ; ˇ/

.x 2 P/:

From the definition of q D .Nred /jP we deduce that ˇ .x/ D x 

xˇ C ˇx ˇˇ

1

ˇ D ˇxˇ ;

and since x and ˇ are both pure quaternions we find ˇ .x/ D ˇxˇ 1

.x 2 P/:

Already we see that if ˇ is the rotation of angle  with axis ˇ, we have ˇ D ˇ D sˇ . Let u 2 O.q; Q/. We can write u as a product of reflections, u D ˇ1    ˇr for ˇ1 ; : : : ; ˇr 2 P, with q.ˇi / ¤ 0, the integer r being even if u is in SO and odd otherwise. We therefore have u.x/ D .1/r ˇ1    ˇr x.ˇ1    ˇr /1

for all x 2 P:

We denote by SO0 .q/ Š SO0 .2; 1/ the connected component of the identity of the special orthogonal group of the quadratic form q on P.

1

2.2 Quaternion Algebras and Arithmetic Groups

39

We now continue with the argument and show that s takes values in SO.q/. Let ˛ 2 Da;b .Q/ and, arguing by contradiction, suppose that s˛ 2 O 3 .q/. Under this hypothesis we would then have, for every x 2 P, s˛ .x/ D ˛x˛ 1 D ˛ 0 x.˛ 0 /1 ;

where ˛ 0 D ˇ1    ˇr :

In other words, x D ˇxˇ 1 , for every x 2 P, with ˇ D ˛ 1 ˛ 0 . As ˇxˇ 1 D x for every x 2 Q, we would then have, for every x in Da;b .Q/: x D ˇxˇ 1 , with ˇ 2 Da;b .Q/ . But x 7! x is an anti-automorphism and x 7! ˇxˇ 1 an automorphism – contradiction. Summarizing, we have a homomorphism s W Da;b .Q/ ! SO.q; Q/, and since the rotations of angle  are in the image and these generate the special orthogonal group, s is surjective. t u Henceforth we denote the isomorphism of Lemma 2.42 by  W D1a;b =f˙1g ! SO0 .q/: The group SO0 .q/ is naturally embedded in GL.3; R/ (take the natural coordinates .x1 ; x2 ; x3 / of P). We let SO0 .q; Z/ be the intersection SO0 .q/ \ GL.3; Z/. The proof of Lemma 2.4 implies that O1 =f˙1g D  1 .SO0 .q; Z//:

(2.3)

The subgroup O1 is therefore discrete in D1a;b and its image, a;b , by ˚ is equally a discrete subgroup of G. Let gij , 1 6 i; j 6 3, be the matrix entries of an element g 2 O.q/  GL.3; R/. By a rational representation of the linear group O.q/ on Rn we mean a homomorphism

W O.q/ ! GL.n; R/ such that for every pair of integers . ; / 2 Œ1; n2 , there exists a polynomial P ; 2 QŒXij j1 6 i; j 6 3 such that for every g 2 O.q/,

.g/ ; D P ; .gij /; where .g/ ; denotes the . ; / matrix entry of .g/ 2 GL.n; R/. Lemma 2.5 Let be a rational representation of the linear group O.q/ on Rn . Then there exists a finite index subgroup of O.q; Z/ which, via the action induced by , leaves invariant the standard lattice Zn .

For a D b D 1 the map obtained in composing  and ˚ 1 induces an isomorphism between PSL.2; R/ and SO0 .2; 1/. This is one of the isomorphisms – called exceptional – which exist only between certain “small” Lie groups (see [57, pp. 518–520]).

2

40

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Proof The coefficients of the polynomials P ; defined by .g/ ; D P ; .gij / are rational. The same is true for the polynomials Q ; defined by

.g/ ;  ı ; D Q ; .gij  ıij /: But the latter have vanishing constant term, for .I3 / D In . Thus if m is the common denominator of all their coefficients, we have Q ; .gij ıij / 2 Z as soon as gij ıij  0 .mod m/, i.e., g  I3 .mod m/. For such a choice of g the coefficients .g/ ; are integral, which assures the invariance of Zn under .g/. Finally, these elements form a finite index subgroup of O.q; Z/ since the quotient injects in GL.n; Z=mZ/. t u

2.2.2 A Subset of the Space of Lattices The map  passes to the quotient as an embedding (continuous for the quotient topologies) of D1a;b =O1 into the space of lattices L3 D GL.3; R/=GL.3; Z/. Denote by M the image of this embedding. The subset M is also the orbit of the standard lattice L0 under the action by left multiplication of the subgroup SO0 .q/  GL.3; R/ on L3 . Lemma 2.6 The subset M is closed in L3 . Proof Let L be a lattice in L3 and assume that there exists a sequence fLi g of lattices in M which converge to L. Each Li is the image by an element of the group SO0 .q/ of the standard lattice L0 D Z3 . The quadratic form q therefore takes integer values on each one. Since it is continuous, the form q must also take integer values on L. Consider now a basis .e1 ; e2 ; e3 / of L. Since each convergent sequence of integers is stationary, there must exist a lattice Li0 in the sequence fLi g with a basis . f1 ; f2 ; f3 / sufficiently close to the basis .e1 ; e2 ; e3 / such that • q. fi / D q.ei / for i D 1; 2; 3, and • q. fi ˙ fj / D q.ei ˙ ej / for i; j D 1; 2; 3. We thus have B.ei ; ej / D B. fi ; fj /, where B is the underlying symmetric bilinear form of the quadratic form q. Taking g 2 GL.3; R/ to satisfy g.ei / D fi , we see that the lattice Li0 can be sent to L by a matrix in GL.3; R/ which leaves the quadratic form q invariant. It follows that the lattice L is contained in the orbit of the standard lattice under the action of the group O.q/. The group SO0 .q/ being of finite index (equal to 4) in O.q/, we conclude that L 2 M. t u

2.2.3 The Cocompactness Criterion The reduced norm being multiplicative, the algebra Da;b .Q/ is a division algebra if and only if .0; 0; 0; 0/ is the unique integral solution of the Diophantine equation

2.2 Quaternion Algebras and Arithmetic Groups

41

x20  ax21  bx22 C abx23 D 0. In this case we shall show that the subset M of L3 is compact.

The Subset M Is Compact According to the preceding subsection, it suffices to show that M is relatively compact. For this we shall apply the Hermite-Mahler criterion. Let us begin by noticing that every lattice L in M is of volume 1. It suffices therefore to verify that the height is bounded uniformly from below on M. Now by hypothesis .0; 0; 0/ is the only integral solution to the Diophantine equation q.x1 ; x2 ; x3 / D 0. The set U D fx 2 R3 j jq.x/j < 1g is therefore an open neighborhood of 0 in R3 intersecting the standard lattice L0 D Z3 only at the origin. But M is equal to the orbit of L0 in L3 under the action of the group SO0 .q/, and since SO0 .q/ preserves the quadratic form q, the latter takes the same set of values on each of the lattices contained in M. The intersection of the open set U with any lattice belonging to M is therefore always reduced to the point 0 2 R3 . In other words, the height stays uniformly bounded from below on M. We may then apply the criterion of Hermite-Mahler to deduce that the subset M of L3 is relatively compact. We have thus shown that the lattice a;b is cocompact in G whenever Da;b .Q/ is a division algebra. Proof of the converse direction Suppose now that there exists an integral solution ¤ .0; 0; 0/ to the Diophantine equation q.x1 ; x2 ; x3 / D ax21 bx22 Cabx23 D 0. Let us show that the quadratic form q is similar over Q to the quadratic form x21  x22 C x23 . Indeed, there exists w 2 P.Q/ – a vector of P with rational coordinates – such that the orthogonal complement w? of w with respect to q contains a non-zero qisotropic rational vector u. Write  D 1=q.w/ and q0 D q. We have q0 .w/ D 1 and q0 .u/ D 0. Since qjw? is non-degenerate, there exists a rational vector v 2 w? such that WD q0 .u; v/ ¤ 0. Note that the expression q0 .v C tu/ D q0 .v/ C 2tq0 .u; v/ is linear in t and has a zero t 2 Q. Replacing v by vCtu if necessary, we can suppose that q.v/ D 0. If x and y are in Q, q0 .xu C yv/ D 2 xy D



x C 2 y 2

2

 

x  2 y 2

2

:

In the basis .w; u  v=.2 /; u C v=.2 // the quadratic form q0 is therefore equal to x21  x22 C x23 .

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Lemma 2.7 If Da;b .Q/ is not a division algebra, the group a;b is commensurable with the group SL.2; Z/. In particular, it is Fuchsian of the first kind and noncocompact. Proof Assume that Da;b .Q/ is not a division algebra. There exists an integral solution .x0 ; x1 ; x2 ; x3 / ¤ .0; 0; 0; 0/ to the Diophantine equation x20  ax21  bx22 C abx23 D 0. We first prove that there exists an integral solution ¤ .0; 0; 0/ to ax21  bx22 C abx23 D 0. If x0 D 0, there is nothing to show. Suppose therefore that x0 ¤ 0. The integers ax21 and bx22 cannot be both the zero. Assume for example that ax21 is non-zero. Since x20  ax21  bx22 C abx23 D 0, we obtain x20  bx22 D a.x21  bx23 /. Then a simple calculation shows that .b.x0 x3 C x1 x2 /; a.x21  bx23 /; x0 x1 C bx2 x3 / is an integer solution to ax21  bx22 C abx23 D 0. If the latter is zero, then ax21 C abx23 D 0 and .x1 ; 0; x3 / is a non-zero integer solution (since x1 ¤ 0) to ax21  bx22 C abx23 D 0. According to the paragraph preceding Lemma 2.7, the quadratic forms q D ax21  bx22 C abx23 and q0 D x21  x22 C x33 are similar over Q. Since multiplying a quadratic form by a non-zero scalar does not change the associated orthogonal group, the groups O.q/ and O.q0 / are conjugate by a matrix belonging to GL.3; Q/. Conjugation by a rational invertible matrix being a rational representation, Lemma 2.5 implies that the image of the group O.q; Z/ in O.q0 / is commensurable with the group O.q0 ; Z/. The isomorphism between SO0 .q0 / and PSL.2; R/ then implies that the groups a;b and 1;1 are commensurable in G. The result follows by observing that 1;1 is commensurable with the group SL.2; Z/. u t The group a;b is therefore commensurable with SL.2; Z/ if and only if ax2  by C abz2 is similar to x2  y2 C z2 . On the other hand 2

ab.ax2  by2 C abz2 / D b.ax/2  a.by/2 C .abz/2 ; so that the quadratic forms ax2  by2 C abz2 and x2  ay2  bz2 are themselves similar. The three first claims of Theorem 2.3 are thus proved.

2.2.4 Commensurability Classes Although not directly used in the rest of the text, the fourth and final claim of Theorem 2.3 will be established in this subsection. In order to be brief, we freely use more advanced algebraic concepts and results; appropriate references are quoted at the end of this chapter. Proofs requiring more background have been put in small typeface.

2.3 Arithmetic Hyperbolic Surfaces

43

We begin by remarking that it follows from the proof of Lemma 2.5 that if the quadratic forms x2  ay2  bz2 and x2  a0 y2  b0 z2 (and hence ax2  by2 C abz2 and a0 x2  b0 y2 C a0 b0 z2 ) are similar, the groups a;b and a0 ;b0 are commensurable. We must now prove the converse. Assume then that the groups a;b and a0 ;b0 are commensurable. Then there exists an R-linear map ˛ W R3 ! R3 which sends the quadratic form q D ax2 by2 Cabz2 to q0 D a0 x2  b0 y2 C a0 b0 z2 and, by conjugation, a finite index subgroup   SO.q; Z/ to a finite index subgroup  0  SO.q0 ; Z/. Let   O.q/ (resp.  0  O.q0 /) be the extension by f˙I3 g of the group  (resp.  0 ). Lemma 2.8 The Q-vector subspace generated by  in M3 .R/ is equal to M3 .Q/. Proof The Q-vector subspace generated by  is a subalgebra of M3 .Q/. According to the Burnside lemma [76, Cor. 3.4] it suffices to show that the space Q3 is a simple Q -module. We first show that the group  is dense in O.q/ Š O.1; 2/ with respect to the Zariski topology. In other words, every polynomial in the matrix entries of matrices in M3 .R/ which vanishes on  should also vanish on O.q/. This can be deduced from the following facts: 1. The closure (in the Zariski topology) of  in O.q/ is a subgroup of O.1; 2/ which is not abelian. 2. The group SO.q/ is minimal among the non-abelian subgroups of O.q/ which are closed with respect to the Zariski topology. The first statement is immediate. The second statement follows from the fact that the group SO.q/ is a connected algebraic group. Its Lie algebra is isomorphic to the Lie algebra of SL.2; R/ which does not contain any proper non-abelian sub Lie algebra. Since  contains f˙I3 g it follows from the preceding facts that  is dense in O.q/ Š O.1; 2/ with respect to the Zariski topology. Now, since the natural representation of O.1; 2/ in R3 is irreducible over C, the t u representation of  in GL.3; C/ is also irreducible.

End of the proof of Theorem 2.3 Since ˛ sends  onto  0 , ˛ sends Q onto Q 0 . But according to Lemma 2.8, we have Q D M3 .Q/, which implies that ˛ D ˛0 , where ˛0 is defined over Q and  2 R . Thus ˛0 sends q to 2 q0 and, since q ¤ 0, the scalar 2 is in Q. This shows that the forms q and q0 are similar over Q, completing the proof of the proof of Theorem 2.3. t u Remark 2.9 Theorem 2.3 implies in particular that there always exists an infinity of solutions to the Diophantine equation x20  ax21  bx22 C abx23 D 1.

2.3 Arithmetic Hyperbolic Surfaces Hyperbolic surfaces are associated with discrete subgroups of SL.2; R/ that are torsion free – those containing no non-trivial element of finite order. We now explain how to eliminate torsion if we have it.

44

2 Arithmetic Hyperbolic Surfaces

2.3.1 Eliminating Torsion The process of eliminating torsion is based on the following theorem of Minkowski. Theorem 2.10 Let n be an integer > 2 and F a finite subgroup of GL.n; Z/. Then F injects in GL.n; Z=`Z/ for every integer ` > 3. Proof We consider the standard embedding Zn  Rn , so that the group F identifies as a subgroup of GL.n; R/. By averaging any choice of positive definite quadratic form on Rn over F we obtain a norm invariant under the action of F. We may normalize so that the non-zero points of Zn closest to 0 are at distance 1; let V  Rn be the subspace generated by these points. By renormalizing the orthogonal complement V ? , we may again assume that the points of the lattice Zn in Rn  V closest to the origin are at distance 1. Since F preserves V and V ? , the resulting quadratic form is still F-invariant. Continuing in this way we obtain an F-invariant norm on Rn such that 1. every vector in Zn is of norm greater than or equal to 1, and 2. the points in Zn of norm 1 generate the space Rn . Now let  2 F be such that  is sent to the identity in GL.n; Z=`Z/ for some integer ` > 1. Viewed as a matrix in Mn .Z/,   In 2 `Mn .Z/. If  ¤ In , there exists an element x 2 Zn of norm 1 such that  x ¤ x. But  x  x is then a non-zero vector 2 `Zn and is thus of norm > `. On the other hand,  x and x both belong to the sphere3 of radius 1. They are thus at distance at most 2 from one another and ` is necessarily less than or equal to 2. t u Let a and b be two positive integers. Given a positive integer N we define the N-th principal congruence subgroup of a;b as the subgroup a;b .N/  a;b given by the image in a;b of the subgroup fx 2 O1 j x  1 2 NDa;b .Z/g  O1 : Note that a;b .N/ is contained in the image in SL.2; R/ of the subgroup   SO0 .q; Z/ \ Ker GL.3; Z/ ! GL.3; Z=NZ/ under the map ˚ ı  1 . We keep the notation from the proof of Theorem 2.3: the quadratic form q is equal to ax2  by2 C abz2 . Corollary 2.11 For every pair of integers .a; b/ > 1 and for every integer N > 3, the subgroup a;b .N/  SL.2; R/ is torsion free. The quotient Xa;b .N/ WD a;b .N/nH

3

Recall that the norm is F-invariant.

2.3 Arithmetic Hyperbolic Surfaces

45

is therefore a hyperbolic surface of finite area. It is compact if and only if the quadratic form x2  ay2  bz2 does not represent 0 over Q.   Proof The subgroup SO0 .q; Z/ \ Ker GL.3; Z/ ! GL.3; Z=NZ/ is torsion free since, according to Theorem 2.10, the kernel   Ker GL.3; Z/ ! GL.3; Z=NZ/ .N > 3/ contains no finite subgroup. The group a;b .N/ (N > 3) is therefore itself torsion free and is clearly of finite index in a;b . Theorem 2.3 then implies Corollary 2.11. t u An arithmetic surface is any surface admitting a cover isometric to a finite cover of one of the surfaces Xa;b .N/ defined in Corollary 2.11. Note that the fourth claim of Theorem 2.3 implies that Xa;b .N/ and Xa0 ;b0 .N 0 / admit two finite isometric covers if and only if x2 ay2 bz2 and x2 a0 y2 b0 z2 are similar as quadratic forms over Q.

2.3.2 The Modular Surface and Its Covers The fundamental example of an arithmetic surface is the modular surface. As we recalled in the introduction, the subset D D fz D x C iy 2 H j jxj < 1=2 and jzj > 1g of the upper half-plane H is a fundamental domain for the action of the modular p group  .1/ D SL.2; Z/. Moreover, i is an elliptic vertex of order 2,  D .1Ci 3/=2 is an elliptic vertex of order 3, 1 is the only cusp (up to equivalence) and the genus of X.1/ is 0. We may add to the above example all of the congruence covers of the modular surface. Let N be an integer > 1. The principal congruence subgroup of level N, denoted  .N/, is the subgroup of the modular group made up of matrices congruent to the identity modulo N, i.e.,    ˇ 10 ˇ .mod N/ :  .N/ D  2 SL.2; Z/ ˇ   01 Theorem 2.12 The group  .N/ is normal in  .1/ D SL.2; Z/ and of index N D Œ .1/ W  .N/ D N 3

Y .1  p2 /: pjN

(2.4)

46

2 Arithmetic Hyperbolic Surfaces

The group  .N/ is of index ( N D

N =2 if N > 2 N

if N D 2

in PSL.2; Z/. It is torsion free as soon as N > 1 and the number of cuspidal equivalence classes is hN D

N : N

Proof Let N > 1. We begin with the following result. Lemma 2.13 The homomorphism W SL.2; Z/ ! SL.2; Z=NZ/ is surjective. Proof Let us show more generally that the homomorphism W SL.m; Z/ ! SL.m; Z=NZ/ is surjective. We first assume that N is a prime power ps . It suffices to show that the group SL.m; Z=ps Z/ is generated by the transvection matrices Im C Eij , i ¤ j, where Eij is the m  m matrix all of whose coefficients are zero except in the i-th row and j-th column, where it is 1. Indeed, the transvections are clearly in the image of . We prove this by induction on m. The case m D 1 is trivial. Assume then that the statement is true for m  1 where m > 1. Let A 2 Mm .Z/ be such that det.A/  1 .mod ps /. The first column of A contains at least one coefficient c which is invertible mod ps ; we shall use it as a pivot for row operations – these are realized by multiplying A by transvection matrices – until we are brought to a matrix whose first column is the vector .1; 0; : : : ; 0/. By operating next on the columns of this new matrix we are led to assume that ! 1 0 AD 0 A0 with A0 2 Mm1 .Z/ and det.A0 /  1 .mod ps /. The result easily follows by induction.

2.3 Arithmetic Hyperbolic Surfaces

47

WeQdeduce the case for arbitrary N from the Chinese Remainder Theorem. If N D k pskk is the prime power decomposition of N for distinct primes pk , we have Z=NZ Š

Y .Z=pskk Z/; k

GL.2; Z=NZ/ Š

Y

GL.2; Z=pskk Z/;

k

SL.2; Z=NZ/ Š

Y

SL.2; Z=pskk Z/:

k

Q From this it follows that the transvection Im C . k¤` pskk /Eij is congruent to the identity modulo pskk for k different from ` and congruent to the transvection Im C Eij modulo ps`` . t u Consider now the kernel K of the surjective morphism GL.2; Z=pskk Z/ ! GL.2; Z=pk Z/: Since K is made up of matrices of M2 .Z=pskk Z/ which are congruent to the identity 4.s 1/ matrix I2 mod pk , the cardinality of K is pk k . Let us calculate the cardinality of GL.2; Z=pk Z/. This group is made up of matrices with coefficients in Z=pk Z, such that the column vectors are linearly independent. There are p2k  1 choices for the first column vector and, once the first has been fixed, there are p2k  pk remaining choices for the second (we exclude the pk multiples of the first vector). The cardinality of GL.2; Z=pk Z/ is therefore equal to .p2k  1/.p2k  pk /. 4.s 1/ The cardinality of GL.2; Z=pskk Z/ is therefore equal to pk k .p2k pk /.p2k 1/ D 4sk s 2 pk .1  p1 k /.1  pk /. Now SL.2; Z=pk Z/ is the kernel of the surjective morphism det W GL.2; Z=pskk Z/ ! .Z=pskk Z/ . The cardinality of SL.2; Z=pskk =pskk Z/ is then k 2 equal to p3s k .1  pk /. As a consequence Œ .1/ W  .N/ D N 3

Y .1  p2 /: pjN

The matrix I2 belongs to the group  .N/ if and only if N D 2. We deduce ( Π.1/ W  .N/ D

N =2; if N > 2; N ;

if N D 2:

Let us now show that if N > 1 the group  .N/ has no elliptic elements. The only fixed points of  .1/ D PSL.2; Z/ in the fundamental hyperbolic triangle D are the two vertices of angle =3 and the point i. The elliptic conjugacy classes in

48

2 Arithmetic Hyperbolic Surfaces

PSL.2; Z/ are therefore represented by   0 1 ; 1 0



 0 1 ; 1 1

  1 1 : 1 0

None of these elements is congruent to the identity modulo N if N > 1. Since  .N/ is normal in  .1/ we obtain, as claimed, that  .N/ is torsion free. Now, if s is a cusp, s is  .1/-equivalent to 1. But

 .N/1

ˇ   1m ˇ m 2 Z ;  .1/1 D ˙ ˇ 0 1 ˇ   1 mN ˇ D  .N/ \  .1/1 D ˙ ˇm2Z ; 0 1

so that Œ .1/1 W  .N/1  D N. In this way  .N/ has exactly N =N equivalence classes of cusps. t u The surfaces X.N/ D  .N/nH are therefore “true” hyperbolic surfaces as soon as N > 1. These are the covers – ramified over the elliptic points – of the modular surface X.1/. The degree of this cover is equal to N . The area of X.N/ is then equal to N times the area of X.1/, namely 8 3 < N Q 2 pjN .1  p /; if N > 2; area.X.N// D 6 : 2; if N D 2:

(2.5)

The surface X.N/ (N > 1) has hN D N =N cusps. We obtain a fundamental domain for the action of  .N/ on H by taking the union of N translates of the triangle D. This induces a triangulation of X.N/ by N triangles. Each of these triangles has two vertices of angle =3 around which are glued 6 triangles and then another vertex of angle 0 around which are glued N triangles. There are therefore N . 13 C N1 / vertices, 32 N sides and N faces. The Euler characteristic of X.N/ is equal to  N

 1 1  : N 6

Now the Euler characteristic is also equal to 2 2gN , where gN is the genus of X.N/. So we obtain 8 2; N (2.6) gN D 24N :0; if N D 2:

2.4 Commentary and References

49

Every subgroup of the modular group containing  .N/ is called a congruence subgroup. The following example is particularly important, 

  ˇ  ˇ 0 .N/ D  2 SL.2; Z/ ˇ   .mod N/ : 0

2.4 Commentary and References § 2.1 The Iwasawa decomposition is discussed in detail in [93, p. 44].

§ 2.2 In general there are two possibilities for a quaternion algebra D over a field F of characteristic zero: either D is a division algebra, or D Š M2 .F/, the algebra of 2  2 matrices over F. See for example [88, Th. 2.1.7] for a proof of this result which we do not use here. The general notion of arithmetic group is a bit technical; the reader can consult [99]. The classification of arithmetic groups described in this reference implies, in particular, that every arithmetic subgroup of SL.2; R/ defined over Q is commensurable to one of the subgroups constructed in Theorem 2.3. One can more generally replace Q by a totally real number field. The construction is similar and treated in [67, 88]. The proof of Theorem 2.3 that we give here is slightly different from that which one usually finds in the literature, as for example in [67]. We reduce the argument to the study of arithmetic subgroups of SO.2; 1/ via the exceptional isomorphism between PSL.2; R/ and SO0 .2; 1/. This way of constructing Fuchsian groups is essentially the same used by Poincaré in his article “Les fonctions fuchsiennes et l’arithmétique,” which appeared in 1887. The groups obtained in this way are in fact the first examples of Fuchsian groups that Poincaré succeeded in constructing. He also used the geometry of the space of lattices L3 previously studied by Hermite, to whom we owe the proof of Theorem 2.1 when n D 3. Before that the geometry of the space L2 had been completely understood by Gauß in his Disquisitiones arithmeticae which contains a very clear exposition of the action of the modular group on the upper half-plane H. The construction of infinitely many commensurability classes of discrete cocompact subgroups in SL.2; R/ is borrowed from [88, pp. 87–88]. Through the exceptional isomorphism the geometry of the orthogonal group of a nondegenerate quadratic form comes into play. A convenient reference on this subject is the book [96]. In particular, [96, Th. 2.6] gives a proof of the fact that the rotations of angle  generate the special orthogonal group. The classification of commensurability classes relies on notions in algebraic geometry and algebraic groups. The books [97] and [16] can serve as good introductions to these notions.

50

2 Arithmetic Hyperbolic Surfaces

§ 2.3 Our study of congruence subgroups of the modular group follows Shimura’s exposition of the topic in [121, §1.6]. One can also have a look at the excellent book of Miyake [92]. These two references contain general computations on the cardinality of equivalence classes of elliptic fixed points of order 2 and 3 for the groups 0 .N/. We have assumed familiarity with the Euler characteristic of a surface and the Euler phi function. For an introduction to these notions the reader can refer to [92, 121] and Serre’s Course in Arithmetic [118], respectively.

2.5 Exercises p Exercise 2.14 Show that the group SL.2; ZŒ 2/ is not a Fuchsian group. Exercise 2.15 1. Show that the quotient SL.2; Z/nSL.2; R/, endowed with the quotient topology, is homeomorphic to the unit tangent bundle of the modular surface SL.2; Z/nH. (This question is answered in § 9.2.) 2. Using the explicit description of the fundamental domain for SL.2; Z/ given in § 1.4.1, deduce from the preceding question that a subset of volume 1 lattices in R2 is relatively compact if and only if the height is uniformly bounded from below on this set by a positive constant. 1 Exercise 2.16 When n D 2, compare the set D of § 1.4.1 and the orbit Ss;t i of the 1 point i 2 H under the action of a Siegel set. Deduce that D.1/  Ss;t i as soon as p s > 2= 3, t > 1=2.

Exercise 2.17 Taking inspiration from the proof of Theorem 2.3, show that there exist infinitely many integer solutions .m; n/ to the “Pell-Fermat” Diophantine equation m2  an2 D 1; where a is an integer such that

p a … N.

Exercise 2.18   N0 . Show that I2 2 0 .N/ and that 1. Let ˛ D 0 1 0 .N/ D ˛ 1  .1/˛ \  .1/:

2.5 Exercises

51

Fig. 2.1 Degree 6 cover of the modular surface

2. Show that the group 0 .N/= .N/ is isomorphic to the subgroup of upper triangular matrices in SL.2; Z=NZ/. Deduce that it is of order N.N/, where  is the Euler totient function which associates with a positive integer N the cardinality of the group of invertible elements in Z=NZ. 3. Deduce from Question 2 that the index of 0 .N/ in  .1/ is given by Π.1/ W 0 .N/ D N

Y .1 C p1 /: pjN

4. Show that the number h of equivalence classes of cusps for 0 .N/ is equal to the number of double classes in 0 .N/n .1/= .1/1 . 5. Let MN be the set of elements .a; c/ 2 .Z=NZ/2 of order N. Show that the map from  .1/ to the set MN defined by   ab 7 ! .a; c/  cd induces a bijective map from 0 .N/n .1/= .1/1 onto MN = , where  is the equivalence relation on MN given by .a; c/  .a0 ; c0 / if .a0 ; c0 /  ˙.ma C nc; m1 c/

.m 2 .Z=NZ/ ; n 2 Z=NZ/:

6. Deduce from the two preceding questions that hD

X djN; d>0

where  is the Euler phi function.

 .gcd .d; .N=d/// ;

52

2 Arithmetic Hyperbolic Surfaces

Exercise 2.19   ab 2 0 .8/. 1. Let cd a. Show that a C d ¤ 0 (use the fact that 1 is not a square modulo 8). b. Show that a and d are odd and thus that a C d ¤ ˙1. c. Deduce that ja C dj > 2 and thus that the image of the group 0 .8/ in PSL.2; R/ is a torsion free subgroup. 2. Deduce from the first question and the preceding exercise that the quotient 0 .8/nH is a non-singular hyperbolic surface of genus 0 with 4 cusps, which is a degree 12 cover of the modular surface. The union of two adjacent polygons in Fig. 2.1, generated by Arnaud Chéritat, forms a fundamental domain for the group 0 .8/.

Chapter 3

Spectral Decomposition

The goal of this chapter is to prove the Spectral Theorem (Theorem 1.6).

3.1 The Laplacian Every element g 2 G D SL.2; R/ defines a linear operator Lg on functions f W H ! C by the formula .Lg f /.z/ D f .g1 z/. A linear operator T on functions f W H ! C is said to be invariant if it commutes with the G-action: Lg T D TLg ;

for all g 2 G:

According to (1.20), the Laplacian  is invariant. All of the invariant differential operators are important but chief among them is the Laplacian, which plays a particular role: a diffeomorphism of H is an isometry if and only if it preserves the Laplacian. In the z D x C iy coordinates of the upper half-plane, the hyperbolic Laplacian is given by  D y

2



@2 @2 C @x2 @y2

 :

(3.1)

According to (1.21), the Laplacian is expressed in the .r; / polar coordinates of the upper half-plane as D

@2 1 @2 1 @   : @r2 tanh r @r .2 sinh r/2 @ 2

(3.2)

Harmonic analysis on H – a hyperbolic analog of the Euclidean Fourier theory – consists in spectrally decomposing the Laplacian. © Springer International Publishing Switzerland 2016 N. Bergeron, The Spectrum of Hyperbolic Surfaces, Universitext, DOI 10.1007/978-3-319-27666-3_3

53

54

3 Spectral Decomposition

The space of functions in question is the Hilbert space L2 .H/ of (classes of) square-integrable functions f W H ! C equipped with the Hilbert space inner product Z h f ; gi D f .z/g.z/ d .z/: H

Here is the hyperbolic area measure, see (1.16). Let H be a separable Hilbert space, i.e., one which admits a countable orthonormal basis. Notice that, if D  H is a fundamental domain for  whose boundary is of zero measure, then the Hilbert space L2 . nH/ is isomorphic to L2 .D/; in particular it is separable. We denote by jj the Hilbert space norm on H. A linear map T W H ! H is a bounded operator if there exists a constant C such that jTxj 6 Cjxj for all x 2 H. The smallest possible constant C is called the operator norm of T, and we write it as jTj. The linear map T is a bounded operator if and only if it is continuous. Since the L2 functions on H are not all differentiable, a differential operator is only defined on a dense subspace of the Hilbert space L2 .H/. We thus need a generalization of the notion of differential operator. We agree to refer to any linear transformation defined on a dense subspace of H as an operator. In other words, an operator is a pair .T; DT / consisting of a dense vector subspace DT  H, called the domain of T, and a linear transformation T W DT ! H. The operator is closed if its graph f. f ; Tf / j f 2 DT g is a closed subspace of the product H  H. It is unbounded if it is not continuous (with respect to the topology on DT induced by that of H).1 In practice one often speaks of “the operator” T, dropping the dependence on the domain of definition; if done in context, this should not lead to any confusion. The Laplacian  is well-defined on the subspace Cc1 .H/ (dense in L2 .H/) consisting of C1 function of compact support in H, although it is not continuous with respect to the L2 topology: the Laplacian is an unbounded operator. An operator .T; DT / is said to be symmetric if hTf ; gi D hf ; Tgi

(3.3)

for all f ; g 2 DT , where h ; i is the Hilbert space inner product on H. Given an operator .T; DT / on H, we write DT  for the space of all g 2 H such that f 7! hTf ; gi is a continuous linear form on DT . For such a g, the corresponding linear form can be extended by continuity to all of H, and one obtains in this way a bounded linear form on H. A classical theorem of Riesz (see, for example, [18, p. 61]) states that every continuous linear form on a Hilbert space H is of the form f 7! hf ; hi for a certain element h 2 H. Thus, if g 2 DT  , there exists a unique element T  g 2 H such that hTf ; gi D hf ; T  gi. The operator .T  ; DT  / is called the adjoint of T. If T is symmetric then DT  DT  .

1

A continuous operator .T; DT / on DT naturally extends to a bounded operator of H to itself.

3.1 The Laplacian

55

The operator T is said to be self-adjoint if DT D DT  and T D T  . A self-adjoint operator is in particular symmetric; it is also closed. For a bounded operator, it is one and the same to be symmetric or to be self-adjoint. An eigenvector of T for the eigenvalue  is a non-zero vector f 2 H such that Tf D f . The scalar  being fixed, the set of eigenvectors of eigenvalue  is called the -eigenspace. If T is a symmetric operator, all of its eigenvalues are real and the eigenspaces corresponding to distinct eigenvalues are orthogonal. From now on we denote by  the unbounded operator .; Cc1 .H// whose underlying Hilbert space is L2 .H/. Let e D

@2 @2 C 2 2 @x @y

be the usual Euclidean Laplacian.2 Let d be the exterior derivative which sends differential 1-forms to 2-forms. Let f and g be two C1 functions defined on a bounded region ˝  C with boundary a C1 (or piecewise C1 ) curve @˝. Define   @f =@x : rf D @f =@y Then we have

   @f  @f dy  dx D .ge f C rf  rg/ dx ^ dy d g @x @y

(3.4)

so that

    @g @f  @g  @f D .ge f  f e g/ dx ^ dy: dy  dx  f dy  dx d g @x @y @x @y

Stokes’ Theorem then implies Green’s formula: Z ˝

.ge f  f e g/ dx ^ dy Z D

    @g @f  @g  @f dy  dx  f dy  dx ; g @x @y @x @y @˝

(3.5)

where the integral along @˝ is taken in the clockwise direction.3

2

The usual Euclidean Laplacian is negative definite. Throughout this text the hyperbolic Laplacian that we consider is the geometric Laplacian, which is positive definite. This accounts for the difference in sign here.

3 The two interior parentheses on the right-hand side of (3.5) are the “normal derivatives” which arise in the general Green’s formula on a Riemannian manifold.

56

3 Spectral Decomposition

Proposition 3.1 The Laplacian  is a symmetric operator on L2 .H/ with domain Cc1 .H/. Proof Let f and g be two C1 functions with support in H. According to Green’s formula (3.5), Z H

.ge f  f e g/ dx ^ dy D

 Z    @g @f  @g  @f dy  dx  f dy  dx ; g @x @y @x @y C

where C is any contour which completely encircles the union of the supports of f and g. The integral on the right-hand side being clearly zero, we find Z

Z ge f dx ^ dy D

H

H

f e g dx ^ dy:

(3.6)

Now one has Z hf ; gi D

H

.f /g

dx ^ dy D y2

Z ge f dx ^ dy; H

and it therefore follows from (3.6) that  is symmetric.

t u

The aim of this chapter is to decompose the Laplacian acting on the Hilbert space L2 . nH/, where  is a Fuchsian group of the first kind. Before we get to that we exhibit – in the next section – a few important eigenfunctions  on H.

3.2 Eigenfunctions of the Laplacian on H Although it is not directly useful for what follows, it is important to point out that the Laplacian is an elliptic operator and hence a weak solution f , locally in L2 , of the equation f D f

. 2 C/

(3.7)

is necessarily C1 – and moreover is real analytic – see, for example, [107, p. 212] for an introduction to the theory of elliptic operators and a proof of their regularizing nature. It is natural therefore to look for solutions to (3.7) among the functions f 2 C1 .H/.

3.2 Eigenfunctions of the Laplacian on H

57

In the z D x C iy coordinates of the upper half-plane, we begin by looking for a solution f of (3.7) which we require to be a function of y only, i.e., constant in x. For s ¤ 1=2, we land immediately on the following two linearly independent solutions: 1 s .y C y1s / 2

and

1 .ys  y1s /; 2s  1

(3.8)

where s.1  s/ D . When s D 1=2, so that  D 1=4, we can take the limiting values of the above expressions to obtain y1=2

and y1=2 log y;

(3.9)

respectively. Of course if s ¤ 1=2, one can obviously take the simpler pair .ys ; y1s /. If one just wants f to be 1-periodic in x, the change of variables f .z/ D e.nx/F.2ny/ (n ¤ 0) reduces the study of the Eq. (3.7) to that of the ordinary differential equation in F F 00 .y/ C .y2  1/F.y/ D 0:

(3.10)

There are two linearly independent solutions to this equation, namely .2 1 y/1=2 Ks1=2 .y/  ey and .2y/1=2 Is1=2 .y/  ey ; as y ! C1, where K .y/ and I .y/ are the standard Bessel functions, see Appendix B. If we assume that f doesn’t grow too fast in the vertical strip fz 2 H j j Re.z/j 6 1=2g; – more precisely, that f .z/ D o.e2y / as y ! 1 – then the second solution is excluded and f .z/ must be a scalar multiple of Ws .nz/. Here Ws denotes the Whittaker function: Ws .z/ D 2y1=2 Ks1=2 .2y/ e.x/:

(3.11)

Note that these functions are cuspidal, meaning that for every y > 0, the average Z

1 0

Ws .z/ dx

is zero. Whittaker functions play the role of the exponential functions in the hyperbolic analog of Fourier inversion. We will not elaborate upon this since the

58

3 Spectral Decomposition

harmonic analysis of interest to us will take place on quotients of the hyperbolic plane by discrete groups of symmetries  under which the Whittaker functions are not in general invariant.

3.2.1 Radial Functions Let F W H ! C be a radial function, i.e., a function depending only on the distance from a fixed point z 2 H. Assume furthermore that F satisfies the equation F D F for a certain complex number . Since, in polar coordinates .r; / about z, the function F does not depend on , it follows from (3.2) that F is a solution to the second-order ordinary differential equation F 00 .r/ C coth rF 0 .r/ C F.r/ D 0:

(3.12)

Given a function f on H, one defines the radialization of f about a point z 2 H, written fzrad , by: Z fzrad .w/ D

f .Tw/ dT; Sz

where Sz is the isotropy subgroup of z in G, and dT is the normalized Haar measure on Sz . We remark that the groups Sz are pairwise conjugate and can be identified with the special orthogonal group SO.2/. It is clear that the function fzrad is radial – about z – and satisfies fzrad .z/ D f .z/. Moreover,  being an invariant operator, it commutes with the radialization. In particular, the radialization of an eigenfunction is again an eigenfunction. The function ys D Im.z/s is an eigenfunction. Let us calculate its radialization Fs in polar coordinates .r; / about the point i and relative to the vector pointing towards infinity. The point z.r; / with coordinates .r; / being .cos er i C sin /=. sin er i C cos /, we find: Fs .r/ D

1 

1 D 

Z Z

 0  0

.Im z.r; //s d (3.13) s

.cosh r C sinh r cos 2 / d :

By construction, the function z 7! Fs . .z; i// is a solution to (3.7) for  D s.1  s/. Since in polar coordinates the function Fs is independent of , it is a solution to (3.12).

3.2 Eigenfunctions of the Laplacian on H

59

Equation (3.12) is completely solved – for  real or complex – by the classical theory. From it we may extract the following proposition.4 Proposition 3.2 Let  D s.1  s/ 2 C. The space of solutions of (3.12) on RC is 2-dimensional and Fs is the unique solution extending continuously to 0 by 1. Proof The 2-dimensionality of the solution space is classical. Moreover, by construction Fs is a solution of (3.12), and it is clear that Fs extends continuously to 0 by 1. To prove the uniqueness, we construct a second solution and show that it does not extend continuously to 0. It will be useful to transform the radial coordinate r by the change of variables u D .cosh r  1/=2. In these new coordinates, Eq. (3.12) becomes u.u C 1/G00 .u/ C .2u C 1/G0 .u/ C s.1  s/G.u/ D 0:

(3.14)

We have . C s.1  s// s1 .1  /s1 . C u/s D s

d s  .1  /s . C u/s1 ; d

(3.15)

where  is the Laplacian applied to the u variable (the variable being constant). Upon integrating (3.15) by parts, we obtain another solution to Eq. (3.12) (or, rather, (3.14)): Gs .u/ D

1 4

Z

1 0

..1  //s1 . C u/s d:

(3.16)

It follows from the following lemma that the function Gs tends toward infinity as u tends toward 0. We obtain in particular a solution which is linearly independent of Fs , and the proposition is proved. t u Lemma 3.3 The integral (3.16) converges absolutely for s in the half-plane Re.s/ > 0. It therefore defines a function u 7! Gs .u/ on RC satisfying Eq. (3.14). Moreover, Gs .u/ D

1 1 log C O.1/; 4 u

as u ! 0.

Proposition 3.2 implies in particular that the functions Fs and F1s are equal for s 2 Œ0; 1; we shall in general assume that s 6 1=2.

4

60

3 Spectral Decomposition

Proof The fact that Gs .u/ satisfies Eq. (3.14) follows immediately from (3.15). Put  D .jsj C 1/u,  D .jsj C 1/1 and D Re.s/. We have Z 4Gs .u/ D



1 0

.1  /  Cu

s1

d D  Cu

Z



Z



C

0

Z

1

C





where Z



 Z D O u

0



0

 1 d

 D O.1/

and Z

1

Z

1

DO





.1  /

1

 d

D O.1/:

For the remaining integral we use the expansion 

.1  /  Cu

s1

s1    u C 2 u C 2 D 1 D1CO uC uC

to obtain Z

 

Z D





D log

d CO  Cu

Z

 

u C 2 d .u C /2



1 uC C O.1/ D log C O.1/: uC u

t u

3.3 Invariant Integral Operators on H Let k W   1; C1Π! C be an even C1 -function. By letting5 k.z; w/ D k. .z; w//, with z; w 2 H, we obtain a C1 -function of z and w which depends only on the distance between them. One calls such a function a point-pair invariant.6 Lemma 3.4 Let k.z; w/ be a point-pair invariant. Then z k.z; w/ D w k.z; w/:

(3.17)

Here the subscript indicates the variable on which one applies the Laplacian.

5

Recall that .z; w/ is the hyperbolic distance between two points z and w in H.

Here the term “invariant” comes from the fact that for every g 2 G and for all z; w 2 H we have k.gz; gw/ D k.z; w/.

6

3.3 Invariant Integral Operators on H

61

Proof In polar coordinates about the point w, we obtain, since k.z; w/ D k.r/, z k.z; w/ D k00 .r/  coth rk0 .r/: But in the polar coordinates centered about z, we obtain the same expression for w k.z; w/. u t A point-pair invariant k defines an invariant integral operator: Z Tk W f 7!

H

k.; w/f .w/ d .w/;

(3.18)

where d is the Riemannian measure. In what follows we assume that the functions k and f are C1 and are chosen in a way so that the integral in (3.18) converges absolutely; concerning this latter assumption, one loses very little if one simply takes k to be of compact support. Theorem 3.5 The invariant integral operators commute with the action of the Laplacian: Tk  D Tk : Proof Let f 2 Cc1 .H/ (the domain of ). According to Proposition 3.1, Z

Z H

k.z; w/.w f .w// d .w/ D

H

.w k.z; w//f .w/ d .w/:

Then, from Lemma 3.4, the integral on the right is equal to Z H

.z k.z; w//f .w/ d .w/:

t u

Lemma 3.6 For  2 C and z 2 H, there exists a unique function ! .z; w/ in w, radial about the point z and such that 1. ! .z; z/ D 1, and 2. w ! .z; w/ D ! .z; w/. In fact, one has ! .z; w/ D Fs . .z; w//; where  D s.1  s/. Proof Let F.r/ D ! .z; w/ where r D .z; w/. The function F satisfies Eq. (3.12). According to Proposition 3.2 the normalization F.0/ D ! .z; z/ D 1 forces F D Fs . t u

62

3 Spectral Decomposition

We remark that if  2 Œ0; 1=4, then s 2 Œ0; 1. Since, according to Proposition 3.2, we have Fs D F1s , we can assume that s 6 1=2. When s < 1=2 the integral Z

 0

.1 C cos.2 //s d

is absolutely convergent. It then follows from the explicit expression (3.13) of Fs that the function ! .z; w/ grows as 1

p

.const/es .z;w/ D .const/e 2 .1

14/ .z;w/

:

(3.19)

Theorem 3.7 Every eigenfunction of  is also an eigenfunction of the invariant integral operators. More precisely, for every compactly supported C1 point-pair invariant k, there exists a function h W C ! C such that if f 2 C1 .H/ is a eigenfunction with eigenvalue  2 C then Tk f D h./f , i.e., Z H

k.z; w/f .w/ d .w/ D h./f .z/:

(3.20)

Proof Let f 2 C1 .H/ be an eigenfunction of the Laplacian with eigenvalue . For all z 2 H, the function fzrad is radial about z and satisfies fzrad .z/ D f .z/. It then follows from Lemma 3.6 that fzrad .w/ D ! .z; w/f .z/: Let us first show that Z Z k.z; w/f .w/ d .w/ D k.z; w/fzrad .w/ d .w/: H

H

Indeed, Z

Z H

k.z; w/fzrad .w/ d .w/ D

Z f .Tw/dT

k.z; w/ d .w/ Z

H

D

Sz

Z dT

Z

Sz

D

Z

H

dT Z

Sz

D

Z

H

dT Z

Sz

D H

H

k.z; w/f .Tw/ d .w/ k.z; T 1 w/f .w/ d .w/ k.z; w/f .w/ d .w/

k.z; w/f .w/ d .w/:

(3.21)

3.3 Invariant Integral Operators on H

63

Here we have written Sz for the isotropy subgroup of z in G and dT for the normalized Haar measure on Sz . We therefore have Z Z k.z; w/f .w/ d .w/ D k.z; w/fzrad .w/ d .w/ H

H

Z

D f .z/

H

k.z; w/! .z; w/ d .w/:

Since k is of compact support, the integral Z h./ WD H

k.z; w/! .z; w/ d .w/

(3.22)

is well-defined. Finally, for a given , this integral doesn’t depend on z: this follows from the transitivity of the G-action on H and the fact that for g 2 G, Z

Z H

k.gz; w/! .gz; w/ d .w/ D Z

H

D H

k.z; g1 w/! .z; g1 w/ d .w/ k.z; w/! .z; w/ d .w/:

t u

The converse of Theorem 3.7 is also true, in the follows precise sense. Theorem 3.8 If f is an eigenfunction for all of the invariant integral operators for which the kernel k belongs to Cc1 .R/, then f is an eigenfunction for . Proof Let f be an eigenfunction for all of the invariant integral operators whose kernel k belongs to Cc1 .R/. (Then f is necessarily C1 .) If f is constantly equal to 0 there is nothing to show. Assume then that f is nonzero at at least one point. It is not difficult to show – for example, by choosing a sequence of smooth approximations to the delta function [18, p. 70]7 – that there exists a point-pair invariant k 2 Cc1 .R/ such that Eq. (3.20) is satisfied with h./ ¤ 0. By applying  to Eq. (3.20), we find Z H

z k.z; w/f .w/ d .w/ D h./f .z/:

But z k.z; w/ is again the compactly supported C1 kernel of an invariant integral operator, and by the hypothesis on f the above integral is therefore equal (as a function of z) to a scalar multiple of f . Since h./ ¤ 0 Theorem 3.8 is proved. u t

7

One could just as well refer to the proof of Lemma 3.29.

64

3 Spectral Decomposition

The condition in Theorem 3.7 that k be of compact support is not essential. It is nevertheless necessary to impose a decay at infinity condition. For example, it will suffice to assume that there is a real number ı > 0 such that jk. /j D O.e .1Cı/ /:

(3.23)

This is a natural condition: one wants the integral Z Z C1 k.z; w/ d .w/ D k. / sinh d

H

0

to converge absolutely.

3.4 The Selberg Transform In this section we seek to calculate the function h of Theorem 3.7. To this end, it is convenient to introduce the function U defined by U.cosh / D k. /;

(3.24)

  jz  wj2 D k.z; w/: U 1C 2 Im z Im w

(3.25)

so that

The function U W Œ1; C1Œ ! C then satisfies: U.t/ D O.t.1Cı/ /. Since  D s.1  s/ takes on all complex values as s runs through C, we can view the function h as a function of a complex parameter r related to s by the equation s D 1=2 C ir. Note that y D Im z. Recall that the function y1=2Cir (r 2 C) is an eigenfunction of the Laplacian  on H with eigenvalue .r2 C 1=4/. By abuse of notation we write again h.r/ D h./ D h.r2 C 1=4/I we’re looking to calculate h.r/ as a function of U. The formula (3.20) applied to the eigenfunction z 7! y1=2Cir implies that Z

k.z; i/y1=2Cir d .z/:

h.r/ D

(3.26)

H

The function h is the Selberg transform S.f / of the function f W z 7! k.z; i/. With the help of (3.22) it may be rewritten as Z S.f /.r/ D

H

f .z/!r2 C1=4 .i; z/ d .z/:

(3.27)

3.4 The Selberg Transform

65

The Selberg transform (3.27) is well-defined if f 2 Cc1 .H/. Nevertheless, on the hyperbolic circle fz 2 H j .z; i/ D g we have8 y D Im z 6 e . Under the rapid decay condition (3.23) of k, the integral (3.26) is therefore absolutely convergent as long as 1 C ı0; 2

j Im rj
1

This can be easily seen if one recalls that this particular hyperbolic circle is a Euclidean circle whose maximal ordinate is on the imaginary axis.

9

Note the change in convention relative to the introduction.

66

3 Spectral Decomposition

defines a compactly supported C1 -function and this transformation is invertible via the formula: ˚.x/ D 

1 

Z

C1 x

dQ.t/ p : tx

Proof First note that Z

C1

˚.x C  2 / 2d D 2 

Q.x/ D 0

Z

C1 0

˚.x C  2 /d:

The function Q is then clearly C1 and of compact support, with derivative Q0 .x/ D 2

Z

C1

0

˚ 0 .x C  2 /d:

(3.31)

We then find that Z ˚.x/ D  Z

C1 0 C1

D 0

D D

4  4 

2 D  D D

1  1 

Z

dŒ˚.x C r2 / 2˚ 0 .x C r2 / rdr

=2

Z

0

Z

0

C1

0

Z

C1

0

Z

x

Z

C1

˚ 0 .x C r2 / rdr d

C1 0

C1

0

Z

C1

˚ 0 .x C 2 C  2 / d d

Q0 .x C 2 / d .d’après (3.31)/ d.Q.x C 2 //  dQ.t/ : p tx

t u

In particular, 1 k.argch t/ D  p 2

Z

C1 t

dg.argch x/ ; p xt

3.5 A Family of Examples: The Heat Kernel

67

and the formula (3.29) inverts to 1 k. / D  p 2 1 D p 2

Z Z

C1 cosh

C1

dg.argch x/ p x  cosh

dg.u/ p : cosh u  cosh

We admit the following proposition, which we include for completeness but which will not be used in the text. Let10 PW.C/ be the space of even holomorphic functions r 7! h.r/ on C such that there exists a real number R (the type of h) with the property that for every N 2 N, ! eRj Im.r/j : h.r/ D ON .1 C jrj/N Proposition 3.11 The Selberg transform S is a bijection of Cc1 .H/ onto PW.C/ whose inverse is given by Z C1 1 1 .S h/.z/ D h.r/!r2 C1=4 .i; z/r tanh.r/ dr: 2 0 Moreover, if h 2 PW.C/ is of type R, then f D S 1 h 2 C1 .H/ has support in the hyperbolic disc of center i and radius R. We already remarked that it is not necessary to assume that k is of compact support in Theorems 3.7 and 3.8. The hypothesis (3.23) is not optimal; it is more convenient to give a condition on h. The most natural one is to ask for the existence of an " > 0 such that h is an even holomorphic function defined on the strip B" D fr 2 C j j Im.r/j < 12 C "g and satisfying h.r/ D O..1 C jrj2 /1" /:

3.5 A Family of Examples: The Heat Kernel Let p W H  H  .0; C1/ ! R be a continuous function of class C2 in the first two variables .z; w/ and of class C1 in the last variable t. One calls p a fundamental solution to the heat equation on H if 1. @p=@t D z p, 2. p.z; w;Rt/ D p.w; z; t/, and 3. limt#0 H p.z; w; t/f .w/ d .w/ D f .z/, locally uniformly in z and for every compactly supported continuous function f .

10

The notation “PW” is in reference to the classical Paley-Wiener theorem, see [106].

68

3 Spectral Decomposition

Our goal now is to construct a fundamental solution to the heat equation on H. This amounts to constructing the kernel – a function of t – which to a pair of points .z; w/ in H associates the value p.z; w; t/. Denote this kernel by k. We shall first proceed formally as a way of guessing an expression for k; then we verify rigorously that this kernel furnishes a fundamental solution to the heat equation on H. As a function of t we can differentiate k. We denote by kP D @k=@t the differentiated kernel. We would like for k to satisfy the equation kP D z k:

(3.32)

Upon integrating (3.32) against an eigenfunction of the Laplacian with eigenvalue .1=4 C r2 /, and using the fact that  is symmetric, one deduces from Theorem 3.7 that the Selberg transform h of k satisfies the ordinary differential equation 1  P h.r/ D C r2 h.r/: 4

(3.33)

Moreover, condition 3 forces h to tend towards the constant function 1 as t tends toward 0. We then expect to have (at least formally) 2

h.r/ D et=4tr : One would then like to take k.z; w/ D .S 1 h/.z/. Having done so, we continue to R C1 write g for the function (3.29). We then expect that h.r/ D 1 eiru g.u/ du and thus – by Fourier inversion – that g should be of the form Z C1 1 g.u/ D eiru h.r/ dr 2 1 Z p 2 p p p et=4 et.r= / e..r= /.u=2 // d.r= / D p 2  R Z p et=4 2 D p etr e.r.u=2 // dr: 2  R It then follows from Lemma 1.1 that et=4 2 g.u/ D p eu =4t : 2 t By Abel inversion one expects that Z C1 2 d.eu =4t / 1 et=4 p k.z; w/ D  p p 2  2 t .z;w/ cosh u  cosh .z; w/ p Z C1 2 2 ueu =4t t=4 D e p du: .4t/3=2 cosh u  cosh .z; w/

.z;w/

3.5 A Family of Examples: The Heat Kernel

69

All of the above computations are formal but they allow us to “guess” the following theorem. Theorem 3.12 The function pH W H  H  .0; C1/ ! R given by p Z C1 2 2 ueu =4t t=4 e p du pH .z; w; t/ D .4t/3=2 cosh u  cosh .z; w/

.z;w/ is a fundamental solution to the heat equation. Proof It suffices to verify the expression that we “guessed” indeed provides a solution. We write pt .z; w/ D pH .z; w; t/. This is a point-pair invariant which corresponds to an even C1 -function, which we again denote by pt W .1; C1/ ! C. Lemma 3.13   2 pt . / D O t1 e =8t : Proof Let us fix > 0. Then Z

2

2

reu =4t 2 du 6 2 e =4t p cosh u  cosh

2

The inequality xex 6 ex

2 =2

Z

2



du 2 p 6 4 e =4t : .u  / sinh

implies

4 e

2 =4t

DO

p  2 =8t  : te

Next, by using cosh u  cosh > 12 .u  /2 and writing r D u  2 , we find Z

C1 2

p ueu =4t p dr 6 2 2 cosh u  cosh

2

Z

C1

du

2

Z p 2 6 2 2 e =t DO

2 =4t

eu

C1

er

2 =4t

dr

0

p  2 =t  : te

t u

70

3 Spectral Decomposition

This lemma allows us to justify all of the formal computations above. One verifies in particular that pH 2 C1 .H  H  .0; C1// is a solution to the equation11   @ pH .z; w; t/ D 0 z C @t

(3.34)

and that Z

C1 0

pt . / sinh d D 1:

(3.35)

Equation (3.34) implies that pH satisfies the first condition in the definition of a solution to the heat equation. The second condition is satisfied by construction. And finally, since pt is positive and since, according to Lemma 3.13, pt ! 0

as t ! 0;

uniformly on compacta avoiding 0, condition 3 follows from (3.35).

t u

We emphasize the fact that for for every t > 0, the kernel pt .z; w/ D pH .z; w; t/ is a point-pair invariant which satisfies the decay property (3.23); the results of the preceding sections therefore apply to it.

3.6 The Laplacian on  nH We are setting our sites on the spectral decomposition of L2 . nH/, where  is a cocompact Fuchsian group or at least of the first kind. We fix then a Fuchsian group of the first kind  . The Laplacian  defines an unbounded operator on L2 . nH/. In this section we prove that this operator is symmetric and positive. The Riemannian metric on H induces a norm on each tangent space and thus on the space of 1-forms on H. One can therefore speak of a bounded 1-form on H. Lemma 3.14 Let ! be a C1 differential 1-form, bounded and  -invariant on H. Then Z d! D 0:  nH

Proof First assume that S D  nH is compact. R Since !R is  -invariant it comes from a 1-form on S. Stokes’ theorem implies that S d! D @S ! D 0. If  is cocompact, the boundary of S is empty and the lemma is proved.

11

One can just as well verify this directly by using the formula for pH .z; w; t/, but it is a bit tedious.

3.6 The Laplacian on  nH

71

The lemma remains valid when  nH is of finite area since ! is bounded whereas the length of a closed horocycle in  nH tends toward 0 as it approaches infinity. u t Lemma 3.15 Let f and g be two functions in C1 . nH/. The function z 7! .yrf .z//  .yrg.z// is  -invariant. Proof Let  D

a b c d

2  , and let w D .z/ D

azCb czCd .

Then

@w @ @w @ @ @ D C D .cz C d/2 : @z @z @w @z @w @w One deduces from this that .w  w/

@ D @w



 @ cz C d .z  z/ : cz C d @z

(3.36)

Denote by L the differential operator L D iy

@ @ @ C y D .z  z/ : @x @y @z

Equation (3.36) implies  cz C d 1 cz C d

.Lf /

 az C b 

h @ i D .w  w/ f .w/ cz C d @w 

@ f .w/ D .z  z/ @z    az C b  DL f : cz C d

One sees then that Lf and Lg – and hence also yrf and yrg – belong to the space of C1 -functions  on H such that  .z/ D

   az C b cz C d  ; cz C d cz C d

  ab 2 : cd

It follows that the function .yrf /  .yrg/ is  -invariant. Let D. nH/ D ff 2 C1 . nH/ j f bounded and f boundedg:

t u

72

3 Spectral Decomposition

It is clear that D. nH/ is dense in L2 . nH/ and we have already remarked that an eigenfunction is necessarily C1 . By the Laplacian on the surface  nH we mean the (unbounded) operator .; D. nH// on the Hilbert space L2 . nH/. Proposition 3.16 The Laplacian  is an (unbounded) symmetric positive operator on L2 . nH/. Proof Let f and g be two functions in D. nH/. Consider the form !Dg

 @f @x

dy 

@f  dx : @y

The 1-form df is  -invariant; moreover, the 2-form df ^

 @f @x

@f  dx D jyrf j2 d .z/ @y

dy 

is  -invariant by Lemma 3.15. Thus the form ! is  -invariant and we can apply Lemma 3.14 to it. We then deduce from (3.4) that Z Z e hf ; gi D  g fdx ^ dy D .rg/  .rf / dx ^ dy; D

D

where D is a fundamental domain for  . In particular hf ; gi D hf ; gi and Z hf ; f i D

jrf j2 dx ^ dy > 0:

t u

D

We now write grad f D yrf : It follows from Lemma 3.15 that the function grad f  grad g is  -invariant and the proof of Proposition 3.16 shows that Z hf ; gi D

grad f  grad g d .z/:

(3.37)

 nH

Proposition 3.16 implies that the eigenvalue  D s.1  s/ of an eigenfunction f 2 D. nH/ is real and non-negative. Thus, either s D 1=2 C it with t 2 R, or 0 6 s 6 1.

3.7 Integral Operators on  nH

73

3.7 Integral Operators on  nH We begin by consider the case where  is a cocompact Fuchsian group. Such a group operates properly discontinuously on the hyperbolic plane H with a compact fundamental domain D that we fix once and for all. We shall realize the spectral decomposition of the Laplacian  in L2 . nH/ with the help of the invariant integral operators. Recall that such an operator is given by a C1 point-pair invariant k.z; w/ D k. .z; w//;

z; w 2 H;

(3.38)

which induces Z .Tk f /.z/ D

H

k.z; w/f .w/ d .w/

for f W H ! C. If we assume that f is  -invariant (in other words, that f descends to a function f W  nH ! C), we can write Z .Tk f /.z/ D

K.z; w/f .w/ d .w/; D

where the kernel K is given by the series K.z; w/ D K .z; w/ D

X

k.z;  w/:

(3.39)

 2

We call such a kernel an automorphic kernel. We shall want to first take care of all convergence issues. 1 We shall always assume that  the C function k satisfies the decay condi  .1Cı/ , for a certain real number ı > 0. tion (3.23): jk. /j D O e Lemma 3.17 Let z 2 D. For m D 0; 1; : : : , we put m D f 2  j m 6 .z;  z/ < m C 1g: Then m is of size jm j D O.em / as m ! C1. Proof Let d be the diameter of D. The subsets .D/,  2 m , are contained in the disk of center z and radius r D m C 1 C d. The area of this disk is equal to 2.cosh r  1/. Since the subsets are disjoint, one immediately deduces the lemma. t u

74

3 Spectral Decomposition

Lemma 3.18 Let k be a C1 point-pair invariant satisfying (3.23). For every compact subset A  H and for all " > 0 there exists a finite subset    such that X jk.z;  w/j < " for all z; w 2 A:  2 

Proof We fix an integer M > 1 and take  D X

mD0 m .

C1 X

jk.z;  w/j 6

 2 

SM

Then

jm jk.m/:

mDMC1

It then follows from (3.23) and from Lemma 3.17 that there exists a constant c, not depending on M, such that X

jk.z;  w/j 6 c

 2 

C1 X

eım :

mDMC1

t u

The result follows by taking M sufficiently large.

In view of Lemmas 3.17 and 3.18 we can give a legitimate sense to the function K defined by the sum (3.39). Recall that a kernel K is said to be symmetric if K.z; w/ D K.w; z/. Lemma 3.19 The automorphic kernel K is bi- -invariant: K. z; ıw/ D K.z; w/

for all ; ı 2 :

It induces a symmetric kernel K W  nH   nH ! C. Proof Since k.z; w/ depends only on the distance, we have k. z; gıw/ D k.z;  1 gıw/. When g runs through  ,  1 gı runs through  as well, and the bi- -invariance of K follows. The symmetry comes from X  2

k.z;  w/ D

X  2

k. w; z/ D

X

k.w;  1 z/:

t u

 2

We now concern ourselves with the differentiability of the automorphic kernels above. In practice, point-pair invariants depend frequently on an additional real parameter – such is the case, for example, for the heat kernel. The following lemma allows one to study the differentiability with respect to this supplementary parameter. We thus let T denote the space of parameters on which a given automorphic kernel will depend. We shall always take T to be an open interval of the real line.

3.7 Integral Operators on  nH

75

Lemma 3.20 Let k D k. ; t/ W .1; C1/T ! C be an even function in the first variable, C1 (as a function of two variables) and whose partial derivatives satisfy k.n;/ . ; t/ D O.e .1Cı/ / on Œ0; 1/  T for a certain strictly positive constant ı. Then K belongs to C1 . nH   nH  T; C/. Idea of the proof Let U  H be an open disk and let z 2 H be a point at non-zero distance from U. Then  D

C1 [

m

mD0

where m D f 2  j m 6 .z;  z/ < m C 1g. According to Lemma 3.17, jm j D O.em /. As in the proof of Lemma 3.18, it suffices to show that for each partial derivative D, there exists constants depending only on U such that DŒk.z;  w/ D O.em.1Cı/ /;

(3.40)

whenever z; w 2 U and  2 m . Here the notation DŒ: means that the derivative is applied to the function .z; w/ 7! k.z;  w/. This derivative is of the form DŒk.z;  w/ D

 X .DT k.j/ /. .z;  w//Dj Π.z;  w/; jD1

where DT is a partial derivative with respect to T, k.j/ D @j k=@ j , and Dj is a partial derivative of order j with respect to the variables in H  H. The result follows by showing12 that each jDj Π.z;  w/j is uniformly bounded for all z; w 2 U and  2. t u

3.7.1 The Heat Kernel The above results apply notably to the family of examples that we have already considered: the heat kernel. In particular, we obtain the following theorem.

12

This is a fundamental property of hyperbolic geometry.

76

3 Spectral Decomposition

Theorem 3.21 Let p.z; w; t/ be the heat kernel of the hyperbolic plane and let P.z; w; t/ D P .z; w; t/ D

X

p.z;  w; t/:

 2

The function P defines an element in C1 . nH nH.0; C1// which, moreover, is a fundamental solution to the heat equation on  nH. Proof Lemma 3.13, and its proof, implies that the function pt satisfies the hypotheses of Lemma 3.20. From this it follows that P belongs to C1 . nH   nH  .0; C1//. It is immediate that the equation z p D @p=@t implies that z P D @P=@t. The symmetry of P results from Lemma 3.19. Let us now verify the initial condition. Let f W  nH ! R be a continuous function, viewed as a  -invariant function on H. Suppose that the boundary of D – the fundamental domain of  – is of zero measure (this is the case, for example, when D is a Dirichlet domain), then Z XZ P.z; w; t/f .w/ d .w/ D p.z;  w; t/f .w/ d .w/ D

D

 2

D

 2

.D/

XZ

p.z;  w; t/f .w/ d .w/

Z

D H

p.z; w; t/f .w/ d .w/;

and the initial condition follows from the corresponding property for p.

t u

3.7.2 The Non-compact Case Contrary to most of the rest of this chapter, we shall assume in this subsection that  is of the first kind but non-cocompact. The fundamental domain then contains cusps, which complicates the situation. We begin by introducing the space C. nH/ of cuspidal functions, that is, the space of bounded C1 functions f W  nH ! C whose constant term is zero at each cusp. Let us explicate this last condition. Let a be a cusp of the group  . According to Lemma 1.3, there exists an element a 2 G such that the group a1  a has a cusp at infinity and that . a1  a /1 D

 ˇ 1m ˇ ˇm2Z : 0 1

3.7 Integral Operators on  nH

77

In particular, any  -invariant function f W H ! C satisfies     f a 10 m1 z D f . a z/; for all m 2 Z. In other words, f ı a .z C m/ D f ı a .z/. One can therefore expand f ı a in a Fourier series, f . a z/ D

X

fan .y/e.nx/

.z D x C iy/;

(3.41)

n

where the coefficients are given by Z fan .y/ D

1 0

f . a z/e.nx/ dx:

We write simply fa for the function fa0 . If f is C1 , then the series (3.41) converges absolutely and uniformly on compacta. One says that f has a vanishing constant term in each cusp if, for every cusp a of  , the function fa is identically zero. We denote by a the parabolic subgroup of  satisfying a1 a a D . a1  a /1 ; it is the stabilizer of the cusp a in  . In this subsection we assume that the point-pair invariant k is C1 and of compact support. Proposition 3.22 The invariant integral operator Tk associated with k sends C. nH/ to itself. Proof It is clear that Tk sends a function bounded by M to a function bounded by the product Mjjkjj1 AD=2 , where jjkjj1 is the supremum norm of k, D is the diameter of the support of k, and AR is the area of the hyperbolic disc of radius R. Let f 2 C. nH/. We compute the constant term of Tk f at the cusp a of  : Z .Tk f /a .y/ D Z

1 0 1

D Z

0

D

Tk f . a

1 t  01

Z H

k. a

1 t  01

Z

1

k.z; w/ Z

H

D H

z/dt

0

 z; w/f .w/ d .w/ dt

f . a

1 t  01

 w/dt

d .w/

k.z; w/fa .Im.w// d .w/:

Thus, if fa is identically zero, the same is true for .Tk f /a .

t u

78

3 Spectral Decomposition

Now consider the automorphic kernel K.z; w/. Again let D be a fundamental domain for  that we fix once and for all in this subsection. The complication relative to the compact case comes from the fact that the kernel K is not bounded on D  D, and this regardless of the size of the support of k. The reason is that, as z and w tend towards the same cusp, the sum (3.39) tends towards infinity. To remedy this problem one subtracts from K.z; w/ the principal parts Ha .z; w/ D

X Z  2a n

1 1

k.z; a

1 t  01

a1  w/dt:

(3.42)

The function Ha .z; w/ is clearly  -invariant in the second variable. One defines the total principal part of the kernel K.z; w/ by summing all of the principal parts Ha .z; w/ corresponding to nonequivalent cusps: H.z; w/ D

X

Ha .z; w/:

(3.43)

a

We write b K.z; w/ D K.z; w/  H.z; w/

(3.44)

for the difference, called the compact part of the kernel K.z; w/. This defines a kernel on DD and therefore an integral operator b T k acting on functions f W D ! C. Proposition 3.23 If f 2 C. nH/ we have Tk f D b Tkf . Proof This is an immediate consequence of the following lemma.

t u

Lemma 3.24 For all z 2 H, the principal part Ha .z; / is bounded, belongs to L2 . nH/ \ C1 . nH/ and is orthogonal to the subspace C. nH/. Proof We begin by showing that Ha .z; / is bounded. Since k. a z; a w/ D k.z; w/, we have Z 1 X k.z; t C w/dt: (3.45) Ha . a z; a w/ D 1   2Bn a a

1

Here the function k is of compact support. Thus the sum and the integral range over a domain where jz  t  wj2 D O.Im.z/ Im.w//, see (1.12). Restricting to this domain we thus have Im.w/ D O.Im.z// (as well as Im.z/ D O.Im.w//), and the integral is an O.Im.z//.

3.7 Integral Operators on  nH

79

Fact Let z 2 H and Y > 0. We have ˇ ˇ ˇf 2 . 1  a /1 n 1  a j Im.z/ > Ygˇ D O .1 C 1=Y/ ; a a

(3.46)

where the implied constant in the O does not depend on z. Proof of the fact The volume of the part of the strip fz D x C iy 2 H j jxj < 1=2g above the horocycle of length log 1=Y is 1=Y. The argument of the Lemma 3.18 can then be modified to prove the fact. In the case of  D SL.2; Z/, we can just as well give an elementary proof: The group  has only one cusp a D 1. By  -invariance, we can assume that z belongs to the closure of the fundamental domain D D fz D x C iy 2 H j jzj > 1; jxj < 1=2g: We therefore have y >

p 3=2. Let D

   2 : cd

Suppose that Im.z/ D

y > Y: jcz C dj2

If c ¤ 0, we have jcz C dj2 > 3c2 =4 > 3=4, so that y>

3Y : 4

p p It follows from this that jcj < . yY/1 and jcx C dj < 4y=3Y. For a fixed c ¤ 0, p this last inequality has at most O. 4y=3Y/ solutions d, whence a total number of pairs .c; d/, with c ¤ 0, bounded by   O y1=2 Y 1=2 y1=2 Y 1=2 D O.Y 1 /: The result follows by adding 1 (the contribution of B), which corresponds to c D 0. t u

80

3 Spectral Decomposition

Since the sum (3.45) ranges over a domain where Im./ D O.Im.z//, it follows from (3.46) that there are at most O.1C1= Im.z// terms in the sum. But each integral is an O.Im.z//, so one gets  Ha . a z; a w/ D O

1C

  1 Im z D O.1 C Im z/: Im z

(3.47)

The function Ha .z; / is in particular bounded. Given a function f 2 C. nH/, unfolding the integral yields  dvdu k.z; u C iv C t/dt f . a .u C iv// 2 v 0 0 1  Z 1  Z C1Z C1 D k.z; t C iv/dt f . a .u C iv//du v 2 dv Z

C1

Z

hHa . a z; :/; f i D

0

1

Z

C1

1

0

D 0; since 0 D fa .v/ D

R1 0

f . a .u C iv//du.

t u

Proposition 3.25 Suppose that D is a polygonal and that its vertices at infinity are pairwise distinct modulo  . Then the kernel b K.z; w/ is bounded on D  D. Proof Since k is of compact support, the non-parabolic elements only contribute a finite number of terms to the sum K.z; w/: the set of non-parabolic  2  for which some z; w 2 D satisfy distH .z;  w/ D O.1/ is finite. We therefore have XX k.z;  w/ C O.1/; K.z; w/ D a  2a

where the first sum ranges over the set of nonequivalent cusps. The upper bound (3.47) moreover implies that, in the sum (3.42), all of the terms, except for the class a , contributes a uniformly bounded amount. (These terms correspond to the O.1/ D O.Im.z/= Im.z// in (3.47).) We deduce that   Z C1 1 t

1 w/dt C O.1/ Ha .z; w/ D k.z; a 01 a 1 so that b K.z; w/ D

X

Ja .z; w/ C O.1/;

a

where Ja .z; w/ D

X  2a

Z k.z;  w/ 

C1 1

k.z; a

  1 t

1 w/dt: 01 a

3.8 Review of Functional Analysis

81

It remains to show that Ja .z; w/ is bounded on D  D. This is a consequence of the Euler-MacLaurin formula Z Z X F.b/ D F.t/dt C .t/dF.t/; b2Z

where

.t/ D t  Œt  1=2. Indeed, this implies that

Ja . a z; a w/ D

X

Z

b2Z

Z

C1

k.z; w C b/ 

k.z; w C t/dt 1

Z

C1

D 1

.t/dk.z; w C t/ D O

C1

 jdk.u/j D O.1/:

0

Since k is of compact support one therefore finds that the function Ja .z; w/ is in fact bounded on H  H. t u

3.8 Review of Functional Analysis In this chapter we aim to show that the spectrum of the Laplacian  is discrete whenever the surface  nH is compact. Since the Laplacian is an unbounded operator, there is no general theorem implying that its spectrum is discrete. We shall get around this problem by deducing the spectral theorem for  from the spectral theorem for certain integral operators commuting with the Laplacian and which are themselves compact operators. In this section we recall the statements of general theorems concerning compact operators. These theorems are classical, see for example [18, Chap. VI], [103, §97]. Let H be a separable Hilbert space whose underlying norm we denote by j  j. An operator T W H ! H is compact if T sends bounded sets to relatively compact sets. Since H is separable, a subset of H is compact if and only if it is sequentially compact. The operator T is therefore compact if and only if, for any sequence of unitary vectors xn 2 H, there exists a subsequence xnk such that the sequence T.xnk / converges. A compact operator is automatically bounded and is therefore continuous. Moreover, if  is a non-zero eigenvalue then the -eigenspace is finitedimensional. Theorem 3.26 (Spectral theorem for compact operators) Let T be a compact self-adjoint operator on a separable Hilbert space H. Then the space H admits an orthonormal basis i (i D 1; 2; 3; : : : ) of eigenvectors of T, say Ti D i i . Moreover, the sequence of eigenvalues i tends towards 0 as i ! C1.

82

3 Spectral Decomposition

Let X be a locally compact space equipped with a positive Borel measure. Assume that H D L2 .X/ is a separable Hilbert space. Let K 2 L2 .X  X/. One calls the operator Z K.x; y/f .y/ dy

.Tf /.x/ D

(3.48)

X

a Hilbert-Schmidt operator. Theorem 3.27 (Hilbert-Schmidt) A Hilbert-Schmidt operator is a compact operator. A Hilbert-Schmidt operator T whose kernel is real and symmetric is selfadjoint. One can thus apply the spectral theorem for self-adjoint compact operators. The image of T in H is generated by its eigenfunctions: there exists a maximal orthonormal system of eigenfunctions 1 ; 2 ; 3 ; : : : of T in H, i.e., hj ; k i D ıjk ;

Tj D j j with j1 j > j2 j >   

Assume now that K is continuous, then every eigenfunction j with j ¤ 0 is continuous and any function f in the image of T can be represented by an absolutely and uniformly convergent series f .x/ D

X

hf ; j ij .x/:

(3.49)

j>1

P A compact self-adjoint operator is a trace class operator if the sum j of its eigenvalues is absolutely convergent. We then call this sum the trace of T, written tr.T/. The following theorem implies, in particular, that a Hilbert-Schmidt operator T on L2 .X/ with X compact and whose kernel K is positive and continuous is trace class. One has then Z tr.T/ D K.x; x/ dx: (3.50) X

Theorem 3.28 (Mercer) Let T be a Hilbert-Schmidt operator with positive kernel K. There exists a maximal orthonormal system 1 ; 2 ; 3 ; : : : of Teigenfunctions in H such that the sequence of corresponding eigenvalues .j / is positive. Moreover, if the kernel K is continuous, it can be represented by an absolutely and uniformly convergent series K.x; y/ D

X j>1

j j .x/j .y/:

(3.51)

3.9 Proof of the Spectral Theorem

83

3.9 Proof of the Spectral Theorem We begin by assuming that the surface  nH is compact. Let k W H  H ! R be a compactly supported C1 point-pair invariant. Denote by K W  nH nH ! R the associated automorphic kernel. Let Tk be the invariant integral operator defined in § 3.7 by Z .Tk '/.z/ D

K.z; w/'.w/ d .w/:  nH

Then the spectral theorem for self-adjoint compact operators applies to this HilbertSchmidt operator, and consequently there exists a maximal orthonormal system 1 ; 2 ; 3 ; : : : of Tk -eigenfunctions in L2 . nH/ such that every function f in the closure of the image of Tk can be expanded as f .z/ D

X

hf ; j ij .z/:

(3.52)

j>1

Moreover, the eigenspace associated to a non-zero Tk -eigenvalue is of finite dimension. Since Tk and  commute and the Laplacian is a symmetric operator, every function f 2 C1 . nH/ \ Im.Tk / admits an expansion (3.52) where the functions j 2 D. nH/ are eigenfunctions of the Laplacian. To prove the spectral theorem, it suffices then to choose K (or rather k) in such a way that the image Tk .C1 . nH// is dense in C1 . nH/. We begin with the following lemma. Lemma 3.29 Let f 2 L2 . nH/ be non-zero. Fix a positive real number ". Then there is an automorphic kernel K (associated to a compactly supported function k) such that jTk f  f j < ", where Tk is the integral operator associated with the kernel K and j  j denotes the L2 -norm. In particular, if " < jf j then Tk f ¤ 0. Proof From the density of continuous functions in L2 . nH/, it suffices to prove the lemma for f continuous. Since S D  nH is compact, the function f is uniformly continuous and there exists a constant  > 0 such that for all z; w 2 H of distance

.z; w/ < , one has jf .w/  f .z/j
1 > 2 >    > 0: It is clear that the function constantly equal to 1 is a solution to the heat equation. Hence it is an eigenfunction of the operators Pt for the eigenvalue 1. Let 'j be an non-constant eigenfunction of P1 . We have d jPt 'j j2 D 2 dt D

Z  nH

22t j

.Pt 'j /.Pt 'j / d

Z

 nH

'j 'j d < 0:

Here we used j > 0 and the fact that if ' is non-constant then Z h'; 'i D

 nH

j grad 'j2 d > 0:

We deduce that jPt 'j j < j'j j so that j < 1. We can now prove the spectral theorem. More precisely, we demonstrate the following theorem. Theorem 3.32 (Spectral theorem) Let S D  nH be a compact hyperbolic surface. The spectral problem ' D ' admits a complete orthonormal system '0 ; '1 ; : : : of C1 -eigenfunctions in L2 .S/. The corresponding eigenvalues 0 ; 1 ; : : : satisfy the following properties 1. 0 D 0 < 1 6 2 6    , n ! C1 as n ! C1. 2. The series C1 X jD1

converges for all real > 1.

 j

3.9 Proof of the Spectral Theorem

87

Proof Keeping the above notation we write j D  log j ;

j D 0; 1; 2; : : :

From the equation 0 D Pt 'j C

@ Pt 'j @t

D etj .'j  j 'j / one deduces that the 'j are eigenfunctions of the Laplacian, and one obtains point 1 of the theorem. The Hilbert-Schmidt theorem then implies that P .z; w; t/ D

1 X

ej t 'j .z/'j .w/;

(3.55)

jD0

in the L2 -sense. Next Mercer’s theorem implies that the series converges absolutely and uniformly for all t > 0. To prove point 2, we must pass from ej t to  and j show that the series C1 X

 j 'j .z/'j .w/

(3.56)

jD1

converges absolutely and uniformly on S  S. The idea is to take ej t , multiply it by t 1 and then integrate. The integral expression for the  -function implies that for > 1, Z

C1 0

t 1 ej t dt D  . / j :

One can therefore consider the functions 'j as eigenfunction of a new invariant integral operator whose kernel K is given by Z K .z; w/ D

C1

0

 P .z; w; t/ 

 1 t 1 dt: area.S/

Let us show that the kernel K is well-defined and continuous on S  S. We begin by remarking that for all > 1, there exists an integer j0 such that, for all j > j0 and for all " 2 .0; 1=2/, one has Z

C1 "

ej t t 1 dt 6 ej t :

88

3 Spectral Decomposition

It then follows from (3.55) that for all " > 0 the function14 Z K ;" .z; w/ D

C1 "

 P .z; w; t/ 

 1 t 1 dt area.S/

is well-defined and continuous on S  S. It satisfies Z lim K ;" .z; w/'j .w/ d .w/ D  . / j 'j .z/ "#0

S

for all z 2 S and for all j > 1. Moreover, it follows from Lemmas 3.13 and 3.17 that P .z; w; t/ 6 .const/t1 for all t < 1. Thus for every ı 2 .0; "/ we have Z ı

"

P .z; w; t/t

1

 " 1 : dt D O

1 

We deduce that K .z; w/ is well-defined on S  S and that K ;" converges uniformly to K as " tends toward 0. Furthermore, we have Z K .z; w/'j .w/ d .w/ D  . / j 'j .z/: S

The functions '1 ; '2 ; : : : then form a complete system of eigenfunction for the compact operator with kernel K . The associated eigenvalues are  . / j , j D 1; 2; : : : In particular, they are positive and Mercer’s theorem implies that the series (3.56) converges uniformly on S  S (and that the sum is equal to K .z; w/). On integrating (3.56) along the diagonal, one obtains point 2 of the theorem. t u In the case where the surface  nH is of finite area but non-compact, we will content ourselves with the following theorem. Theorem 3.33 Let S D  nH be a hyperbolic surface of finite area. The spectral problem for cuspidal functions ' D ' .' 2 C. nH// admits a complete orthonormal system of eigenfunctions. Proof The statement to be established is just one part of a more general spectral theorem for such surfaces. The proof is a straightforward generalization of the analogous one for the compact case (cf. the passage directly following the proof of Proposition 3.30): one simply replaces the space L2 . nH/ by the closure of C. nH/ and the operators Tk by the operators b T k associated to the kernel b K defined 14

p Note that '0 is the function constantly equal to 1= area.S/.

3.10 The Minimax Principle

89

in Eq. (3.44). Then one uses Propositions 3.23 and 3.25; note that b T k sends an L2 function to a bounded C1 -function. Finally, we show that the first eigenvalue is strictly positive. This comes from the fact that the functions f 2 C. nH/ are bounded. The equation f D 0 then implies that f on H is harmonic – in the Euclidean sense – and bounded, and thus constant. But a non-zero constant function cannot be cuspidal. t u

3.10 The Minimax Principle A better geometric understanding of the spectrum of the Laplacian comes from the minimal principle that we now describe. Let S D  nH be a compact hyperbolic surface. For k D 0; 1; : : : ; we write k .S/ for the k-th eigenvalue of the Laplacian on S. Theorem 3.34 (Minimal principle) 1. Let f0 ; : : : ; fk 2 C1 .S/ be k C 1 functions of L2 -norm equal to 1 whose supports have measure zero pairwise intersections. Then Z

j grad fj .z/j2 d .z/:

k .S/ 6 max

06j6k

S

2. Let M D N1 [    [ Nk be a partition into relatively compact subsets of strictly positive measure and having measure zero pairwise intersections. Put Z

j grad f .z/j2 d .z/;

.Nj / D inf Nj

where f runs over the set of C1 -functions such that Z

Z

2

jf .z/j d .z/ D 1

f .z/ d .z/ D 0:

and

Nj

Nj

Then k .S/ > min .Nj /: 16j6k

Proof Let '0 ; '1 ; : : : be a complete orthonormal system of C1 -eigenfunctions with corresponding eigenvalues 0 .S/; 1 .S/; : : : Since the fj in the first part of the theorem are linearly independent, one can form a linear combination f D ˇ0 f0 C    C ˇk fk of L2 -norm equal to 1 and orthogonal to the functions '0 ; : : : ; 'k1 . Let ˛j be the “Fourier coefficient” Z ˛j D f 'j d : S

90

3 Spectral Decomposition

For all m > k, we have Z

2

j grad f .z/j d .z/  S

m X

j˛j j2 j

jDk

Z

D

Z  Pm

f f d  S

jDk

S

˛j 'j

P

m jDk

 ˛j j 'j d

Z     P Pm f m d D ˛ '  f  ˛ ' j j j j jDk jDk Z

S

D

j grad.f  S

Pm jDk

˛j 'j /j2 d > 0:

Here j .S/ > k .S/, and according to Parseval’s theorem C1 X

j˛j j2 D 1:

jDk

Since the pairwise supports of the fj meet each other in measure zero sets and the L2 -norm of f is 1, we have k X

jˇi j2 D 1;

iD0

from which one deduces that Z

j grad f .z/j2 d .z/

k .S/ 6 S

6

k X

jˇi j2

Z

j grad fj .z/j2 d .z/ S

jD0

Z

j grad fj .z/j2 d .z/:

6 max

06j6k

S

This proves the first point of the theorem. We proceed to the proof of the second point. Let j be the characteristic function of the subset Nj  S. Consider a linear combination ' D ˛0 '0 C    C ˛k 'k of L2 norm equal to 1 and orthogonal to the functions 1 ; : : : ; k . Then, for j D 1; : : : ; k, Z ' d D 0: Nj

3.10 The Minimax Principle

91

By definition of .Nj /, we then have Z

j grad 'j2 d > .Nj / Nj

and so (since

P R j Nj

Z

j'.z/j2 d .z/

Nj

j'.z/j2 d .z/ D 1) Z

j grad 'j2 d > min .Nj /: 16j6k

S

On the other hand, since Z j'.z/j2 d .z/ D j˛0 j2 C    C j˛k j2 D 1; S

we have Z

j grad 'j2 d D S

Z ''d D S

k X

j .S/j˛j j2 6 k .S/:

t u

jD0

3.10.1 Small Eigenvalues I: Geometric Existence Criterion Proposition 3.35 Let S be a compact hyperbolic surface which is the union S of two compact connected subsets A and B which intersect along a finite union kiD1 i of simple closed geodesics. Let `.i / be the lengths of the i and Pk

hD

`.i / : min.area.A/; area.B// iD1

Let " be a positive real number such that the "-neighborhood of each geodesic i is embedded in S. Then there exists a constant C D C."/ depending only on " such that 1 .S/ 6 C.h C h2 /: Proof According to the minimax principle, to majorize 1 .S/ it suffices to majorize R the quotient S j grad f j2 d =jf j2L2 .S/ for a well chosen test function f with support in A or B. Suppose, for example, Pk hD

`.i / : area.A/ iD1

92

Let X D

3 Spectral Decomposition

Sk iD1

i be the boundary of A. Let A.t/ D fz 2 A j dist.z; X/ 6 tg

and for sufficiently small t put 8 < 1 dist.z; X/ f .z/ D t :1

if z 2 A.t/; if z 2 A  A.t/:

The function f satisfies j grad f .z/j2 6 t2 for all z 2 A.t/, and j grad f .z/j D 0 for all z 2 A  A.t/. We now estimate the area of A.t/. In Fermi coordinates (see Appendix A) and for t sufficiently small (t "), we find area.A.t// 6

k X

Z `.i /

iD1

t 0

cosh d D

k X

`.i / sinh t:

iD1

For t D min."; argsh.1=2h//, we thus have Z S

j grad f j2 d =jf j2L2 .S/ 6 6

Pk

`.i / sinh t P t2 .area.A/  kiD1 `.i / sinh t/ iD1

h sinh t  h sinh t/

t2 .1

6 C.h C h2 /; for a constant C depending only on ".

t u

Corollary 3.36 Let S be a compact hyperbolic surface. For every real " > 0, the surface S admits a finite cover S0 whose first non-zero eigenvalue satisfies 1 .S0 / 6 ": Proof The Euler characteristic .S/ of the compact hyperbolic surface S is negative. Since .S/ D 2  2g where g is the genus of S, we have g > 2. In particular, g > 1 and there exists a simple closed curve  in S such that S   is connected. Upon replacing  by another curve homotopic to it, if necessary, we can assume that  is a closed geodesic. Corresponding to  there is a surjective homomorphism from the fundamental group of S onto Z: take the algebraic intersection of a loop with  . We form SN , the cyclic cover of degree 2N above S. This is the surface obtained by gluing 2N copies of S   boundary to boundary in a circular arrangement. There then exists two lifts 1 and 2 of  in SN such that the union 1 [ 2 cuts the surface SN into two pieces A and B, each consisting of N fundamental domains for

3.10 The Minimax Principle

93

the action of the transformation group of the cover of SN over S. Proposition 3.35 then implies the existence of a constant C independent of N such that  1 .SN / 6 C

 4`. /2 2`. / C 2 : N area.S/ N area.S/2 t u

We deduce the corollary by letting N tend towards infinity.

3.10.2 Small Eigenvalues II: The Selberg Conjecture Conjecture 3.37 (The Selberg conjecture) Let a, b, and N be positive integers. Then the first non-zero eigenvalue of the Laplacian on C.a;b .N/nH/ satisfies 1 .a;b .N/nH/ > 1=4: In as much as there should exist some uniform lower bound on 1 , Selberg’s conjecture states that congruence covers of arithmetic surfaces are, in a geometric sense, opposite to the case of cyclic covers explored in the preceding subsection. The precise value of 1=4 in the above conjecture lies deeper; it is a reflection of the underlying arithmetic defining these surfaces. We shall return to this point later. In the case of the modular surface, the conjecture has been proved for its congruence subgroups of small level. Theorem 3.38 Let  D SL.2; Z/. Then 1 . nH/ >

1 3 2 > : 2 4

Proof Let D denote the standard fundamental domain of the modular group. Let f 2 C. nH/ have L2 -norm 1 and satisfy Z

j grad f j2 d :

1 . nH/ D  WD D

Write F D fz j jxj < 1=2; y >

p 3=2g:

The subset F is contained in the union of D with its translate by thus have Z 2 >

C1 p

3=2

Z

1 0

j grad f .z/j2 d .z/:

 0 1  1 0

2  . We

94

3 Spectral Decomposition

The function f .z/ being 1-periodic in x, we can expand it in its Fourier series f .z/ D

X

cn .y/e.nx/:

n¤0

Hence grad f .z/ D

y

! P 2nc .y/e.nx/ n n¤0 P ; y n¤0 c0n .y/e.nx/

and Z

C1 p

3=2

Z

1 0

2

Z

j grad f .z/j d .z/ >

C1

X

p 3=2

n¤0

Z

C1

> 3 2 > 3 2 > 3 2

p 3=2

Z Z

j2ncn .y/j2 dy X n¤0

C1 Z p 3=2

jcn .y/j2 y2 dy

0

1

jf .z/j2 d .z/

jf .z/j2 d .z/: D

From this one deduces that 2 > 3 2 .

t u

Remark 3.39 In [15], the authors compute to more than 1000 decimal places the first non-zero eigenvalue 1 D 1 . nH/: 1 91; 14134533635527808180 : : : In [14] the Selberg conjecture is verified for several congruence subgroups of small level.

3.11 Commentary and References The main result of this chapter is Theorem 3.32. Two classical references concerning this chapter are the books by Iwaniec and Buser [24, 63]. We have borrowed a lot from their presentations, following in large part Iwaniec’s book.

3.11 Commentary and References

§ 3.1 Unbounded operators are frequently encountered in mathematical physics and have been widely studied; we refer the reader to the books of Rudin [107], Dunford and Schwartz [38], or Taylor [127] for the general theory. It is a general fact that on every Riemannian manifold the Laplacian is a symmetric operator. This can be deduced from Green’s theorem, see for example [41, p. 184]. In our exposition, with its focus on surfaces, we have opted to reproduced the “elementary” proof to be found in [20, p. 133]; this proof uses only the classical Stokes’ theorem (in the plane).

§ 3.2 Lemma 3.3 is taken from [63, Lem. 1.7] and Proposition 3.11 is proved in [63, (1.64’)].

§ 3.5 One can study the heat equation on any Riemannian manifold. The book [8] proves, for example, the existence and uniqueness of solutions on compact manifolds. The book by Davies [33] is a good reference for the case of the hyperbolic plane. The Riemannian manifold H being non-compact, a fundamental solution to the heat equation is not necessarily unique. One can find in the literature [33] good upper bounds on the heat kernel for general spaces. Lemma 3.13 is taken from [24, Lem. 7.4.26].

§ 3.6 The Laplacian defines a symmetric positive operator on any compact Riemannian manifold, see for example [8, p. 53]. Here again we have opted for an “elementary” proof close to that which one finds in the Bump’s book [20] and which uses only Stokes’ theorem for the hyperbolic surface S D  nH. A classical theorem of Friedrichs implies that every positive symmetric operator admits a self-adjoint extension. Since D . nH/ is dense in L2 . nH/ and the area of the surface  nH is finite, it follows from Proposition 3.16 that .; D . nH// admits a unique selfadjoint extension to L2 . nH/. It is this extension that one classically refers to as the Laplacian.

§ 3.7 Lemma 3.20 is extracted from [24, §7.5] which gives more details. In the study of integral operators on finite area surfaces we have followed Iwaniec [63, §4.2].

95

96

3 Spectral Decomposition

§ 3.9 The proof of Lemma 3.29 and the proof of the first part of the spectral theorem generalize easily to show that if G is a locally compact topological group and  a discrete cocompact subgroup of G, the space L2 . nG/ decomposes into a (Hilbert) direct sum of invariant and irreducible subspaces under the right-regular representation of G. In the non-compact case, there is a priori no reason for the cuspidal space C . nH/ to be different from f0g. It is in fact still an open question to decide in which cases C . nH/ ¤ f0g. With the help of the trace formula, Selberg showed that this is in fact the case whenever  is commensurable with SL.2; Z/ (see Chap. 5). In the next chapter we present a recent alternative proof of this fact due to Lindenstrauss and Venkatesh. On the basis of these arithmetic examples, it was for a long time believed that C . nH/ is always non-zero and in fact of infinite dimension. The remarkable works of Phillips-Sarnak [98], and their continuations by Deshouillers-Iwaniec [36], Wolpert [141], and Luo [85] (among others), showed that under a widely believed conjecture on eigenvalue multiplicities one has C . nH/ D f0g for generic  . We refer the reader to the above cited works for the precise meaning of “generic”. The aforementioned conjecture is that the multiplicity of an eigenvalue  for the group 0 .N/ is ON .1/, or at least ON .log. C 1//; it is considered out of reach.

§ 3.10 The minimax principle is reproduced from [24, §8.2]. Proposition 3.35 is an easy case of the Buser inequality [23]. A first fundamental link between an isoperimetric constant – closely related to the constant h of Proposition 3.35 – and the first non-zero eigenvalue 1 is a famous inequality due to Cheeger [26]. In Proposition 3.35, the constant C can be chosen uniformly, see [23]. Theorem 3.38 is due to Roelcke [105].

3.12 Exercises Exercise 3.40

    @ @ @ @ and @ D 12 @x . Verify  i @y C i @y 1. Show that  D 4y2 @@, where @ D 12 @x the equality (1.20). 2. Show that in polar coordinates .r; / of the hyperbolic plane (see Appendix A), we have D

@2 1 @ 1 @2  :  2 2 @r tanh r @r .2 sinh r/ @ 2

3.12 Exercises

97

Exercise 3.41 Let k be a non-negative integer. The Maaß operators are the following differential operators on C1 .H/: Rk D .z  z/

@ k C ; @z 2

Lk D .z  z/

k @  : @z 2

We define the weight k Laplacian by the formula  2  @ @ @2 2 k D y C 2 C iky : @x2 @y @x 1. Show that k D LkC2 Rk 

    k k k k 1C D Rk2 Lk C 1 : 2 2 2 2

2. Show that Rk ı k D kC2 ı Rk and Lk ı k D k2 ı Lk . Exercise 3.42 For all k 2 Z, one defines an action of G on C1 .H/ by  f jk g D

cz C d jcz C dj

k   az C b ; f cz C d

gD

  ab : cd

1. Show that if f 2 C1 .H/ and g 2 G, then .Rk f /jkC2 g D Rk .f jk g/ and .Lk f /jk2 g D Lk .f jk g/: 2. Deduce from the preceding question and the previous exercise that .k f /jk g D k .f jk g/: Exercise 3.43 Show that the weight k Laplacian is a symmetric operator on L2 .H/ with domain Cc1 .H/. Exercise 3.44 Let  be a cocompact Fuchsian group. Assume that I 2  . Let  be a homomorphism from  to C . We denote by C1 . nH; ; k/ the space of C1 -functions on H satisfying       az C b cz C d k ab 2 : ; D f . /f .z/ D cd jcz C dj cz C d

98

3 Spectral Decomposition

1. Show that C1 . nH; ; k/ D f0g if .I/ ¤ .1/k . In what follows we shall always assume that .I/ D .1/k . 2. Let f ; g 2 C1 . nH; ; k/. Show that f g is a  -invariant function and that the expression Z dxdy hf ; gi D f .z/g.z/ 2 y  nH defines a scalar product on C1 . nH; ; k/. We shall denote by L2 . nH; ; k/ the Hilbert space completion of C1 . nH; ; k/ with respect to this scalar product. 3. Deduce from the preceding exercise that the operators Rk and Lk send C1 . nH; ; k/ to C1 . nH; ; k C 2/ and C1 . nH; ; k  2/, resp. 4. Let f 2 C1 . nH; ; k/ and g 2 C1 . nH; ; k C 2/. Show that hRk f ; gi D hf ; LkC2 gi: 5. Deduce that k defines a symmetric operator on the Hilbert space L2 . nH; ; k/. Exercise 3.45 We keep the notation from the preceding exercise. A holomorphic modular form of weight k with respect to  is a holomorphic function f W H ! C such that   ab 2 : f . z/ D . /.cz C d/k f .z/;  D cd 1. Show that if f is a holomorphic modular form of k with respect to  , then the function z 7! yk=2 f .z/ is an eigenfunction of the weight k Laplacian with eigenvalue 2k .1  2k /, and that f lies in the kernel of Lk . 2. Conversely, show that if g 2 C1 . nH; ; k/ is an eigenfunction for k and lies in the kernel of Lk , then the function z7!yk=2 g.z/ is a holomorphic modular form of weight k with respect to  . Exercise 3.46 Show that if k 2 N and if  is an eigenvalue of k , then either  D 2` .1  2` /, where ` is an integer between 1 and k having the same parity as k, or  > 0, and in fact  > 14 if k is odd.

Chapter 4

Maaß Forms

In this chapter we consider the particular case of congruence covers Y.N/ D  .N/nH or Y0 .N/ D 0 .N/nH of the modular surface. In the first sections we pay particular attention to the case of the modular group SL.2; Z/. Let us recall that a Maaß form is an eigenfunction of the Laplacian on a hyperbolic surface  nH, where   SL.2; Z/ is a congruence subgroup, that grows at most polynomially in the cusps of  . As we have already remarked, it is not obvious that non-constant cuspidal – or even square integrable – Maaß forms exist. In this chapter we show that Maaß forms do indeed exist, and plentifully so. We then recall how Maaß came to explicitly construct such functions from quadratic extensions of Q.

4.1 Eisenstein Series for SL.2 ; Z/ In this section  D SL.2; Z/. Fix z D x C iy 2 H and consider the series in the s-variable given by the expression1: E.z; s/ D

ys 1 X 2 c;d2Z jcz C dj2s .c;d/D1

1 D 2

1

The coefficient

1 2

X

(4.1) .Im. z// : s

 21 n

comes from the fact that the element I 2 SL.2; Z/ acts trivially on H.

© Springer International Publishing Switzerland 2016 N. Bergeron, The Spectrum of Hyperbolic Surfaces, Universitext, DOI 10.1007/978-3-319-27666-3_4

99

100

4 Maaß Forms

One passes from the first to the second equality by associating with each pair of relatively prime integers .c; d/ the set of matrices in  whose second row is .c; d/, this being a coset in 1 n . Lemma 4.1 The series (4.1) is absolutely convergent on Re.s/ > 1. The sum of this series, E.z; s/, is called the Eisenstein series of the modular group  . It satisfies E..z/; s/ D E.z; s/;

for all  2 :

Proof It suffices to show that the expression (4.1) is absolutely convergent. Fix z0 2 D D fz 2 H j jxj < 1=2; jzj > 1g and let ı be a positive real number such that the (hyperbolic) ball B.z0 ; ı/ is contained in D. The domain D and the strip S D fz 2 H j jxj < 1=2g are the respective fundamental domains for the groups  and 1 . Thus there exists a set of coset representatives E   for 1 n such that [

.D/ D S:

 2E

Each ball B. z0 ; ı/ D .B.z0 ; ı// ( 2 E) is contained in .D/. In particular, these balls are pairwise disjoint and, if T is a real number sufficiently large for B.z0 ; ı/  fz D x C iy 2 C j y 6 Tg; then all of the sets B. z0 ; ı/ ( 2 E) are contained in fz 2 H j jxj < 1=2; y 6 Tg. Besides, each  z0 , except possibly z0 itself, has imaginary part less than 1 since it lies in S  D. If two points z D x C iy and z0 D x0 C iy0 in H are at (hyperbolic) distance less than or equal to 2ı, we have ye2ı 6 y0 6 ye2ı . Hence there exists a constant C D C.ı/ such that, if z00 D x00 C iy00 2 H satisfies y00 6 1, then for every z D x C iy 2 B.z00 ; ı/ we have y00 6 Cy. We deduce that for a real number s strictly greater than 1: X

y. z0 /s 6 ys0 C Cs

 21 n

X  21 n  ¤1

6 ys0 C 6 ys0 C

1 vol.B.z0 ; ı//

Cs vol.B.z0 ; ı//

Z

1=2 1=2

Z

Z

T

ys 0

ys  B.z0 ;ı/

dx dy ; y2

dx dy ; y2

s s1

CT : .s  1/ vol.B.z0 ; ı//

The series (4.1) is therefore absolutely convergent. The automorphy follows immediately from the definition. t u

4.1 Eisenstein Series for SL.2; Z/

101

The fundamental properties of the Eisenstein series are its analytic continuation and functional equation. As for the Riemann zeta function, it is more natural to complete the Eisenstein series, which we do by writing E .z; s/ D  s  .s/

ys 1 X 2 m;n2Z jmz C nj2s

(4.2)

.m;n/¤0

D  s  .s/.2s/E.z; s/: The passage from the first to the second equality comes from the fact that summing over all non-zero pairs .m; n/ 2 Z2 is the same thing as summing over all positive integers N and all pairs of relatively prime integers .c; d/ by taking .m; n/ D .Nc; Nd/. Of course  is the Riemann zeta function. Theorem 4.2 The function E .z; s/, defined by (4.2) for Re.s/ > 1, admits a meromorphic continuation to all s 2 C; it is holomorphic everywhere except at s D 0 and s D 1, where it has simple poles. The completed Eisenstein series satisfies the functional equation E .z; s/ D E .z; 1  s/:

(4.3)

Moreover, it is bounded in vertical strips 1 6 Re.s/ 6 2 ( 1 ; 2 2 R) for s uniformly away from the poles. Remark 4.3 Since the  function and the  function both admit meromorphic continuations to all of C (see Exercise 1.16), it follows from Theorem 4.2 that the function E.z; s/ admits a meromorphic continuation to all s 2 C. We shall see in § 4.2.3 that this continuation is holomorphic on the half-plane Re.s/ > 1=2. Proof As in the case of the Riemann zeta function (see Exercise 1.16), we shall use the so-called “theta series” given by the formula X

.t/.D .z; t// D

2 t=y

ejmzCnj

;

.m;n/2Z2

for t > 0. It is immediate that E .z; s/ D D

1 2 1 2

Z Z

C1

..t/  1/ts

0 1 0

..t/  1/ts

dt t

1 dt C t 2

D E0 .z; s/ C E1 .z; s/:

Z

C1 1

..t/  1/ts

dt t

102

4 Maaß Forms

Since y and t are positive, the function .t/  1 is of rapid decay and the function E1 .z; s/ is both holomorphic on the entire s-plane and bounded in vertical strips. Now consider the function R2 ! C: 2 t=y

fz;t .u1 ; u2 / D eju1 zCu2 j

D eŒ.u1 xCu2 /

2 C.u y/2 t=y 1

:

2

2

2

It follows from Lemma 1.1 that the function u D .u1 ; u2 / 7! ejuj D e.u1 Cu2 / is its own Fourier transform. One then has Z 2 2 b eŒ.u1 xCu2 / C.u1 y/ t=y e.h; ui/ du fz;t .1 ; 2 / D Z D

R2

2

R2

ejvj e.htA1 ; vi/j det.A/j1 dv;

where r   t x1 AD y y0 and v D Au. From this one deduces that 2 2 1 jtA1 j2 D t1 eŒ.y2 / C.1 x2 / =ty : fb z;t .1 ; 2 / D t e

Hence, the Poisson summation formula (see the introduction) applied to fz;t implies that .t/ D t1 .t1 /. But the change of variables t 7! t1 then shows that if Re.s/ < 0, E1 .z; s/

1 D 2 D D

1 2 1 2

Z Z Z

1 0 1 0 1 0

.t.t/  1/ts .t/t1s

dt ; t

1 dt C ; t 2s

.t.t/  1/t1s

D E0 .z; 1  s/ C

1 1 dt C C ; t 2  2s 2s

1 1 C : 2  2s 2s

We find that the function E0 .z; s/ – and thus E .z; s/ as well – admits a meromorphic continuation to all s 2 C, and that E .z; s/ D E1 .z; s/ C E1 .z; 1  s/ C

1 1  : 2s  2 2s

t u

4.1 Eisenstein Series for SL.2; Z/

103

The following paragraph is a first aside on the links between the Eisenstein series and the Riemann zeta function.

4.1.1 Eisenstein Series and the Riemann Zeta Function I Let s be a complex number with real part strictly larger than 1. The Eisenstein series E.z; s/ is invariant under the transformation z 7! z C 1. As a result, we can expand E.z; s/ in a Fourier series, E.z; s/ D

X

ar .y; s/e.rx/:

(4.4)

r2Z

We calculate the Fourier coefficients Z ar .y; s/ D

1

E.x C iy; s/e.rx/ dx

0

or rather the coefficients ar .y; s/

Z

1

D 0

E .x C iy; s/e.rx/ dx;

via the first expression for E .z; s/ in (4.2). We begin by considering the contribution of the terms with m D 0 in the completed Eisenstein series. Such a term is independent of x and therefore doesn’t contribute to a0 . Since n and n contribute equally, their joint contribution is  s  .s/ys

C1 X

n2s D  s  .s/.2s/ys :

(4.5)

nD1

This is one part of a0 . Let us now calculate the contribution of the terms with m ¤ 0 to each of the ar . Since .m; n/ and .m; n/ contribute equally, we can group them together and assume that m > 0. The contribution to the coefficient ar is therefore equal to

s

  .s/y

s

C1 X C1 X

Z

mD1 nD1 0

1

Œ.mx C n/2 C m2 y2 s e.rx/ dx

D  s  .s/ys

C1 X

X Z

mD1 n2Z=mZ

C1 1

Œ.mx C n/2 C m2 y2 s e.rx/ dx:

104

4 Maaß Forms

The change of variables x 7! x  n=m transforms this last expression into  s  .s/ys

C1 X

X

m2s

mD1

Z

C1

e.rn=m/ 1

n2Z=mZ

Œx2 C y2 s e.rx/ dx;

which, since (

X

e.rn=m/ D

n2Z=mZ

m

if mjr;

0

else,

is equal to  s  .s/ys

X

m12s

Z

C1 1

mjr

Œx2 C y2 s e.rx/ dx:

(4.6)

We distinguish two cases according to whether r is zero or not. Assume first that r D 0. The condition mjr in (4.6) is then empty and, using the integral expression (1.26) of the  function, we find that (4.6) is equal to  y s 

Z  .s/

C1

1

2

Z

2 s

Œx C y  dx D

C1

Z

C1

e Z

1 C1

Z

0 C1

D 0

1

t



ty .x2 C y2 /

et.x

2 Cy2 /=y

ts dx

s

dt dx t

dt ; t

where we have interchanged the order of integration (the integral is absolutely convergent for Re.s/ > 1=2) and applied the change of variable t 7! t.x2 C y2 /=y. But it is well known that r Z C1 y tx2 =y : e dx D t 1 Inserting this expression into the last integral and combining the result with Eq. (4.5), we obtain the following expression for the constant term: a0 .y; s/ D  s  .s/.2s/ys C  sC1=2  .s  1=2/ .2s  1/y1s : Now assume that r ¤ 0. Then ar .y; s/

D 12s .jrj/

 y s 

Z  .s/

C1 1

Œx2 C y2 s e.rx/ dx;

(4.7)

4.1 Eisenstein Series for SL.2; Z/

105

where X

12s .r/ D

m12s :

mjr

But analogously to the case r D 0,  y s 

Z  .s/

C1 1

Œx2 C y2 s e.rx/ dx D

Z C1 C1 t.x2 Cy2 /=y s 2irx

Z

e

0

te

dx

1

dt : t

According to Lemma 1.1, Z

C1

e

tx2 =y

r e.rx/ dx D

1

y yr2 =t e : t

One thus obtains Z

C1 Z C1 0

et.x

2 Cy2 /=y

1

dt t Z p D y

ts e2irx dx

C1

1 tC.jrj1 t/1 /=2

e2jrjy.jrj

ts1=2

0

dt : t

We recognize this last expression – see Appendix B – as p 2jrjs1=2 y Ks1=2 .2jrjy/; where K denotes the Bessel K function, Z K .y/ D

C1

ey cosh  cosh./d D

0

1 2

Z

C1

ey.tCt

1 /=2

0

t

dt : t

It follows from all of this that if r ¤ 0, p ar .y; s/ D 2jrjs1=2 12s .jrj/ y Ks1=2 .2jrjy/:

(4.8)

We summarize these computations in the following proposition. Proposition 4.4 Assume that Re.s/ > 1. Then the Eisenstein series E .z; s/ admits a Fourier expansion of the form E .z; s/ D .2s/ys C .2s  1/y1s X p C2 jrjs1=2 12s .jrj/ y Ks1=2 .2jrjy/e.rx/; r2Z

106

4 Maaß Forms

where .s/ D  s=2  compacta. In particular,

s 2

.s/ and the right-hand sum converges uniformly on

E.z; s/ D ys C '.s/y1s C 2

X jrjs1=2 12s .jrj/ p y Ks1=2 .2jrjy/e.rx/; .2s/ 

r2Z

where '.s/ D .2s  1/=.2s/. We immediate remark that the meromorphic continuation and functional equation of the Riemann zeta function follow from the analogous results for the Eisenstein series (Theorem 4.2) and from the computation of the constant term (4.7) that we can rewrite as a0 .y; s/ D .2s/ys C .2  2s/y1s :

(4.9)

From this discussion we shall principally retain the fact that the a0 Fourier coefficient of the Eisenstein series (see (4.4)) satisfies a0 .y; s/ D ys C '.s/y1s ;

(4.10)

where '.s/ D .2  2s/=.2s/. The function ' admits a meromorphic continuation to the entire complex plane, holomorphic for Re.s/ > 1=2. It follows from the functional equation of the Riemann zeta function that '.s/'.1  s/ D 1. Finally, '.s/ D '.s/. Since s D 1  s on the line Re.s/ D 1=2, one finds ˇ 1 ˇ ˇ ˇ C it ˇ D 1 for all t 2 R: ˇ' 2

(4.11)

4.2 Eisenstein Series and the Spectrum of the Laplacian In this section we shall always take  D SL.2; Z/. Recall then that D D fz D x C iy 2 H j jxj < 1=2; jzj > 1g is a fundamental domain for  . Since the function z 7! Im.z/s is a Laplacian eigenfunction with eigenvalue  D s.1  s/, and  commutes with the  action, the Eisenstein series is also a Laplacian eigenfunction with eigenvalue . In other words, if Re.s/ > 1, E.z; s/ D s.1  s/E.z; s/:

4.2 Eisenstein Series and the Spectrum of the Laplacian

107

If follows from the Fourier expansion that the function z 7! E.z; s/ is never square integrable2 on D. Nevertheless, for s 2 12 C iR, the expression (4.10) shows (at least formally – the function ' could have a singularity) that each series E.z; s/ is “almost L2 -integrable” on D. We now show that these series allow one to describe the orthocomplement of the cuspidal spectrum: the continuous spectrum. Let us begin by truncating the series E.z; s/ in such a way as to render L2 integrable. Let be a C1 function with support in .0; C1/. The series E.zj / D

1 2

X

.Im. z//

(4.12)

 21 n

is called an incomplete Eisenstein series. The series (4.12) defines a  -invariant function z 7! E.zj / from H to C, bounded and in fact lying in Cc1 . nH/; in particular it is L2 -integrable on D. On the other hand, the incomplete Eisenstein series are not eigenfunctions of the Laplacian. Via Mellin inversion, we can nevertheless write them as weighted integrals of Eisenstein series. Let us make a brief detour to recall the Mellin transform, the multiplicative analog of the Fourier transform.

4.2.1 Mellin Inversion Formula, Phragmén-Lindelöf Principle be a continuous function on .0; C1/ and  its Mellin

Lemma 4.5 Let transform:

Z  .s/ WD

C1

.y/ys

0

dy : y

If non-empty, the domain of absolute convergence of  .s/ is a vertical strip 1 < Re.s/ < 2 ( 1 ; 2 2 Œ1; C1). For every 2 . 1 ; 2 / and 0 < y < C1, 1 .y/ D 2i

Z

C1 1

 . C ir/y ir dr:

Proof If the integral Z

1 0

2

.y/y

s dy

y

 Z resp. 1

C1

.y/y

When Re.s/ > 1, this is obvious since s.1  s/ doesn’t lie in RC .

s dy

y



108

4 Maaß Forms

converges absolutely for a certain value of s, it remains absolutely convergent in the half-plane Re.s/ > Re. / (resp. Re.s/ < Re. /). The domain of absolute convergence of  .s/ is thus a vertical strip. To continue we relate Mellin inversion to Fourier inversion. Let .v/ D .ev /e v . For 2 . 1 ; 2 /, the function is continuous and belongs to the space L1 .R/ \ L2 .R/. The change of variables y D ev in the definition of  gives the expression Z  . C ir/ D

C1 1

.v/e

ivr

dv;

which, by Fourier inversion, is equivalent to

.v/ D

1 2i

Z

C1 1

 . C ir/eivr dr:

We obtain the Mellin inversion formula by multiplying this last expression by e v and putting y D ev . t u Lemma 4.6 (Phragmén-Lindelöf principle) Let f .s/ be a holomorphic function in the upper part of a vertical strip

1 6 Re.s/ 6 2 ;

Im.s/ > c;

 ˛ such that f . C it/ D O et for a real number ˛ > 0 when 1 6 6 2 . Suppose that f . C it/ D O.tM / for D 1 and D 2 . Then f . C it/ D O.tM / uniformly in 2 Œ 1 ; 2 . Proof We write s D C it D rei . Replacing f by the function s 7! f .s/=sM if necessary, we can assume that M D 0. Moreover, it is enough to consider the function f in the half-plane t > t1 for large t1 , in such a way that the argument of s stays close to =2 as runs through the interval Œ 1 ; 2 . Let m be an integer congruent to 2 mod 4 and strictly greater than ˛. Then sm D rm .cos m C i sin m /; and m is close to  mod 2 so that cos.m / is negative. We fix a positive real number " and consider the function g" .s/ D g.s/ D f .s/e"s : m

Then there exists a constant B such that for t sufficiently large one has ˛

jg.s/j 6 Bet e"r

m cos m

:

4.2 Eisenstein Series and the Spectrum of the Laplacian

109

There is a real number T" (we may take T" greater than t1 ) such that for all in the interval Π1 ; 2  and for all t greater than T" , jg. C it/j 6 B. We fix a real number t2 greater than T" . By hypothesis on f the function f is bounded on the vertical lines Re.s/ D 1 and Re.s/ D 2 . And by definition of t1 and m, if Im.s/  t1 and m

2 Π1 ; 2  then cos m is negative so that we have e"r cos m 6 1. Changing B if necessary we can thus assume that jgj is bounded above by B on the boundary of the rectangle defined by the conditions t1 6 Im.s/ 6 t2 and 1 6 6 2 . According to the maximum modulus principle jg.s/j is then bounded above by B in all of the rectangle. One deduces that j f .s/j 6 Be"r

m cos m

in this rectangle and thus (letting t2 tend to 1) for all s satisfying the conditions

2 Π1 ; 2  and Im.s/ > t1 . By letting " tend towards 0, we find j f .s/j 6 B for all s such that 2 Π1 ; 2  and t > t1 and the lemma is proved.

t u

In order to apply the Phragmén-Lindelöf principle we shall need the following proposition. A holomorphic function f .s/ is said to be of order  if

C"

f .s/ D O.ejsj

/

for all " > 0. Proposition 4.7 The function s 7! s.s  1/.s/ is (holomorphic and) of order 1. Proof By virtue of the functional equation (1.27), it suffices to consider the halfplane Re.s/ > 1=2. We already know that this function is holomorphic. Moreover, it is well known – this is Stirling’s formula – that when Re.s/ > 1=2 and jsj ! C1,  .s/ 

p p 2 es ss1=2 D 2 esC.s1=2/ log s :

(4.13)

In particular, there exists a constant M > 0 such that  .s/ D O.eMjsj log jsj /. The following lemma then allows us to conclude that when Re.s/ > 1=2 and jsj ! C1, 1C"

s.s  1/ s=2  .s=2/.s/ D O.jsj3 e 2 jsj log jsj / D O.ejsj M

for all " > 0.

t u

Lemma 4.8 We have .s/  for Re.s/ > 1=2.

/;

1 D O.jsj/ s1

110

4 Maaß Forms

Proof For Re.s/ > 1, C1

.s/ 

X 1 D ns  s1 nD1 D

C1 Z nC1 X

Z

C1

xs dx

1

(4.14) .ns  xs / dx:

n

nD1

The expression under the integral in (4.14) is bounded above in absolute value by ˇZ x ˇ ˇ ˇ jns  xs j D ˇˇ sts1 dtˇˇ 6 jsjn Re.s/1 : n

We deduce that ˇ ˇ ˇ ˇ ˇ.s/  1 ˇ 6 jsj.Re.s/ C 1/: ˇ s  1ˇ 1 The expression (4.14) thus furnishes an analytic continuation of .s/  s1 to all 1 of the half-plane Re.s/ > 0. In particular, j.s/  s1 j 6 jsj.3=2/ for Re.s/ > 1=2. t u

It follows from Proposition 4.7 that the function s 7! .s  1/.s/ is entire and of order 1. Indeed, s.s  1/.s/ and 1=s .s=2/ are both entire and of order 1.

4.2.2 The Space of Incomplete Eisenstein Series We now apply the Mellin inversion formula to a function in .0; C1/. We begin by observing that the integral b.s/ WD

Z

C1 0

with compact support

.y/ys1 dy

(4.15)

converges absolutely for all s. The Mellin inversion formula then implies that 1 .y/ D 2i

Z . /

ys b.s/ds

(4.16)

where . / denotes the vertical line Re.s/ D . The integral (4.16) is independent of the real number : this of course follows from the Mellin inversion formula but one can also see this more directly. Given two real numbers 1 < 2 , an iterated

4.2 Eisenstein Series and the Spectrum of the Laplacian

111

integration by parts shows that on the vertical strip 1 6 Re.s/ 6 2 we have b.s/ D O..jsj C 1/A /;

for every constant A > 0:

(4.17)

Let T be real and arbitrary and consider the line integral 1 2i

Z C

ys b.s/ds;

where C is the rectangle with vertices 1 iT, 1 CiT, 2 iT and 2 CiT, taken in the direct orientation. According to the Cauchy integration formula, since s 7! ys b.s/ is holomorphic, we have 1 2i

Z C

ys b.s/ds D 0:

The rapid decay (4.17) of b in vertical strips (take A D 2) implies that the contributions to the integral of each of the two horizontal sides H1 and H2 are majorized by 

Z

1 O 2

2

1



2

y .1 C jt C iTj/ dt D O.y 2 T 1 /: t

By letting T tend towards C1 (with the other parameters being fixed), we deduce that these two contributions disappear and it remains 1 2i

Z . 1 /

ys b.s/ds D

1 2i

Z . 2 /

ys b.s/ds:

We shall make regular use of this method of contour shift along a vertical line. The key point is that the integrand decays rapidly at infinity in vertical strips, see (4.17). We shall encounter cases where the contour C encircles a finite number – bounded for all T – of poles of the integrand. One must then take into account the residues of the integrand of these poles. It follows from this expression that for > 1, E.zj / D

1 2i

Z . /

E.z; s/b.s/ds:

This is formally obvious by summing over 1 n the identity 1 .Im. z// D 2i

Z . /

Im. z/s b.s/ds:

(4.18)

112

4 Maaß Forms

This identity is valid for every real . The question is to justify the convergence of the sum and the interchange of the sum and integral. But according to Lemma 4.1 the series E.z; s/ is absolutely convergent for D Re.s/ > 1 and jE.z; s/j 6 E.z; /. It then follows from (4.17) that the integral (4.18) converges absolutely as soon as

> 1. Again we can show directly – by shifting contours – that (4.18) is independent of > 1. We shall in fact shift the contour to the vertical line D 1=2 so that we integrate b against the “unitary Eisenstein series”. Along the way we encounter a pole of E.z; s/.3 Let us from now on write E. nH/ for the space of incomplete Eisenstein series. Proposition 4.9 The Hilbert space L2 . nH/ is the orthogonal sum L2 . nH/ D C. nH/ ˚ E. nH/ of the closures of C. nH/ and E. nH/. Proof We conserve the notation from the proof of Lemma 4.1; recall in particular that the strip S D fz 2 H j jxj < 1=2g is a fundamental domain for 1 . Consider a function f 2 L2 . nH/. It follows from the Cauchy-Schwarz inequality and the finiteness of the area of D that j f j is integrable on D. Thus h f ; E.j /i D D

1 2

Z D

Z 1X 2 Z

X

f .z/

 2E C1

D 0

.Im. z// d .z/

 21 n

Z

f .z/ .y/ d .z/ D

D

f .z/ .y/ d .z/ S

Z

1=2

1=2

 f .z/ dx

.y/y2 dy:

The inner integral is the constant term f0 .y/ of the Fourier expansion of f .4 We thus obtain5 Z h f ; E.j /i D

0

C1

f0 .y/ .y/y2 dy:

(4.19)

We shall see that this explains why the constant functions belong to the subspace E . nH/  L2 . nH/ in the decomposition of Proposition 4.9 below.

3

4

Note that according to Fubini’s theorem we can speak of the constant term (as a measurable function) f0 of any integrable function f on D. The subspace C . nH/ then consists of functions whose constant term is almost everywhere equal to 0.

5

This is no problem at zero since

is of compact support.

4.2 Eisenstein Series and the Spectrum of the Laplacian

113

Assume now that f is orthogonal to the subspace E. nH/. Then the integral (4.19) is zero for all functions of compact support in .0; C1/. This last property is equivalent to the condition that f 2 C. nH/. t u It is immediate that the Laplacian  preserves the subspace C. nH/ and E. nH/. We have already verified that  has discrete spectrum in C. nH/. In the rest of this section we show that, in E. nH/, the spectrum of  is continuous, with the exception of the zero eigenvalue corresponding to the constant functions. We thus have to implement the meromorphic continuation of the Eisenstein series in (4.18) in such a way as to reduce to the case of D 1=2. We begin by showing that the Eisenstein series E.z; s/ is analytic on the vertical line Re.s/ D 1=2.

4.2.3 Regularity of E.z; s/ on the Vertical Line Re.s/ D 1=2 If follows from Theorem 4.2 (and from the meromorphic continuation of the Riemann zeta function) that the Eisenstein series E.z; s/ admits a meromorphic continuation to the entire complex plane, with only one pole on the half-plane Re.s/ > 1=2 at s D 1. In this subsection we show that the Eisenstein series remains analytic on the line Re.s/ D 1=2. We deduce this from the following lemma which expresses what is known as the Maaß-Selberg relations. Given a real number Y > 0, write EY .z; s/ for the truncated Eisenstein series. This is a  -invariant function equal to E.z; s/ when z 2 D has imaginary part 6 Y and equal to E.z; s/  ys  '.s/y1s if z 2 D has imaginary part > Y. Lemma 4.10 (Maaß-Selberg) Let > 1=2 be a real number and s D C ir and s0 D C ir0 two complex numbers on the line . / whose imaginary parts r and r0 are not opposite each other. Then hEY .; s/; EY .; s0 /i D 

1 0 '.  ir0 /Y i.rCr / 0 i.r C r /

1 1 0 0 '. C ir/Y i.rCr / C Y 2 1Ci.rr / i.r C r0 / 2  1 C i.r  r0 / 

1 0 '. C ir/'.  ir0 /Y 12 i.rr / : 0 2  1 C i.r  r /

Proof Note first that hEY .; s/; EY .; s0 /i D

Z EY .z; s/EY .z; s0 / d .z/ Z

D

EY .z; s/EY .z; s0 / d .z/

D D

114

4 Maaß Forms

and that EY D E for z 2 DY , the subset of D consisting of complex numbers of imaginary part 6 Y. Write  D s.1  s/ and 0 D s0 .1  s0 / for the corresponding eigenvalues of the Eisenstein series E.z; s/ and E.z; s0 /. According to Green’s formula (3.5) and the Fourier expansions of the Eisenstein series, .  0 /

Z z2DY

E.z; s/E.z; s0 / d .z/ Z D .E.z; s0 /E.z; s/  E.z; s/E.z; s0 // d .z/ Z

z2DY 1

D 0

D

X

 @  E .x C iY; s/ E.x C iY; s0 / @y  @  E.x C iY; s/ E .x C iY; s0 / dx @y

.an .Y; s0 /a0n .Y; s/  a0n .Y; s0 /an .Y; s//:

n2Z

But according to the differential equation of the Whittaker functions (see (3.10)),  d an .y;s0 /a0n .y; s/  a0n .y; s0 /an .y; s/ dy D an .y; s0 /a00n .y; s/  a00n .y; s0 /an .y; s/ D .1  y2 /an .y; s0 /an .y; s/  .1  0 y2 /an .y; s0 /an .y; s/ D .0  /y2 an .y; s0 /an .y; s/: If n ¤ 0, this last expression is of exponential decay as y ! C1, so one can then integrate it over the set of z 2 D of imaginary part > Y. One obtains in this way an .Y; s0 /a0n .Y; s/  a0n .Y; s0 /an .Y; s/ D .0  /

Z

C1

an .y; s0 /an .y; s/y2 dy:

Y

By summing over all n ¤ 0, this gives .  0 /

Z z2DY

E.z; s/E.z; s0 / d .z/ D a0 .Y; s0 /a00 .Y; s/  a00 .Y; s0 /a0 .Y; s/ Z C1 Z 1 C .0  / EY .z; s/EY .z; s0 / d .z/: Y

0

4.2 Eisenstein Series and the Spectrum of the Laplacian

115

We can rewrite this as hEY .; s/; EY .; s0 /i D

 1  0 /a0 .Y; s/  a0 .Y; s0 /a .Y; s/ : a .Y; s 0 0 0 0   0

We deduce that hEY .; s/; EY .; s0 /i  1 a0 .Y;  ir0 /a00 .Y; C ir/ i.r C r0 /.1  2  i.r  r0 //   a00 .Y;  ir0 /a0 .Y; C ir/  n 1

ir0 0 1 Cir0 Y D C '.  ir /Y i.r C r0 /.1  2  i.r  r0 //    . C ir/Y 1Cir C .1   ir/'. C ir/Y  ir    Y Cir C '. C ir/Y 1 ir o  0 0  .  ir0 /Y 1ir C .1  C ir0 /'.  ir0 /Y  Cir   n 0 0  .1  2  i.r  r0 // '. C ir/Y i.rCr /  '.  ir0 /Y i.rCr / o  0 0 C i.r C r0 / Y 2 1Ci.rr /  '. C ir/'.  ir0 /Y 12 1i.rr /

D

D

1 1 0 0 '.  ir0 /Y i.rCr /  '. C ir/Y i.rCr / i.r C r0 / i.r C r0 / C

1 0 Y 2 1Ci.rr / 0 2  1 C i.r  r /



1 0 '. C ir/'.  ir0 /Y 12 i.rr / : 2  1 C i.r  r0 /

t u

Given > 1=2 and r ¤ 0, we in particular find that if s0 D C ir,   jEY .; s0 /j2 D .2  1/1 Y 2 1  r1 Im '. C ir/Y 2ir  .2  1/1 Y 12 j'. C ir/j2 :

(4.20)

We can now prove the following theorem. Theorem 4.11 The function s 7! E.z; s/ has no pole on the line Re.s/ D 1=2. Proof According to (4.11), j'. C ir/j2 ! 1

116

4 Maaß Forms

as tends to 1=2. It then follows from (4.20) that jEY .; C ir/j remains bounded as ! 1=2. But ( E.z; s/ D

EY .z; s/

if y 6 Y;

E .z; s/ C y C '.s/y Y

s

1s

if y > Y:

The function s 7! E.; s/ therefore has no pole on the line Re.s/ D 1=2.

t u

4.2.4 Eisenstein Series and the Riemann Zeta Function II The r-th Fourier coefficient of E.z; s/ takes the form ar .y; s/ D 2

jrjs1=2 12s .jrj/ p y Ks1=2 .2jrjy/: .2s/

We may therefore deduce from Theorem 4.11 the following well-known result. Corollary 4.12 (Hadamard-De la Vallée Poussin) The Riemann zeta function .s/ does not vanish on the line Re.s/ D 1. The prime number theorem is a corollary of the theorem of Hadamard-De la Vallée Poussin. Theorem 4.13 (Prime Number Theorem) One has the asymptotic .x/ WD jfp 2 N prime 6 xgj 

x log x

as x 2 RC tends toward C1. Proof For Re.s/ > 1, we have 

X log p  0 .s/ D .s/ ps  1 p prime D

X log p p

ps

„ ƒ‚ … ˚.s/

C

X p

„

log p :  1/ ƒ‚ …

ps .ps

converges for Re.s/>1=2

The theorem of Hadamard-De la Vallée Poussin then implies that the function s 7! ˚.s/ extends to the full half-plane Re.s/ > 1 with a simple pole at s D 1

4.2 Eisenstein Series and the Spectrum of the Laplacian

117

of residue 1. But ˚.s/ D

X log p ps

p

Z

C1

Ds 1

Z

C1

D 1

d .x/ xs

.x/ dx Ds xsC1

Z

C1 0

est .et /dt;

where .x/ D

X

log p:

p6x

We shall admit the following theorem due to Ikehara. Theorem 4.14 Let ' be a positive increasing function on RC such that the integral Z f .s/ D

C1 0

est '.t/dt .s D C i/

converges for > 1. Assume furthermore that there exists a constant A and a function  7! g./ such that A D g./ s1

lim f .s/ 

!1C

uniformly on compacta  2 Œa; a. Then '.t/  Aet : Applied to the function '.t/ D .et /, the Theorem 4.14 implies .x/  x: Finally, for " > 0, we have .x/ log x D

X

log x >

p6x

X

log p D .x/

p6x

>

X

log p

x1" 6p6x

> .1  "/

X

log x

x1" 6p6x



> .1  "/ log x .x/ C O.x1" / : From this we deduce the claim that .x/  x= log x. Let us now return to the problem of the spectral decomposition on  nH.

t u

118

4 Maaß Forms

4.2.5 The Eisenstein Transform We denote by Cc1 ..0; C1// the subspace of compactly supported C1 functions in the Hilbert space L2 ..0; C1// endowed with the scalar product 1 h f ; gi D 2

Z

C1 0

f .r/g.r/ dr:

The Eisenstein transform is the map E W Cc1 ..0; C1// ! C1 . nH/ defined by 1 .Ef /.z/ D 2

Z

C1

f .r/E.z; 1=2 C ir/ dr:

0

(4.21)

The computation (4.10) of the constant term of E.z; s/ shows that the Eisenstein series E.z; 1=2 C ir/ are almost L2 on D. Upon integrating by parts in the r variable we obtain a slightly better upper bound on the Eisenstein transform, namely .Ef /.z/ D O.y1=2 .log y/1 /

(4.22)

as y tends toward infinity in the cusp. The gain by a logarithmic factor implies that the image of the Eisenstein transform is in L2 . nH/, i.e., E W Cc1 ..0; C1// ! L2 . nH/: Proposition 4.15 The Eisenstein transform E is an isometry from Cc1 ..0; C1// into L2 . nH/. Proof We again denote by EY .z; s/ the truncated Eisenstein series. This is the  -invariant function equal to E.z; s/ if z 2 D has imaginary part 6 Y and to E.z; s/  ys  '.s/y1s if z 2 D has imaginary part > Y. Consider the truncated Eisenstein transform .EY f /.z/ D

1 2

Z

C1 0

f .r/EY .z; 1=2 C ir/ dr:

For all z 2 D we have .EY f /.z/ D .Ef /.z/ if y.z/ 6 Y and, according to (4.22),  .EY f /.z/ D .Ef /.z/ C O

y.z/1=2 log y.z/



4.2 Eisenstein Series and the Spectrum of the Laplacian

119

in general. In particular, j.E  EY /f j D O..log Y/1=2 /: Moreover, jEY f j2 D

1 .2/2

Z

C1

0

Z

C1 0

f .r0 /f .r/hEY .; 1=2 C ir0 /; EY .; 1=2 C ir/idrdr0 ;

and according to Lemma 4.10, hEY .; 12 C ir0 /; EY .; 12 C ir/i D

  1 0 1  '. 12 C ir0 /'. 12  ir/ Y i.rr / i.r0  r/   0 1 0 Y i.r r/  Y i.rr / C 0 i.r  r/ 

1 0 '. 1 C ir0 /Y i.rCr / i.r C r0 / 2

C

1 0 '. 12  ir/Y i.rCr / : 0 i.r C r /

The integral expression of jEY f j2 then decomposes as the sum of four integrals corresponding to the four terms on the right-hand side of the above equality. We can again integrate by parts (in the r variable) in three of the four integrals to gain a factor of log Y. We leave only the second integral untouched for the moment, yielding 1 jE f j D .2/2 Y

2

Z

Z C1 C1 0

0

0

Y i.r r/  Y i.rr / drdr0 C O..log Y/1 /: f .r /f .r/ i.r0  r/

0

0

Now we change variables from r0 to u D r0  r and then from u to v D u log Y. We get 1 jE f j D .2/2 Y

2

D

1 .2/2

Z Z

Z

C1 0

f .r/ C1

0

f .r/

Z

C1 r

f .r C u/2 sin.u log Y/

du dr C O..log Y/1 / u

C1

f .r C v= log Y/2 sin.v/

r log Y

dv dr C O..log Y/1 /: v

120

4 Maaß Forms

Since f is C1 and compactly supported in .0; C1/, we have Z

C1

dv D f .r C v= log Y/2 sin.v/ v r log Y

Z

C1

f .r C v= log Y/2 sin.v/

1

D f .r/

Z

C1 1

2 sin.v/

dv v

dv C O..log Y/1 / v

D 2f .r/ C O..log Y/1 /; where the constant in the O depends on f but not on r. Finally, 1 jE f j D 2 Y

2

Z

C1 0

f .r/f .r/ dr C O..log Y/1 /:

We conclude the proof of Proposition 4.15 by letting Y tend towards C1.

t u

Proposition 4.15 allows us to extend E to an isometry from L2 .RC / onto (the closure of) its image Im.E/  L2 . nH/. This subspace is invariant under the action of the Laplacian . Indeed, we have E D EM; where .Mf /.r/ D .r2 C 1=4/f .r/: The proof of Lemma 4.10 – more precisely, the computation of hEY .; s/; EY .; s/i – shows that, if f 2 C. nH/ is a Laplacian eigenfunction6 with eigenvalue ..r0 /2 C 1=4/, then h f ; EY .; 1=2 C ir/i D 0 (the constant term of f is trivial). The image of the Eisenstein transform is thus orthogonal to the space C. nH/ of cusp forms. It is moreover orthogonal to the constant functions since the eigenvalue 0 is strictly less than 1=4 6 1=4 C r2 for r 2 R. We deduce that C  1 ˚ Im.E/  E. nH/: We are going to show that the space of constant functions C  1 and the image of the Eisenstein transform generate a dense subspace of E. nH/.

6

Since f is cuspidal, f Y D f .

4.2 Eisenstein Series and the Spectrum of the Laplacian

121

4.2.6 The Spectral Theorem for the Modular Surface Since the incomplete Eisenstein series are of compact support in  nH, we can in particular form the “scalar product”7 hE.j /; E.; 1=2 C ir/i: We may use the analog of the formula (4.19) and the Fourier expansion of the Eisenstein series to see that hE.j /; E.; 1=2 C ir/i Z C1  1=2ir  D y C '.1=2  ir/y1=2Cir .y/y2 dy 0

D b.1=2 C ir/ C '.1=2  ir/b.1=2  ir/:

(4.23)

We put f .r/ D b.1=2 C ir/ C '.1=2  ir/b.1=2  ir/. Thus we have 1 4

Z hE.j /; E.; 1=2 C ir/iE.z; 1=2 C ir/ dr D E.f /:

(4.24)

R

The modular surface is of area =3. We then denote by u0 the constant function p equal to 3= and by .uj /, for j > 1, an orthonormal basis of the cuspidal space C. nH/, ordered by increasing eigenvalues j . Thus uj D j uj and 0 D 0 < 1 6 2 6    6 n    Theorem 4.16 Let  D SL.2; Z/. For all u 2 C. nH/ ˚ E. nH/, we have X

1 u.z/ D hu; ujiuj .z/ C 4 j>0

Z hu; E.; 1=2 C ir/iE.z; 1=2 C ir/ dr

(4.25)

R

in the L2 sense. If, moreover, u belongs to the domain D. nH/ of , the series (4.25) converges absolutely and uniformly on compacta. Finally, we have the Plancherel formula jujL2 . nH/ D

X j>0

7

1 jhu; uj ij C 4 2

Z

jhu; E.; 1=2 C ir/ij2 dr:

R

This is an abuse of notation: the Eisenstein series E.; 1=2 C ir/ is not square integrable.

(4.26)

122

4 Maaß Forms

Remark 4.17 As for the Fourier transform, the Plancherel formula allows one to extend the spectral decomposition to all of L2 . nH/. The notation hu; E.; 1=2Cir/i then becomes dangerous: the integral no longer converges! Proof We can assume u 2 E. nH/, so that u is equal to an incomplete Eisenstein series E.zj /, where 2 Cc1 ..0; C1//. The integral (4.18) converges absolutely for > 1. We then shift the integration contour to the left, stopping at D 1=2. To do this we need to control the growth of E.z; s/ in vertical strips in s. We shall apply the Phragmén-Lindelöf principle. Let " be a positive real number. We have already remarked that jE.z; 1C"Cit/j 6 E.z; 1 C "/, which is a strictly positive constant. The functional equation of the zeta function along with Stirling’s formula (4.13) together imply that ˇ ˇ j.2" C 2it/j  j.1 C 2"  2it/j ˇ

t ˇˇ1=2C2" ˇ 

as jtj tends toward infinity. It then follows from Theorem 4.2 – and again from Stirling’s formula – that jE.z; "  it/j D Oz;" .jtj1=2C" / as jtj tends toward infinity. As for the Riemann zeta function, the Eisenstein series E.z; s/ is of order 1. The Phragmén-Lindelöf principle then applies. We deduce that the Eisenstein series E.z; s/ remains of polynomial growth (away from the poles) in the vertical strip " 6 Re.s/ 6 1 C ". Since, moreover, the function b is of rapid decay in vertical strips (see (4.17)), we may shift the contour to D 1=2. We pass the unique pole at s D 1 of E.z; s/. The residue of E.z; s/ at s D 1 is =2 .1/.2/ D 3=, and one finds r Z 3 b 1 b.s/E.z; s/ds: .1/u0 C E.zj / D (4.27)  2i .1=2/ Finally, since the Eisenstein series E.z; s/, on the line D 1=2, are orthogonal to the constant functions and since the area of the modular surface is =3, we may rewrite the first term above as r 3 b .1/ D hE.j /; u0 i: (4.28)  The expansion (4.27) is not sufficient to prove Theorem 4.16: the coefficient b.s/ is not the projection of E.j / onto E.; s/. To obtain the desired decomposition we use (4.23) and the functional equation (4.3), written as E.z; 1  s/ D '.1  s/E.z; s/:

4.3 Existence of Cusp Forms

123

These two expressions imply hE.j /; E.; 1=2 C ir/iE.z; 1=2 C ir/ D b.1=2 C ir/E.z; 1=2 C ir/ C b.1=2  ir/E.z; 1=2  ir/: Upon integrating this last expression over R, we finally obtain Z Z 1 b.s/E.z; s/ds D 1 hE.j /; E.; 1=2 C ir/iE.z; 1=2 C ir/ dr; 2i .1=2/ 4 R which is nothing other than the projection onto the Eisenstein series. It remains then to establish the Plancherel formula. This follows from (4.23), (4.24) and Proposition 4.15: ˇ ˇ2 Z ˇ 1 ˇ ˇ ˇ hE.j /; E.; 1=2 C ir/iE.z; 1=2 C ir/ dr D jE.f /j2L2 . nH/ ˇ 4 ˇ2 R L . nH/ Z C1 1 D j f j2 2 0 Z C1 1 D jhE.j /; E.; 1=2 C ir/ij2 dr 2 0 Z 1 D jhE.j /; E.; 1=2 C ir/ij2 dr: 4 R (The last equality is obtained by using the functional equation E.z; 1  s/ D '.1  s/E.z; s/ and the unitarity of ' along the line D 1=2.) t u

4.3 Existence of Cusp Forms We have already remarked that it is not at all obvious that C. nH/ should not in general be reduced to f0g. A profound theorem of Selberg affirms that dim C. nH/ D C1 as soon as  is a finite index subgroup of SL.2; Z/. It suffices of course to show this for  D SL.2; Z/. This is the goal of this section.

4.3.1 Automorphic Wave Equation Just as for the heat equation considered in Chap. 3, Theorem 4.16 allows one to construct solutions u D u.z; t/ 2 C1 . nH  R/ to the automorphic wave equation u @2 u C u D ; 2 @t 4

(4.29)

124

4 Maaß Forms

with initial conditions @u .z; 0/ D 0; @t

u.z; 0/ D f .z/;

f 2 Cc1 . nH/:

(4.30)

We can think of a solution to (4.29) as the amplitude of a wave which propagates on the hyperbolic surface  nH. For all t 2 R we can define a linear operator Ut of Cc1 . nH/ in L2 . nH/ which, with a function f 2 Cc1 . nH/, associates the unique solution u D u.z; t/ of (4.29) such that u.; 0/ D f and ut .; 0/ D 0. Formally, we want p Ut D cos.t   1=4/: One can define Ut rigorously by Ut f D h f ; u0 i cosh.t=2/u0 C

X j>1

C

1 4

h f ; uj i cos.rj t/uj

Z h f ; E.; 1=2 C ir/i cos.rt/E.; 1=2 C ir/ dr: R

Here we have retained the notation of Theorem 4.16 and have written f1=4 C rj2 g for the spectrum of  in the cuspidal space C. nH/. The operator Ut is self-adjoint. Similarly to the case of the heat kernel, we now seek to write Ut in the form of an invariant integral operator Z Tk W f 7!

H

k.; w/f .w/ d .w/

such that lim Tk f D f ; t#0

lim.Tk f /t D 0; t#0

locally uniformly and for all f 2 Cc1 . nH/. The kernel k, which depends on t, should in particular satisfy the equation 1 ktt D z k C k: 4

(4.31)

By integrating (4.31) against an eigenfunction of the Laplacian with eigenvalue .1=4 C r2 /, while using the fact that the Laplacian is a symmetric operator, we deduce from Theorem 3.7 that the Selberg transform h of k satisfies the ordinary differential equation  1 1 htt .r/ D  C r2 h.r/ C h.r/ D r2 h.r/: 4 4

(4.32)

4.3 Existence of Cusp Forms

125

We thus expect (at least formally) to have h.r/ D cos.rt/: The function h lies in the space PW.C/ (cf. § 3.4) and is of type t. It follows from Proposition 3.11 that k.i; z/ D .S 1 h/.z/ Z C1 1 D .eirt C eirt /!r2 C1=4 .i; z/r tanh.r/ dr: 4 0

(4.33)

(Recall that z 7! !r2 C1=4 .i; z/ is the unique radial Laplacian eigenfunction of eigenvalue .r2 C 1=4/ whose value at i is 1.) Theorem 4.18 Given f 2 Cc1 . nH/, the solution u.z; t/ to Eq. (4.29) satisfying the initial conditions (4.30) can be written 1 u.z; t/ D 4

Z Z H

C1 0

.eirt C eirt /!r2 C1=4 .z; w/r tanh.r/f .w/ drd .w/:

Given z and t, the value of u.z; t/ is independent of the values of f at points w such that .z; w/ 6 jtj. Proof We leave as an exercise the verification of the first part of the theorem. For the second part, note that the above integral has a kernel k given by (4.33). But, since h is of type t, Proposition 3.11 implies that 7! k. / has support contained in the ball of radius 6 jtj. This establishes the second claim. t u

4.3.2 Construction of Cusp Forms We begin by noting that – at least formally – we have 2Ut E1=2Cir D .eirt C eirt /E1=2Cir : A lovely observation of Lindenstrauss and Venkatesh relates this identity to the existence, for every prime number p, of Hecke operators Tp , acting on functions on  nH. By definition, one has ! p1  X 1 zCk : f Tp f .z/ D p f .pz/ C p p kD0

(4.34)

126

4 Maaß Forms

We shall describe the Hecke operators in more detail later in Chaps. 7 and 9. For the moment we simply record the facts that each operator Tp is self-adjoint and commutes with the Laplacian, and that the Eisenstein series E1=2Cir .z/ are eigenfunctions of the Tp . In fact, we have Tp E1=2Cir D .pir C pir /E1=2Cir :

(4.35)

We thus obtain the following result. Proposition 4.19 For all f 2 E. nH/, Tp f D 2Ulog p f :

(4.36)

The two operators Tp and Ulog p being self-adjoint, Proposition 4.19 implies the following corollary. Corollary 4.20 For all f 2 L2 . nH/, ŒTp  2Ulog p f 2 C. nH/. To show that C. nH/ is not reduced to f0g, it then remains to find a function f 2 L2 . nH/ which does not satisfy (4.36). We shall now in fact show that there are “many” such functions, by constructing them “high in the cusp”. Given a positive real number R we write ˝R D 1 nfz j Im.z/ > Rg: When R is strictly greater than 1 the projection of ˝R to  nH is injective. We can then think of the space L2 .˝R / as a Hilbert subspace of L2 . nH/. Given a non-zero integer n and a real number R strictly greater than 1, we write Vn;R for the subspace of Cc1 .˝R / consisting of functions of the form f .z/ D h.y/e.nx/. Assume that R > et . Since the solutions to Eq. (4.29) propagate at a speed less than 1, for all f 2 Vn;R , the function Ut f (t > 0) has compact support contained in the set of points whose distance to ˝R is smaller than t. The support of Ut f is therefore contained in ˝et R . Each domain ˝R is preserved by the action by translations of the group R=Z and this induced an action .s  f /.x C iy/ D f .x C s C iy/;

s 2 R=Z;

on the space C01 .˝R /. This action commutes with the Laplacian and thus with the operator Ut as well. Finally, the subspaces Vn;R of C01 .˝R / correspond to the proper subspaces of this action relative to the characters s 7! e.ns/. Each operator Ut thus preserves the spaces Vn;R , for Ut Vn;R  Vn;et R ;

R > et :

(4.37)

4.4 Hyperbolic Periods of Eisenstein Series

127

On the other hand, it follows directly from (4.34) that, for R > p, we have ( Tp Vn;R 

Vpn;R=p

if p 6 jn

Vpn;R=p ˚ Vn=p;pR

if pjn:

(4.38)

Since the spaces Vn;R and Vn0 ;R0 are orthogonal for n ¤ n0 , it follows from (4.37) and (4.38) that Tp Vn;R and Ulog p Vn;R (where n ¤ 0, R > p) are orthogonal. The following theorem is now almost immediate. Theorem 4.21 For all n ¤ 0, R > p, the map .Ulog p  Tp / sends Vn;R injectively into C. nH/. In particular, there exists an infinity of linearly independent Maaß cusp forms for SL.2; Z/ (and thus for every Fuchsian subgroup  SL.2; Z/). Proof The operator Tp is injective on Vn;R : this can be seen by inspecting the action of Tp on the Fourier coefficients, see Lemma 7.6. But if f 2 Vn;R and Tp f are nonzero, so is .Tp  Ulog p /f as soon as n ¤ 0 and R is taken strictly greater than p. u t

4.4 Hyperbolic Periods of Eisenstein Series We have already related the parabolic periodicity E.z C 1; s/ D E.z; s/ of the Eisenstein series to the study of the Riemann zeta function. But E.z; s/, being  invariant, a other periods as well; in this section we study its periodicity along closed primitive geodesics in  nH. This will allow us to show that certain L-functions defined by Hecke satisfy a functional equation.

4.4.1 Primitive Geodesics on the Modular Surface A hyperbolic element ı 2  is conjugated in SL.2; R/ to a diagonal matrix 

t 0 0 t1



where t is a real number of absolute value jtj > 1. The trace of ı is thus equal to t C t1 and one calls the number N.ı/ D t2 the norm of ı. One says that ı is primitive if it is not a non-trivial power of another element in  . The axis (taken with its natural orientation) of a primitive hyperbolic element ı projects on a closed (oriented) geodesic of length log N.ı/ in the modular surface  nH. The different conjugates of ı in  give the same closed (oriented) geodesic. Every closed geodesic on  nH is the projection of the axis of some hyperbolic element in  . All of the closed geodesics on  nH are thus obtained in the way just described.

128

4 Maaß Forms

One can associate with a hyperbolic matrix ıD

  ab 2 SL.2; Z/ cd

.ja C dj > 2/

an integral quadratic form Qı .x; y/ D cx2 C .d  a/xy  by2 of discriminant D D .a C d/2  4 > 0. Two matrices sharing the same trace are conjugate in SL.2; Z/ if and only if the corresponding quadratic forms are SL.2; Z/-equivalent, where SL.2; Z/ acts on binary quadratic forms via the formula .g; Q/ 7! Q ı g1 :

(4.39)

Conversely one may associate with every integral quadratic form Q.x; y/ D Ax2 C Bxy C Cy2 whose discriminant D D B2  4AC > 0 is not a perfect square the oriented geodesic . 1 ; 2 / of H which joins the two (real) roots p B C D 1 D 2A

p B  D and 2 D 2A

(4.40)

of the degree two equation Q.z; 1/ D 0. The subgroup of SL.2; R/ fixing Q under the action (4.39) is, by definition, the special orthogonal group SO.Q/; it is a Lie group of dimension 1, with two connected components (the stabilizers of the two sheets of the hyperbola Q.x; y/ D 1). One can show that   tBu ˇ Cu ˇ 2 2 2 SO.Q/ D ˙ ˇ u; t 2 R; t  Du D 4 : Au tCBu 2 The intersection SO.Q; Z/ D SO.Q/ \ SL.2; Z/ then contains all of the matrices such that t and u are integral; this is in fact an equality when A, B and C are relatively prime. It follows from the proof of Theorem 2.3 (see Exercise 2.17) that the group SO.Q; Z/ is (discrete and) cocompact in SO.Q/. Modulo ˙I it is therefore an infinite cyclic group generated by ı D ıQ D

 t0 Bu0 2

Au0

Cu0 t0 CBu0 2

 ;

(4.41)

4.4 Hyperbolic Periods of Eisenstein Series

129

where t0 and u0 are positive and together form a fundamental solution to the Pell equation t2  Du2 D 4. (Note that we can always assume that this generator is of positive trace.) The element ıQ 2  is primitive and its axis is the geodesic . 1 ; 2 /; the projection of this to  nH is a closed oriented geodesic of length  t C pDu  0 0 2 log : 2

(4.42)

p We put "D D .t0 C Du0 /=2. We have therefore constructed a map Q 7! ıQ which, to a quadratic form of positive discriminant D which is not a perfect square, associates a primitive hyperbolic element in  . Since the image of this map does not change if we replace Q by another quadratic form which is proportional to it by a positive scalar, we obtain more naturally a map from primitive quadratic forms, i.e., those such that .A; B; C/ D 1, towards the primitive elements of  . The map which to ı associates the unique integral primitive quadratic form positively proportional to Qı is clearly an inverse map. We finally remark that these maps commute with the action of PSL.2; Z/ by (4.39) on the space of quadratic forms and by conjugation on primitive elements. We deduce the following proposition. Proposition 4.22 The lengths of closed oriented geodesics in the modular surface  nH are the numbers f2 log "D j D 2 N; D  0; 1 .mod 4/;

p D … N2 g:

Each length appears with finite non-zero multiplicity equal to h.D/, the number of SL.2; Z/-classes of primitive integral quadratic forms of discriminant D. Remark 4.23 (A little bit of algebraic number theory) p For every square free integer d > 0, we can form the real quadratic extension Q. d/=Q; let Od be the ring of integers and ˛ 7! ˛ 0 the Galois involution. The minimal polynomial of p ˛ D a C b d, b ¤ 0, a; b 2 Q, is p p .X  .a C b d//.X  .a  b d// D X 2  2aX C .a2  b2 d/: Thus ˛ is an algebraic integer if and only if 2a 2 Z;

a2  b2 d 2 Z:

130

4 Maaß Forms

It follows easily from this that (

p p ZΠd D fm C n d j m; n 2 Zg Od D p fm C n.1 C d/=2 j m; n 2 Zg

if d  2; 3 .mod 4/; if d  1 .mod 4/:

The norm Nd .x; y/ given by p p .x C y d/.x C y d/0 D x2  dy2 ;  p  p 0 x C y 1C2 d x C y 1C2 d D x2 C xy C

if d  2; 3 .mod 4/; 1d 2 4 y ;

if d  1 .mod 4/;

then defines an integral quadratic form of discriminant ( DD

4d

if d  2; 3 .mod 4/;

d

if d  1 .mod 4/I

p by definition this is in fact the discriminant of the number field Q. d/. In general, if Q is an integral quadratic form of discriminant  the discriminant D p of the number field Q. / divides  and the quotient =D is the square of an integer; this integer is equal to 1pif and only if the form Q is primitive. p A fractional ideal a of Q. d/ is an Od -module contained in Q. d/. Two non-zero fractional ideals are narrowly equivalent if their quotient is a principal p fractional ideal generated by a totally positive element of Q. d/. In our case, an p element 2 Q. d/ is totally positive if it is of positive norm N. / D 0 . Let .a1 ; a2 / be a basis of a viewed as a Z-module. Permuting a1 and a2 if necessary we then have ˇ ˇ ˇ a1 a2 ˇ p ˇ ˇ (4.43) ˇ a0 a0 ˇ D D Na; 1 2 where Na denotes the norm of a, i.e., the multiplicative function which coincides with the index ŒOd W a on integral ideals. To show (4.43) we note that, upon multiplying a by an integer if necessary, we can assume that a is integral; writing Od D Z C !Z, we then have ˇ ˇ ˇ ˇ2 ˇ a 1 a 2 ˇ2 ˇ ˇ ˇ ˇ D ŒOd W a2 ˇ 1 ! ˇ D Na2 D: ˇ a0 a0 ˇ ˇ 1 !0 ˇ 1 2 We put Qa1 ;a2 .x; y/ D Na1 .a1 x C a2 y/.a1 x C a2 y/0 :

4.4 Hyperbolic Periods of Eisenstein Series

131

Since a1 , a2 and a1 C a2 all lie in a their norms are divisible by Na. Thus Naja1 a01 ; a1 a02 C a01 a2 ; a2 a02 and the form Qa1 ;a2 is integral of discriminant D. A simple computation shows that the SL.2; Z/-equivalence class of Qa1 ;a2 depends only on the narrow equivalence class of a and that the map Œa 7! ŒQa1 ;a2  defines a bijection of this quotient onto the set of equivalence classes of integral binary quadratic forms of discriminant p D. The number h.D/ is therefore equal to the narrow class number of the field Q. d/. The term “narrow” reflects the fact that we consider only the oriented geodesics (or that we consider only the upper, and not also the lower, half-plane). Starting from the primitive form Q.x; y/ D Ax2 C Bxy C Cy2 , and recalling the notation in (4.40), we can take as a fractional ideal the Z-module generated by 1 and 2 . This fractional ideal is primitive (not divisible by a rational integer n > 1) and of norm A1 . Geometrically, the matrix  WD

  1  2 1  1

(4.44)

sends the geodesic . 1 ; 2 / onto .0; 1/, the matrix ı 1 is therefore diagonal and equal to   "D 0 : 0 "1 D The various primitive elements ıD hence correspond to the fundamental solution p of the Pell equation t2  Du2 D 4 (in other words, a fundamental unit in Q. d/) p viewed according to different Z-bases of Q. d/ which themselves correspond to the different narrow ideal classes. p An important case – the only case when the narrow class number of Q. d/ is equal to 1 – is the case where the fractional ideal is Od . The corresponding quadratic form is Nd . p For p simplicity suppose that d  2; 3 .mod 4/; the associated geodesic is then . d; d/ and ˇ  x dy ˇ 2 2 SO.Qd / D ˙ ˇ x; y 2 R; x  dy D 1 : y x 

The intersection of this group with SL.2; Z/ consists of those matrices with x and y integral. To conclude this subsection, let us show how to code the geodesics in  nH in such a way as to recover the classical solution via continued fractions of the PellFermat equation: x2  dy2 D 1.

132

4 Maaß Forms

Coding Geodesics Consider the geodesics .p=q; p0=q0 /

.p; q; p0 ; q0 2 Z; pq0  p0 q D ˙1; 1 D 1=0/:

(4.45)

They do not intersect. To see this, first note that they are the images of .0; 1/ by elements in SL.2; Z/. It then suffices to verify that if gD

  ab 2 SL.2; Z/ cd

is different from ˙I then g..0; 1//, when distinct from .0; 1/, does not is a parabolic intersect it. This is obvious if a or b is zero, in which case g or g 01 1 0 element fixing 0 or 1 and g maps .0; 1/ onto itself or onto a disjoint geodesic. Now if ab ¤ 0, the ratio g.0/ 1 D1 g.1/ ad is positive. The endpoints p=q; p0=q0 thus have the same sign so that .p=q; p0=q0 / lies either strictly to the right or left of .0; 1/. We conclude from the above discussion that the images under SL.2; Z/ of the ideal geodesic triangle 0 in H with vertices 0, 1 and 1 define a triangulation of the hyperbolic plane H. We call it the Farey triangulation. It is represented in Fig. 4.1, which was realized by Arnaud Chéritat. Remark 4.24 We have GL.2; Z/ D SL.2; Z/ [ SL.2; Z/ where   1 0

D : 0 1

Fig. 4.1 Farey’s triangulation

4.4 Hyperbolic Periods of Eisenstein Series

133

One can therefore extend the action of SL.2; Z/ on H by having act on H by

W z 7! z: In this way the group GL.2; Z/ acts on H while preserving the Farey triangulation. The element reverses the orientation of H, and the group SL.2; Z/ coincides with the subgroup of elements of GL.2; Z/ which preserve orientation. An oriented geodesic  in H defines a sequence of segments cut out by its intersection points with the Farey triangulation. Each such segment joins two sides of a triangle, and one can thus associate with it the vertex p=q 2 Q [ f1g at which these two sides intersect. An excursion of the geodesic  is then defined to be the largest sequence of segments having some fixed vertex p=q. Since the group SL.2; Z/ preserves the Farey triangulation, it sends an excursion to an excursion. The size of an excursion is the number of segments composing it. Then to every oriented geodesic  there corresponds a bi-infinite sequence .: : : ; n1 ; n0 ; n1 ; n2 ; : : : /, uniquely defined modulo shifts, which one can think of as the “coding” of  . It is defined as the sequence of sizes of successive excursions of  . This is a sequence of positive integers, which a priori can be finite on the right or left. This happens precisely when at least one endpoint lies in Q [ 1, in which case we simply add zeros to make the sequence bi-infinite.8 With this background on geodesic excursions complete, we now return to the p p geodesic . d; d/ and its p relation p to Pell’s equation. For simplicity, rather thanp consider the geodesic . d; d/, we prefer to look at its translate 0 WD p p p . d C Œ d; d C Œ d/. This changes nothing when we pass top the quotient. p Let us consider the sequence of excursions of the geodesic  . Since  d C Œ d 2 0 p p .1; 0/ and d C Œ d > 1, we form the sequence of excursions beginning with that associated with the vertex p=q D 1. Denote by b0 its size. The first excursion is associated with the vertex b0 . The projective transformation corresponding to the matrix   b0 1 2 GL.2; Z/ 1 0 preserves the Farey triangulation, reverses the orientation of H and sends the triple .1; 0; 1/ to .b0 ; 1; b0 C1/. The size of the excursion at b0 is b1 . The next excursion of 0 is thus associated with the vertex   1 b0 1  b1 D b0 C : 1 0 b1

If the geodesic is one of the .p=q; p0 =q0 / defining the triangulation, its coding is simply .: : : ; 0; 1; 0; : : : /.

8

134

4 Maaß Forms

Iterating on n one sees that in general the n-th excursion is associated with the vertex pn1 =qn1 D Œb0 ; b1 ; : : : ; bn1 ; 0; : : :p , the rational number obtained by p truncating the continued fraction expansion9 of d C Œ d to order n  1. We now note that the primitive element ı 2  which generates the stabilizer of 0 in  preserves 0 , the orientation of H, as well as the Farey triangulation, but it acts non-trivially on 0 . The sequence .bn /n>0pis therefore periodic. We have just shown that the continued fraction expansion of d is periodic. Denote by n0 the period of the sequence .bn /n>0 . The generator of the stabilizer of 0 in GL.2; Z/ sends the point 1 to pn0 1 =qn0 1 . The subgroup p p of GL.2; Z/ consisting of elements fixing the oriented geodesic . d; d/ is therefore generated by the elements   x0 dy0 ˙ y0 x0 where p x0 pn 1 D 0  Œ d: y0 qn0 1 We deduce that ıD

  x0 dy0 y0 x0

or

 2 x0 dy0 y0 x0

according to whether x20  dy20 is equal to 1 or 1. Example 4.25 In the case d D 2, we have p 2D1C

1

: 1 2C p p The period n0 is therefore equal to 1: 2 C Œ 2 D Œ2; 2; : : :  and p0 =q0 D b0 D 2 so that x0 =y0 D 1, i.e., x0 D y0 D 1, and 2C

ıD

9

 2   12 34 : D 11 23

Recall that the continued fraction expansion of ˇ is ˇ D b0 C

1 D Œb0 ; b1 ; : : : ; 1 b1 C b2 C:::

where the .bi / form a sequence of integers (a finite sequence, if ˇ 2 Q) with b0 2 Z and bi > 0 for i > 0.

4.4 Hyperbolic Periods of Eisenstein Series

135

4.4.2 Hyperbolic Fourier Series of E.z; s/ We conserve the notation of the previous subsection. Let ı D ıQ 2  be a primitive hyperbolic element and write D for the discriminant of the corresponding quadratic form Q. We again write "D for the maximal eigenvalue of ı and  for the matrix (4.44) which diagonalizes ı. Since the Eisenstein series E.z; s/ are  -invariant, they are in particular ıinvariant and the function z 7! E. 1 z; s/ is invariant by the homothety z 7! "2D z: E. 1 z; s/ D E.ı 1 z; s/ D E. 1 ."2D z/; s/: The restriction of the completed Eisenstein series z 7! E . 1 z; s/ (Re.s/ > 1) to the geodesic .0; 1/ D fiv j v > 0g can therefore be expanded into a Fourier series, X E . 1 .iv/; s/ D bk .s/v ik=log "D (4.46) k2Z

where 1 bk .s/ D 2 log "D

Z

"2D 1

E . 1 .iv/; s/v ik=log "D

dv : v

Since  1 .iv/ D

i 1 v C 2 iv C 1

has imaginary part v. 2  1 / ; 1 C v2 we have E . 1 .iv/; s/ D  s  .s/

X m;n2Z .m;n/¤.0;0/

v s . 2  1 /s : ..m 1 C n/2 v 2 C .m 2 C n/2 /s

We deduce that bk .s/ D

 s  .s/. 2  1 /s 2 log "D 

X ˇ¤0

ˇ ˇik=log "D Z "2 jˇ=ˇ0 j  s D ˇˇˇ v dv N.ˇ/s ˇˇ 0 ˇˇ v ik=log "D ; 2C1 0 ˇ v v jˇ=ˇ j

136

4 Maaß Forms

where the sum runs over all of the non-zero elements of the Z-module generated by 1 and 2 : ˇ D p m 2 C n (and ˇ 0 D m 1 C n). Note then that these elements lie in an ideal b of Q. D/ whosepclass corresponds to the quadratic form Q and which moreover satisfies 2  1 D DNb. p Letp"0 be the positive fundamental unit in the quadratic field Q. D/. Every unit in Q. D/ is, up to multiplication by ˙1, an integer power of "0 . Note that "D D "0 or "20 according10 to whether p "0 is of norm 1 or 1. Given a ˇ 2 Q. D/ we again write .ˇ/ p for the principal (fractional) ideal generated by ˇ. Two elements ˇ1 ;p ˇ2 2 Q. D/ generate the same ideal if and only if their quotient is a unit in Q. D/. We have ˇ ˇ ˇ ˇ ˇ "0 ˇ ˇ ˇ ˇ ˇ D j"2 j ˇ ˇ ˇ ˇ 0 ˇ 0ˇ ˇ "0 ˇ 0 ˇ ˇ 0 and since j"20 ji=log "D is equal to 1 or 1 according to whether "D D "0 or "20 , we find ( ˇ ˇ ˇ "0 ˇ ˇi =log "D if "D D "0 ; jˇ=ˇ 0 ji=log "D ˇ ˇ D ˇ "0 ˇ 0 ˇ 0 i=log "D if "D D "20 :  jˇ=ˇ j 0 The two cases can be treated in the same way, although the latter is slightly more delicate. Since we will need it in the following section, we content ourselves with dealing with this second case only. Assume then for the rest of this subsection that "D D "20 . In the expression for bk we replace the sum over non-zero elements ˇ 2 b by a sum over ideals .ˇ/ and integers m 2 Z. Furthermore, we separate the powers of "0 according to their parity. In this way we find p ˇ ˇik=log "D ˇ ˇ  s  .s/. DNb/s X s ˇ ˇ ˇ bk .s/ D N.ˇ/ ˇ 0 ˇ 2 log "D ˇ 

bj.ˇ/¤0 ˇ 0 2m ˇ ˇˇ " ˇ 0 ˇ "2D ˇˇ 2m ˇ ˇ" ˇ 2m ˇ 0 ˇ ˇ0 " ˇ : ˇ 0 ˇ m2Z ˇ ˇ"2m ˇ

8 j Re./j, and in this case we have Z

C1

K .y/ys

0

s C   s    dy D 2s2   : y 2 2

(4.50)

Proof We use the integral expression 1 2

K .y/ D

Z

C1 0

 y  dt exp  .t C t1 / t 2 t

which can be found in Appendix B. The right-hand side of (4.50) is then equal to 1 2

Z

C1 0

Z

C1 0

 y  dy dt : exp  .t C t1 / t ys 2 y t

Using the change of variables u D 12 ty, v D 12 t1 y, the above integral becomes Z 2

C1

Z

C1

s2 0

0

euv u.sC/=2v .s/=2

du dv : u v

140

4 Maaß Forms

This integral is absolutely convergent for Re.s/ > j Re./j and decomposes as a product of two  -functions. u t By Mellin inversion we find that for > j Re./j and y 2 RC , K .y/ D

1 2i

Z . /

2s2 

s C   s     ys ds: 2 2

Moreover, for sufficiently large the sum (4.49) is absolutely convergent. We deduce that Z p 1 .2 2/s .s; k /ys ds 2i . / Z  s  it   s C it  1 k k  L.s; k /ys ds  s  D 2i . / 2 2 Z  s  it   s C it  X 1 k k k .ˇ/ 2s2  D4  .2N.ˇ/y/s ds 2i . / 2 2 .ˇ/¤0

D4

X

k .ˇ/Kitk .2N.ˇ/y/:

.ˇ/¤0

This leads us to introduce the absolutely convergent series11 k .iy/ D

X

p k .ˇ/ y Kitk .2N.ˇ/y/:

.ˇ/¤0

The above calculation implies that for Re.s/ sufficiently large, Z

C1 0

k .iy/ys1=2

1 p dy D .2 2/s .s; k / y 4

and, by Mellin inversion, p Z p y .2 2/s .s; k /ys ds; k .iy/ D 8i . /

(4.51)

again for sufficiently large. The functional equation of the Hecke L-function (Theorem 4.26) will then allow us to prove the following theorem, a particular case of an important theorem of Maaß.

11

It is natural to include the factor of

p y in order to obtain a sum of Whittaker functions.

4.5 Explicit Construction of Maaß Forms

141

Theorem 4.28 (Maaß) Let k be a non-zero integer. The series k .z/ D

X

p k .˛/ y Kitk .2N.˛/y/ cos.2N.˛/x/;

.˛/¤0

where tk D k=log "0 and z D x C iyp 2 H, is absolutely convergent. The sum here runs over all non-zero ideals .˛/ of ZŒ 2. The series defines a non-zero Maaß cusp form in C. .8/nH/ with eigenvalue k D 1=4 C tk2 . Proof We begin by remarking that since the Whittaker functions Witk .z/ D p 2 yKitk .2y/e.x/ are themselves Laplacian eigenfunctions with eigenvalue k (see (3.11)) the same is true for k . Next we study the group 0 .8/. Recall (see Exercise 2.19) that the surface 0 .8/nH is non-singular, has 4 cusps and is a degree 12 cover of the modular surface. Topologically, this surface is homeomorphic to a sphere minus 4 points. Its fundamental group is the free group on three generators, that we can take to correspond to the three loops encircling three distinct cusps. The parabolic elements       5 2 10 11 and ; 8 3 81 01 generate the stabilizers of three distinct equivalence classes of cusps of 0 .8/. The group 0 .8/ is therefore the extension by f˙1g of the free product of these three elements,         5 2 10 11 1 0 : ; ; ; 0 .8/ D 8 3 81 01 0 1 We shall show that k is almost 0 .8/-invariant, i.e., invariant up to a character. It suffices to do so for the last three generators. By construction, it is clear that k is invariant by translation z 7! z C 1. The two lemmas that follow treat the two other generators. Lemma 4.29 We have k .z/ D k .1=8z/ : In particular, k is invariant under the action of

1 0 81 .

Proof Shifting the integration contour in (4.51) with the help of the PhragménLindelöf principle, and invoking the functional equation of .s; k /, we find p Z 1=8y Ci1 p s .2 2/ .s; k /.8y/s ds D k .i=8y/ : k .iy/ D 8i

i1

142

4 Maaß Forms

Here we implicitly used the fact that k is non-zero and thus .s; k / has no poles. When k D 0 one obtains an additional term coming from a residue. Note that k is in particular non-zero.12 We have shown that k .iy/ D k .i=8y/. The lemma can be deduced from this by observing that f .z/ D k .z/  k .1=8z/ is a Laplacian eigenfunction on H which, along with its partial derivative @f =@x, vanishes on the imaginary axis. An induction argument then shows that all partial derivatives of f vanish on the imaginary axis and thus, since f is real analytic, f D 0. Finally, from the equality p  p     0 1= 8 11 0 1= 8 1 0 p p D 01 8 0 8 1  8 0



one deduces the invariance of k under the action of

1 0 81

t u

.

The rest of the proof consists in proving the almost invariance of k under the action of the remaining parabolic generator of 0 .8/. As in the above proof, the key point is a conjugacy relation between these parabolic generators within SL.2; R/. Lemma 4.30 We have k

 5 2   8 3 z D k .z/:

p p Proof An ideal .ˇ/ of even norm is divisible by the ideal . 2/: .ˇ/ D . 2/.˛/. p Moreover, k . 2/ D 1, so that k .z C 1=2/ C k .z/ D 2

X

p k .ˇ/ y Kitk .2N.ˇ/y/ cos.2N.ˇ/x/

.ˇ/¤0 N.ˇ/ even

D2

X

p p k . 2 ˛/ y Kitk .2N.˛/2y/ cos.2N.˛/2x/

.˛/¤0

p X k p D 2  .˛/ 2y Kitk .2N.˛/2y/ cos.2N.˛/2x/ .˛/¤0

p D 2 k .2z/; and thus k .z C 1=2/ D

p 2 k .2z/  k .z/

D k .z  1=2/:

(4.52)

Indeed, one checks easily that the function L.s; k / is non-zero by writing it as an Euler product in the region of absolute convergence.

12

4.5 Explicit Construction of Maaß Forms

143

Now the matrix identity       1 1=2 1 1=2 1 0 5 2 ; D 0 1 81 0 1 8 3 implies that         1 1 1=2 5 2 10 : zC z D k k 0 1 8 3 81 2 We then deduce from (4.52) that k

        p 1 1=2 5 2 10 z z D 2 k 2 0 1 8 3 81      10 1 1=2  k z : 81 0 1

But it follows from the invariance of k under the action of k

1 0 81

(4.53)

that

     1 1=2 10 z D k .z  1=2/: 0 1 81

Next, Lemma 4.29 implies that      p 1 1=2 10 z 2 k 2 0 1 81        p 1 1=2 20 10 z D 2 k 0 1 01 41 p       p 1 1=2 20 1 1=2 p0 1= 8 D 2 k z ; 8 0 0 1 01 0 1 since p  p      0 1= 8 1 1=2 0 1= 8 10 p p : D 0 1 8 0 8 0 41

(4.54)

144

4 Maaß Forms

Thus the identity (4.52) yields p 2 k

p       1 1=2 20 1 1=2 p0 1= 8 z 0 1 01 8 0 0 1 p       1 1=2 20 0 1= 8 z D 2k 2 p 0 1 01 8 0 p      p 1 1=2 20 p0 1= 8  2 k z : 0 1 01 8 0

According to Lemma 4.29, p       1 1=2 20 0 1= 8 k 2 p z 0 1 01 8 0 D k

p     1 1=2 p0 1= 8 z 0 1 8 0

D k .z  1=2/ and p      1 1=2 20 0 1= 8 p k z D k .2z  1/ D k .2z/: 0 1 01 8 0 We deduce that      p p 1 1=2 10 z D 2k .z  1=2/  2 k .2z/: 2 k 2 0 1 81

(4.55)

Finally, using (4.53), (4.54) and (4.55) we obtain    p 5 2 k z D 2k .z  1=2/  2 k .2z/  k .z  1=2/ 8 3 D k .z/;

where we once again invoked (4.52):

As the matrices         5 2 10 11 1 0 and ; ; 8 3 81 01 0 1

t u

4.6 Commentary and References

145

generate the group 0 .8/, we conclude that for all  2 0 .8/, k . z/ D . /k .z/;

(4.56)

where  

ab cd



D .d/ D .1/.d ( D

2 1/=8

.mod 2/

1

if d  ˙1 .mod 8/

1

if d  ˙3 .mod 8/:

Since the character  is trivial on  .8/, we deduce finally from (4.56) that k is  .8/-invariant. It remains to verify that k is cuspidal. Since the Bessel functions decay rapidly as y goes to infinity, the series defining k vanishes at infinity. The same is true at 0 according to Lemma 4.29, at 1=2 according to (4.52), and at 1=4 again from Lemma 4.29. But the points 0, 1=2, 1=4, and 1 represent the four distinct classes of cusps modulo 0 .8/. The form k is therefore L2 and vanishes at each cusp; this implies that it is cuspidal. t u The surface  .8/nH admits then an explicit sequence of eigenvalues. No explicit non-zero eigenvalue is known for the modular surface, nor does one expect any explicit description.

4.6 Commentary and References In this chapter we showed by two completely different methods that there exist “many” Maaß cusp forms. The first method – non constructive – is due to Lindenstrauss and Venkatesh [83]. Suitably extended this allows one to quantify “how many” and thereby obtain a “Weyl law”. We shall provide a proof of this quantitative result in the next chapter by following the original approach of Selberg. The second method is constructive and is due to Maaß himself. Indeed, in [87] Maaß explicitly constructs Maaß cusp forms starting from quadratic extensions of Q.

§ 4.1 The Eisenstein series E.z; s/ play a fundamental role in the two approaches. For the computation of their Fourier coefficients we followed the book of Bump [20, §1.6], the first chapter of which nicely completes this chapter.

146

4 Maaß Forms

§ 4.2 The principal source of this section is the book of Iwaniec [63]. The link between the Eisenstein series and the Prime Number Theorem is instructive. Note that Ikehara’s theorem does not contain any arithmetic, see for example [75, p. 305] for a proof. The arithmetic is therefore hidden in the theorem of Hadamard-De la Vallée Poussin (Corollary 4.12). Nevertheless, it is possible to give a purely spectral proof of the analytic continuation and the functional equation of the Eisenstein series; the proof of Corollary 4.12 does not require any use of arithmetic. Recall that the Riemann Hypothesis states that the  function does not vanish in the half plane Re.s/ > 1=2. By working with nonarithmetic surfaces one can show that the spectral method above does not give more than Corollary 4.12. In other words, one can find zeros arbitrarily close to the line Re.s/ D 1. We refer the reader to [111] for more details on this question. Any improvement – at least via this method – of the theorem of Hadamard-De la Vallée Poussin would require taking into account the arithmetic of the surface (in this case the modular surface). Iwaniec [63, Chap. 7] proves the analog of Theorem 4.16 for every finite area hyperbolic surface. The idea of the proof is the same. Here one has to introduce an Eisenstein series for every cusp. Moreover, the contour shift to D 1=2 in general passes by several poles of these Eisenstein series. These poles correspond to eigenvalues between 0 and 1=4, the latter furnishing the residual spectrum. The number of residual eigenvalues appearing in the spectrum of surfaces obtained from finite index subgroups of SL.2; Z/ is unbounded.13 But these groups are not of congruence type. One can in fact show (see [63, Th. 11.3]) that for congruence subgroups of SL.2; Z/ the residual spectrum is always f0g.

§ 4.3 The method that we use to show the existence of an infinity of Maaß cusp forms is not the original one of Selberg that we shall describe in the next chapter. We have instead followed the recent proof of Lindenstrauss and Venkatesh [83] which has the advantage of being directly generalizable to a very wide setting. In the wave equation the term u=4 usually does not appear. It is nevertheless natural for the hyperbolic Laplacian, see the appendix to the first section of the book by Lax and Phillips [79]. We refer the reader to this text for the detailed theory of the automorphic wave equation. It follows from Theorem 4.18 that the solutions to (4.29) (the waves) propagate at unit speed. This phenomenon is general to the so-called “hyperbolic”14 equations, see [127, Th. 6.1].

Starting from a genus 1 subgroup of SL.2; Z/ one can indeed form cyclic covers of genus 1 and m cusps. As m gets large, it is possible to show (see [22]) that these covers have m  1 small eigenvalues. On the other hand Huxley [60] (see also Otal [94] as well as Otal and Rosas [95]) proves that for a genus 1 surface the cuspidal spectrum is contained in 1=4; C1Œ. 13

14

In this context the term “hyperbolic” has a different meaning from the hyperbolic geometry that we consider in this book.

4.6 Commentary and References

147

§ 4.4 The idea to study hyperbolic periods of Eisenstein series is due to Siegel [123] and Zagier [143]. Wielonsky [140] extended their calculation to a much wider context, using the adelic language, thereby recovering a classical formula of Hecke concerning the zeta functions of number fields. For the classical theory of Hecke we refer the reader to the book of Lang [75, Chap. XIII]. We treat here the most elementary case – in the non-adelic set-up – of Wielonsky’s formula by following, essentially, the presentation of Goldfeld in [46, §3.2], see also the beautiful book of Siegel [123]. A nice reference for coding geodesics on hyperbolic surfaces is the article of Series [7]. One can also consult the book by Dal’bo [30], which has more details. Theorem 4.26 is a particular case of a theorem of Hecke [51, 52] (see as well [20, 75]).

§ 4.5 In [87], Maaß showed how to explicitly construct Maaß cusp forms by forming certain theta series associated with real quadratic extensions of Q. Theorem 4.28 is a particular case of Maaß’s theorem. The latter invokes the “converse theorem” of Hecke and Weil (see [20]) which gives a more conceptual framework for the calculations we present in this section. We have nevertheless preferred to restrict ourselves to the more “elementary” case of the p extension Q. 2/=Q. In the following paragraph we state a more general result that we shall not prove.

Maaß Forms and Algebraic Number Theory Theorem 4.28 explicates a first link between Maaß forms and algebraic number theory. Maaß’s theorem is more general: notably, it implies the following theorem that we shall admit, see [19] for an introduction to this theorem, to the Galois representations that we shall discuss below and more generally to the Langlands program. Theorem 4.31 Let K be a real quadratic field of discriminant D. Let  be a character of the ideal class group15 of K. The series X p  .z/ D .a/ y K0 .2N.a/y/ cos.2N.ax/ a¤0

is absolutely convergent for z D x C iy 2 H. It defines an eigenfunction of the Laplacian with eigenvalue 1=4 such that for all  D ac db 2 0 .D/,  .z/ D D .d/ .z/;

15

p In the case of the field Q. 2/ such a character is necessarily trivial. We write it as 0 .

148

4 Maaß Forms

p where D is the Dirichlet character associated with the quadratic extension Q. D/=Q. Moreover,  is cuspidal if and only if the character  is not real. In the latter case, the form  defines a Maaß cusp form in C . .D/nH/ of eigenvalue 1=4! The Selberg conjecture is thus optimal when one allows for a finite level. The construction of Maaß is essentially the only known method for explicitly writing down Maaß cusp forms; one should note that the Fourier coefficients of such Maaß forms are algebraic. One conjectures that every Maaß form of eigenvalue 1=4 should be “algebraic”. More precisely, such a form should arise from an even Artin representation W Gal.Q=Q/ ! GL.2; C/: we now briefly explain what we mean by this.

Galois Representations and Maaß Forms Given a Galois extension K of Q and an irreducible representation16 W Gal.K=Q/ ! GL.2; C/, we can associate to each prime number p unramified in K the Frobenius conjugacy class Frobp in Gal.K=Q/. Artin then defines the function Y det.I  .Frobp /ps /1 L.s; / D p

Y D .1  tr. .Frobp //ps C det. .Frobp //p2s /1 p

DW

C1 X

(4.57)

 .n/ns :

nD1

(One must pay attention to the definition of the local Euler factors at ramified primes.) According to a conjecture of Artin, verified by Langlands [77] and Tunnell [130] in a large number of cases, the function L.s; / extends meromorphically to the complex s-plane, and is in fact entire when

is non-trivial. When is even – that is to say, that det. / is 1 when we evaluate it on complex conjugation – the series .z/ D

C1 X

 .n/y1=2 K0 .2ny/ cos.2nx/

(4.58)

nD1

should then define a Maaß form for a certain surface Y.N/ D  .N/nH of eigenvalue 1=4. Then even representations with dihedral image in PGL.2; C/ should correspond to real quadratic quadratic fields. We refer the reader to Taylor’s article [128] or the book [19] for more details on these Galois theoretic aspects.

16

See § 6.3.1 for a rapid introduction to the theory of representations of finite groups.

4.7 Exercises

149

The use of a Hecke operator in the proof of the existence of Maaß cusp forms already suffices to show the link with arithmetic. We shall again see the essential role of arithmetic in the original proof of Selberg that we describe in the following section. It is therefore of no doubt – in the eyes of a specialist – that Maaß forms are of an arithmetic nature (despite the suspected transcendental nature of the Fourier coefficients of those Maaß forms not coming from even Galois representations). We shall furthermore see, in Chap. 7, the full interest of treating Maaß forms on the same footing as the more classical arithmetic objects such as holomorphic modular forms.

4.7 Exercises Exercise 4.32 Show that the expression u.z; t/ D

1 4

Z Z H

C1 0

.eirt C eirt /!r2 C1=4 .z; w/r tanh.r/f .w/ drd .w/

defines a solution to Eq. (4.29). Exercise 4.33 Let ı 2 SL.2; Z/ be a hyperbolic matrix. Show that the discriminant of the integral quadratic form Qı is not a perfect square. Exercise 4.34 Let Q.x; y/ D Ax2 C Bxy C Cy2 be an integral quadratic form of discriminant D. Show that ˇ   tBu Cu ˇ 2 2 2 u; t 2 R; t  Du D 4 : SO.Q/ D ˙ ˇ Au tCBu 2 Exercise 4.35 (Positive definite quadratic forms) 1. Show that the map z 2 H 7! Qz .u; v/ D y1 juz C vj2 allows one to identify – in an SL.2; R/-equivariant way – the space of positive definite quadratic forms considered up to homothety with the half-plane H. 2. Show that the series X .m;n/2Z2 f.0;0/g

1 ; Qz .m; n/s

(4.59)

where s is a complex number with real part > 1, is absolutely convergent and equal to 2E.z; s/.

150

4 Maaß Forms

3. Let d be a negative integer such that d is not a square. Consider the set Qd of positive definite binary quadratic forms Q.x; y/ D ax2 C bxy C cy2 D Œa; b; c with a; b; c 2 Z, b2  4ac D d, and a > 0. Show that to every equivalence class in SL.2; Z/nQd one can associate a point on the modular surface SL.2; Z/nH. Deduce from this that the series Q .s/ D

X .m;n/2Z2 f.0;0/g

D

X .m;n/2Z2 f.0;0/g

1 Q.m; n/s 1 ; .am2 C bmn C cn2 /s

does not depend on the SL.2; Z/-equivalence class of Q. 4. Using the properness of the action of SL2 .Z/ on H, show that the quotient SL2 .Z/nQd is finite. We call its cardinality h.d/ the class number. Exercise 4.36 (Class number formula) We keep the notation of the preceding exercise but from now on we assume that d D p where p is a prime number > 3 and p  3 mod 4. We write h D h.p/ for the number of SL.2; Z/equivalence classes of primitive integral positive definite quadratic forms Q.x; y/ D ax2 C bxy C cy2 of discriminant b2  4ac D p. We fix a set S of h representatives of these primitive positive definite quadratic forms. 1. For every integer n > 0 and for all Q 2 S, we put R.n; Q/ D #f.x; y/ 2 Z2 j Q.x; y/ D ng and R.n/ D

X

R.n; Q/:

Q2S

h By noting that R.n/ is equal to twice the number of norm n ideals of Z and by decomposing n in its prime factorization, show that R.n/ D 2

X m mjn

p

:

2. Show that .2/s  .s/Q .s/ D

Z

1 0

  dy ; ys Q .iy/  1 y

p i 1C p 2

4.7 Exercises

151

where Q .z/ D

X

qam

2 CbmnCcn2

;

with q D e2iz

.z 2 H/:

.m;n/2Z2

3. Using the Poisson summation formula, show that the function Q satisfies   1 i Q .z/ D p Q  2 pz 4pz

.z 2 H/:

(4.60)

4. Deduce that 1 Q .iy/  p ; 2 py

as y ! 0:

5. Show that X

Q .z/ D h C

Q2S

1 X

R.n/qn :

nD1

6. Using the first question, deduce that X Q2S

Q .z/ D h C 2

1   X m mD1

p

qm : 1  qm

7. Finally, letting z D iy tend toward 0, deduce from (4.61) that one has 1   1 X m 1 h D : p 2 p  mD1 p m

(4.61)

Chapter 5

The Trace Formula

In this chapter we derive the Selberg trace formula. We begin by describing it in a general framework which renders transparent the analogy with the Poisson summation formula recalled in the introduction. Having done so, the remaining work consists in explicating the general formula in the case of compact hyperbolic surfaces and then for the modular surface.

5.1 The Selberg Trace Formula I: General Framework Let G be a locally compact topological group and  a cocompact discrete subgroup in G. Suppose that G is unimodular, meaning that every left-invariant measure on G is also right-invariant. Henceforth we make the further assumption that G is a closed unimodular subgroup of GL.n; R/. Let H be the Hilbert space H D L2 . nG/: We put a G-module structure on H by having G act by the right-regular representation R on H given by .R.g/'/.x/ D '.xg/ .x; g 2 G; ' 2 H/: Let f 2 Cc1 .G/. We put Z f .g/R.g/ dg:

R. f / D G

© Springer International Publishing Switzerland 2016 N. Bergeron, The Spectrum of Hyperbolic Surfaces, Universitext, DOI 10.1007/978-3-319-27666-3_5

153

154

5 The Trace Formula

Then R. f / defines an operator on H as follows: Z f .g/.R.g/'/.x/ dg .' 2 H/

.R. f /'/.x/ D Z

G

Z

G

D

f .g/'.xg/ dg f .x1 g/'.g/ dg;

D G

where the last equality results from the left-invariance of dg. Fubini’s theorem then implies that X

Z .R. f /'/.x/ D

 nG

 2

X

Z D  nG

 f .x1  g/ '.g/ dg

 2

Z D

 f .x1  g/'. g/ dg

K.x; g/'.g/ dg;  nG

where K.x; y/ D

X

f .x1  y/

.x; y 2 G/:

 2

This sum – finite for all fixed x and y – is called the Selberg kernel. The following lemma collects a few immediate properties of the Selberg kernel. Lemma 5.1 The Selberg kernel satisfies the following properties: 1. K.1 x; 2 y/ D K.x; y/, .1 ; 2 2  /. 2. The function K is C1 on the compact manifold  nG   nG. 3. The operator R. f / is an integral operator with kernel K.x; y/. The operator R. f / is therefore a Hilbert-Schmidt operator, see Theorem 3.27. We admit that it is also a trace class operator with Z tr R. f / D

K.x; x/ dx:

(5.1)

 nG

One can deduce from all of the above a first formulation of the Selberg trace formula: Z Z X tr R. f / D K.x; x/ dx D f .x1  x/ dx; (5.2)  nG

 nG  2

5.1 The Selberg Trace Formula I: General Framework

155

where f 2 Cc1 .G/. This formula can take other more convenient forms, which we shall derive in the following sections. Denoting by f g the conjugacy class of an element  in  , and by  D f1 2  j 1  11 D  g the centralizer of  in  , we have Z X X tr R. f / D f .x1 ı 1  ıx/ dx:  nG

f g ı2 n

The first sum runs over the set of conjugacy classes in  . The function f being of compact support, it vanishes on all but finitely many conjugacy classes f g. Thus tr R. f / D

XZ

X  nG

f g

f .x1 ı 1  ıx/ dx:

ı2 n

Replacing the inner sum by an integral against a counting measure dı, we obtain tr R. f / D

XZ

Z  nG

f g

 n

f .x1 ı 1  ıx/ dı dx:

The change of variables t D ıx thus implies tr R. f / D

XZ f g

 nG

f .t1  t/dt:

Denote now by G D fu 2 G j  u D u g the centralizer of  in G. Then the above equality can be written as tr R. f / D

XZ f g

Z G nG

 nG

f .t1 u1  ut/ du dt;

where u 2 G . We finally obtain a new form of the Selberg trace formula, by putting x D ut. Namely, tr R. f / D

X

Z vol. nG /

f g

G nG

f .x1  x/dx:

(5.3)

It is not clear that we may always make the above interchanges of sums and integrals. One must essentially verify that the integral Z fG . / D

G nG

f .x1  x/ dx

156

5 The Trace Formula

is convergent for f 2 Cc1 .G/. This is in fact the case, as can be deduce from Fubini’s theorem and the fact that, if G1  G2  G are unimodular groups, a right-invariant measure on G1 nG can be decomposed as a product of right-invariant measures on G2 nG and G1 nG2 . The integral fG . / is an orbital integral. This name is justified by the fact that the quotient G nG is homeomorphic to the orbit G   D fx1  x j x 2 Gg under the action of G on itself by conjugation. We know that the isotropy subgroup G is unimodular, so that the measure dx on the quotient space is indeed G-invariant. The right-hand side of (5.3) is the geometric side of the trace formula1 ; to understand it one must understand these orbital integrals. The subgroup  being cocompact, the representation R decomposes into a direct sum of irreducible unitary representations. Hence R Š 1 ˚ 1 ˚    ˚ 2 ˚ 2 ˚ : : : M m ; Š m1 1 ˚ m2 2 ˚    Š 2b G where m 2 f0; 1; 2; : : : g and b G is the unitary dual of G, i.e., the set of isomorphism classes of irreducible unitary representations of G. We can write this isomorphism as R.g/ Š

M

m .g/

.g 2 G/:

2b G

Integrating both sides against f 2 Cc .G/, we obtain R. f / Š

M

m . f /

and

tr R. f / D

2b G

X

m tr.. f //:

2b G

One can thus finally write a third version of the Selberg trace formula, whose lefthand side is called the spectral side of the trace formula, X

m tr.. f // D

2b G

X f g

Z vol. nG /

G nG

f .x1  x/dx;

where f 2 Cc1 .G/.

1

Why one uses this terminology will become clear in the next section.

(5.4)

5.2 The Selberg Trace Formula II: The Case of Compact Surfaces

157

5.2 The Selberg Trace Formula II: The Case of Compact Surfaces Let S D  nH be a compact hyperbolic surface; recall that all of the non-trivial elements of  D  =f˙Ig are hyperbolic. To simplify, we assume in this section that I 62  .

5.2.1 The Pretrace Formula Let k W .1; C1/ ! C be an even C1 function of compact support (or satisfying the decay condition at infinity (3.23)). Let f be an L2 function on S. We can consider f as a function on H, L2 on compacta and automorphic with respect to  , i.e., f . z/ D f .z/. Conversely, such a function obviously induces an L2 function on S. Consider the operator Z f 7! H

k.; w/f .w/ d .w/

(5.5)

on the Hilbert space L2 .S/. Here we make the same abuse of notation as in § 3.3: the kernel k.z; w/ is equal to k. .z; w//. Let D be a fundamental domain for  whose boundary is of zero measure. The images of the translations of D by  tesselate the plane H, and we can write Z

Z H

k.z; w/f .w/ d .w/ D

K.z; w/f .w/ d .w/; D

P where as usual we write K.z; w/ D  2 k.z;  w/. We have already verified that the function K is well-defined, C1 , and bi-automorphic, i.e., K.1 z; 2 w/ D K.z; w/ for all 1 ; 2 2  . The Hilbert-Schmidt theorem (Theorem 3.27) thus implies that the operator Z K.z; w/f .w/ d .w/

f 7!

(5.6)

S

is a compact operator on L2 .S/. The action of K on a function f 2 L2 .S/ can be calculated with the help of (5.5), if one lifts f to the hyperbolic plane H. Let '0 ; '1 ; '2 ; : : : be a complete orthonormal system of (real) eigenfunctions of the Laplacian on S, corresponding to the sequence of eigenvalues 0 D 0 < 1 6 2 6   

158

5 The Trace Formula

By viewing the functions 'j as automorphic Laplacian eigenfunctions on H, Theorem 3.7 implies Z H

k.z; w/'j .w/ d .w/ D h.rj /'j .z/;

where rj is always one of the two roots of 1=4 C rj2 D j . This can be rewritten as Z K.z; w/'j .w/ d .w/ D h.rj /'j .z/: S

In other words, the functions 'j are also eigenfunctions of the operator (5.6) and, since the system f'j g is complete, K.z; w/ D

X

h.rj /'j .z/'j .w/;

(5.7)

j

where the convergence is in L2 .S  S/. We have therefore found that X

k.z;  w/ D

X

 2

h.rj /'j .z/'j .w/:

(5.8)

j>0

Lemma 5.2 The operator (5.5) is trace class. Proof We can apply the Laplacian to the function z 7! K.z; w/ and thereby obtain another automorphic kernel K. It follows from (5.7) that .K/.z; w/ D

X

h.rj /j 'j .z/'j .w/;

j

where the convergence is in L2 .S  S/. Since the kernel K is continuous, the associated operator is a Hilbert-Schmidt operator and thus X

jh.rj /j j2 < C1:

(5.9)

j

P Since the sum j 2 converges, one finally deduces from the Cauchy-Schwarz j inequality and from (5.9) that the series X

h.rj /

j

is absolutely convergent.

t u

5.2 The Selberg Trace Formula II: The Case of Compact Surfaces

159

Remark 5.3 Lemma 5.2 can also be obtained as a consequence of the rapid decay of the function h. According to the proof of the spectral theorem (Theorem 3.32) the series C1 X

 j 'j .z/'j .w/

jD1

is absolutely uniformly convergent on S  S as soon as > 1. We find that Eq. (5.7) is moreover valid in the sense of pointwise convergence. As in § 3.2 it will be more convenient to replace the function k by the function U given by (3.25). The equality X X U.cosh .z;  w// D h.rj /'j .z/'j .w/; (5.10)  2

j>0

where h is given by Lemma 3.9, is called the pretrace formula.

5.2.2 The Geometric Side of the Trace Formula Put z D w in either of the two forms of the pretrace formula and integrate it along S, so as to eliminate the eigenfunctions. We prefer to work with the formula (5.10) and this over the universal cover H. We then integrate over a fundamental domain D of  . We obtain X X0 Z h.rj / D AU.1/ C U.cosh .z;  z// d .z/; (5.11) 

j

D

P where A is the (hyperbolic) area of D and the symbol 0 means that the sum runs over non-identity elements  2  . We now group the terms of the right-hand side of (5.11) by conjugacy classes in  . Thus we let f g be the conjugacy class of  in  . We are led to consider sums of the form X Z U.cosh .z;  z// d .z/:  2f0 g

D

A typical term of this last sum is Z D

U.cosh .z; 11 0 1 z// d .z/ D

Z U.cosh .1 z; 0 1 z// d .z/ Z

D

D 1 D

U.cosh .z; 0 z// d .z/:

160

5 The Trace Formula

On the other hand, if 11 0 1 D 21 0 2 then 1 21 2 0 , the centralizer of 0 in  ; in other words, 2 D  0 1 with  0 2 0 . We can therefore replace the sum on the right-hand side of (5.11) by X Z

U.cosh .z;  z// d .z/;

(5.12)

D

where the sum now runs over a set of representatives of conjugacy classes other S than the identity class in  and D D 1 1 D, where 1 describes the set of representatives of the orbits of the left-action of  on  . It is immediate that D is a fundamental domain for the (non-compact) surface  nH, and that the integral in (5.12) is independent of the choice of such a fundamental domain (at least if the boundary is not too pathological, for example piecewise C1 ). We must now make a convenient choice of fundamental domain for  . In order to do so we should better understand the group  .

5.2.3 Contribution of Hyperbolic Elements Each element  2   fIg is hyperbolic since  is torsion-free (and so does not contain any elliptic elements) and cocompact (and so does not contain any parabolic elements); there exists then a unique  -invariant geodesic a in H, the axis of  . Write `. / for the displacement distance of  , i.e., the real number ` satisfying

.z;  z/ D ` for all z 2 a ;

.z;  z/ > ` for all z 2 H  a :

If a is parametrized by arclength and if the orientation of a goes from z to  z (z 2 a ), we have .a .t// D a .t C `. //;

t 2 R:

The displacement distance is given by `. / D 2 arccosh .tr. /=2/ :

(5.13)

Since conjugate elements have the same displacement distance, the latter notion is a well-defined invariant of a conjugacy class in  . In fact it corresponds to the length of the closed geodesic associated to  in S. We shall call an element  2   fIg primitive if it cannot be written as a non-trivial power of an element in  . The corresponding geodesic is called a prime geodesic. Such a geodesic cannot be obtained by winding m times, with m > 2, around a shorter geodesic.

5.2 The Selberg Trace Formula II: The Case of Compact Surfaces

161

Lemma 5.4 For all  2   fIg there exists a unique primitive element ı 2  such that  D ı m for a certain m > 1. The elements ı n , n 2 Z, are pairwise non-conjugate in  and the centralizer of  in  is  D fı n j n 2 Zg: Proof Let a be the axis of  . Denote by Z the subgroup of  which fixes the axis a . Then Z acts freely and properly discontinuously on a . The restriction Zja is therefore a discrete subgroup of R and there exists ı 2 Z such that `.ı/ > 0 and `.ı/ 6 `.˛/ for all ˛ 2 Z  fIg. It follows from all of this that for all ˛ 2 Z, there exists an integer n 2 Z such that ˛ja D .ıja /n . Since .ıja /n D .ı n /ja and since  acts freely, we deduce that ˛ D ı n . In particular, there exists a unique m 2 Z such that  D ı m and, replacing ı by ı 1 if necessary, we can assume that m > 0. The uniqueness of ı comes from the fact that such an element naturally preserves the axis a and thus belongs to the group Z. It remains to show that the group Z is nothing other than the centralizer of  in  . But if ˛ 2  , by the uniqueness of the axis of  it is clear that ˛ preserves a and thus lies in Z. t u Fix then a generator ı of  such that  D ı m for an integer m > 0. Conjugating ı in SL.2; R/ if necessary, we can assume that ı acts on H by a homothety z 7! pz for some positive p ¤ 1. Then, replacing ı by its inverse if necessary, we can assume that p > 1. Then log p is the hyperbolic distance from i to pi, and is therefore the displacement distance `.ı/ of ı. The group  is the cyclic group generated by ı, and a fundamental domain for its action on H is given by the horizontal strip 1 < y < p. We thus obtain Z

Z U.cosh .z;  z// d .z/ D D

Z

pZ

D 1

Z

C1

1 p

D 1

Z

C1 1

C1

C1 1

U.cosh .z; pm z//

log p `

D

log p g.m log p/ 2`

D jpm=2  pm=2 j1 g.m log p/ log p;

dx dy y2

.where 2` D jpm=2  pm=2 j/

U.1 C 2`2 .x2 C 1// dx

U.1 C 2u/ du p u  `2

D

`2

1

Z

  U 1 C 2 .`jzj=y/2 y2 dx dy

y1 dy Z

p

162

5 The Trace Formula

where as usual (see (3.29)) g.u/ D

p Z 2

C1

p

juj

k. / sinh

d : cosh  cosh u

Remark 5.5 The function g is the Fourier transform of the function h given by Lemma 3.9. Using the Abel transform, we can moreover recover the function U (or k) from the function g via the formula 1 U.1 C 2u/ D  

Z

C1

.v  u/1=2 dq.v/;

u

where q.v/ D

p p  1  g 2 log. v C 1 C v/ : 2

If U (or k) is of compact support, the function g W .1; C1/ ! C is an even C1 function of compact support. The converse holds as well.

5.2.4 The Trace Formula We can now finally prove the Selberg trace formula. In this formula, if  is a closed geodesic on S, we denote by N D e`. / the norm of  and by . / the length . / D `.0 /, where 0 is the unique oriented prime geodesic satisfying  D 0m for an integer m > 1. Theorem 5.6 Let S be a compact hyperbolic surface and G.S/ the set of closed oriented geodesics on S. For each j 2 N we fix a complex number rj so that the j D 1=4 C rj2 describe the Laplacian spectrum on S. Let g W .1; C1/ ! C be an even C1 function of compact support and h D b g its Fourier transform. Then, C1 X iD0

h.ri / D

area.S/ 4

Z

C1

rh.r/ tanh.r/ dr C 1

X

. /

1=2  2G.S/ N

1=2

 N

g.log N /:

The series on both sides are absolutely convergent. Proof Since g is of compact support, there exists a kernel k of compact support such that g comes from k via (3.30). The function h is the Fourier transform of g and so lies in S.R/; the function h is also the Selberg transform of k, see Lemma 3.9. We can apply formula (5.11).

5.2 The Selberg Trace Formula II: The Case of Compact Surfaces

163

Since g is of compact support and h is of rapid decay, the sum and integral on the right-hand side present no convergence issues. Moreover, Lemma 5.2 implies that the sum on the left-hand side is itself absolutely convergent. It follows from § 5.2.3 that in (5.11) the sum X0 Z U.cosh .z;  z// d .z/ D



is equal to X

. /

1=2  2G.S/ N

g.log N /:

1=2

 N

The area A is the area of the surface S. Then U.1/ D  D

1 

Z

C1

0

1 2

Z

q0 .v/ p dv v

C1

g0 .u/ du: sinh u=2

0

Now, since g is even so is h and g.u/ D

1 2

Z

C1

eiru h.r/ dr D

1

1 

Z

C1 0

.cos ru/h.r/ dr;

u 2 R:

Since h lies in the Schwartz space, we can differentiate g under the integration. Doing so, we find g0 .u/ D 

1 

Z

C1

rh.r/.sin ru/ dr: 0

The function .r; u/ 7! .sinh.u=2//1 rh.r/ sin ru is integrable on the product space .0; C1/  .0; C1/ and one obtains U.1/ D

1 2 2

Z

Z

C1

C1

rh.r/ 0

0

sin ru du dr: sinh u=2

Now, X 1 2 u=2 D u=2 D 2e enu sinh.u=2/ e  eu=2 n>0

164

5 The Trace Formula

so that Z

C1 0

X Z C1 sin ru e.2nC1/u=2 sin.ru/ du du D 2 sinh u=2 0 n>0 D2

X n>0

D

4r 4r2 C .2n C 1/2

X n2Z

4r2

4r : C .2n C 1/2

f y ./ D The Fourier transform of fy .x/ D eyjxj , where y is a positive parameter, is b 2y=. 2 C y2 /. The Poisson summation formula then implies X n2Z

X r D er2jnj ein r2 C .n C 1=2/2 n2Z D

1  e2r D  tanh.r/: 1 C e2r

Finally, we obtain U.1/ D

1 2

Z

C1

rh.r/ tanh.r/ dr:

t u

0

Let " > 0. Put B" D fr 2 C j j Im rj
0 and write Y;1 .z; w/ D Kcont

1 4

Z

C1 1

h.r/EY .z; 1=2 C ir/EY .w; 1=2 C ir/ dr;

where EY denotes the truncated Eisenstein series, see § 4.2.3. The kernel Kcont then decomposes as a sum of three terms Y;1 Y;2 Y;3 .z; w/ C Kcont .z; w/ C Kcont .z; w/; Kcont .z; w/ D Kcont

where Y;2 Kcont .z; w/

D

1 4

Z

C1 1

  ıY .z/ıY .w/ Im.z/1=2Cir C '.1=2 C ir/ Im.z/1=2ir    Im.w/1=2Cir C '.1=2 C ir/ Im.w/1=2ir h.r/ dr

with ıY .z/ D 1 if z 2 D has imaginary part > Y and ıY .z/ D 0 if z 2 D has imaginary part < Y. In the following lemmas we study the contributions of each of these three terms. Y;1 is a Hilbert-Schmidt operator. Lemma 5.11 The integral operator with kernel Kcont

Proof It suffices to bound its norm in L2 .D  D/. We begin by observing Z DD

Y;1 jKcont .z; w/j2 d .z/ d .w/

6

1 4

1=2 Z

C1 1

jh.r/j2

Z

jEY .z; 1=2 C ir/j2 d .z/ dr: D

As tends toward 1=2, we have '. C ir/ D '.1=2 C ir/ C .  1=2/' 0.1=2 C ir/ C o.  1=2/ and '. C ir/'.  ir/ D 1 C .2  1/' 0 .1=2 C ir/'.1=2 C ir/1 C o.  1=2/: (In the last expression we used the fact that '.s/'.1  s/ D 1 in order to replace '.1=2  ir/ by '.1=2 C ir/1 .) By passing to the limit ! 1=2, Lemma 4.10

5.3 The Selberg Trace Formula III: The Case of SL.2; Z/

169

implies that Z

 1  '.1=2  ir/Y 2ir  '.1=2 C ir/Y 2ir 2ir '0 C 2 log Y  .1=2 C ir/ ' '0 D  .1=2 C ir/ C OY .1/; '

jEY .z; 1=2 C ir/j2 d .z/ D D

for any real r. The function h being of rapid decay, it suffices to show that the integral 1 4

Z

C1 1

' 0 .1=2 C ir/ jh.r/j2 dr '

(5.16)

converges. Now '.s/ D .2  2s/=.2s/ and Proposition 4.7 implies that j log '.1=2 C ir/j D O.jrj1C" / for all " > 0. In particular Z

T

T

' 0 .1=2 C ir/ dr D O.T 2 / '

for T large enough and a simple integration by parts finally implies that the integral (5.16) converges. t u Lemma 5.12 We have Y;2 Y;4 Kcont .z; w/ D ıY .z/ıY .w/K0 .z; w/ C Kcont .z; w/; Y;4 where Kcont is the kernel of a Hilbert-Schmidt operator and

Z K0 .z; w/ D Moreover, the integral Y ! C1.

R D

C1

k.z; w C t/dt: 1

Y;4 Kcont .z; z/ d .z/ is absolutely convergent and tends to 0 as

Proof Suppose that y D Im.z/ and  D Im.w/ are both greater than Y. By Y;2 .z; w/ we obtain four terms, two of which give expanding the kernel Kcont Z C1 Z C1   1 1 1=2 ir ir 1=2 .y/ .y/ .y=/ C .=y/ h.r/ dr D .y=/ir h.r/ dr 4 2 1 1 since h is even. But Z K0 .z; w/ D

Z

C1

C1

k.z; w C t/dt D 1

1

k.y;  C t/dt

170

5 The Trace Formula

  .y  /2 C t2 dt D U 1C 2y 1  Z C1  t2 D U cosh.log.y=// C dt 2y 1 p Z C1 D .y/1=2 2 U.cosh.log.y=// C  2 /d Z

C1

1

p Z D .y/1=2 2

cosh.log.y=//

p Z 1=2 D .y=/ 2

log.y=/

D .y=/

U.u/ p du u  cosh.log.y=//

k. / sinh

d

p cosh  cosh.log.y=//

1=2

g.log.y=// Z C1 1 D .y/1=2 h.r/.y=/ir dr: 2 1

The first two terms above thus contribute ıY .z/ıY .w/K0 .z; w/: The two other terms can be treated in the same way. Consider, for example, 1 4

Z

C1

1

'.1=2 C ir/.y/1=2ir h.r/ dr:

(5.17)

Let " be a positive real number strictly less than 1=2 such that the function ' stays bounded in the strip 1=2 6 Re.s/ 6 1=2 C ". Since h is of rapid decay, we can shift the integration contour to Im.r/ D " to find that the integral (5.17) is O..y/1=2" /. The last two terms thus produce a contribution of O.ıY ./ıY .y/.y/1=2" / and the lemma follows. t u Y;3 is of Hilbert-Schmidt type. Lemma 5.13 The integral with kernel Kcont R operator Y;3 Moreover, the integral D Kcont .z; z/ d .z/ converges absolutely and tends to 0 as Y ! C1. Y;3 Proof We treat all the terms present in Kcont in the same way. Consider, for example,

Z ıY .z/

C1 1

Im.z/1=2ir '.1=2 C ir/EY .w; 1=2 C ir/h.r/ dr:

(5.18)

5.3 The Selberg Trace Formula III: The Case of SL.2; Z/

171

The proof of Lemma 5.11 implies that this integral converges absolutely; we can moreover shift the contour integral in Z

h.r/ Im.z/1=2ir '.1=2 C ir/EY .w; 1=2  ir/ dr

ıY .z/ Im.r/D0

to, for example, Im.r/ D 3=2. In doing so we encounter the pole at s D 1 of the Eisenstein series E. Note nevertheless that '.s/E.w; 1  s/ D E.w; s/ and thus that '.s/EY .w; 1  s/ D EY .w; s/. In particular, the function s 7! '.s/EY .w; 1  s/ has no poles in the half-plane Re.s/ > 1=2. In light of this, the integral (5.18) becomes Z

h.r/ Im.z/1=2ir EY .w; 1=2 C ir/ dr:

ıY .z/ Im.r/D3=2

But EY .w; s/ D O.1/ along the line Re.s/ D 2 and the integral isR then O.1= Im.z//. Y;3 In particular, the kernel is Hilbert-Schmidt and the integral D Kcont .z; z/ d .z/ converges absolutely and tends to 0 as Y ! C1. t u To conclude, observe that it follows from the proof of Proposition 3.25 that KY .z; w/ WD K.z; w/  ıY .z/ıY .w/K0 .z; w/ is a Hilbert-Schmidt kernel. According to Lemmas 5.11, 5.12, and 5.13, the decomposition Y;1 Y;3 Y;4 C Kcont C Kcont / K  Kcont D KY  .Kcont

is a sum of Hilbert-Schmidt kernels. This ends the proof of Theorem 5.10.

t u

5.3.3 The Spectral Side of the Trace Formula Proposition 5.14 The operator with kernel K  Kcont is of trace class. Proof Since the function h is of rapid decay, it suffices to show the following lemma. t u Lemma 5.15 We have jfj j jrj j 6 Tgj D O.T 2 /; for T large enough. Here j D 1=4 C rj2 runs through the (discrete) spectrum of  in Im.E/? .

172

5 The Trace Formula

Proof The sum (5.15) converges absolutely and uniformly on compacta. We deduce that for all Y > 0, Z Z Z X 2 2 jh.rj /j juj .w/j d .w/ 6 jK.z; w/j2 d .z/ d .w/: (5.19) j

DY

DY

D

It remains then to choose a kernel k in such a way as to bound the number of rj of absolute value less than a given constant T. Take for k the characteristic function of a small ball centered at 0. In fact, it shall be more convenient in what follows to use a function of the form U.cosh / D k. /, and we take for U the characteristic function of a small ball centered at 1 and of radius ı. (Such a kernel is not smooth, if one prefers one can always approximate it by compactly supported C1 functions.) We begin by evaluating the Selberg transform h of k. Recall that Z h.r/ D k.i; z/ys d .z/; H

where s D 1=2 C ir. Taking s D 0, we obtain Z h.i=2/ D H

k.i; z/ d .z/ D 2ı

(here 2ı is the area of a hyperbolic disk whose radius r is given by r sinh.r=2/ D

1 .cosh r  1/ D 2

r

ı /: 2

Recall furthermore that 1 jz  ij2 .cosh .z; i/  1/ D u.z; i/ D ; 2 4y where y D Im.z/. If 1 ı .cosh .z; i/  1/ 6 ; 2 2 p we then have jy  1j 6 2yı. Suppose that ı 6 1=4. Then y 6 2 and so jy  1j 6 p z 2 ı. Moreover, for all z 2 C of modulus less than p 1, we have je  1j 6 2jzj. We 1 deduce that for ı sufficiently small and jsj 6 .4 2ı/ , p jys  1j 6 4jsj 2ı:

5.3 The Selberg Trace Formula III: The Case of SL.2; Z/

173

From this it follows that p jh.r/  h.i=2/j 6 4j1=2 C irj 2ıh.i=2/; and finally 2ı 6 jh.r/j 6 6ı; p if j1=2 C irj 6 .8 2ı/1 . Now, we have Z Z 1 X jK.z; w/j2 d .z/ D k. 0 z; w/k. 0 z;  w/ d .z/ 4 0 D D ; 2 Z 1X D k.z; w/k.z;  w/ d .z/: 2  2 H

(5.20)

By definition of the kernel k, we must have u.z; w/ 6 ı=2 and u.z;  w/ 6 ı=2, which implies, via the triangle inequality for the hyperbolic metric and the inequality cosh.a C b/ 6 2 cosh a cosh b  1, that u.w;  w/ 6 ı.ı C 2/. Put Nı .w/ D jf 2  j u.w;  w/ 6 ı.ı C 2/gj: Applying the Cauchy-Schwarz inequality to (5.20) and recalling that k is a characteristic function, we obtain the inequality Z Z 1 1 2 jK.z; w/j d .z/ 6 Nı .w/ k.i; z/2 d .z/ D Nı .w/h.i=2/: 2 2 D H We shall prove in Lemma 5.16 that p Nı .w/ D O. ıY C 1/ for ı 2 .0; 1 and w 2 DY . Keeping only the first sum in the left-hand side of p inequality (5.19) and restricting the summation to rj such that j1=2 C irj j 6 .8 2ı/1 we finally see that .2ı/2

Z

X p j1=2Cirj j 1, are cuspidal and thus decay rapidly in the cusp. More precisely, Z DDY

juj .w/j2 d .w/ D O.jrj jY 2 /:

(5.22)

We shall admit this last estimate, sending the interested reader to the references at the end of this chapter. The lemma is then deduced from (5.21) and (5.22) by taking T D Y. t u Lemma 5.16 Let z D x C iy 2 D and ı 2 .0; 1. We have Nı .z/ D O

p

 ıy C 1 ;

where the implied constant is universal. Proof If u. z; z/ 6 ı.ı C 2/ then u. z; z/ 6 3ı. Since ı is smaller than 1, the orbit   z contains only a finite number – independent of z – of points whose imaginary part is smaller than y. Thus Nı .z/ D jfk 2 Z j u.z; z C k/ 6 3ıgj C O.1/: Finally, u.z; z C k/ D k2 =4y2 and we obtain the stated bound.

t u

It follows from Proposition 5.14 and Lemmas 5.12 and 5.13 that X j

Z h.rj / D

.K.z; z/  Kcont .z; z// d .z/ D

(5.23)

Z D lim

Y!C1 D

.KY .z; z/ 

Y;1 Kcont .z; z// d .z/;

where the eigenvalues j D 1=4 C rj2 form the discrete spectrum of  in Im.E/? . The left-hand side of (5.23) is thus analogous to the spectral side in the trace formula for a compact surface. We now endeavor to understand the various terms on the right-hand side. Y;1 We begin with the term involving Kcont : Z D

Y;1 Kcont .z; z/ d .z/

1 D 4

Z

Z

C1

jEY .z; 1=2 C ir/j2 d .z/ dr:

h.r/ 1

D

The inner integral is calculated in the proof of Lemma 5.11 and we obtain 1 4

Z

C1

1

 '0 h.r/ 2 log Y  .1=2 C ir/ '

5.3 The Selberg Trace Formula III: The Case of SL.2; Z/

175

 1 2ir 2ir .'.1=2  ir/Y  '.1=2 C ir/Y C / dr 2ir   Z C1 1 2ir '.1=2  ir/  '.1=2 C ir/ h.r/ dr Y D 4 1 2ir   Z C1 1 sin.r log Y/ h.r/ dr C '.1=2 C ir/ 4 1 2r Z C1 Z C1 1 1 '0 C h.r/ dr log Y  h.r/ .1=2 C ir/ dr: 2 1 4 1 ' The first integral tends to 0 as Y goes to C1 according to the Riemann-Lebesgue R C1 lemma. Since 1 .sin.x/=x/ dx D , a simple change of variables shows that the second integral tends toward 1 '.1=2/h.0/ 4 as Y ! C1. We therefore obtain the following proposition. Proposition 5.17 We have Z KY .z; z/ d .z/ D D

X j

1 h.rj / C 4

Z

C1 1

' 0 .s/ .1=2 C ir/h.r/ dr '.s/

1 C h.0/'.1=2/ C g.0/ log Y C o.1/; 4 as Y ! C1. (Here as in the rest of the text, h D b g.)

5.3.4 The Geometric Side of the Trace Formula: Parabolic Term Contribution It remains to calculate Z Z KY .z; z/ d .z/ D D

Z K.z; z/ d .z/ C DY

.K.z; z/  K0 .z; z// d .z/: Im.z/>Y

The proof of Proposition 3.25 implies that the second integral is o.1/ as Y goes to C1. We have only to calculate Z K.z; z/ d .z/: DY

176

5 The Trace Formula

But K.z; z/ D

X

k.z;  z/

 2

and for Im.z/ sufficiently large, we have k.z;  z/ ¤ 0 if and only if  2 1 . We thus find XZ  2

DY

k.z;  z/ d .z/ Z

X

D

X

k.z;  z/ d .z/ C D

 jf g\1 D¿

 jf g\1 ¤¿

Z DY

k.z;  z/ d .z/:

The first side can be treated as in the compact case. Any (non-trivial) parabolic conjugacy class is a power of the conjugacy class of   11 0 D : 01 We can therefore write Z

X  jf g\1 ¤¿

DY

k.z;  z/ d .z/ Z D

k.z; z/ d .z/ C DY

X

X Z

`2Z  21 n

DY

k.z;  1 0 z/:

The first term can be treated as in the compact case. Since the area of D is equal to =3, it tends to 1 12

Z

C1

rh.r/ tanh.r/ dr 1

as Y goes to C1. Then for every integer ` > 1, X Z  21 n

DY

k.z;  1 0` z/

Z D 1 nHY

k.z; 0` z/ d .z/

Z D 1 nHY

k.z; z C `/ d .z/;

where HY D fz 2 H j Im.z/ 6 Yg:

5.3 The Selberg Trace Formula III: The Case of SL.2; Z/

177

This is the contribution of the parabolic terms to the trace formula; we calculate it in the following proposition. Proposition 5.18 We have C1 XZ

k.z; z C `/ d .z/

1 nHY

`D1

1 1 D g.0/ log Y  g.0/ log 2 C h.0/  4 2

Z

C1

h.r/ 1

0 .1 C ir/ dr C o.1/ 

as Y ! C1. Proof We begin by remarking that given an integer ` ¤ 0, a change of variables analogous to the one made in the proof of Lemma 5.12 implies that Z

Z k.z; z C `/ d .z/ D

1 nHY

Z

1 0

Y

0 Y

D 0

D

Z

k. .z; z C `// d .z/

  k arg cosh.1 C `2 =2y2 / y2 dy

1 p j`j 2

Z

C1

arg cosh.1C`2 =2Y 2 /

k. / sinh

p d : cosh  1

We deduce that C1 XZ `D1

1 nHY

k.z; z C `/ d .z/

D

p Z 2

D

p Z 2

C1 arg cosh.1C1=2Y 2 / C1 arg cosh.1C1=2Y 2 /

k. / sinh

p cosh  1



X p p 16`< 2 Y cosh 1

 1 d

`

p p k. / sinh  log. 2 Y cosh  1/ C  p cosh  1  C O.Y 1 .cosh  1/1=2 / d

D L.Y/ C O.Y 1 log Y/; where p Z L.Y/ D 2

C1 0

 p k. / sinh  log.Y 2.cosh  1// C  d

p cosh  1

178

5 The Trace Formula

and  is the Euler constant. It remains then to explicate L.Y/. Recall first of all that p Z 2

C1 0

k. / sinh

d D g.0/: p cosh  1

Thus, Z L.Y/ D g.0/.log.2Y/ C  / C 2

C1

k. / cosh. =2/ log.sinh. =2//d : 0

But the Abel inversion formula implies that Z C1 1 dg.t/ k. / D  p p cosh t  cosh

2

Z C1 dg.t/ 1 p D 2 2

sinh .t=2/  sinh2 . =2/ from which we deduce that Z 2

C1

k. / cosh. =2/ log.sinh. =2//d

0

Z

D

1 

D

1 2

C1

Z

C1

cosh. =2/ log.sinh. =2// p dg.t/d

0

sinh2 .t=2/  sinh2 . =2/  Z C1  Z t 1 log.sinh2 . =2// d.sinh2 . =2// D r dg.t/ 2 0 sinh2 . =2/ 0 sinh2 .t=2/  1 sinh2 . =2/ Z

C1 0

Z

v 0

 log u du dq.v/; p u.v  u/

where u D sinh2 . =2/, v D sinh2 .t=2/, and g.t/ D q.v/. But 

Z

 log u du dq.v/ p u.v  u/ 0 0  Z C1  Z 1 log.wv/ 1 dw dq.v/ p D 2 0 w.1  w/ 0 Z 1  Z C1  Z 1 log w dw 1 1 q.0/ p dw  p log.v/dq.v/ D 2 2 w.1  w/ w.1  w/ 0 0 0 Z C1 log.sinh.t=2// dg.t/: D g.0/ log 2 

1 2

Z

C1

v

0

5.3 The Selberg Trace Formula III: The Case of SL.2; Z/

179

In this way, Z L.Y/ D g.0/.log Y C  / 

C1

log.sinh.t=2// dg.t/:

(5.24)

0

We can either content ourselves with the expression (5.24) or rewrite it in the stated form 1 1 L.Y/ D g.0/ log Y  g.0/ log 2 C h.0/  4 2

Z

C1

h.t/ 1

0 .1 C it/dt;  t u

see [63, §10.3].

5.3.5 The Trace Formula R The hyperbolic and elliptic terms contribute to the “trace” DY K.z; z/ d .z/ in a similar way as for the compact case. By passing to the limit Y ! C1 Propositions 5.17 and 5.18 imply the trace formula for the group SL.2; Z/. Theorem 5.19 Let  D SL.2; Z/. Let .h; g/ be an admissible pair of functions. Let j D 1=4 C rj2 , j 2 N, be the set of (discrete) eigenvalues of the hyperbolic Laplacian in L2 . nH/, counted with multiplicity. Then C1 X

1 h.rj / D 12 jD0

Z

C1

rh.r/ tanh.r/ dr 1

C

X P

C

`D1

g .2` arccosh .t=2// sinh .` arccosh .t=2//

Z C1 2 1 er 1X e2`r=3 dr C h.r/ dr 2r 1Ce 3 sin .`=3/ 1 1 C e2r 1 `D1 

0 Z C1 0  1 .1=2 C ir/ C .1 C ir/ dr h.r/ C g.0/ log.=2/  2 1   1 2

Z

arccosh .t=2/

C1 X

C1

h.r/

C2

C1 X nD1

.n/ g.2 log n/: n

180

5 The Trace Formula

Here P denotes the set of primitive hyperbolic conjugacy classes in PSL.2; Z/, t the displacement distance of an element in P, and (

log. p/ if n D pk with p prime and k 2 N ;

.n/ D

0

otherwise.

Proof There are exactly two elliptic conjugacy classes in PSL.2; Z/ one of order 2 and the other of order 3. The terms involving log Y in Propositions 5.17 and 5.18 cancel each other out. Passing to the limit Y ! C1, these two propositions thus imply the equality C1 X

h.rj / D

jD0

C

X

1 12

Z

C1

rh.r/ tanh.r/ dr 1

arccosh .t=2/

`D1

P

Z

2

C

1X 1 3 sin .`=3/

1 4

Z

C1 1

1 g .2` arccosh .t=2// C sinh .` arccosh .t=2// 2

C1

h.r/ 1

`D1

C

C1 X

e2`r=3 1 dr  2r 1Ce 2

Z

Z

C1

h.r/ 1 C1

h.r/ 1

er dr 1 C e2r

0 .1 C ir/ dr 

1 '0 .1=2 C ir/h.r/ dr  log.2/g.0/ C .1  '.1=2//h.0/: ' 4

(5.25)

Recall now that we have '.s/ D  2s1

 .1  s/ .2  2s/  .s/ .2s/

and

C1 X .n/  0 .s/ D .s/ ns nD1

for Re.s/ > 1. A simple calculation with the logarithmic derivative of ' then implies that 1 4

Z

C1

'0 .1=2 C ir/h.r/ dr 1 ' Z C1 C1 X .n/ 1 0 g.2 log n/: D g.0/ log./  h.r/ .1=2 C ir/ dr C 2 2 1  n nD1

This concludes the proof of the theorem.

t u

Remark 5.20 We can in the same way develop a trace formula for an arbitrary Fuchsian group of the first kind, see [63]. The case of the group 0 .N/ is particularly interesting: one has the following theorem.

5.4 Applications

181

Theorem 5.21 Let N > 1 be a square-free integer and let .h; g/ be an admissible function pair. Denote by j D 1=4 C rj2 , j 2 N, the set of (discrete) eigenvalues of the hyperbolic Laplacian in L2 .0 .N/nH/, counted with multiplicity. Then C1 X

AN h.rj / D 4 jD0

Z

C1

rh.r/ tanh.r/ dr 1

X

C1 X

g .2` arccosh .t=2// sinh .` arccosh .t=2// P `D1 Z C1 X m1 X 1 e2`r=m C h.r/ dr m sin .`=m/ 1 1 C e2r E

C

arccosh .t=2/

`D1

C2

!.N/

n

1 g.0/ log.=2/  2 C2

Z

C1 1

C1 X .n/ nD1

n

 0 0 .1=2 C ir/ C .1 C ir/ dr h.r/  

g.2 log n/ 

X

C1 X

pjN; p premier kD0

o log p g.2k log p/ : pk

Here AN is the area of a fundamental domain for the action of 0 .N/ on H, P (resp. E) is the set of primitive hyperbolic (resp. elliptic) conjugacy classes in 0 .N/=f˙Ig, t is the displacement distance of an element in P, m is the order of an element in E, and !.N/ is the number of prime divisors of N.

5.4 Applications We now give two types of applications of the trace formulas explicated above.

5.4.1 The Weyl Law Let S D  nH be a compact hyperbolic surface. Let us apply the Selberg trace 2 2 formula to the admissible pair h.r/ D eır and g.x/ D .4ı/1=2 ex =4ı , where ı

182

5 The Trace Formula

is a positive parameter. According to Theorem 5.6 we have C1 X jD0

e

ırj2

area.S/ D 4

Z

C1

2

reır tanh.r/ dr

1

C .4ı/1=2

X

. /

1=2  2G.S/ N



1=2 N

e`. /

2 =4ı

:

(5.26)

Lemma 5.22 The number of closed geodesics on S of length 6 L is O.eL / for L large enough. Proof Fix a base point p 2 H and consider the  -orbit of p in H. For each point in this orbit one may construct the corresponding Dirichlet fundamental domain, thereby obtaining a tiling of H. For a real number L > 0, let N.L/ be the number of fundamental domains meeting the hyperbolic disk of radius L C 2 diam.S/ centered at p. Note that the number of closed geodesics on S of length less than L is bounded above by N.L/. Since the area of the disk is O.eL /, we have N.L/ D O.eL / and Lemma 5.22 follows. t u According to Lemma 5.22, the second term in the right-hand side of (5.26) is 2 bounded by O.ı 1=2 e`0 =8ı /, where `0 is the length of the smallest geodesic, and converges then to 0 as ı ! 0. An integration by parts gives Z

C1

2

reır tanh.r/ dr D

1

1 C O.1/: ı

Hence one obtains C1 X

2

eırj D

jD0

area.S/ .1 C O.1// 4ı

(5.27)

for ı small enough. Tauberian theorems then allow one to relate the asymptotic expansion of the partial sums of a series to the asymptotic expansion of its Abel transform. In particular, a Tauberian theorem of Karamata (see the references at the end of this chapter) allows one to deduce from (5.27) the following theorem – known as a Weyl law. Theorem 5.23 As T ! C1, jfj j 0 6 j 6 Tgj 

area.S/ T: 4

5.4 Applications

183

When  D SL.2; Z/ the trace formula (5.25) replaces Theorem 5.6. Although we skip the details here, formula (5.25) allows one to deduce C1 X jD0

e

ırj2

1  4

Z

C1 1

'0 2 .1=2 C ir/ eır dr ' 

area.S/ 4

Z

C1

2

reır tanh.r/ dr;

1

as ı tends toward 0. (Recall that '.s/ D .2  2s/=.2s/, where  is the completed Riemann zeta function.) Just as in the proof of Theorem 5.23, it follows from the theorem of Karamata that jfj > 0 j j 6 Tgj 

1 4

Z

T T

T2 '0 .1=2 C ir/ dr  ; ' 12

(5.28)

as T ! C1. Taking into account the explicit expression for '.s/ in this case and in particular the fact that '.s/ is a holomorphic function of order 1, we have, for any " > 0, ˇ Z ˇ 1 ˇ ˇ 4

T T

ˇ ˇ '0 .1=2 C ir/ drˇˇ D O.T 1C" /; '

for T large enough. We thus obtain the following result. Theorem 5.24 Let j denote the (discrete) eigenvalues of the hyperbolic Laplacian in L2 .SL.2; Z/nH/, counted with multiplicity. Then jfj j 0 6 j 6 Tgj 

T ; 12

as T ! C1.

5.4.2 The Prime Geodesic Theorem We return now to the case of a compact surface S D  nH and write .x/ for the number of prime geodesics  on S such that N D e`. / 6 x. Let 1 sj D C 2

r

1  j ; 4

for j 6

1 ; 4

where as usual the j are the Laplacian eigenvalues on S. Selberg [115] highlighted the analogy between his trace formula and the “explicit formulas” in number theory relating prime numbers to zeros of the Riemann zeta function. In this analogy

184

5 The Trace Formula

the prime numbers correspond to the lengths of prime geodesics. He developed this analogy by proving the following theorem, which is reminiscent of the prime number theorem recalled in the preceding chapter. Theorem 5.25 For x large enough, X

.x/ D li.x/ C

li.xsk / C O.x3=4 = log x/:

1>sj >3=4

Here li denotes the logarithmic integral: Z

x

li.x/ D 2

d  x= log x; log 

x ! 1:

Proof We shall give asymptotic expansions of various auxiliary functions closely related to .x/. We begin by considering the function X

H.T/ D

`. /6T

X

D

 1 log N 2 cosh  1=2 1=2 2 N  N . /

. /.1 C N1 /.1  N1 /1 ;

`. /6T

where N D e`. / and . / as usual denotes the length of the unique oriented prime geodesic 0 such that  D 0m for an integer m > 1. Put ET .˛/ D ˛ 1 eT˛ . Lemma 5.26 For T large enough we have H.T/ D ET .1/ C

X

ET .sj / C O.e3T=4 /:

1>sj >3=4

We admit momentarily the lemma and show how it can be used to deduce the theorem. It is clear that X H.T/ D . /.1 C N1 /.1 C O.N1 // `. /6T

D

X

. / C O

 X

`. /6T

`. /6T

Put .T/ D

X `. /6T

. /:

. /N1

 :

(5.29)

5.4 Applications

185

Since only a finite number of N are less than a fixed constant and since – according to Lemma 5.26 – H.T/ tends to infinity with T, it follows from (5.29) that H.T/ D

.T/ C o. .T//

as T tends toward C1. Lemma 5.26 implies that C1. We deduce that X

Z

. /N1 D

T 0

`. /6T

(5.30)

.T/  ET .1/ as T tends toward

ex d .x/ D O.T/:

Lemma 5.26 and identity (5.29) then imply X

.T/ D ET .1/ C

ET .sj / C O.e3T=4 /:

(5.31)

1>sj >3=4

Consider now the function 

X

#.T/ D

 . / D

`. /6T  prime

X

`. /:

`. /6T  prime

It is immediate that .T/ D #.T/ C #.T=2/ C    C #.T=k/;

(5.32)

where k is of order T since (5.32) is satisfied as soon as k > T=`1 , where `1 denotes the length of the shortest closed geodesic on S. It follows from (5.31) and (5.32) that #.T/ D ET .1/ C

X

ET .sj / C O.e3T=4 /:

(5.33)

1>sj >3=4

Now observe that .x/ D that

R log x ı

.x/ D li.x/ C

T 1 d#.T/ if ı < `1 ; hence it follows from (5.33) X

li.xsk / C O.x3=4 = log x/;

(5.34)

1>sj >3=4

where li.x/ D

Rx 2

dt= log t. This concludes the proof of the theorem.

t u

Proof of Lemma 5.26 We now apply the Selberg trace formula to well-chosen test functions. We begin by defining a family of functions g"T .

186

5 The Trace Formula

1. Let ŒT;T be the characteristic function of the interval ŒT; T in R and gT the function defined by gT .x/ D 2 cosh.x=2/ŒT;T .x/. 2. Let ' be an even non-negative C1 function with support contained in Œ1; 1 R1 and such that 1 '.x/ dx D 1. Given a real number " > 0, write R " '" for '" .x/ D "1 '.x="/. The function '" is supported in Œ"; " and we have " '" .x/ dx D 1; the functions '" thus approximate the Dirac mass at 0 as " tends to 0. 3. Finally, we put g"T .x/

Z D .gT  '" /.x/ D 2

C1 1

cosh..x  y/=2/ŒT;T .x  y/'" .y/ dy:

For any "; T > 0, the function g"T is even, C1 , and of compact support. We can therefore apply the Selberg trace formula to it. The corresponding function h in the trace formula (the Fourier transform of g"T ) shall be denoted h"T . R C1 For the Fourier transform b f .r/ D 1 eirx f .x/ dx, we have f1  f2 D b f 2 and f 1b b f D f if f is even. A direct calculation then shows that 2 b

1

Z b gT .r/ D

C1 1

Z

D2 Z

T

eirx gT .x/ dx eirx cosh.x=2/ dx

T T

D

e.1=2ir/x C e.1=2Cir/x dx

T

D S.1=2 C ir/ C S.1=2  ir/; where S.w/ D 2w1 sinh.Tw/ with the convention S.0/ D 2T. The function h"T is then given by ' " .r/: h"T .r/ D .S.1=2 C ir/ C S.1=2  ir//b We now define functions H" , which will be approximations to H.T/, by letting H" .T/ D

X

. /

1=2  2G.S/ N

1=2

 N

g"T .log N /:

5.4 Applications

187

Observe that for all " > 0, H" .T  "/ 6 H.T/ 6 H" .T C "/. On the other hand, the trace formula implies H" .T/ D

C1 X

h"T .ri / 

iD0

D

X

h"T .ri /

Z

area.S/ 4 Z

1

C1

C

i

0

C1

rh"T .r/ tanh.r/ dr (5.35)

h"T .r/ dm.r/;

P where  denotes the finite sum over the purely imaginary rj , and dm.r/ is the P measure on Œ0; C1/ given by dN.r/  area.S/ 0 m.1 ;  . p//jh.r1 /j : We must now choose a good kernel k and apply formula (6.7). Consider then the kernel kR .z; w/ D R . .z; w//;

6.2 Multiplicity of the First Eigenvalue

197

where R is a positive real number to be chosen a posteriori and R is the characteristic function of the interval Œ0; R. The kernel kR is not C1 , but this is not a problem here (we could instead approximate it by a smooth kernel). By approximating kR by a compactly supported piecewise affine function equal to 1 on the interval Œ0; R, we can show (see [63, (12.8)]) that1 : hR .r/ D  1=2

 .s  1=2/ sR e C O.eR=2 /;  .s C 1/

as R ! C1 (where  D 1=4 C r2 D s.1  s/ and s 2 .1=2; 1/ as usual). It is natural to introduce the kernel Z R .z; w/ D kR .z; x/kR .x; w/ d .x/

(6.8)

(6.9)

H

and the associated automorphic kernel KR .z; w/ D

X

R .z;  w/:

 2 . p/

We thus have Z KR .z; w/ D

KR .z; x/KR .x; w/ d .x/ Sp

and Z Sp Sp

Z

2

jKR .z; w/j d .z/ d .w/ D

KR .z; z/ d .z/ Sp

D

X

Z

X

R .ız;  ız/ d .z/

 2 . p/ ı2 . p/n

S1

X Z

D ΠW  . p/

 2 . p/

1

R .z;  z/ d .z/; S1

Note that the expression (3.19) immediately implies that there exists a constant c ¤ 0 such that Z 2 hR .r/ D ! .i; z/ d .z/  cesR ; B.i;R/

as R ! C1. This asymptotic is sufficient for what follows.

6 Multiplicity of 1 and the Selberg Conjecture

198

since  . p/ is normal in  . Lemma 6.6 which follows and the compactness of S1 allows one to estimate the last integral: for any real number R > 0 and for all  2  , (

Z R .z;  z/ d .z/ D S1

O.eR e .i; i/=2 /

if .i;  i/ 6 2.R C diam.S1 //

0

else:

Let ı D 2 diam.S1 /. We thus obtain Z Sp Sp

jKR .z; w/j2 d .z/ d .w/  D O ΠW  . p/eR

 e .i; i/=2 :

X

(6.10)

 2 . p/; .i; i/62RCı

Since the constant ı depends only on S1 , an integration by parts allows one to deduce from Theorem 6.2 that for all " > 0, X

e

 2 . p/; .i; i/62RCı

 .i; i/=2

Z

2RCı

D

e

t=2



X

d

0

 2 . p/; .i; i/6t

Z

2R

DO

e

t=2

0

Z

 1 

N.e ;  . p//dt t=2

2R .1C"/t

 e.1C"/t=2 C C 1 et=2 dt D O" p3 p2 0  Z 2R  .1=2C"/t   e e"t=2 t=2 dt C 2 Ce D O" p3 p 0  e.1C2"/R  e"R D O" C C 1 ; p3 p2 e



for sufficiently large R. We deduce from (6.7), (6.8) and (6.10) that for all " > 0,   .1C2"/R  e"R .12s/R e m.1 ;  . p// D O" Π. p/ W  e C 2 C1 ; p3 p

(6.11)

for sufficiently large R. Lemma 6.5 If p is sufficiently large, the quotient  = . p/ is isomorphic to the finite group SL.2; Fp /. Proof We can suppose that a and b are prime to p. Thus Da;b .Z/=pDa;b .Z/ D Da;b .Fp /:

6.2 Multiplicity of the First Eigenvalue

199

We admit the fact (see the references at the end of this chapter) that the equation x20  ax21  bx22 D 0 has a non-trivial solution modulo p. Exchanging the roles of a and b if necessary we can assume that there exist x0 ; x1 2 Fp such that b D x20  ax21 modulo p. Consider the map ˚ from Da;b .Fp / to M2 .Fp / defined on basis elements of Da;b .Fp / by     01 10 ; ; i 7! 1 7! a0 01 (6.12)     ax1 x0 x0 x1 ; k 7! : j 7! ax1 x0 ax0 ax1 One can verify that ˚.i/2 D ˚.a/, ˚.j/2 D ˚.b/ and ˚.i/˚.j/ D ˚.j/˚.i/ and that the matrices in (6.12) are linearly independent mod p. The map ˚ then induces an algebra isomorphism from Da;b .Fp / onto M2 .Fp / and therefore an isomorphism between Da;b .Fp /1 and SL.2; Fp /. Finally, as in the case of SL.2; Z/, we can deduce from the approximation theorem [92, Th. 5.2.10] for quaternion algebras that the map Da;b .Z/1 ! Da;b .Fp /1 induced by reduction mod p is surjective for sufficiently large p. The composition with ˚ then identifies the quotient  = . p/ with the group SL.2; Fp /. t u From now on we assume that p is sufficiently large for the conclusion of Lemma 6.5 to hold. In particular, Œ W  . p/ D p3 .1  p2 / and (6.11) can be rewritten as   m.1 ;  . p// D O" e.12s/R .e.1C2"/R C pe"2R C p3 / : Finally, taking R such that eR D p3 , we obtain the claimed upper bound. Lemma 6.6 Fix R > 0. Then for z; w 2 H, jR .z; w/j 6 4eR .z;w/=2 ; as long as .z; w/ 6 2R; otherwise this expression is equal to 0. Proof It follows immediately from the definition of R that R .z; w/ D area.B.z; R/ \ B.w; R//;

(6.13) t u

6 Multiplicity of 1 and the Selberg Conjecture

200

where we consider the hyperbolic area and B.z; R/ denotes the hyperbolic ball of center z and hyperbolic radius R. In particular, it is clear that FR .z; w/ D 0 if

.z; w/ > 2R. Now suppose that .z; w/ 6 2R and write E D B.z; R/ \ B.w; R/. We can assume that z D i and that w lies on the vertical half-geodesic going from i to infinity, so that it suffices to study the function R . / D R . .i; w//. In polar coordinates about i relative to the upward pointing vector based at i, denote by c.r; / the distance of the point with coordinates .r; / to the point with coordinates . ; 0/. Hyperbolic trigonometry (see (1.15)) states that cosh c.r; / D cosh cosh r  cos sinh sinh r: Consider the angle ˛ 2 Œ0; =2 defined by the relation cos ˛ D D

cosh cosh R  cosh R sinh sinh R tanh. =2/ cosh R.cosh  1/ D : sinh sinh R tanh R

Denote by v the point with polar coordinates . =2; 0/. For a point p 2 @B.i; R/ \ B.w; R/ let .r; / be its polar coordinates. We have c.r; / R and 2 Œ˛; ˛. By considering the triangle with vertices i, v and .r; / – colored in grey in Fig. 6.1 – we find cosh .v; p/ D cosh. =2/ cosh R  cos sinh. =2/ sinh R 6 cosh. =2/ cosh R  cos ˛ sinh. =2/ sinh R   tanh. =2/ sinh. =2/ sinh R 6 cosh. =2/ cosh R  tanh R 6

cosh R : cosh. =2/

p

z

Fig. 6.1 The triangle .zpv/

α

v

α

w

6.2 Multiplicity of the First Eigenvalue

201

By symmetry in z and w, we finally deduce that the set E is included in a hyperbolic ball of radius T (here B.v; T/) defined by the relation cosh T D

cosh R : cosh. =2/

The area of such a ball is 4 sinh2 .T=2/ D 2.cosh T  1/ D 2



 cosh R  1 6 4eR =2 : cosh. =2/ t u

This concludes the proof of Lemma 6.6.

In the above proof we assumed that Sp was compact. It remains then to treat the case where  is the modular group. Proof when Sp is non-compact In the above proof we used the compactness of S on two occasions. We now show how to bypass this hypothesis in each of the two cases. Let k be a point-pair invariant of the form R as before. Let K again denote the associated automorphic kernel and h the Selberg transform of k. As in the compact case, the integral operator with kernel K is a positive operator on L2 . . p/nH/. The orthogonal projection onto the subspace L2 . . p/nH/ and orthogonal to the subspace generated by the Laplacian eigenfunctions with eigenvalue 1 in C. . p/nH/ is also a positive operator. The operator obtained by composing these two commuting operators is therefore itself positive; it is moreover an integral operator with kernel B.z; w/ D K.z; w/ 

X

h.1 /'j .z/'j .w/:

j D1

By considering (for fixed z) the sequence of functions ( fn .w/ D

1 if .z; w/ 6 1=n; 0 if .z; w/ > 1=n;

we have B.z; z/ D lim hBfn ; fn i > 0: n!C1

In other words, K.z; z/ 

X j D1

h.1 /j'j .z/j2 > 0:

(6.14)

6 Multiplicity of 1 and the Selberg Conjecture

202

Compared to the compact case, the situation is complicated by the presence of R cusps: the integral Sp K.z; z/ d .z/ is divergent. We again alleviate this problem by truncating a fundamental domain for  . p/ in H. The fundamental domain D D fz D x C iy 2 H j jxj 6 1=2 and jzj > 1g for  decomposes, for all Y > 1, as a union D D DY [ P.Y/ where P.Y/ is the fundamental domain of a cusp P.Y/ D fz D x C iy j jxj 6 1=2 and y > Yg based on the horocycle hY D fz D x C iy j y D Yg: A cuspidal region subdivides into a collar of width log 2 and the closure of a contiguous cuspidal region. More precisely (and again at the level of the fundamental domain), P.Y/ D C.Y/ [ P.2Y/; where C.Y/ D fz D x C iy j jxj 6 1=2 and Y < y < 2Yg is the subset of points of P.Y/ lying between the horocycles hY and h2Y (see Fig. 6.2). Fig. 6.2 The collar C.Y/ 2Y C(Y ) Y

6.2 Multiplicity of the First Eigenvalue

203

We obtain an analogous decomposition of  . p/nH by considering the covering map  . p/nH !  nH. Fix a real number Y strictly greater than 1. Let ˝p be the compact subset of  . p/nH obtained by taking the union of the heart – by which we mean the preimage of DY – in  . p/nH and the collars of different cusps. Note that ˝p depends on the choice of Y. We are going to compare the L2 -norm of a Laplacian eigenfunction on  . p/nH with the L2 norm of its restriction to ˝p . Let f then be a Laplacian eigenfunction in C. . p/nH/ with eigenvalue . Let P.Y/ be the fundamental domain of a cusp that we’ll take to be of the form P.Y/ D fz D x C iy j 0 < x < q; y > Yg so that its collar is C.Y/ D fx C iy j 0 < x < q; Y < y < 2Yg: Lemma 6.7 Let s0 > 1=2 be a real number. Then there exists a constant Cs0 > 0 such that for all Y > 0, s 2 .s0 ; 1/ and any function f as above with  D s.1  s/, we have R 2Y R q

2 Y 0 j f .z/j d .z/ > Cs0 : R C1 R q 2 d .z/ j f .z/j 2Y 0

Proof Conjugating the group  . p/ in G if necessary, we can assume that q D 1. The function f then satisfies f .z C 1/ D f .z/, and we can therefore expand it in its Fourier series X fn .y/e.nx/; f .z/ D n

where Z fn .y/ D

1 0

f .z/e.nx/:

Since f is C1 , the Fourier series above converges absolutely and uniformly on compacta, and since f 2 C. . p/nH/, the constant term f0 is constantly equal to 0. Since f has eigenvalue , the function F.y/ D fn .y=.2n// is a solution to the differential equation (3.10). Since moreover f is cuspidal, it is bounded (by definition); in particular f .z/ D o.e2y / as y ! 1. It follows from the general solution of equation (3.10) that the function z 7! fn .y/e.nx/ is a multiple of the Whittaker function z 7! Ws .nz/ (see (3.11)). We define fO .n/ by the identity fn .y/e.nx/ D fO .n/Ws .nz/;

8z D x C iy 2 H:

6 Multiplicity of 1 and the Selberg Conjecture

204

We then have the following expansion X fO .n/Ws .nz/: f .z/ D

(6.15)

n¤0

Parseval’s identity takes the form X

j fO .n/Ws .iny/j2 D

Z

n¤0

1 0

for all y > 0. Thus Z Z 1Z 1 X j f .z/j2 d .z/ D j fO .n/j2 0

Y

j f .x C iy/j2 dx;

1

jWs .iny/j2 y2 dy:

(6.16)

Y

n¤0

We put  D s  1=2. According to the expression (3.11), Lemma 6.7 is a consequence of the following result regarding Bessel functions, the proof of which we omit: for Y > 0 and  2 .0 ; 1=2/ we have R 2Y RY1 2Y

.K2 .x/=x/ dx .K2 .x/=x/ dx

> C0 > 0:

(6.17)

(Note that the Bessel function K is real for  real.)

t u

Let us come back to the proof of Theorem 6.4. As in the compact case one deduces from (6.8) that there exists a constant c > 0 such that hR .r1 / > ce2s1 R as R ! C1. On integrating inequality (6.14) over ˝p we get Z

X ˝p  D j 1

 Z 2s1 R j'j .z/j d .z/ D O e 2

˝p

 K.z; z/ d .z/ :

Swapping the sum and integral, we may apply Lemma 6.7 to the functions 'j to arrive at (modifying the constant if necessary) Z

X  . p/nH  D j 1

Z

2

j'j .z/j d .z/ D O

X ˝p  D j 1

 j'j .z/j d .z/ : 2

Since Z m.1 ;  . p// D

X  . p/nH  D j 1

j'j .z/j2 d .z/;

6.3 Representation Theory of PSL.2; Z=pZ/

205

we have therefore shown that  Z 2s1 R m.1 ;  . p// D O e

˝p

 K.z; z/ d .z/ :

(6.18)

Finally, similarly as in the compact case but substituting the compactness of DY for that of S, one shows that   Z X R  .i; i/=2 K.z; z/ d .z/ D O Œ . p/ W  e e : ˝p

 2 . p/; .i; i/62R

The conclusion of the proof now proceeds precisely as in the compact case.

t u

6.3 Representation Theory of PSL.2 ; Z=pZ/ The goal of this section is to prove the following theorem. Theorem 6.8 Let p > 5 be a prime. The dimension of a non-trivial linear representation of PSL.2; Z=pZ/ is > . p  1/=2. For the proof we shall use a few basic facts in the representation theory of finite groups. We shall thus begin by recalling the necessary results.

6.3.1 Review of the Representation Theory of Finite Groups We start by recalling that a representation of a finite group G is a pair .; V/ where V is a complex vector space V and  is a homomorphism from G into GL.V/. The dimension of the representation .; V/ is the dimension of V. We shall often simply write  for a representation of G. A subspace W  V is called an invariant subspace if for all g 2 G one has .g/.W/ D W. A representation .; V/ with V ¤ f0g is said to be irreducible if it admits no non-trivial invariant subspace, i.e., no invariant subspace different from f0g and V. Every representation .; V/ of G with V ¤ f0g is equivalent to a direct sum of irreducible representations of G. Let .; V/ and . ; W/ be two representations of G. Denote by HomG .; / the set of linear maps T W V ! W which are G-equivariant: T.g/ D .g/T

.g 2 G/:

Then .; V/ and . ; W/ are said to be equivalent if there exists an invertible linear map in HomG .; /.

6 Multiplicity of 1 and the Selberg Conjecture

206

Lemma 6.9 (Schur’s Lemma) Let .; V/ and . ; W/ be two finite-dimensional irreducible representations of G. Then ( dimC HomG .; / D

0

if  and are not equivalentI

1

if  and are equivalent:

Starting from two representations .; V/ and . ; W/ we can naturally form the representations  ˚ in the vector space direct sum V ˚ W and  ˝ in the tensor product V ˝ W. Moreover, given .; V/ we can form the dual representation   in the dual vector space V  of V. The character of a representation  is the function  W G ! C; g 7! tr..g//. That characters are useful can be seen in the following theorem which, together with Schur’s lemma, shows that the characters of irreducible representations of G form an orthonormal system with respect to the Hermitian scalar product h f1 ; f2 iG D

1 X f1 .g/f2 .g/; jGj g2G

defined on the space of functions G ! C. Theorem 6.10 Let .; V/ and . ; W/ be two representations of the finite group G. Then h ;  iG D dimC HomG .; /: Remark 6.11 Note that it follows in particular from Theorem 6.10 and Schur’s lemma that the representation .; V/ is irreducible if and only if h ;  iG D 1. We shall use Theorem 6.10 several times in this form. The left regular representation G of the group in the vector space CG of functions G ! C is given by .G .g/f /.x/ D f .g1 x/: Its character G satisfies ( G .g/ D

jGj if g D 1 0

else:

Theorem 6.10 and Schur’s lemma then imply that an irreducible representation

of G appears dim times in the decomposition of G into irreducible subrepresentations. Thus X G D .dim / ;

6.3 Representation Theory of PSL.2; Z=pZ/

207

where runs through the set of equivalence classes of irreducible representations of G. Theorem 6.10 implies the degree formula jGj D

X

.dim /2 ;



where runs through the set of equivalence classes of irreducible representations of G. We shall now use these general facts to prove Theorem 6.8.

6.3.2 Proof of Theorem 6.8 Consider the affine group B on Fp D Z=pZ, i.e., the group of transformations z 7! ˛z C ˇ

.˛ 2 Fp ; ˇ 2 Fp /

of Fp . The group B acts on the space W of complex functions on Fp by .L.b/f /.z/ D f .b1 z/: Consider the subspace n o P W0 D f W Fp ! C j z2Fp f .z/ D 0 of W. Then B preserves W0 ; write L0 for the representation of B on W0 . Lemma 6.12 The linear representation L0 of B on W0 is irreducible of dimension p  1. Proof In the canonical basis of W, one calculates 1 X W˝W .b/ D 2: jBj b2B Thus hW ; W iB D D

1 X W .b/2 jBj b2B 1 X W˝W .b/ jBj b2B

D 2:

6 Multiplicity of 1 and the Selberg Conjecture

208

Now observe that the representation W of B decomposes into a direct sum of the trivial representation (in the subspace of constant functions) and the representation W0 . Hence W D 1 C W0 and 2 D h1 C W0 ; 1 C W0 iB D 1 C 2h1; W0 iB C hW0 ; W0 iB : This in turn forces hW0 ; W0 iB D 1 and implies (using Theorem 6.10 and Schur’s lemma) that the representation W0 (of dimension p  1) is irreducible. t u Let B0 be the stabilizer in PSL.2; Fp / of the line generated by the vector .1; 0/, i.e., the point at infinity in P1 .Fp /; the group B0 can also be viewed as the image in PSL.2; Fp / of the subgroup  ˇ a b ˇ  a 2 F ; b 2 F ˇ p p 0 a1 of SL.2; Fp /. The action of B0 on Fp is then given by   a b : 'A .z/ D a z C ab; where A D 0 a1 2

We may identify B0 with an index 2 subgroup of B. Writing ˛ W B ! Fp I

.z 7! az C b/ 7! a;

2  we have B0 D ˛ 1 .F2 p /, where Fp denotes the subgroup of squares in Fp .

Proposition 6.13 Let p be an odd prime. There exists exactly . p C 3/=2 irreducible pairwise nonequivalent representations of B0 , consisting of • . p  1/=2 group homomorphisms B0 ! C , factoring through ˛jB0 , • two nonequivalent representations 1 and 2 , both of dimension . p  1/=2. Proof Since F2 p is abelian of order . p1/=2 it admits exactly . p1/=2 irreducible representations, all of dimension 1, given by the homomorphisms 1 ; : : : ; . p1/=2

6.3 Representation Theory of PSL.2; Z=pZ/

209

 of F2 p into C . Composing them with ˛jB0 we obtain . p  1/=2 homomorphisms 1 ı ˛jB0 ; : : : ; . p1/=2 ı ˛jB0 from B0 into C . Continuing, we restrict the representation L0 to the group B0 with the aim of showing that it breaks up into a direct sum of two irreducible nonequivalent representations 1 and 2 , both of dimension . p  1/=2. We begin by recalling that the homomorphisms

ec W Fp ! C I

z 7! e .cz=p/

.c 2 Fp /

form a basis of W. In particular, the ec , where c 2 Fp , form a basis of the subspace W0 of W. Note that if g 2 B is given by g.z/ D az C b, then .L0 .g/ec /.z/ D ec .g1 z/ D ec



zb a



  cb De  ec=a .z/: ap

Denote by W1 (resp. W2 ) the subspace of W0 generated by the ec , with c 2 F2 p (resp. c 2 Fp  F2 p ). The above formula implies that the subspaces W1 and W2 are left invariant by the restriction of the representation L0 to the group B0 . Observe moreover that dimC W1 D dimC W2 D

p1 : 2

We shall show that the representations 1 and 2 of B0 in W1 and W2 respectively are irreducible. For this we first notice that in the decomposition W0 D W1 ˚ W2 , we have 8  < 1 .g/ 0 if g 2 B0 I L0 .g/ D  0  2 .g/ : 0 if g 2 B  B0 :  0

This implies the following identity on characters (  1 .g/ C  2 .g/ if g 2 B0 I L0 .g/ D 0 if g 2 B  B0 : According to Lemma 6.12 the representation is irreducible, we have (Theorem 6.10 and Schur’s lemma): 1 D hL0 ; L0 iB D D

1 X jL0 .g/j2 jBj g2B

1 X j 1 .g/ C  2 .g/j2 2jB0 j g2B 0

D

1 h C  2 ;  1 C  2 iB0 ; 2 1

210

6 Multiplicity of 1 and the Selberg Conjecture

which implies (again using Theorem 6.10 and Schur’s lemma) that 1 and 2 are irreducible and nonequivalent. The degree formula p1 2 p. p  1/ jB0 j D D 1 C2 2 2



p1 2

2

finally implies that we have obtained all of the representations of B0 . This concludes the proof of Proposition 6.13. t u Proof of Theorem 6.8 Let  be a non-trivial representation of PSL.2; Fp / on Cn . The restriction jB0 decomposes as a direct sum of irreducible representations of B0 . Since p > 5, the group PSL.2; Fp / is simple. The representation jB0 is therefore faithful, meaning that for every non-trivial element b 2 B0 , the transformation .b/ is different from the identity. But the . p  1/=2 representations of dimension 1 of B0 are trivial on fz 7! zCbg  B0 , so that one of the representations 1 ; 2 must appear in jB0 . We deduce that n is greater than or equal to . p  1/=2. t u

6.4 Lower Bound on the First Non-zero Eigenvalue We can now finish the proof of the principal result of this chapter. Proof of Theorem 6.1 We must show that if the first non-zero eigenvalue 1 of the Laplacian on  . p/nH (for large p) is strictly smaller than 5=36 then 1 is a Laplacian eigenvalue for  nH. Suppose then that 1 D s1 .1s1 / with s1 2 .5=6; 1/. According to Theorem 6.4, for any " > 0 we have   m.1 ;  . p// D O" p6.1s1 /C" : On the other hand, since  . p/ is normal in  , we have a group action  = . p/ on  . p/nH by deck transformations on the cover  . p/nH !  nH. From Lemma 6.5 it follows that for large enough p we have  = . p/ Š PSL.2; Fp /: The automorphisms of the cover  . p/nH !  nH then induce a group action PSL.2; Fp / on the functions on  . p/nH. This action preserves the 1 eigenspace; we write V.1 / for the corresponding (finite-dimensional) representation.

6.5 Comments and References

211

Arguing by contradiction, assume that 1 is not in the spectrum of  nH. The representation V.1 / cannot contain the trivial representation and Theorem 6.8 then states that its dimension is greater than or equal to . p  1/=2 (as soon as p is larger than 5). For any positive " we thus have   p1 6 m.1 ;  . p// D O" p6.1s1 /C" ; 2 for large enough p. This contradicts our hypothesis, according to which s1 is strictly larger than 5=6. t u

6.5 Comments and References The proof of Theorem 6.1 described in this chapter is based on an idea of Kazhdan, developed by Sarnak and Xue in [113].

§ 6.1 For the Dirichlet unit theorem the reader can consult for example [75]. The fact that the equation ax21  bx22 C abx23 D 0 admits a non-trivial solution mod p when a and b are prime to p follows from the ChevalleyWarning theorem (see [118, Chap. I Th. 3]) according to which every quadratic form in at least 3 variables over the finite field with p elements Fp has a non-trivial zero. The extension of the proof of Theorem 6.4 to the case of non-compact surfaces is nontrivial and was the topic of the thesis of Gamburd. We have followed his article [42]. In particular, the reader can find the proof of the lower bound (6.17) in [42, Claim 4.1].

§ 6.3 The proof of Theorem 6.8 is reproduced in the book of Davidoff, Sarnak and Valette [32]. All of the results from the representation theory of finite groups summarized in the first paragraph are proved in detail in [32]. The book of Serre [119] is the standard reference on the subject. One can find a proof of the simplicity of the group PSL.2; Fp / (p > 5) in [96, Th. 4.1].

Chapter 7

L-Functions and the Selberg Conjecture

Let N be a square-free integer. Recall that  0 .N/ D

 ˇ ab ˇ 2 SL.2; Z/ ˇ Njc : cd

The goal of this chapter is to prove, using methods in analytic number theory, the following approximation to the Selberg conjecture. Theorem 7.1 We have 1 .0 .N/nH/ >

5 : 36

Let  D 0 .N/ and consider a Maaß cusp form f 2 C. nH/ such that f D .1=4   2 /f

. 2 Œ0; 1=2/ [ iR/:

We must show that Re./ 6 1=3. Since   11 2 ; 01 the function f is 1-periodic: f .z C 1/ D f .z/. It can therefore be expanded in a Fourier series in the variable x D Re.z/, see (6.15): f .z/ D

X

p ar y K .2jrjy/e.rx/:

(7.1)

r2Z; r¤0

(The coefficient a0 is zero since f is cuspidal.)

© Springer International Publishing Switzerland 2016 N. Bergeron, The Spectrum of Hyperbolic Surfaces, Universitext, DOI 10.1007/978-3-319-27666-3_7

213

214

7 L-Functions and the Selberg Conjecture

The work of Maaß teaches us to think of the “Fourier coefficients” ar of f as arithmetical quantities. This point of view is particularly justified by the behavior of the L-functions attached to them. The study of these L-functions is rather similar to that of the Riemann zeta function (see Exercise 1.16), the role of the theta function being played now by the Maaß form. The L-function, once completed, satisfies a functional equation and contains the “spectral information” . We shall see that the Riemann hypothesis for the Rankin-Selberg L-function implies the Selberg theorem. The method then consists in proving a weak (average) version of the Riemann hypothesis in this special case.

7.1 The L-Function Attached to a Maaß Form The antiholomorphic involution  W x C iy 7! x C iy induces an isometry of H. Since it commutes with the Laplacian one can freely suppose that f is either even ( f ı  D f ) or odd ( f ı  D f ). If f is even (resp. odd), its Fourier coefficients satisfy ar D ar (resp. ar D ar ); we shall then write " D 0 (resp. 1). In either case, one defines the L-function associated with f by L.s; f / D

C1 X nD1

an : ns

(7.2)

The following lemma is due to Hecke. Lemma 7.2 The Fourier coefficients of f satisfy an D O.n1=2 /; for large enough n. In particular, the series (7.2) is absolutely convergent in the half-plane Re.s/ > 3=2. Proof Since f is cuspidal, it is bounded by a certain constant C. For all y > 0, we thus have ˇ Z 1 ˇZ 1 ˇ ˇ ˇ f .x C iy/e.nx/ dxˇˇ 6 j f .x C iy/jdx 6 C: ˇ 0

0

But Z

1 0

p f .x C iy/e.nx/ dx D an y K .2jnjy/

and the lemma is proved by taking y D 1=n. On average one can prove the following bound.

t u

7.1 The L-Function Attached to a Maaß Form

215

Lemma 7.3 For x sufficiently large we have X

jam j2 D Of .x/:

16m6x

Proof Since f is a cusp form, its Fourier coefficient a0 is zero and Parseval’s identity implies X

2

jam j2 yjK .2my/j2 6

Z

1 0

16m6x

j f .x C iy/j2 dx:

We deduce from this that for any positive real number Y we have 2

X

jam j2

Z

C1

jK .2my/j2

Y

16m6x

dy 6 y

Z

C1

Z

Y

1 0

j f .z/j2 d .z/:

(7.3)

Since f is bounded, Z

C1

Z

Y

1

0

j f .z/j2 d .z/ D O .1=Y/ :

Taking Y D 1=x in (7.3) we obtain the stated bound.

t u

Remark 7.4 In Sect. 7.6 we prove a non-trivial bound on individual Fourier coefficients of Maaß cusp forms that cannot be deduced from the above lemma. See Theorem 7.42. The element p   0 1= N

D p 2 SL.2; R/ N 0 normalizes  D 0 .N/. Define fQ .z/ D f . z/I this function again defines a Maaß cusp form such that fQ D .1=4   2 /fQ . Proposition 7.5 Let . f ; s/ D  s 

s C " C   s C "     L. f ; s/: 2 2

(7.4)

216

7 L-Functions and the Selberg Conjecture

Then . f ; s/ extends to a holomorphic function on the entire complex plane C. Moreover . f ; s/ D .1/" N 1=2s .fQ ; 1  s/: Proof We first prove Proposition 7.5 in the case where f is even (" D 0). According to Lemma 4.27, for Re.s/ sufficiently large we have Z .s; f / D 2

C1

0

f .iy/ys1=2

dy : y

Given a real number ˛ > 0, we break up this last integral so as to obtain Z

Z C1 dy dy C2 f .iy/ys1=2 y y 0 ˛ Z C1 Z C1   du dy i C2 .Nu/sC1=2 D2 f f .iy/ys1=2 Nu u y 1=.N˛/ ˛ Z C1 Z C1 du dy C2 f .iy/ys1=2 D2 fQ .iu/.Nu/sC1=2 u y 1=.N˛/ ˛ Z C1 Z C1 du dy D 2N 1=2s fQ .iu/usC1=2 C 2 f .iy/ys1=2 : u y 1=.N˛/ ˛

.s; f / D 2

˛

f .iy/ys1=2

The analytic continuation of .s; f / comes as an immediate consequence; the functional equation can be obtained by taking ˛ D N 1=2 . If f is odd (" D 1), note that Z .s; f / D

C1

g.iy/ysC1=2 0

dy ; y

where C1

g.z/ D

X 1 @f p .z/ D an n y K .2iny/ cos.2nx/: 4i @x nD1

We have g .i=Nu/ D .1/Nu2

@fQ .iu/; @x

and the functional equation can be deduced as above.

t u

According to Maaß, the L-functions associated with Maaß forms, despite the likely transcendental nature of their Fourier coefficients, should correspond to

7.2 Hecke Operators and Applications

217

“arithmetic” L-functions (associated, for example, with Hecke characters of real quadratic extensions of Q, see § 4.5). We shall approach them in this light, keeping in mind the various properties (some conjectural) of the Riemann zeta function. The Riemann hypothesis, generalized to L.s; f /, implies that the function s 7! L.s; f / should never vanish on the half-plane Re.s/ > 1=2. But according to Proposition 7.5 the function s 7! .s; f / is everywhere holomorphic. The  -factors which arise in the definition of .s; f / should therefore not admit any poles in the half plane Re.s/ > 1=2. Assuming f even, and recalling that f D 1=4   2 f , these  -factors are precisely 

s    2

and 

s C   : 2

These have poles at  and , respectively. Thus jj 6 1=2 for  real. It follows that 1 .0 .N/nH/ must be. . . greater than or equal to 0! Admittedly, this is not a very interesting conclusion. It is tempting, though, to consider more general Dirichlet series1 of the form C1 X nD1

jan j2k ns

for k D 1; 2; : : : to deduce better lower bounds on the first non-zero eigenvalue of the Laplacian. This is the approach we follow in this chapter. We begin by noting that the “arithmetic” nature of the coefficients an is perhaps less surprising if we relate them to the Hecke operators (4.34).

7.2 Hecke Operators and Applications 7.2.1 Hecke Operators The Hecke operators are associated with “correspondences” on arithmetic hyperbolic surfaces. We define a degree r correspondence C on a Riemannian manifold X as a map C W X ! .X      X/=Sr z 7! fz1 ; : : : ; zr g;

1

This idea, due to Langlands, was used by Deligne in his proof [35] of the Weil conjectures.

218

7 L-Functions and the Selberg Conjecture

where Sr is the permutation group on r elements and each function z 7! zi .z/ is a local isometry.2 Let   SL.2; R/ be a Fuchsian group. Let ˛ 2 GL.2; R/, having positive determinant, be such that ˛ D  \ ˛ 1  ˛ is of finite index in  and ˛ 1  ˛. We associate with ˛ the correspondence C˛ on  nH given by C˛ W  z 7! f ˛ı1 z; : : : ;  ˛ır zg; where the ı1 ; : : : ; ır form a set of representatives of the classes ˛ n in such a way that  n ˛ D f ˛ı1 ; : : : ;  ˛ır g : To such a correspondence there corresponds a Hecke operator T˛ W L2 . nH/ ! L2 . nH/;

T˛ f .z/ D

r X

f .˛ıj z/:

jD1

The operator T˛ is well-defined; moreover, since each ˛ıj acts by isometries on H, the operator T˛ commutes with the Laplacian. The T˛ generate an algebra. When ˛ belongs to  , the operator T˛ is trivial. But the algebra of Hecke operators is infinite as soon as the commensurator Com. / D f˛ 2 SL.2; R/ j ˛ is of finite index in  and ˛ 1  ˛g is infinite modulo  . Set  D 0 .N/ for an integer N greater than or equal to 1 and take ˛ 2 GL.2; Q/ to have positive determinant. Then the group ˛ 1  ˛ contains a congruence subgroup  .M/ for some large enough integer M. From this one deduces that ˛ is of finite index in  (and ˛ 1  ˛). For every integer n > 1 we consider the operator defined by X

T˛ ;

det ˛Dn

which clearly lies in the algebra generated by the Hecke operators of  nH. This operator can also be obtained from the set   ˇ ab ˇ Hn D ˇ a; b; c; d 2 Z; ad  bc D n; Njc; .a; N/ D 1 ; cd

2

The functions zj are not individually globally defined but the entire set of such zj is.

(7.5)

7.2 Hecke Operators and Applications

219

for it is obvious that  Hn D Hn  D Hn and, analogously to the quotient  n ˛ considered above, the quotient  nHn defines a correspondence on  nH. We leave as an exercise the verification that Hn D

[ d1 [ adDn bD0 .a;N/D1

  ab  0d

(disjoint union):

(7.6)

The set Hn depends on N but if n and N are relatively prime the relation (7.6) is independent of N. In this case we can speak of “the” correspondence defined by Hn even while varying the level N. For notational convenience, we shall write Hn for the above set as well as the correspondence it defines. Given an integer n greater than or equal to 1, we denote by TnN (or just Tn if the context is sufficiently clear or if .n; N/ D 1) the operator associated with the p modular correspondence determined by Hn and renormalized by 1= n. To simplify, we shall write p1n Hn for this latter correspondence. It suffices in fact to define the Hecke operators by the following formulas, see (4.34), and to study their action on the Fourier coefficients of Maaß forms: d1 1 X X  az C b  .TnN f /.z/ D p : f d n adDn bD0

(7.7)

.a;N/D1

Given two positive integers m and n, the operators TnN and TmN commute. We leave as an exercise the verification that X N TnN ı TmN D Tmn=d 2:

(7.8)

0 1, .TnN f /.z/ D

X X r2Z

arn=a2

 p y K .2jrjy/e.rx/:

aj.n;r/ .a;N/D1 a>1

Proof Let n be an integer > 1. Then d1 1 X X  az C b  f .Tn f /.z/ D p d n adDn bD0 .a;N/D1

D

X X r2Z adDn .a;N/D1 d>1

! d1 1X p ar y K .2jrjay=d/e.rax=d/ e.rb=d/ : d bD0

The last sum is trivial except when d divides r in which case it equals 1. Note finally that d=a D n=a2 , from which the lemma follows directly. t u Let f be a non-zero Maaß cusp form, even or odd, and an eigenfunction of the Hecke operators: Tn f D .n/f , .n; N/ D 1. If its first Fourier coefficient a1 is nonzero, by multiplying f by a non-zero scalar if necessary we can assume without loss of generality that a1 D 1. It then follows from Lemma 7.6 that, for .n; N/ D 1: an D .n/ and an D .1/" .n/:

(7.9)

The expected multiplicative properties of the Fourier coefficients (at least when .n; N/ D 1) can be read off from those of the Hecke operators, see (7.8) and (7.12). But it is possible, in general, that a1 is zero. If this is the case, if h is a cusp form for a group 0 .M/ 0 .N/ with MjN, each Maaß form f .z/ D h.Dz/, where DMjN, lies in C.0 .N/nH/ and only the Fourier coefficients an , for n  0 .mod D/, can be non-zero. If M < N, such a Maaß form ( f .z/ D h.Dz/) is called an oldform. In a certain sense the form h.Dz/ is linked more to the group 0 .M/ than the group 0 .N/. The idea of Atkin-Lehner theory, which we shall adapt to the context of Maaß forms in the next section, is that by “eliminating” these forms coming from smaller levels, the simultaneous diagonalization by all Hecke operators becomes possible; moreover, the common eigenspaces are of dimension 1. From now on we write Cold .0 .N/nH/ for the subspace of C.0 .N/nH/ generated by oldforms and Cprim .0 .N/nH/ its orthogonal complement. We call the latter

7.2 Hecke Operators and Applications

221

the space of newforms. The Hecke operators Tn , for .n; N/ D 1, obviously preserve the subspace Cold .0 .N/nH/, since they commute with the operator f .z/ 7! f .Dz/ for all DjN. They also preserve the subspace Cprim .0 .N/nH/, since they are selfadjoint. There exists then a basis of Maaß forms in Cprim .0 .N/nH/ consisting of eigenforms of the Tn , .n; N/ D 1, even or odd; such a form is called a newform.

7.2.2 Atkin-Lehner Theory The main interest in introducing these newforms is to prove the multiplicativity of the Fourier coefficients, which is based on the following proposition. Proposition 7.7 Let f .z/ D

X

p ar y K .2jrjy/e.rx/

r2Z

be a non-zero Maaß cusp (a0 D 0) newform in C.0 .N/nH/. Then a1 ¤ 0. Proof Suppose the opposite holds. Thus we assume that f .z/ D

X

p ar y K .2jrjy/e.rx/

r2Z

is an even or odd non-zero Maaß cusp form (a0 D 0) which is a eigenfunction for the Hecke operators Tn , .n; N/ D 1, and for which a1 D 0. We shall show that f is a linear combination of oldforms. Since a1 D 0, it follows from Lemma 7.6 that for every integer n relatively prime to N the coefficient an is zero. The last of the next four lemmas (each an intermediate step in the proof) implies the existence, for every prime divisor p of N, of forms fp 2 C.0 .N=p/nH/ such that f .z/ D

X

fp . pz/:

pjN

This concludes the proof of the proposition.

t u

The following lemmas constitute the basic statements of Atkin-Lehner theory. Lemma 7.8 Let ` be an integer greater than or equal to 1 and f a 1-periodic C1 function on H such that z 7! f .`z/ defines a Maaß form in C.0 .N/nH/. Then 1. if N D D`, the function f lies in C.0 .D/nH/; 2. if ` 6 jN, then f is zero.

222

7 L-Functions and the Selberg Conjecture

Proof Arguing inductively on the number of prime divisors of ` one can reduce to the case when ` is a prime number. We begin by showing that f lies in C.0 .N/nH/. Put g.z/ D f .`z/, 

`0 01

ı` D



and  0 D

  ˇ ab ˇ 2 0 .N/ ˇ `jb : cd

Since ı`1  0 ı`  0 .N/, g is ı`1  0 ı` -invariant and f is therefore  0 -invariant. Now f is moreover invariant by z 7! z C 1 and so is necessarily invariant under   the action of the whole group 0 .N/ (which is generated by  0 and the matrix 10 11 ). Since g is cuspidal, the form f indeed lies in C.0 .N/nH/. Finally, the form f is invariant under the action of the groups 0 .N/ and ı` 0 .N/ı`1 . According to whether ` divides N or not, the group generated by 0 .N/  1 0 and the matrix N=` 1 is discrete or not. In the first case, the generated group is 0 .N=`/ and f then lies in C.0 .D/nH/. In the second case the generated group is necessarily dense in SL.2; R/ and so f is zero. t u Lemma 7.9 Let f .z/ D

X

p ar y K .2jrjy/e.rx/

r2Z

be a Maaß form in C.0 .N/nH/ and p a prime number. If g.z/ D

X

p ar y K .2jrjy/e.rx/

.r;p/D1

then for p dividing N the form g is in C.0 .Np/nH/ and otherwise g is in C.0 .Np2 /nH/. Proof Replacing N by pN if necessary, we can assume that p divides N. It then follows from Lemma 7.6 that X p .Tp f /.z/ D arp y K .2jrjy/e.rx/: r2Z

But since Tp f belongs to C.0 .N/nH/, the form .Tp f /. pz/ D

X

p arp y K .2jrpjy/e.rpx/

r2Z

lies in C.0 .Np/nH/. From this one deduces that g.z/ D f .z/  .Tp f /. pz/ lies in C.0 .Np/nH/.

t u

7.2 Hecke Operators and Applications

223

Lemma 7.10 Let ` be a squarefree integer strictly greater than 1 and f .z/ D

X

p ar y K .2jrjy/e.rx/

r2Z

a Maaß form in C.0 .N/nH/. If for every integer r prime to ` the coefficient ar is zero then X f .z/ D gp . pz/; pj`

where the sum runs over all prime divisors of ` and gp 2 C.0 .N`2 /nH/ (gp 2 C.0 .N`/nH/, if `jN). Proof The proof is by induction on the number of prime factors of `. Suppose first that ` is prime. The function g.z/ D f .z=`/ then satisfies the conditions of Lemma 7.8, so that either g lies in C.0 .N=`/nH/ or f and g are both zero, according to whether ` divides N or not. The last case in degenerate. In the first case, we have a fortiori that g lies in C.0 .N`/nH/ and f .z/ D g.`z/: Suppose now that ` is a product of at least two prime numbers and that the conclusion of the lemma holds for all of the proper divisors of `. Let p be a prime factor of ` and set `0 D `=p. Define X

h.z/ D

p ar y K .2jrjy/e.rx/:

.r;p/D1

According to Lemma 7.9, h lies in C.0 .Np2 /nH/. We write f .z/  h.z/ D

X

p br y K .2jrjy/e.rx/ 2 C.0 .Np2 /nH/:

r2Z

If n and p are relatively prime, then the coefficient bn is zero. The first step in our induction – applied with N replaced by Np2 – then implies that gp .z/ WD f .z=p/  h.z=p/ belongs to C.0 .Np/nH/. Thus f .z/ D gp . pz/ C h.z/;

(7.10)

where h satisfies the conditions of the lemma with Np2 and `0 in place of N and `, respectively. By the induction hypothesis, we can find functions gq in

224

7 L-Functions and the Selberg Conjecture

C.0 .Np2 `02 /nH/ for every prime factor `0 such that h.z/ D

X

gq .qz/:

qj`0

Since Np2 `02 D N`2 , if we insert this expression into (7.10) this proves the lemma for ` not dividing N. If ` does divide N, it follows immediately from the above proof that and Lemma 7.9 that one can take the gp 2 C.0 .N`/nH/, so that Lemma 7.10 is completely proved. t u Lemma 7.11 Let ` be an integer greater than or equal to 1 and f .z/ D

X

p ar y K .2jrjy/e.rx/

r2Z

a Maaß form in C.0 .N/nH/. Suppose ar is equal to 0 for every integer r relatively prime to `. 1. If .`; N/ D 1, then f is zero. 2. If .`; N/ ¤ 1, there exists Maaß forms fp in C.0 .N=p/nH/, where p runs through the set of prime factors of .`; N/, such that f .z/ D

X

fp . pz/:

pj.`;N/

Proof As the hypothesis of the lemma depends only on the largest squarefree divisor of `, we can assume ` to be squarefree. We proceed by induction on the number of prime factors of `. If ` is prime, it suffices to apply Lemma 7.8 to f .z=`/. Suppose then that the result is true for all proper divisors of `. Let p be a prime divisor of ` and set `0 D `=p. We write f D g C h where g.z/ D

X

p ar y K .2jrjy/e.rx/;

.r;`0 /D1

and h.z/ D

X

p ar y K .2jrjy/e.rx/:

.r;`0 /¤1

According to Lemma 7.9 the form g belongs to C.0 .N`02 /nH/ and thus so does h.

7.2 Hecke Operators and Applications

225

If p does not divide N then p does not divide N`02 either, and since the r-th Fourier coefficient of g is zero if r is prime to p, Lemma 7.8 implies that g is zero. Hence X

f .z/ D h.z/ D

p ar y K .2jrjy/e.rx/

.r;`0 /¤1

satisfies the hypotheses of the lemma with ` replaced by `0 . The conclusion therefore holds by the induction hypothesis. Now assume that p divides N. Let gp .z/ D g.z=p/. Lemma 7.8 implies that gp lies in C.0 .N`02 =p/nH/. The quotient 0 .N`02 /

1 0 0p

n0 .N`02 /

1 0 0p

0 .N`02 =p/

is finite; denote by 1 ; : : : ; d a system of representatives of 0 .N`02 =p/. The double class T D 0 .N`02 /

1 0 0p

0 .N`02 =p/

then operates on the 0 .N`02 /-invariant function g by3 : d d 1 X  1 0   X g 0 p i z D gp .i z/: .gjT/.z/ D p p iD1 iD1

Since gp lies in C.0 .N`02 =p/nH/, we obtain gp .z/ D

p p .gjT/.z/: d

One can then define a function fp by fp .z/ D relatively prime we can replace T by T 0 D 0 .N/

1 0 0p

p

p d . f jT/.z/.

Since p and `0 are

0 .N=p/

in the expression for fp . We deduce that fp lies in C.0 .N=p/nH/. We have, by definition of gp , f .z/  fp . pz/ D f .z/  fp . pz/  g.z/ C gp . pz/ D h.z/ 

p p .hjT/. pz/: d

(7.11)

The factor p1=2 is there to simplify the expression for the action of T on the Fourier expansion, similarly to the appearance of n1=2 in the definition of Tn .

3

226

7 L-Functions and the Selberg Conjecture

Since the r-th Fourier coefficient of h is zero if r is prime to `0 , Lemma 7.10 allows us to write X h.z/ D hq .qz/ .hq 2 C.0 .N`03 /nH//: qj`0

For every prime divisor q of `0 , since p does not divide `0 , we have hjT D hjTq , where this time Tq D 0 .N`03 q/

1 0 0p

0 .N`03 q=p/:

It follows that .hjT/.z/ D

X

.hq jT 00 /.qz/;

qj`0

where T 00 D 0 .N`03 /

1 0 0p

0 .N`03 =p/:

This implies in particular that the r-th Fourier coefficient of hjT is zero if r and `0 are relatively prime. As this is also the case for h, the expression (7.11) implies that f  fp satisfies the hypotheses of the theorem with ` replaced by `0 . The induction hypothesis then implies that f .z/  fp . pz/ D

X

fq .qz/

. fq 2 C.0 .N=q/nH//;

q

where the sum runs over all prime factors q of `0 . This proves Lemma 7.11.

t u

Thanks to the above, we can always normalize a cuspidal newform f so that a1 D 1. In this case, we will say that f is normalized. Theorem 7.12 (Multiplicity 1) The space Cprim .0 .N/nH/ admits a basis of normalized newforms. If f 2 Cprim .0 .N/nH/ and f 0 2 Cprim .0 .N 0 /nH/, with N 0 jN, are two newforms of the same parity having the same eigenvalues for the Laplacian and the Hecke operators Tn for all integers .n; N/ D 1, then N D N 0 and f D f 0 . Proof We begin by noting that if f and f 0 are two normalized newforms in Cprim .0 .N/nH/, of the same parity and lying in the same eigenspaces for the Laplacian and the Hecke operators Tn , with .n; N/ D 1, then Lemma 7.11(2) applied to f  f 0 with ` D N implies that f  f 0 is an oldform and thus zero, so that f D f 0 . Having fixed some Laplacian eigenvalue  there exists a finite number of normalized newforms in Cprim .0 .N/nH/ having  as their joint Laplacian eigenvalue. Moreover, it follows from the preceding paragraph that these forms are pairwise orthogonal. By induction on the level N we can then show that there exists an

7.2 Hecke Operators and Applications

227

orthogonal basis of C.0 .N/nH/ made up of functions of the form z 7! h.Dz/, where h is a newform in Cprim .0 .M/nH/ and DM divides N. The theorem then follows from the following lemma, which is of interest in and of itself. t u Lemma 7.13 Let f 2 Cprim .0 .N/nH/ be a normalized newform and let f 0 2 C.0 .N/nH/, of the same parity as f , be an eigenfunction of the Laplacian and of the Hecke operators Tn , for all .n; N/ D 1, with the same eigenvalues as f . Then f 0 is a multiple of f . Proof Expanding f 0 in an orthogonal basis and renormalizing if necessary, we can assume that f 0 .z/ D h.Dz/ where h is a normalized newform in Cprim .0 .M/nH/, where DM divides N. In particular, the form h lies in C.0 .N/nH/ and has the same eigenvalues as f 0 for the Laplacian and the operators Tn , .n; N/ D 1; it is an oldform if M < N. It then follows from Lemma 7.11 that f h lies in Cold .0 .N/nH/. Since f is a newform, the form h cannot be an oldform; we thus have M D N (and D D 1) and the first part of the proof implies f D h D f 0 . t u The next result we prove is that a newform, remarkably enough, is an eigenfunction for all of the Hecke operators, and not just the Tn for .n; N/ D 1. Proposition 7.14 Let f be a non-zero cuspidal Maaß newform in Cprim .0 .N/nH/. Then f is an eigenfunction for all the Hecke operators. Proof Let m be a positive integer (not necessarily prime to N). Then Tm f is of the same parity as f and is an eigenfunction for the Laplacian and the operators Tn , for all .n; N/ D 1, with the same eigenvalues as f . The proposition then follows from the multiplicity 1 theorem. t u

7.2.3 Multiplicative Properties of Fourier Coefficients It follows from (7.9) and Proposition 7.14 that the Fourier coefficients an of a normalized newform f are the eigenvalues of the Hecke operators4 Tn . Lemma 7.6 and the relation (7.8) can then be used to show am an D

X

amn=d2

if .n; N/ D 1 (7.12)

dj.m;n/

am ap D amp

4

if pjN:

This way of thinking of the Fourier coefficients an allows one to generalize the work of Hecke, by dispensing with Fourier theory and hence the necessity of having a surface with cusps.

228

7 L-Functions and the Selberg Conjecture

We return to the function L.s; f / and, taking our cue from Maaß, think of it as an L-function coming from arithmetic (Riemann zeta function, Artin L-function. . . ). The cusp form f plays the role, for this L-function, of the theta series .x/ D P n2 x2 , for the Riemann zeta function (see Exercise 1.16). When f is a n¤0 e normalized newform we again have at our disposal an Euler product decomposition, as follows. Theorem 7.15 Let f .z/ D

X

p ar y K .2jrjy/e.rx/

r2Z

be a cuspidal (a0 D 0) normalized Maaß newform in Cprim .0 .N/nH/. Then L.s; f / D

Y Y .1  ap ps /1  .1  ap ps C p2s /1 ; p6 jN

pjN

in the domain of absolute convergence. Proof It follows from the multiplicativity (7.12) of the coefficients that L.s; f / D

C1 X nD1

Y an D ns p

 C1 X

a p pr

rs

 :

rD0

It then remains to calculate the series arising from the right-hand side. Equation (7.12) implies, when p 6 jN, that aprC1  ap apr C apr1 D 0 and thus that 2

.1  ap X C X /

 C1 X

 apr X

r

D 1:

rD0

In the same way, we find that .1  ap X/

 C1 X

 apr X r

D 1;

rD0

if pjN. Setting X D ps in the last two equalities we obtain the claimed expression. t u

7.2 Hecke Operators and Applications

229

When f is a cuspidal normalized Maaß newform in Cprim .0 .N/nH/, the functional equation of Proposition 7.5 takes on an even more satisfying form due to the following lemma. Lemma 7.16 Let f be a cuspidal normalized Maaß newform in Cprim .0 .N/nH/. Then fQ is a multiple of f by a complex number of modulus 1. Proof The matrix  WN D

0 1 N 0



normalizes the group 0 .N/; in particular it sends C.0 .N/nH/ into itself. The inverse of the correspondence Hn (see (7.5)) is .Hn /1 D .Nn/1 WN Hn WN : Writing Hn D

S i

0 .N/˛i as the disjoint union (7.6) we thereby obtain .Hn /1 D

[

.Nn/1 0 .N/WN ˛i WN

i

and the adjoint of the Hecke operator Tn is given by the formula .Tn / D WN ı Tn ı WN : But if n is prime to N, Tn is self-adjoint. The matrix WN thus commutes with Tn and Lemma 7.13 implies that fQ D f ı WN is a multiple of f . Since these two Maaß forms have the same L2 norm, the multiplicative coefficient must have modulus 1. t u We shall need an analog of this lemma for a specific prime divisor of N. Let p be a prime dividing N and ˛ the largest integer such that p˛ jN. Fix three integers x, y and z such that p2˛ x  yNz D p˛ . We write Wp D N Wp D

  ˛ p x y : Nz p˛

The matrix Wp normalizes 0 .N/ and therefore defines an operator from C.0 .N/nH/ into itself. Lemma 7.17 Let f be a normalized cuspidal Maaß newform in C.0 .N/nH/. Then f ı Wp D p f where p 2 C is of modulus 1.

230

7 L-Functions and the Selberg Conjecture

Proof This again follows from the fact that the operator Wp commutes with the Hecke operators Tn for .n; N/ D 1. u t

7.3 Dirichlet Characters and Twisted Maaß Forms 7.3.1 Dirichlet Characters Let q be a positive integer. Let 0 be a character of the finite multiplicative group .Z=qZ/ , i.e., a group homomorphism with values into the unit circle. A mod q Dirichlet character is a q-periodic map  W Z ! C obtained from a character 0 as above by extension to Z: ( .a/ D

0

if .a; q/ ¤ 1;

0 .a mod q/

if .a; q/ D 1:

Note in particular that  is multiplicative. If d divides q, the group Z=qZ surjects onto Z=dZ and therefore .Z=qZ/ surjects onto .Z=dZ/ . The Dirichlet character  is said to be primitive mod q if there is no proper divisor d of q such that 0 comes from a character of .Z=dZ/ . A character mod a prime number is primitive as soon as it is non-trivial, in other words when it is not the character5 triv constantly equal to 1 on .Z=qZ/ . To simplify the statements and proofs in the following subsections we shall suppose P that q is prime and consider a non-trivial character  mod q. We write q ./ D `D1 .`/e .`=q/ for the Gauß sum associated with . Lemma 7.18 For all integers n 2 Z, ./  .n/ D

q X

.`/e .`n=q/ :

`D1

Proof We distinguish two cases according to whether n is prime to q or not. If n is prime to q, then j.n/j D 1 and the change of variable ` 7! `n transforms the Gauß sum ./ into ./ D

q X `D1

.`n/e .`n=q/ D .n/

q X

.`/e .`n=q/ ;

`D1

which, multiplied by .n/1 D .n/, is the claimed identity.

5

One should be careful here that the character triv depends on q.

7.3 Dirichlet Characters and Twisted Maaß Forms

231

Suppose now that q divides n. Then .n/ D 0 and it suffices to show that the expression in the lemma vanishes identically. Write n D qn0 . We then have q X

.`/e .`n=q/ D

q X

`D1

.`/:

`D1

This last expression is indeed zero since  is not the trivial character.

t u

Recall that X

( .`/ D



q  1 if `  1 .mod q/ 0

(7.13)

else,

where the sum runs over all characters  mod q. Regarding the Gauß sums ./ and the Jacobi sums J.1 ; 2 / D

q X

1 .`/2 .1  `/;

`D1

we shall need the following classical lemma. Lemma 7.19 We have 1. .triv / D 1, J.; / D .1/; p 2. ././ D .1/q, j./j D q; 3. J.1 ; 2 / D .1 /.2 /=.1 2 /, if 2 ¤ 1 . Proof One verifies immediately the first point. We then calculate .1 /.2 / D

q q X X

1 .`/2 .k/e ..k C `/=q/

`D1 kD1

D

q X

e .n=q/

nD1

D

X

q X

1 .`/2 .n  `/

`D1

e .n=q/ 1 .n/2 .n/

n2.Z=qZ/

q X

1 .n1 `/2 .1  n1 `/

`D1

C D .1 2 /J.1 ; 2 / C 1 .1/

q X `D1

q X `D1

.1 2 /.`/:

1 .`/2 .`/

232

7 L-Functions and the Selberg Conjecture

The second sum in the last expression is zero unless 1 2 D triv , in other words when  D 1 D 2 . In this last case, ././ D .triv /J.; / C .1/.q  1/ D .1/q: Finally, since ./ D .1/./, the lemma is completely proved.

t u

In the following section we are going to form the product – referred to as the Rankin-Selberg product – of a Maaß form f by the same form f twisted by a Dirichlet character. We must begin by extending the theory of Maaß forms – including the Eisenstein series – to the case of forms twisted by a character.

7.3.2 Maaß Forms Twisted by a Character Let f .z/ D

X

p ar y K .2jrjy/e.rx/

r2Z

be a Maaß form in C.0 .N/nH/. Let  be a primitive Dirichlet character mod a prime q, where q − N. We define the twist of f by the character  by the following sum: X p f .z/ D ar .r/ y K .2jrjy/e.rx/: r2Z

Lemma 7.20 For all  2 0 .Nq2 /, we have f . z/ D . /2 f .z/; where, by abuse of notation, we write  

 ab D .d/: cd

Proof According to Lemma 7.19, the Gauß sum ./ is non-zero. It then follows directly from Lemma 7.18 that   1 X .`/  f z C `=q : ./ q

f .z/ D

`D1

(7.14)

7.3 Dirichlet Characters and Twisted Maaß Forms

233

If  D .  d / 2 0 .Nq2 / then     2 1 `=q 1 d `=q  2 0 .N/  : 0 1 0 1 In other words, for all z 2 H the points .z/ C `=q and z C d2 `=q are congruent modulo 0 .N/. We deduce from this that f . z/ D ./1

q X

  .`/  f z C d 2 `=q

`D1

D ./1

q X

  .`d2 /.d/2  f z C d2 `=q

`D1

D .d/2 f .z/:

t u

Recall that fQ D f ı where .z/ D 1=Nz. Lemma 7.21 We have   ./ Q f 1=Nq2 z D .N/ f .z/: ./ Proof As in the proof of Lemma 7.20 we have ./f



0 1=Nq q 0

X

q   X      0 1=Nq q ` z D z .`/f 0q q 0 `D1

q

D

.`/fQ



`D1

X

D

.`/fQ

0 1 N 0



  q `   0 1=Nq   z 0 q q 0

q r   q r   0q z N` s

.`;q/D1



where r D r.`/ and s D s.`/ are such that qs  rN` D 1 and thus .`/ D .N/.r/: X  q r   q r   D .N/ .r/fQ N` s 0q z



.r;q/D1

D .N/

X

.r/fQ

 q r   Q 0 q z (since f 2 C.0 .N/nH//

.r;q/D1

D .N/./fQ :

t u

234

7 L-Functions and the Selberg Conjecture

By averaging (7.14) with the help of (7.13) – and by keeping track of the trivial character triv – we obtain the following useful lemma. Lemma 7.22 Let q be a prime and let u be an integer not divisible by q. Then f .z C u=q/ D

o 1 nP  .u/./f .z/ C .qfq .z/  f .z// ; q1

where the sum runs over all the primitive characters mod q and fq .z/ D

X

p arq y K .2jrjqy/e.rqx/:

r2Z

7.3.3 Twisted Eisenstein Series Assume from now on that  is even, i.e., .1/ D 1 and that q is prime to N. The character  extends to an imprimitive character mod Nq2 that we shall denote 0 ; it is given by ( 0

 .a/ D

.a/

if .a; N/ D 1;

0

else:

Let z D x C iy 2 H. Consider the twisted Eisenstein series E.z; s; / D

1 2

1 D 2

X

.d/

.c;d/D1; Nq2 jc

X

ys jcz C dj2s (7.15)

. /.Im. z// : s

 21 n0 .Nq2 /

One passes from the first equality to the second by associating with each pair of relatively prime integers .c; d/ the set6 of matrices in 0 .Nq2 / whose bottom row is .c; d/. Each such matrix represents a unique class in 1 n0 .Nq2 /. The proof of Lemma 4.1 implies the following lemma. Lemma 7.23 The series (7.15) is absolutely convergent on Re.s/ > 1 and satisfies E..z/; s; / D . /E.z; s; /;

6

for all  2 0 .Nq2 /:

The coefficient 1=2 again comes from the fact that the element I 2 SL.2; Z/ acts trivially on H.

7.3 Dirichlet Characters and Twisted Maaß Forms

235

The fundamental property of these Eisenstein series is again the existence of an analytic continuation and a functional equation. The associated completed Eisenstein series are  s  .s/L.2s; 0 /E.z; s; / 1 X 0 ys D  s  .s/  .n/ 2 2 m;n2Z jmNq z C nj2s D  s  .s/

X 1X ys .d/.d/ .n/ 2 jmNq2 z C ndj2s m;n2Z djN

X 1X .y=d/s D   .s/ .d/.d/ds .n/ 2 jmNq2 z=d C nj2s m;n2Z

(7.16)

s

D

X

djN

.d/.d/ds E .z=d; s; /;

djN

where is the Möbius function ( .n/ equals 0 if n has a square factor and equals .1/r if n is the product of r distinct primes), E .z; s; / D  s  .s/

1 X ys .n/ 2 2 m;n2Z jmNq z C nj2s

(7.17)

and L.s; 0 / is the ordinary Dirichlet series L.s; 0 / D

X 0 .n/ n>1

ns

D

Y pjN

1

. p/  L.s; /: ps

Theorem 7.24 The function E .z; s; /, defined by (7.17) for Re.s/ > 1, holomorphically continues to all s 2 C. It moreover satisfies the functional equation E .z; s; / D ./N 12s q25s E .1=N 2 q3 z; 1  s; /;

(7.18)

remains bounded on vertical strips 1 6 Re.s/ 6 2 ( 1 ; 2 2 R) and satisfies E.x C iy; s/ D O.y / where D Re.s/.

as y ! 1;

(7.19)

236

7 L-Functions and the Selberg Conjecture

Proof The proof is essentially the same as that of Theorem 4.2, only here we consider the twisted theta function X 2  .t; z/ D .n/ejmzCnj t=y ; .m;n/2Z2

where t > 0 and z 2 H. Lemma 7.25 The theta function  .t; z/ satisfies the functional equation  .t; z/ D

./  .1=qt; q=z/: qt

Proof To prove the lemma we will need to make use of a twisted Poisson summation formula. Let us show that for any rapidly decreasing function f W R2 ! C we have X

.n/f .m; n/ D

.m;n/2Z2

./ q

X

.n/fO.m; n=q/:

(7.20)

.m;n/2Z2

To prove (7.20) note that according to Lemma 7.18, ./ X .`/e .n`=q/ : q q

.n/ D

`D1

The left-hand side of (7.20) is therefore equal to the sum

P

.m;n/2Z2 f1 .m; n/,

where

./ X .`/e .u2 `=q/ f .u1 ; u2 / q q

f1 .u1 ; u2 / D

`D1

is a rapidly decreasing function to which we can apply the usual Poisson summation formula. One finds that q ./ X .`/fO .1 ; 2 C `=q/ : fO1 .1 ; 2 / D q `D1

The left-hand side of (7.20) is therefore equal to q ./ X X .`/fO .m; n C `=q/ : q 2 `D1 .m;n/2Z

Since .`/ D .qnC`/, using the fact that qnC` is a parametrization of the integers as ` runs through the set f1; : : : ; qg, we conclude that this last sum is indeed equal to the right-hand side of (7.20).

7.3 Dirichlet Characters and Twisted Maaß Forms

237

Recall now that if 2 t=y

fz;t .u1 ; u2 / D eju1 zCu2 j then

1 Œ.y2 /2 C.1 x2 /2 =ty fb : z;t .1 ; 2 / D t e

The twisted Poisson summation formula then implies that  .t; z/ D D

D

D

./ qt ./ qt ./ qt

X

.n/eŒ.ny/

2 C.qmxn/2 =tq2 y

.m;n/2Z2

X

2 =tq2 y

.n/ejnzqmj

.m;n/2Z2

X

.n/ejnqmz

1 j2 jzj2 =tq2 y

.m;n/2Z2

./  .1=qt; q=z/; qt

where we have used the fact that the imaginary part of q=z is qy=jzj2 .

t u

This time we get 1 E .z; s; / D 2 

Z Z

C1 0

 .Nq2 t; Nq2 z/ts

1=Nq2

D

1 2

D

E0 .z; s; /

0

dt t

 .Nq2 t; Nq2 z/ts C

1 dt C t 2

Z

C1 1=Nq2

 .Nq2 t; Nq2 z/ts

dt t

E1 .z; s; /:

Since for every z 2 H the function t 7!  .Nq2 t; Nq2 z/ is of rapid decay, the function E1 .z; s/ is holomorphic on the entire complex s-plane and bounded on vertical strips. Lemma 7.25 and the change of variables t 7! N 2 q4 t1 implies moreover that if Re.s/ > 0, E1 .z; s; / D

./ 2Nq3

Z

C1

1=Nq2

./N 12s D 2q4s1

 .1=Nq3 t; 1=Nqz/ts1

Z

1=Nq2 0

dt t

 .Nqt; 1=Nqz/t1s

dt : t

238

7 L-Functions and the Selberg Conjecture

Finally, changing variables t 7! t=q gives E1 .z; s; / D

./N 12s 2q5s2

Z

1=Nq3

 .Nq2 t; Nq2 .1=N 2 q3 z//t1s

0

dt : t

(7.21)

This last expression implies that the function s 7!

E0 .1=N 2 q3 z; s; /

Z

1 D 2

1=Nq2

 .Nq2 t; Nq2 .1=N 2 q3 z//ts

0

dt t

holomorphically continues7 to all s 2 C. Replacing  by  and z by 1=N 2 q3 z we find that the function E0 .z; s; /, and thus E .z; s; /, admits a holomorphic continuation to all s 2 C, and that 1 E .z; s; / D 2 

Z

C1 0

 .Nq2 t; Nq2 z/ts

D ./N 12s q25s D ./N

1 2

Z

C1

0

dt t

 .Nq2 t; Nq2 .1=N 2 q3 z//t1s

dt t

12s 25s 

q

E .1=N 2 q3 z; 1  s; /:

We deduce from this the stated functional equation and the fact that E .z; s; / is bounded in vertical strips. The asymptotic (7.19) can be proved by calculating the Fourier expansion of E .z; s; / associated with the cusp at infinity. Using the expression E .z; s; / D  s  .s/L.2s; /ys C

C1 XX

.n/

mD1 n2Z s

D   .s/L.2s; /y  C

y 2 N q4

ys jmNq2 z C nj2s

s

s C1 X mD1

mNq2

m

2s

X rD1

.r/

X ˇˇ ˇd C z C d2Z

r ˇˇ2s ˇ ; mNq2

The fact that one integrates only from 0 to 1=Nq3 in (7.21) is not a problem: the integral is defined for all s 2 C away from 0.

7

7.4 Rankin-Selberg L-Functions

239

the computation is similar to the one carried out for the classical Eisenstein series in § 4.1. This time we use X ˇˇ ˇd C z C d2Z

r ˇˇ2s  .2s  1/ D 2 j2yj12s ˇ mNq2  .s/2 C

  2 s X ˇˇ y ˇˇ1=2s Ks1=2 .2jnjy/ e n.x C r=mNq2 / : ˇ ˇ  .s/ n n¤0

The Fourier expansion of E .z; s; / has constant term  s  .s/L.2s; /ys and what remains is a sum of terms involving Bessel functions which decay rapidly in y; see Appendix B. The bound (7.19) follows immediately from these considerations. t u We note that Theorem 7.24 remains true when  D triv as long as we substitute 1 for ./.

7.4 Rankin-Selberg L-Functions We fix two normalized cuspidal (a0 D b0 D 0) Maaß newforms f .z/ D

X

p ar y K .2jrjy/e.rx/

and g.z/ D

r2Z

X

p br y K 0 .2jrjy/e.rx/

r2Z

of the same parity " in Cprim .0 .N/nH/. In particular, f (resp. g) is an eigenfunction for the Laplacian with eigenvalue 1=4   2 (resp. 1=4   02 ). We fix furthermore an even Dirichlet character , i.e., .1/ D 1, which we require to be primitive to a prime modulus q which does not divide N; to simplify further we shall assume that N is square-free.8 We now consider the Rankin-Selberg L-function L.s; f  g/ defined by L.s; f  g/ D L.2s; 02 /

C1 X nD1

.n/

an bn : ns

(7.22)

The hypotheses that f and g be of the same parity, that  should be even, and that q should be prime are not necessary. They only serve to simplify the presentation.

8

240

7 L-Functions and the Selberg Conjecture

It follows from Lemma 7.2 that this series converges absolutely in the half-plane Re.s/ > 2. Rankin and Selberg succeeded in showing that this series admits a meromorphic continuation to the complex s-plane and satisfies a certain functional equation. The proof is based on the study of the twisted Eisenstein series treated in the previous section. Recall that one says a function  W H ! C is of rapid decay at infinity if .x C iy/ D O.yA / for all A > 0 as y ! C1: Since Bessel functions decay rapidly (see Appendix B), it follows from its Fourier expansion that a Maaß cusp form is of rapid decay at infinity; the same is therefore true of the product function z 7! f .z/g.z/ and it follows from Theorem 7.24 that the integral Z E.z; s; 2 /f .z/g.z/ d .z/ 0 .Nq2 /nH

converges absolutely on Re.s/ > 1. It could happen that 2 D triv but, as we remarked at the end of the previous section, this is not a problem. The Rankin-Selberg method is based on the following proposition. Proposition 7.26 For Re.s/ > 1 we have Z 0

.Nq2 /nH

E.z; s; 2 /f .z/g.z/ d .z/ D .1/"

 C1 Y  X  s an bn s ˙  ˙ 0  .s/1 .n/ s :  4 2 n nD1

(On the right-hand side, the product has four factors, corresponding to all possible combinations of signs.) Proof Since Re.s/ > 1, we may unfold the integral to obtain Z 0

.Nq2 /nH

E.z; s; 2 /f .z/g.z/ d .z/ D D

1 2 1 2 Z

 21 n0 .Nq2 /

X  21 n0 .Nq2 / C1

Z

1

D 0

Z

X

0

0 .Nq2 /nH

2 . /.Im. z//s f .z/g.z/ d .z/

Z 0 .Nq2 /nH

.Im. z//s f . z/g. z/ d .z/

ys f .x C iy/g.x C iy/y1 dx

dy : y

(One goes from the second to the third equality by using Lemma 7.20.)

(7.23)

7.4 Rankin-Selberg L-Functions

241

Replacing f and g by their Fourier expansions, we get Z

1 0

f .x C iy/g.x C iy/ dx Dy

X Z n;m2Z

Dy

X

1 0

.n/an bm K .2jnjy/K 0 .2jmjy/e..n C m/x/ dx

.n/an bn K .2jnjy/K 0 .2jnjy/:

n2Z

From this it is clear that C1 Z 1

Z 0

0

ys f .x C iy/g.x C iy/y1 dx D .1/"

X

Z .n/an bn

n2Z

dy y

C1 0

K .2jnjy/K 0 .2jnjy/ys

dy : y

(7.24)

Similarly to the proof of Lemma 4.27, one calculates the Mellin transform of the product of two Bessel functions in the next lemma. Lemma 7.27 The integral Z

C1

0

K .y/K 0 .y/ys

Y  s ˙  ˙ 0  dy D 2s3  .s/1  y 2

(7.25)

is absolutely convergent on Re.s/ > j Re./j C j Re. 0 /j. Proof The Mellin inversion formula, since it is nothing other than a rewriting of Fourier inversion, transforms the product of two functions into the convolution product of their Mellin inverses. It then follows from Lemma 4.27 that the left-hand side of (7.25) equals 1 2i

Z

Ci1

i1

u C   u     2 2 s  u C  s  u    du  2su2  2 2 Z 2s3 =2Ci1     D  u  uC 2i =2i1 2 2 s C  s    u   u du: 2 2

2u2 

242

7 L-Functions and the Selberg Conjecture

The stated formula is then a consequence of the Barnes lemma – see [138, §14.5] – according to which Z . /

 .s  ˛/ .s  ˇ/ .s C  / .s C ı/ds D

 .˛ C  / .˛ C ı/ .ˇ C  / .ˇ C ı/  .˛ C ˇ C  C ı/

where the line Re.s/ D is to the right of the poles of  .s  ˛/ and  .s  ˇ/ and to the left of the poles of  .s C  / and  .s C ı/. t u Here j Re./j and j Re. 0 /j 6 1=2 and one finds that for Re.s/ > 1, Z

C1

dy y Y  s ˙  ˙ 0  1 : D .jnj/s  .s/1  8 2

K .2jnjy/K 0 .2jnjy/ys

0

(7.26)

The identities (7.23), (7.24) and (7.26) finally imply9 : Z 0 .Nq2 /nH

E.z; s; 2 /f .z/g.z/ d .z/  Y  1 an bn s ˙  ˙ 0  X D .1/"  s  .s/1 .n/ s  8 2 jnj n2Z D .1/

"

s

4

 .s/

1

Y



 s ˙  ˙  0  C1 X 2

nD1

.n/

an bn : ns

t u

Let D0 .Nq2 / be a fundamental domain for 0 .Nq2 / inside the strip jxj 6 1=2. According to Proposition 7.26, for Re.s/ > 1,  2s Y  s ˙  ˙  0  L.s; f  g/  4 2 Z X D .1/" .d/2 .d/ds djN

9

D0

.Nq2 /

E .z=d; s; 2 /f .z/g.z/ d .z/:

One uses here the fact that f and g are of the same parity.

(7.27)

7.4 Rankin-Selberg L-Functions

243

For a prime divisor p of N, note that10 p does not divide Nq2 =p. We fix integers u, v and w such that p2 u  Nq2 vw D p and write 

 px y Wp D : Nq2 z p

(7.28)

It follows from Lemma 7.17 that f ı Wp D p f

and g ı Wp D p g;

where p and p are two complex numbers of modulus 1. Theorem 7.28 Let N be a square-free positive integer. The function .s; f  g/ D

 q 2s Y 



 s ˙  ˙  0  2 Y  s 1  L.s; f  g/; .1  2 . p/p1 1 p / p pjN

defined for sufficiently large Re.s/, admits a meromorphic continuation to the entire plane of s 2 C; it is everywhere holomorphic (since  is primitive) and satisfies a functional equation 

./ .s; f  g/ D w. f ; g/ .N/ p q 2

4

N 12s .1  s; fQ  gQ /;

where w. f ; g/ is a complex number of modulus 1, independently of . Proof The next two lemmas allow one to rewrite .s; f  g/ in a more convenient form. Lemma 7.29 We have f ı Wp D . p/p f : Proof According to Lemma 7.18 we have 1 X .`/  f .z C `=q/ : ./ q

f .z/ D

`D1

10

Here we use the fact that N is square-free!

244

7 L-Functions and the Selberg Conjecture

A simple matrix calculation moreover implies that     1 `p=q 1 `=q Wp 2 0 .N/Wp 0 1 0 1 for ` D 1; : : : ; q. We see then that   1 X .`/  f Wp .z/ C `=q ./ q

f ı Wp .z/ D

`D1

1 X .`p/  . f ı Wp / .z C `p=q/ D . p/ ./ q

`D1

1 X .`/  f .z C `=q/ ./ q

D . p/p

`D1

D . p/p f :

t u

Lemma 7.30 Let d be a divisor of N and p a prime dividing N but not d. Then Z D0 .Nq2 /

E .z=dp; s; 2 /f .z/g.z/ d .z/ D . p/p1 1 p

Z D0

.Nq2 /

E .z=d; s; 2 /f .z/g.z/ d .z/:

Proof The matrix Wp normalizes the group 0 .Nq2 / and a simple (but laborious) computation using the definition (7.28) allows us to verify E .z=dp; s; 2 / D E .z=d; s; 2 / ı Wp : The change of variables z 7! Wp .z/ thus implies the lemma. It follows from Lemma 7.30 that for any divisor d of N, we have Z D0 .Nq2 /

E .z=d; s; 2 /f .z/g.z/ d .z/ D .N=d/

p p

pj.N=d/

Z 

Y

D0 .Nq2 /

E .z=N; s; 2 /f .z/g.z/ d .z/:

t u

7.4 Rankin-Selberg L-Functions

245

On the other hand, we have X X Y Y s .d/.d/2 ds p1 1 .d/ 2 . p/p1 1 p D p p djN

pjd

djN

pjd

Y s D .1  2 . p/p1 1 p p /: pjN

Formula (7.27) can be rewritten in the form .s; f  g/ D .1/" 4q2s .N/

Y

 p p

pjN

Z 

D0

.Nq2 /

E .z=N; s; 2 /f .z/g.z/ d .z/:

(7.29)

The first part of Theorem 7.28 (meromorphic continuation and holomorphy) is then a consequence of Theorem 7.24. To prove the functional equation of Theorem 7.28 we now hit (7.29) with the functional equation of the Eisenstein series E given by Theorem 7.24. Carrying this out in detail will occupy the rest of the proof. Let I denote the integral on the right-hand side of (7.29). The function equation of E implies I D .2 /N 12s q25s 2

D . /N

12s 25s

q

Z D0 .Nq2 /

E .1=Nq3 z; 1  s; 2 /f .z/g.z/ d .z/

(7.30)

J

with Z JD

D0 .Nq2 /

E .z=q; 1  s; 2 /f .1=Nq2 z/g.1=Nq2z/ d .z/:

(7.31)

Here we have made the change of variables z 7! 1=Nq2 z while using the fact that  p  0 1=.q N/ p normalizes 0 .Nq2 /. 0 q N The denominator q, in the Eisenstein series of (7.31), is a bit annoying; the following lemma allows us to get rid of it. Lemma 7.31 We have s 

2

q E .z=q; s;  / D

X u mod q

u 1 0 E .z; s;  / ı : Nq 1 

2



246

7 L-Functions and the Selberg Conjecture

Proof Since Im



 1 0 u Nq 1

 z D

Im.z/ ; jNquz C 1j2

we have 

1 0 E .z; s;  /ı Nq 1 

2

u

D  s  .s/

1 X 2 ys  .n/ 2 2 m;n2Z jmNq z C nNquz C nj2s

D  s  .s/

1 X 2 ys  .n/ : 2 m;n2Z j.mq C nu/Nq2 .z=q/ C nj2s t u

The lemma can be deduced directly from a summation on u modulo q. It follows from Lemma 7.31 that q1s J D

X Z 2 u mod q D0 .Nq /

E .z; 1  s; 2 / ı



1 0 Nq 1

u

 f .1=Nq2 z/g.1=Nq2 z/ d .z/:

(7.32)

Now 0 .Nq2 / contains 1 .Nq2 ; q/ D

  ˇ ab ˇ 2 0 .Nq2 / ˇ a  d  1 .mod Nq2 / and b  0 .mod q/ cd

as a subgroup of finite index equal to11 q.Nq2 /. Let D1 .Nq2 ; q/ denote a fundamental domain for 1 .Nq2 ; q/. We then have u  X Z qs1 1 0 2  E .z; 1  s;  / ı JD Nq 1 q.Nq2 / u mod q D1 .Nq2 ;q/  f .1=Nq2 z/g.1=Nq2 z/ d .z/:

11

Here  denotes the Euler totient function.

(7.33)

7.4 Rankin-Selberg L-Functions

247

Lemma 7.32 The integral u 1 0 E .z; 1  s;  / ı f .1=Nq2 z/g.1=Nq2 z/ d .z/ 2 Nq 1 D1 .Nq ;q/

Z



2



(7.34)

is equal to Z

    u u 1 1 g  2 C d .z/: E .z; 1  s; 2 /f  2 C Nq z q Nq z q D1 .Nq2 ;q/

Proof The element 

1 0 Nq 1

u (7.35)

normalizes 1 .Nq2 ; q/. Now the integral (7.34) runs over a fundamental domain for this group. The change of variable induced by the matrix (7.35) then transforms the integral (7.34) into Z D1 .Nq2 ;q/

E .z; 1  s; 2 /f



0 1 Nq2 0



1 0 Nuq 1

  z

g



0 1 Nq2 0



1 0 Nuq 1

  z d .z/:

0 1 Nq2 0

  z d .z/;

Finally, this last integral equals Z D1 .Nq2 ;q/

E .z; 1  s; 2 /f

  1 u=q 0 1

0 1 Nq2 0

  z g

  1 u=q 0 1

which is precisely the stated expression.

t u

Lemma 7.33 We have # " X q f .z C u=q/g.z C u=q/ D f .z/g .z/ ; q1  u mod q X

where the sum runs over all characters  modulo q, with the convention that if  is the trivial character, f D f and g D g  gq (see Lemma 7.22).

248

7 L-Functions and the Selberg Conjecture

Proof Lemma 7.22 implies that for u D 1; : : : ; q  1, f .zCu=q/g.z C u=q/ nX 1 1 .u/2 .u/. 1 /. 2 /f1 .z/g2 .z/ D .q  1/2  ;  1 2 X

.u/./ f .z/.qgq .z/  g.z//  f .z/g .z/ C 

o  f .z/.qgq .z/  g.z// ;

where the sums run over all primitive characters modulo q. Summing over u, we find (using Lemmas 7.18 and 7.19): q1 X

f .z C u=q/g.z C u=q/ D

uD1

X  q f .z/g .z/  f .z/g.z/; q1 

where the sum runs over all characters  mod q, with the convention described in the lemma. Since f and g are 1-periodic, we deduce the lemma. t u Starting from the expression (7.33), Lemmas 7.32 and 7.33 imply JD

qs X q1 

Z

 1   1  f  2 g  2 Nq z Nq z D0 .Nq2 /  E .z; 1  s; 2 / d .z/;

(7.36)

the sum again running over all characters  modulo q. Lemma 7.34 The integral Z

 1   1  f  2 g  2 E .z; 1  s; 2 / d .z/ Nq z Nq z D0 .Nq2 /

(7.37)

equals .N/

.././/2 q2

Z D0 .Nq2 /

fQ .z/Qg.z/E .z; 1  s; 2 / d .z/:

Proof Since the characters  and  are both primitive, Lemma 7.21 implies that the integral (7.37) is equal to .N/

././ ././

Z D0 .Nq2 /

fQ .z/Qg .z/E .z; 1  s; 2 / d .z/:

7.4 Rankin-Selberg L-Functions

249

Next, Lemma 7.19 implies .././/2 ././ D : ././ q2 The proof of Proposition 7.26 finally implies Z D0

.Nq2 /

fQ .z/Qg .z/E .z; 1  s; 2 / d .z/ Z D

D0 .Nq2 /

fQ .z/Qg.z/E .z; 1  s; 2 / d .z/;

whence the lemma. It remains to treat the cases where  is trivial or equal to . Suppose for example that  D ; then we have 1 X f .z/ D f .z/  fq .z/ D triv .`/f .z C `=q/ : .triv / q

`D1

The proof of Lemma 7.21 implies  1  . f  qfq /  2 D .fQ  qfQq /.z/: Nq z

(7.38)

Since  1  Q 2 f  D f .q z/; Nqz2 Eq. (7.38) can be rewritten as  1  1 1 fq  2 D fQq .z/  fQ .z/ C fQ .q2 z/: Nq z q q We conclude in the same way as before. Lemma 7.35 We have   q  1 ./ 4 1 X 2 ../.// D .2 /: p q2  q q

(7.39) t u

250

7 L-Functions and the Selberg Conjecture

Proof From Lemma 7.19 and the hypothesis that  is even we have 1 X 1 X .././/2 D 2 2 q  q 

"

 ! X  !#2 q1 q1 X k ` ./.k/e .`/e q q kD1 `D1

!2  q1 q1 X 1 X X k.1 C `k1 / 1 D 2 .k/ .`k /e q  kD1 q `D1

X 1 X .k / .k2 / 1 q2 k D1 k D1 q

q

D

1



X q



e

`1 ;`2 D1

2

 Œk1 .1  `1 / C k2 .1  `2 / X .`1 `2 /; q 

where the last sum is zero if `1 `2  1 .mod q/ and equals q  1 otherwise. Thus q q X  .1  `/Œk1  k2 `1   X q1 X D .k1 / .k2 / e ././ q2 k D1 q k D1

X ././ 

1

.`;q/D1

2

  q1 X X .1  `/k2 `1 D .k /e  2 q2 k D1 q q

.`;q/D1

2

1  .1  `/k 1 A .k1 /e @ q k1 D1 „ ƒ‚ … 0



q X

D.1`/ ./

  q X X q1 .1  `/k`1 D ./ .1  `/ .k/e  q2 q kD1 .`;q/D1 „ ƒ‚ …

D..1`/`1 / ./D.1`/.`/ ./

D

q1 ./2 J.; 2 /: q2 t u

We end the proof by appealing to Lemma 7.19. Starting from the expression (7.36) we can use Lemmas 7.34 and 7.35 to show 

JDq

s1

./ .N/ p q

4

.2 / Z  D0

.Nq2 /

fQ .z/Qg.z/E .z; 1  s; 2 / d .z/:

(7.40)

7.4 Rankin-Selberg L-Functions

251

An inductive argument using Lemma 7.30 again implies, as for (7.29), that Z D0

.Nq2 /

fQ .z/Qg.z/E .z; 1  s; 2 / d .z/ D .N/

Y

Z  p p

pjN

D0

.Nq2 /

fQ .z/Qg.z/E .z=N; 1  s; 2 / d .z/:

Finally, the identities (7.29), (7.30) and (7.40) together imply that there is a complex number w. f ; g/ – which we could write down explicitly if needed – of modulus 1 such that   ./ 4 12s 2 .s; f  g/ D w. f ; g/ .N/ p N .1  s; fQ  gQ /: t u q In addition to the above properties, Selberg noticed that L.s; f  g/ admits an Euler product decomposition. To see this, we begin by factoring the Hecke polynomials (for p not dividing N): 1  ap X C X 2 a D .1  ˛1 . p/X/.1  ˛2 . p/X/; and 1  bp X C X 2 D .1  ˇ1 . p/X/.1  ˇ2 . p/X/: Here ap and bp are the p Fourier coefficients of f and g and ˛i . p/; ˛i . p/ (i D 1; 2) lie in C. Theorem 7.36 We have L.s; f  g/ D

2 Y YY .1  . p/ap bp ps /1 .1  . p/˛i . p/ˇj . p/ps/1 : p−N i;jD1

pjN

Proof First let p be a prime divisor of N. Then X

. pk /apk bpk X k D

k>0

X .. p/ap bp /k X k k>0

D

1 : 1  . p/ap bp X

Now let p be a prime such that p − N. It follows from the proof of Proposition 7.15 that for k > 0, apk D

˛1kC1  ˛2kC1 ˛1  ˛2

and bpk D

ˇ1kC1  ˇ2kC1 : ˇ1  ˇ2

252

7 L-Functions and the Selberg Conjecture

Then X k>0

. pk /apk bpk X k D

2 X ˛i ˇj 1 .1/iCj .˛1  ˛2 /.ˇ1  ˇ2 / i;jD1 1  . p/˛i ˇj X

D Q2

1  2 . p/X 2

i;jD1 .1

 . p/˛i ˇj X/

:

From the two preceding cases one immediately deduces that L.s; f  g/ indeed admits the stated Euler product decomposition, valid for Re.s/ sufficiently large. u t Remark 7.37 Given a cuspidal Maaß newform f 2 C.0 .N/nH/; the generalized Riemann hypothesis applied to L.s; ff / would imply L.s; f f / ¤ 0 for all s in the half-plane Re.s/ > 1=2. But if   f D 1=4   2 f the function .s; f  f / has at most simple poles at s D 0 and s D 1. Now 2 j Re./j is a pole of  ..s  2 j Re./j/=2/ and all other local factors in .s; f  f / are nonzero for Re.s/ > 1=2. We must have 2j Re./j 6 1=2. It follows that 1 .0 .N/nH/ must be greater than or equal to 3=16. In view of the above remark, the “right method” (proposed by Langlands) to prove the Selberg conjecture would be to use the generalized Rankin-Selberg Lfunction L.s; f  f      f  f /: ƒ‚ … „ n times

We can in fact define such a function by its Euler product. We can then naturally complete this Euler product at infinity with powers of  which are polynomial in s and with explicit  -functions, in which  ..s  2nj Re./j/=2/ would appear. This completed L-function should again satisfy a functional equation. The first pole, starting from the right, of these  -factors would then be at 2nj Re./j and (without having to use the Riemann hypothesis) we could conclude that j Re./j 6 1=2n (since the L-function is non-vanishing to the right of the line Re.s/ D 1). Letting n go to infinity, we would then obtain the Selberg conjecture. Now we do not in fact know how to extend the method of Rankin-Selberg to an arbitrary product of Maaß forms, nor can we prove the Riemann hypothesis for Rankin-Selberg L-functions. What we can do is consider a family of RankinSelberg L-functions each having the same  -factor. This is the reason we have been interested in the behavior of L-functions under twisting by Dirichlet characters. In

7.5 The Luo-Rudnick-Sarnak Theorem

253

the following section we shall show that a suitably strong approximation to the Riemann hypothesis is true “on average” over Rankin-Selberg L-functions twisted by Dirichlet characters. This result – due to Luo, Rudnick and Sarnak – suffices to show Theorem 7.1.

7.5 The Luo-Rudnick-Sarnak Theorem We again fix a normalized cuspidal (a0 D 0) Maaß newform f .z/ D

X

p ar y K .2jrjy/e.rx/;

r2Z

in Cprim .0 .N/nH/, where N is squarefree. We again denote the roots of the Hecke polynomial 1  ap X C X 2 by ˛1 . p/ and ˛2 . p/. Let  be an even primitive Dirichlet character of prime conductor q not dividing N. We write  s  2i Im./   s C 2i Im./    2 2   s  2 Re./   s C 2 Re./  Y   .1  2 . p/jp j2 ps /1 ; 2 2

L1 .s; f  f / D

 q 2s



pjN

where p is the complex number of modulus 1 defined by f ı Wp D p f . The product of L.s; f  f / by L1 .s; f  f / is equal to the completed L-function .s; f  f / which enjoys the properties stated in Theorem 7.28. We deduce that L.s; f  f / is holomorphic on the entire complex plane since L1 never vanishes. The following theorem, due to Luo, Rudnick and Sarnak, is the key step towards a uniform lower bound on 1 .0 .N// via L-functions. Theorem 7.38 Fix a real number ˇ > 2=3. There exists an infinite set of even primitive Dirichlet characters of prime modulus not dividing N such that L.ˇ; f  f / ¤ 0: Proof According to Theorem 7.36, the function L.s; f  f / can be written as an Euler product and it follows from Lemma 7.2 that the factors .1  . p/ap bp ps /1 and .1  . p/˛i . p/˛j . p/ps /1 are non-zero for s D ˇ > 1; the same is thus true for L.s; f  f /. We can assume then that 2=3 < ˇ < 1 and Theorem 7.38 is an immediate consequence of the following proposition. t u

254

7 L-Functions and the Selberg Conjecture

Proposition 7.39 Let ˇ be a real number lying strictly between 2=3 and 1. For every " > 0, there exists a constant c D c. f ; ˇ/ > 0 such that j

X

L.ˇ; f  f /j > cq1" ;

.q/

as q ! C1. Here the sum runs over all primitive even Dirichlet characters  of prime modulus q. Proof Fix a primitive even Dirichlet character  of prime conductor q not dividing N. The only tool that we have at our disposal is the functional equation of L.s; f  f / that we now rewrite in the form L.s; f  f / D w. f ; f /2 .N/



./ p q

4   q 24s 12s N G1 .s/L.1  s; f  f /; 

(7.41)

where G1 .s/ D





1s2i Im./ 2







 

s2i Im./ 2

 

1sC2i Im./ 2



sC2i Im./ 2

 

 

 



1s2 Re./ 2 s2 Re./ 2







 



1sC2 Re./ 2

sC2 Re./ 2





Y .1  2 . p/jp j2 p.1s/ /1 pjN

.1  2 . p/jp j2 ps /1

:

The proof of the proposition then consists in deriving from (7.41) a manageable expression of L.ˇ; f  f /. This is not immediate since ˇ lies outside of the region of absolute convergence. Nevertheless, there exists a standard method in analytic number theory which addresses this problem: the “approximate functional equation”. This method is based on the knowledge of the functional equation and shifting integration contours. In this case we obtain the following lemma. Put ˇ0 D 2j Re./j:

(7.42)

Lemma 7.40 There exist two real-valued C1 -functions V1 , V2 satisfying V1 .y/; V2 .y/ D OA .yA /

as y ! C1;

V1 .y/ D 1 C OA .yA /; V2 .y/ D O" .1 C y1ˇ0 ˇ" /;

as y ! 0;

(7.43)

7.5 The Luo-Rudnick-Sarnak Theorem

255

for all A > 0 and " > 0 and such that for all parameters Y > 1 we have

L.ˇ; f  f / D

C1 X nD1



n jan j2 .n/V 1 nˇ Y

    C1 w. f ; f /2 .N/ X jan j2 nY 4 ./ 4 : .n/ V p 2 .q2 N/2ˇ1  24ˇ nD1 n1ˇ q q4 N 2

(7.44)

Proof We introduce a C1 test function V W R ! R which is non-negative, compactly supported in ŒA; B  RC , and such that Z

C1 0

V.y/ dy D 1: y

Its Mellin transform e D V.s/

Z

C1

V.y/ys 0

dy y

is a holomorphic function which decays rapidly in vertical strips. From Lemma 7.3 we deduce that the series L.s; f  f / D

C1 X nD1

.n/

jan j2 ns

converges absolutely for Re.s/ > 1. For any Y > 1 we can then consider the integral 1 2i

Z

ds e V.s/L.s C ˇ; f  f /Y s s .2/  s Z C1 X jan j2 1 Y ds e : V.s/ .n/ ˇ D n 2i n s .2/ nD1

I.ˇ; Y/ D

We let V1 .y/ D

1 2i

Z .2/

e s ds V.s/y s

so that I.ˇ; Y/ D

C1 X nD1

.n/

jan j2  n  : V1 nˇ Y

(7.45)

256

7 L-Functions and the Selberg Conjecture

We immediate observe that by Mellin inversion, Z

C1

V1 .y/ D y

V.x/ dx; x

and thus ( 0 6 V1 .y/ 6 1

and V1 .y/ D

1

if 0 < y 6 A;

0

if y > B:

Since L.s; f f / is bounded in vertical strips, we can shift the integration contour

D 2 to the left to reach D 1. Along the way we pick up a pole at s D 0, which gives 1 I.ˇ; Y/ D L.ˇ; f  f / C 2i

Z

ds e V.s/L.s C ˇ; f  f /Y s : s .1/

(7.46)

The functional equation (7.41) implies that the integral on the right-hand side of (7.46) is equal to 

./ w. f ; f / .N/ p q 2



4 

1 2i

Z

Nq2 2

12ˇ

e V.s/ .1/



 4Y q4 N 2

s G1 .s C ˇ/L.1  ˇ  s; f  f /

ds : s

Changing variables s 7! s transforms this last expression into

w. f ; f /2 .N/

  12ˇ ./ 4 Nq2 p q 2  4 s Z 1 ds  Y e G1 .s C ˇ/L.1  ˇ C s; fQ  fQ / V.s/ 4 2 2i .1/ q N s



which is then equal to w. f ; f /2 .N/



./ p q

4 

Nq2 2

12ˇ C1 X nD1

.n/

jan j2  nY 4  V2 4 2 ; n1ˇ q N

where we have put, for y > 0, 1 V2 .y/ D 2i

Z

s ds e : V.s/G 1 .s C ˇ/y s .1/

(7.47)

7.5 The Luo-Rudnick-Sarnak Theorem

257

Note that if A > 1, by shifting the integration contour to the right of D 1 to

D A, we obtain V2 .y/ D OA .yA /

as y ! C1;

e decays since G1 is of at most polynomial growth in vertical strips whereas V.s/ rapidly. We now show that V2 .y/ D O" .1 C y1ˇ0 ˇ" /

as y ! 0:

If ˇ0 C ˇ  1 > 0, this is obvious. Otherwise, we shift contours to the left; as the first pole (of G1 .s C ˇ/) we cross is at ˇ0 C ˇ  1, the conclusion follows at once. We conclude the proof of the lemma by putting (7.45) equal to (7.46), after having replaced the integral on the right-hand side of (7.46) by (7.47). We get L.ˇ; f  f / D

C1 X nD1

n jan j2 .n/V 1 nˇ Y

 w. f ; f /2 .N/



./ p q

4 

Nq2 2

12ˇ C1 X nD1

.n/

jan j2  nY 4  V2 4 2 ; n1ˇ q N t u

as claimed.

We can now average the approximate functional equation over the set of even Dirichlet characters which are non-trivial modulo a prime q (we assume that q is sufficiently large so that q stays prime to N). The sum X L.ˇ; f  f / 

decomposes into a sum T1 C T2 of two terms corresponding to the terms in the approximate functional equation. Thus T1 D

X C1 X jan j2  nD1

nˇ

.n/V1

n Y

(7.48)

and     C1 nY 4 w. f ; f / 4ˇ X jan j2 X 2 4 : T2 D 2 14ˇ q  .N/.n/./ V2 . N/ n1ˇ q4 N 2  nD1

(7.49)

258

7 L-Functions and the Selberg Conjecture

We begin by evaluating the term T1 . It is not difficult to deduce from (7.13) that 8 ˆ 0 ˆ < X q1 .n/ D 1 ˆ ˆ : 2 ¤triv even 1

if n  0 .mod q/ if n  ˙1 .mod q/

(7.50)

else:

We thus obtain T1 D

q1 2

X n˙1 .mod q/

jan j2  n  X jan j2  V1 V1 .n=Y/ : nˇ Y nˇ

(7.51)

q−n

All terms in the first sum are non-negative. Hence one may obtain a lower bound on the first sum by considering only the contribution from n D 1 term: q1 2

X n˙1 .mod q/

q1 jan j2 V1 .1=Y/ : V1 .n=Y/ > ˇ n 2

(7.52)

The right-hand side of (7.52) is equal to .q  1/=2 for Y sufficiently large. On the other hand, one has X jan j2 q−n

nˇ

V1 .n=Y/ 6

C1 X nD1

jan j2 V1 .n=Y/ : nˇ

We deduce from Lemma 7.3 that for Y large enough C1 X nD1

  jan j2 jV1 .n=Y/j D Of Y 1ˇ ˇ n

and thus that T1 >

  q1  Of Y 1ˇ : 2

(7.53)

We finally arrive at the contribution from T2 . Since for primitive  we have p j./j D q, we easily bound the inner sum of (7.49) as X ¤triv even

.nN 02 /./4 D O.q3 /:

(7.54)

7.5 The Luo-Rudnick-Sarnak Theorem

259

We deduce that ˇ ˇ  C1 X jan j2 ˇ  nY 4 ˇ  34ˇ ˇ ˇ V jT2 j D O q 1ˇ ˇ 2 4N 2 ˇ n q nD1   ˇ Z C1 ˇ  ˇ rY 4 ˇˇ ˇ dr 34ˇ ˇ DO q ˇV 2 q 4 N 2 ˇ r r 1   D O q3 Y ˇ ; whence it follows that   jT2 j D O q3 Y ˇ ;

(7.55)

for large enough q and Y. The upper bounds (7.53) and (7.55) imply the inequality j

X

L.ˇ; f  f /j >



q1  Of .Y 1ˇ C q3 Y ˇ /; 2

(7.56)

for large enough q and Y. Taking Y  q3 , we obtain Proposition 7.39 (recall that 2=3 < ˇ < 1). t u We now show that Theorem 7.38 implies Theorem 7.1. Proof of Theorem 7.1 We restrict ourselves to cusp forms. Let X p ar y K .2jrjy/e.rx/ 2 C.0 .N/nH/ f .z/ D r2Z

satisfy f D .1=4   2 /f . Suppose that  is real and strictly greater than 1=3; we seek to obtain a contradiction. Let ˇ D 2jj: In view of the decomposition of C.0 .N/nH/ into an orthogonal sum of spaces of oldforms and newforms, we can safely assume that f is a newform. We can moreover take f to be normalized. According to Theorem 7.38, there then exists a non-trivial even Dirichlet character of prime modulus q not dividing N such that L.ˇ; f  f / ¤ 0: According to Theorem 7.28, the function .s; f  f / D L1 .s; f  f /L.s; f  f / is entire whereas ˇ is a pole of L1 .s; f  f /. Contradiction.

t u

260

7 L-Functions and the Selberg Conjecture

7.6 Bounds on Fourier Coefficients The initial motivation of Rankin and Selberg to prove the existence of an analytic continuation of Rankin-Selberg L-functions was the study of the Fourier coefficients of modular forms. Their method applies equally well to the study of the Fourier coefficients of Maaß forms. Let X p f .z/ D ar y K .2jrjy/e.rx/ r2Z

be a cuspidal (a0 D 0) normalized Maaß newform in Cprim .0 .N/nH/. The following conjecture is referred to as the Ramanujan-Petersson conjecture; it should be compared with the “trivial” bound of Hecke (Lemma 7.2). Conjecture 7.41 For every " > 0, the sequence of Fourier coefficients of f satisfies jan j D O" .n" /; for n sufficiently large. Any progress towards this conjecture is significant and rich in arithmetic applications. We shall see such an application in Chap. 9. In this section we give a proof of the following bound, under the simplifying hypothesis that N D 1. Theorem 7.42 For any " > 0, the sequence of Fourier coefficients of f satisfies jan j D O" .n3=10C" /; for n sufficiently large. Proof in the case N D 1 The theorem is a corollary of the following lemma – due to Landau – whose proof we shall admit. P Lemma 7.43 Let L.s/ D n bn ns be a Dirichlet series with non-negative coefficients bn which converges on Re.s/ > 1. Assume that L.s/ admits a meromorphic continuation to all of C with at most a simple pole at s D 1 in the right half-plane Re.s/ > 1=2. Assume moreover that L is of finite order in the half-plane Re.s/ > 1 and satisfies a functional equation of the form qs L1 .s/L.s/ D wq1s L1 .1  s/L.1  s/; where w and q are positive constants and L1 .s/ D

d Y iD1

 .˛i s C ˇi /;

7.6 Bounds on Fourier Coefficients

261

with d > 1 and ˛i > 0, ˇi 2 C for i D 1; : : : ; d. Then for all " > 0 we have X

 21  bn D Cx C O";L x 2C1 C" ;

n6x

where the constant C > 0 depends only on L and  D

Pd iD1

˛i .

We now apply the lemma to the function L.s; f  f / D .2s/

C1 X jan j2 nD1

ns

:

Note that Theorem 7.28 applies here as well. This time the above L-function admits two simple poles at s D 0 and 1. It satisfies the functional equation  2s L1 .s/L.s; f  f / D  2.1s/ L1 .1  s/L.1  s; f  f / with L1 .s/ D 

 s 2  s  2   s C 2   :  2 2 2

In order to appealPto Landau’s lemma, we must take into account the factor .2s/ in 2 s front of the sum C1 nD1 jan j n . Observe that .2s/

C1 X

jan j2 ns D

nD1

C1 X

bn ns ;

nD1

where .bn /n2N is a sequence of positive reals. Here the sequence .bn /n2N is obtained by Dirichlet convolution of the characteristic function of the squares 2 with the sequence .jan j2 /n2N : X 2 .k/jad j2 : bn D kdDn

We then deduce from Landau’s lemma that X bn D Cx C O";L .x3=5C" /: n6x

Now 2 .1/ is non-zero so the function 2 is invertible under Dirichlet convolution; let 2 be its inverse. Then we have C1

X 1 D 2 .n/ns .2s/ nD1

262

and

7 L-Functions and the Selberg Conjecture

X

jan j2 D

n6x

XX

2 .d/bk

n6x kdDn

D

X

2 .d/

d6x

X

bk

k6x=d

  X x 3=5C" 3=5" DC 2 .d/ C O";L x 2 .d/d d d6x d6x X

D

C x C O";L .x3=5C" /: .2/

Taking x D n and x D n  1 in the last inequality, we obtain jan j2 D O";L .n3=5C" /, whence the theorem. t u

7.7 Comments and References The proof of Theorem 7.1 that we gave in this chapter is due to Luo, Rudnick and Sarnak [84]. In fact their theorem is much more general, since it treats all congruence groups and in fact applies to cuspidal automorphic representations of GL.n/. In particular, it is not necessary to assume that N is squarefree; and we could just as well replace 0 .N/ by  .N/. We can also improve the lower bound from 5=36.D 1=4  .1=3/2 / to 4=25.D 1=4  .1=2  1=5/2 /, as we shall explain below. The original proof of Selberg gives the lower bound of 3=16. There are nevertheless several advantages to the above method: (1) it can be adapted to any number field, (2) it extends to the groups GL.n/. Finally, this method arises in the recent proof of Kim and  7 2 . Sarnak [68] of the best known approximation to the Selberg conjecture, namely 14  64 At present the article [11] of Blomer and Brumley contains the more complete results. We have limited ourselves to the case treated here in order to simplify the exposition, but the argument generalizes in a natural way.

§ 7.1 A deep theorem of Margulis [1, 89] states that if  is a Fuchsian group, then Com. /= is infinite if and only if  comes from a quaternion algebra (possibly defined over a finite extension of Q). In this case the commensurator is dense in G.

§ 7.2 Equality (7.6) is proved in [92, (4.5.25)]. It allows one to recover the “classical” Hecke operators defined, for example, in [118, Chap. VI §5]. On the other hand, the equality (7.8) is proved in [92, Th. 4.5.13].

7.7 Comments and References

263

Atkin-Lehner theory is developed in [5] in the case of modular forms. We have followed the book of Miyake [92], transposing the statements there into the context of Maaß forms. The proofs are completely identical in the two cases: Lemma 7.8 reproduces [92, Th. 4.6.4], Lemma 7.9 reproduces [92, Lem. 4.6.5], Lemma 7.9 reproduces [92, Lem. 4.6.7] and Lemma 7.11 reproduces [92, Th. 4.6.8]. Theorem 7.12 is due to Atkin and Lehner. It is not necessary to assume that N 0 divides N nor that f and f 0 lie in the same Laplacian eigenspace; it suffices to assume that f and f 0 share the same Tp eigenvalue for all but a finite number of primes p. However, to operate under such weak assumptions seems to require passing to the study of L-functions and the use of non-trivial estimates on the Fourier coefficients of Maaß forms. We refer the reader to [5] for more details in the case of modular forms.

§ 7.3 Lemmas 7.18 and 7.19 are classical, due to Gauß and Jacobi. For more details one can consult [31, 70].

§ 7.4 In formula (7.16), the character  is primitive. We can show that the function L.s; / has a “good functional equation”, see [20, 31, 75]. Theorem 7.28 is remarkable: even though we do not know how to realize the RankinSelberg L-function as the L-function of an automorphic form, we can nevertheless prove the existence of a meromorphic continuation and functional equation. This important theorem – in the case of modular forms – is due to Rankin [100] and Selberg [114] when N D 1. The extension to arbitrary level is more delicate and requires the use of newforms. Rather than aim for maximal generality, we have stated and proved the result which was needed in the proof of Theorem 7.1. The extension to arbitrary level of the theorem of Rankin-Selberg was obtained by Winnie Li [81] in the classical language (and again for modular forms) and by Jacquet [66] in the adelic language. Theorem 7.28 square factors Q remains true if N has s 1 and the or if  is odd: one must simply adjust the factor pjN .1  2 . p/p1 1 p p / computations are a bit more intricate, see [81]. The right point of view for approaching the general case is the theory of representations of adelic groups, see [27, 66] or [9].

§ 7.5 The proof of Proposition 7.39 is based on the method of the “approximate functional equation”, see [46, 49, 61, 64, 78]. The upper bound (7.54) is not optimal. In Appendix C, Valentin Blomer and Farrell Brumley explain more generally how to relate the sums Sn .r; q/ D

X

 .mod q/

.r/ ./n

264

7 L-Functions and the Selberg Conjecture

(here q; n are two integers with q; n > 2, r is an integer relatively prime to q, and the asterisk indicates that the sum is taken over primitive characters) to certain exponential sums called “hyper-Kloosterman sums”. A theorem of Deligne furnishes an optimal upper bound on these sums and allows one to replace the bound (7.54) by X

.nN 02 / ./4 D O.q5=2 /:

¤triv even

In Theorem 7.38 one can then replace 2=3 by 1  2=5; this is in fact the exact result that Luo, Rudnick and Sarnak prove. The proof of Deligne’s theorem is extremely difficult and passes through mountains of algebraic geometry. It is natural to ask whether it is absolutely necessary to appeal to such a powerful result. Indeed, one can in fact refine the argument of Luo, Rudnick and Sarnak without recourse to the Deligne bounds. In Appendix C Blomer and Brumley give an elementary proof of the upper bound on hyper-Kloosterman sums when q is a non-trivial power of a prime. It suffices then to go through the above proof, taking q to be a non-trivial power of a prime to arrive at a proof which avoids the “mountains of algebraic geometry” (however beautiful such peaks may be) we made reference to earlier. In this way, one can prove the theorem of Luo, Rudnick and Sarnak independently of that of Deligne.

§ 7.6 It is rather the analog of Conjecture 7.41 for modular forms which is classically referred to as the “Ramanujan-Petersson conjecture”. This version of the conjecture was moreover proved by Deligne as a consequence of his proof of the Weil conjectures, see [35]. The proof of Theorem 7.42 goes back, in principle, to Rankin and Selberg. The first general versions are due to Serre, Moreno-Shahidi and Murty, and of several other authors; see [21] for a brief summary of the literature. Lemma 7.43 is an improvement of the famous result known as Landau’s lemma according to which a Dirichlet series with non-negative coefficients converges absolutely to the right of its first pole; see [118, Chap. VI Prop. 7]. The improvement we present is also due to Landau [74], see [10] as well. Theorem 7.42 remains true for an arbitrary congruence subgroup. One must work a bit harder, however; see [21]. We shall see in Chap. 9 that Conjecture 7.41 is in fact a p-adic analog of the Selberg conjecture. We can moreover give a proof of Theorem 7.42 which is essentially identical to the proof of Theorem 7.1: Write Lp .s; f  f / D L.s; f  f / 

2 Y

.1  . p/˛i . p/ˇj . p/ps /

i;jD1

for the (partial) L-function, where we have removed the Euler factor at p. As before, one must show that if ˇ is a real number strictly greater than 1  25 , then there exists an infinite set of primitive Dirichlet characters such that . p/ D 1 and Lp .ˇ; f  f / ¤ 0 (analogously to Theorem 7.38). See Luo, Rudnick and Sarnak in [86].

7.8 Exercises

265

7.8 Exercises Exercise 7.44 Show that [ d1 [

Hn D

adDn bD0 .a;N/D1



  ab 0d

(disjoint union):

Exercise 7.45 Show that TnN ı TmN D

X 0 0; thus b D ˙1. This establishes the second inclusion. t u It follows from Lemma 8.1 that we can assume, changing bases for O if necessary, that the unit 1 2 A is a basis element, say e1 D 1. For i D 2; 3; 4, we then have (see (2.1) and (2.2)): e2i D Nred .ei /  1 C tr.ei /  ei I as a consequence, tr.ei / and Nred .ei / are integers and the equality ei D tr.ei /  ei implies that ei 2 O. This proves Lemma 8.2 Every order O of A is stable by conjugation: O D O. The discriminant of a basis .e1 ; e2 ; e3 ; e4 / of A over Q is the determinant of the matrix of traces of pairwise products of basis elements: disc.e1 ; e2 ; e3 ; e4 / D det..tr.ei ej //16i;j64 / 2 Q: For example, for the standard basis .1; i; j; k/ of A D Da;b .Q/, we find 0

1 2 0 0 0 B0 2a 0 0 C 4 2 2 C disc.1; i; j; k/ D det B @0 0 2b 0 A D 2 a b : 0 0 0 2ab

8.1 Arithmetic of Quaternion Algebras

269

Let .e1 ; e2 ; e3 ; e4 / and .f1 ; f2 ; f3 ; f4 / be two bases and write P for the change of basis matrix from the first to the second. The matrix of the bilinear form .x; y/7! tr.xy/ in the first basis is .tr.ei ej //i;j . Its matrix in the basis .f1 ; f2 ; f3 ; f4 / is therefore equal to t P.tr.ei ej //i;j P; but it is also .tr.fi fj //i;j . From this we deduce disc.f1 ; f2 ; f3 ; f4 / D disc.e1 ; e2 ; e3 ; e4 /.det P/2 :

(8.1)

If O D fx1 e1 C x2 e2 C x3 e3 C x4 e4 j x1 ; : : : ; x4 2 Zg is an order of A, then ei ej 2 O for i; j D 1; : : : ; 4 and Lemmas 8.1 and 8.2 imply that tr.ei ej / D ei ej C ei ej 2 O \ Q D Z: Consequently, disc.e1 ; : : : ; e4 / is an integer which does not depend on the choice of basis of O; denote it by disc.O/. According to the preceding arguments, if A D Da;b .Q/ we have disc.O/ D disc.e1 ; : : : ; e4 / D disc.1; i; j; k/.det P/2 D 24 a2 b2 .det P/2 ; where P is the change of basis matrix from .1; i; j; k/ to .e1 ; : : : ; e4 /. We can then write disc.O/ D r2 , where r D 22 ab.det P/. As disc.O/ 2 Z and det P 2 Q, we must have r 2 Z; we call this integer the reduced discriminant of O, denoted d.O/. By definition we have disc.O/ D d.O/2 : The following proposition follows immediately from (8.1). Proposition 8.3 Let O and O0 be two orders of A. If O O0 , then d.O/ divides d.O0 /. Moreover, if O O0 and d.O/ D d.O0 /, then O D O0 . As every integer has a finite number of divisors, the preceding proposition shows that an increasing sequence of orders is necessarily stationary. From this it follows that every order is contained in a maximal order, i.e., an order which is not strictly contained in any other one. The preceding results generalize to the case where one replaces the base ring Z by a principal ideal domain, such as the ring of p-adic integers Zp , or in the case where the algebra A is replaced by a division algebra of arbitrary finite dimension, in particular by a field. We shall briefly return to the case of quadratic extensions of Q in the following subsection. Let us now consider a quaternion algebra over the ring of p-adic integers (we refer the reader to Sect. 8.5 at the end of this chapter for references pertaining to these notions). The ring of p-adic integers is principal and its unique (up to multiplication by units) irreducible element is p. Moreover, it

270

8 Jacquet-Langlands Correspondence

contains the ring Z, so that its fraction field Qp contains the field Q. Every rational quaternion algebra A D Da;b .Q/ embeds naturally in a p-adic quaternion algebra A ,! Ap D Da;b .Qp / and if O D fx1 e1 C x2 e2 C x3 e3 C x4 e4 j x1 ; : : : ; x4 2 Zg is an order of A, then Op D fx1 e1 C x2 e2 C x3 e3 C x4 e4 j x1 ; : : : ; x4 2 Zp g is an order of Ap over Zp . As .e1 ; : : : ; e4 / is simultaneously a basis of O over Z and a basis of Op over Zp , it is clear that d.O/ is a representative of d.Op / (this latter element being well-defined up to multiplication by a unit in Zp ). In what follows, we admit the classification of p-adic quaternion algebras: up to isomorphism, there is a unique quaternion division algebra over Qp . Moreover, this quaternion division algebra contains a unique maximal order: the set of elements whose norm has positive p-adic valuation; the reduced discriminant of this order is p. From this it follows that if p is a prime not dividing the reduced discriminant d.O/, the algebra Ap is a matrix algebra; indeed, this algebra contains the order Op whose reduced discriminant is invertible in Zp , and thus it cannot be a division algebra. In addition, if O is a maximal order of A, then Op is a maximal order of Ap for every prime p. It follows that the discriminant d.O/ of a maximal order O is the product of the primes p such that Ap is a division algebra; as a result, all the maximal orders of A have the same reduced discriminant, that we shall call (by abuse of language) the discriminant of the algebra A. We admit that this discriminant is always the product of an even number of primes.

8.1.2 Orders in Quadratic Extensions of Q p Recall that a quadratic field extension of Q is of the form Q. D/=Q for D a fundamental discriminant, i.e., an integer ( DD

4d

if d  2; 3 .mod 4/;

d

if d  1 .mod 4/;

where d ¤ 0; 1 is squarefree. p An order in K D Q. D/ is necessarily of the form ˝.n/ D Z C n!Z;

p DC D where ! D 2

(8.2)

8.1 Arithmetic of Quaternion Algebras

271

andpn D 1; 2; 3; : : : The order ˝.1/ is in particular the unique maximal order p of Q. D/. Since Œ˝.1/ W ˝.n/ D n, we call ˝.n/ the order of index n of Q. D/. The discriminant of ˝.1/ is equal to D, the discriminant of the field K. Let p be a prime. The algebra Kp D K ˝ Qp is of dimension 2 over Qp . It is therefore either isomorphic to the product Qp  Qp or equal to a quadratic field extension of Qp . In the latter case the ideal generated by .p/ in the ring of integers of Kp is of the form pe , where p is the unique maximal ideal in the ring of integers of Kp and the ramification index e is equal to 1 or 2. We say that Kp is a ramified extension if e D 2 and unramified otherwise. For an even  prime p it is convenient to describe Kp with the help of the Legendre symbol p . This is the Dirichlet character of modulus p induced by the character of .Z=pZ/ equal to 1 on squares and 1 otherwise. When p D 2 we may extend the Legendre symbol to fundamental discriminants by putting D 2

D

8 ˆ ˆ 0 C if D < 0



Y

˝p



p

is finite, equal in fact to h.˝/. Theorem 8.6 Let A be a quaternion algebra over Q such that A ˝Q R Š M2 .R/. Let O be an order in A such that Nred .Op / D Zp for all primes p. Then A embeds diagonally in AA and, relative to the coordinatewise product  in AA , we have  Y    C Op : AA D A  GL .2; R/  p

8.2 Optimal Embeddings of Quadratic Fields Henceforth we shall assume that we are in one of the two following cases: Case I.

The quaternion algebra A is equal to D1;1 .Q/ D M2 .Q/ and O D M.N/ D

  ˇ ab ˇ 2 M2 .Z/ ˇ c  0 .mod N/ ; cd

for a certain squarefree integer N > 1. Case II. The quaternion algebra A is a division algebra Da;b .Q/ (a; b > 0) of discriminant dA > 1 and O is a maximal order of A. Note that in the first case the order is not maximal if N D 1. Let K be a quadratic field extension of Q of discriminant D and ˝ an order in K. Suppose that there exists an embedding  W K ,! A. The order ˝ is said to be optimally embedded in O relative to  if .˝/ D O \ .K/. Two optimal embeddings 1 and 2 are conjugated by  if 1 .˝/ D  2 .˝/ 1 . In what follows we write O1 D SL.1; O/ for the elements of norm 1 in O. Example 8.7 The map 8 p < Q. 2/ ! M2 .Q/ W : a C bp2 7!  a 2b  b a p realizes an optimal embedding of ZŒ 2 in the order M2 .Z/.

274

8 Jacquet-Langlands Correspondence

p More generally if D is a fundamental discriminant, let  W Q. D/ ! M2 .Q/ be the map which sends 1 to the identity matrix and ! – defined in (8.2) – to 0 .!/ D 1

DD2 4

!

D

:

p Then  realizes an optimal embedding of the maximal order ˝.1/  Q. D/ in M2 .Z/. An optimal embedding does not always exist. The following theorem gives, amongst other things, an existence criterion. Theorem 8.8 The number E.˝; O1 / of optimal embeddings of ˝ modulo conjugation by elements of O1 D SL.1; O/ is given by the formulas  1

Case I: E.˝; O / D h.˝/   Case II: E.˝; O1 / D h.˝/ 

2 if D < 0 1 if D > 0 2 if D < 0 1 if D > 0

 

   ˝ 1 C pjN p

Q

   ˝ 1  ; pjd.O/ p

Q

where 

˝ p

 D

8 ˆ ˆ 0:

8.2 Optimal Embeddings of Quadratic Fields

275

For almost every prime p, the algebra Ap is unramified and therefore isomorphic to the 2  2 matrix algebra over Qp . We begin then by calculating Ep .˝; O / when Ap D M2 .Qp /. Let p be a prime. Suppose that Ap D M2 .Qp / and   ˇ ab ˇ  Op D 2 M2 .Zp / ˇ c 2 p Zp cd for a certain integer  > 0. Let p be an embedding of Kp in Ap . As an order in Kp , the set ˝p is a free Zp -module, and one can then write ˝p D Zp Œ˛

with ˛ 2 Kp :

Let us write f˛ .X/ D X 2  tX C n .t; n 2 Zp / for the minimal polynomial of ˛. Lemma 8.9 The embedding p defines an optimal embedding of ˝p if and only if there exists  2 E.˛; ˝p / D f 2 Zp j f˛ ./  0 .mod p /g and ( up 2 N.Op / D

Qp Op Qp Op

[



0 1 p 0



if  D 0 Qp Op

if  > 0;

such that up p .˛/u1 p

  1 : D ˛ WD f˛ ./ t   

Proof We write   ab p .˛/ D : cd The embedding p defines an optimal embedding of ˝p if and only if for every integer r > 1, the set .Zp C p .˛// \ pr Op is empty. This last property is equivalent to the fact that the elements b, a  d and p c are relatively prime integers. It is clear that, if  2 E.˛; ˝p /, ˛ 2 M2 .Zp /

and ˛ … Zp C pM2 .Zp /;

276

8 Jacquet-Langlands Correspondence

so that 1   0 1 0 1 ; ˛  p 0 p 0



lies in M2 .Zp / but not in Zp C pM2 .Zp /. We easily deduce that the condition of the lemma is satisfied; let us show that it is necessary. Assume then that b, a  d and p c are relatively prime. Note that f˛ .a/ D bc. If b is a unit, put   10 : up D 0b the Then  D a lies in E.˛; ˝p / and up p .˛/u1 p D ˛ . We must therefore reduce   argument to the case where b is a unit. If p c is a unit, we conjugate p .˛/ by p0 01 .   Otherwise we conjugate p .˛/ by 10 11 , which replaces b by .a C c/ C b C d. The latter element is a unit, since p does not divide ad. This establishes Lemma 8.9. u t Let F.˛; ˝p / denote the set f 2 Zp j f˛ ./  0 .mod pC1 /g; if t2  4n  0 .mod p/ and  > 1, and the empty set otherwise. Lemma 8.10 The number of optimal embeddings of ˝p modulo inner automorphisms induced by Op is equal to the sum of the cardinality of the image of E.˛; ˝p / in Z=p Z and the cardinality of the image of F.˛; ˝p / in Z=p Z. In particular, if  D 0 this number is equal to 1. Proof Let 

 0 1 wD  : p 0 To count the equivalence classes modulo conjugation by Op we begin by observing that, according to Lemma 8.9, two optimal embeddings p and p of ˝p are equivalent modulo N.Op / if and only if p is equivalent to p or to wp w1 modulo Op . According to Lemma 8.9, it remains then to determine when ˛ and ˛ 0 (resp. ˛ and w˛ 0 w1 ), with ;  0 2 E.˛; ˝p /, are equivalent modulo Op . We start by noting that if    0 .mod p /, we have  1 0 2 Op   0 1

 up D

8.2 Optimal Embeddings of Quadratic Fields

277

and up ˛ u1 p

 0   1 D D ˛ 0 : 

Conversely, suppose that ˛ is equivalent to ˛ 0 mod Op . Every element of Op being upper triangular mod p , the same is true of ˛ . Thus if up 2 Op conjugates ˛ 0  to ˛ , the matrix up ˛ u1 p has the same diagonal entries mod p as ˛ . We deduce 0  that    .mod p /. Finally, ˛ is equivalent to ˛ 0 ”    0 .mod p /:

(8.4)

We now show in the same way that if  > 1, ˛ is equivalent to w˛ 0 w1 (   t   0 .mod p / and f˛ . 0 / 6 0 .mod pC1 / ”   t   0 .mod p /

if pj.t2  4n/; else.

(8.5)

If p f˛ . 0 / is a unit, it follows from Lemma 8.9 (and its proof) that the matrix w˛ 0 w1 is equivalent to  t  0 1 : f˛ . 0 /  0



In this way, according to (8.4) ˛ is equivalent to w˛ 0 w1 mod Op if and only if  D t   0 .mod p /. If now p f˛ . 0 / is not a unit, we put up D

  1b ; 01

with b 2 Zp to be chosen later. Modulo p , we have up w˛ 0 w1 u1 p

  t   0 b.2 0  t/  p f˛ . 0 / D :  

Thus if 2 0  t is a unit, which is to say that t2  4n is a unit, we can choose b in such a way that up w˛ 0 w1 u1 p is again equivalent to 

t  0 1 f˛ . 0 /  0



278

8 Jacquet-Langlands Correspondence

mod Op . Finally, suppose that p f˛ . 0 / and t2 4n are not units. Observing that Op is generated modulo p by diagonal matrices and matrices of the form   1b ; 01 we deduce that for all up 2 Op , if up w˛ 0 w1 u1 p D .xij /, x12 is never a unit, and thus w˛ 0 w1 cannot be equivalent to ˛ mod Op . The first part of the lemma follows from (8.4) and (8.5). Note finally that if  D 0 it is clear that E.˛; ˝p / is not empty and that the number of optimal embeddings of ˝p modulo inner automorphisms induced by Op is equal to 1. t u Let us come back to the global count and denote now by ˚ W  7! .p /p2f1;2;3;5;::: g the map which to any optimal embedding  of ˝ associates the sequence of optimal embeddings of ˝p , obtained by censoring by Qp ; two sequences .1p /p2f1;2;3;5;::: g , .2p /p2f1;2;3;5;::: g are said to be equivalent, written .1p /p2f1;2;3;5;::: g  .2p /p2f1;2;3;5;::: g ; if there exists a sequence u D .up /p2f1;2;3;5;::: g 2 U D GLC .2; R/ 

Y

Op

p

such that for p 2 f1; 2; 3; 5; : : : g, 2p D up 1p u1 p : Lemma 8.11 We have E.˝; O1 / D h.˝/ 

Y

Ep .˝; O /:

p2f1;2;3;5;::: g

Proof We fix an embedding  W K ,! A (which, recall, we have assumed to exist). We begin by showing that the strong approximation theorem (Theorem 8.6) implies that ˚ is surjective. Indeed, fix a sequence .p /p2f1;2;3;5;::: g of optimal embeddings of ˝p . Since, according to Lemma 8.10, for almost all p, the embeddings p and p are conjugate by an element of Op , and more generally p and p are conjugate by

8.2 Optimal Embeddings of Quadratic Fields

279

elements in Ap , Theorem 8.6 implies that there exists an invertible element  2 A such that ˚.  1 /  .p /p2f1;2;3;5;::: g I the embedding   1 is necessarily an optimal embedding of ˝. Suppose now that  is an optimal embedding of ˝. Given an element  2 A , we have ˚.  1 /  ˚./ ”  2 .U  AK / \ A (here A and AK are viewed as embedded in AA ). Thus Y

E.˝; O1 / D C 

Ep .˝; O /;

p2f1;2;3;5;::: g

where the constant C is given by the formula ˇ ˇ C D ˇO1 n.U  AK / \ A =K  ˇ : Given t1 ; t2 2 K  , Theorem 8.6 implies that there exists two elements u1 ; u2 2 U such that u1 t1 and u2 t2 2 .U  AK / \ A : It then follows from O1 D U \ A that O1 u1 t1 AK D O1 u2 t2 AK if and only if t1 t21 2 .AK \ U/  .AK \ A / D .AK \ U/  K  : Therefore C D jAK =.AK \ U/  K  j D jK  nAK =.AK \ U/j : But AK

 \U D

RC if D > 0 C if D < 0

and the lemma follows from Theorem 8.5.



Y

Op ;

p

t u

280

8 Jacquet-Langlands Correspondence

The following lemmas allow us to complete the proof of Theorem 8.8. Lemma 8.12 Let p be a prime such that Kp is a quadratic field extension of Qp , i.e., such that .D=p/ ¤ 1. Let Ap be a division algebra and Op a maximal order of Ap . If ˝p is a maximal order of Kp , then ( 

Ep .˝; O / D

2

if Kp =Qp is unramified

1

if Kp =Qp is (totally) ramified:

If ˝p is not maximal, Ep .˝; O / D 0. Proof Suppose that Kp D Qp Œ˛ is contained in Ap . The set of optimal embeddings of ˝p is in bijection with a subset of the conjugation class of ˛ in Ap , equal to C.˛; ˝p / D fh˛h1 j h 2 Ap ; Qp Œh˛h1  \ Op D h˝p h1 g: Note that the unique maximal order Op of Ap coincides with the set of elements of Ap whose reduced norm and trace both belong to Zp . The conjugacy class of an element ˛ in Ap thus meets Op if and only if ˛ itself belongs to Op . Furthermore, if ˛ is a non-scalar element of Op , the intersection Qp Œ˛\Op is the unique maximal order of Qp Œ˛. Hence one sees that if ˝p is not maximal C.˛; ˝p / is empty; there is therefore no optimal embedding of ˝p . Suppose now that ˝p is maximal. There then exists an element in C.˛; ˝p /, which is in fact unique up to conjugation by an element in Ap . Note that there exists an element p 2 Ap of reduced norm p such that Ap D

1 [

pn Op :

nD0

According to whether p belongs to Qp Œ˛ or not, we deduce ( Ap D

Qp Œ˛Op ;

if p ramifies in Qp Œ˛;

Qp Œ˛Op [ p Qp Œ˛Op ; else.

The lemma then follows immediately. Lemma 8.13 Let p be a prime. Let Ap D M2 .Qp / and   ˇ ab ˇ Op D 2 M2 .Zp / ˇ c 2 pZp : cd

t u

8.2 Optimal Embeddings of Quadratic Fields

281

Then Ep .˝; O / D

8 1 (case II) and the congruence subgroup 0 .N/ of SL.2; Z/ for squarefree N > 1 (case I). We begin with case II. Let  2 O,  … Q, such that tr. / D t and Nred . / D n. The subalgebra QŒ   A is commutative and of dimension 2 over Q; it is a quadratic field extension K of Q, completely determined by t and n. The inclusion QŒ   A induces an embedding of K into A. We admit that two such embeddings are conjugate by an element of A . The set of embeddings of K into A is therefore in bijection with the conjugacy class f gA of  in A. Consider now the conjugacy classes fıgO1 relative to O1 . For t; n 2 Z, we denote by E.t; n; O1 / the number of conjugacy classes fıgO1 with ı 2 O, tr.ı/ D t and Nred .ı/ D n. Lemma 8.14 Let  2 O,  … Q, such that tr. / D t and Nred . / D n. Then E.t; n; O1 / D

X

E.˝; O1 /;

˝ ZŒ 

where the sum runs over the set of orders in QŒ  which contain  . Proof If ˛ 2 A , then QŒ  \ ˛ 1 O˛ is an order of QŒ . We thus have f gA D

[

C.; ˝/;

(8.9)

˝

where the union is taken over the set of orders of QŒ  and ˚  C.; ˝/ D ˛ ˛ 1 j ˛ 2 A ; QŒ  \ ˛ 1 O˛ D ˝ : Note moreover that the union (8.9) is disjoint. To see this, suppose that ˛ ˛ 1 D ˇˇ 1 (˛; ˇ 2 A ). Then ˛ 1 ˇ commutes with  and QŒ; ˛ 1 ˇ is an algebra of dimension 2 or 4 over Q (it contains QŒ , which itself is of dimension 2 over Q).

8.3 The Trace Formula

283

From the commutativity of QŒ; ˛ 1 ˇ we have ˛ 1 ˇ 2 QŒ  and thus   QŒ  \ ˇ 1 Oˇ D .˛ 1 ˇ/ QŒ  \ ˇ 1 Oˇ .˛ 1 ˇ/1 D QŒ  \ ˛ 1 O˛: It follows from the definition that the set C.; ˝/ meets O if and only if  is contained in the order ˝. But the elements of C.; ˝/ correspond exactly to the embeddings of K into A relative to which the order ˝ is optimally embedded in O. Considering these embeddings modulo conjugation by O1 , we obtain the claimed expression. t u The isomorphism between A1 and M2 .R/ induces an embedding O ! M2 .R/. According to Theorem 2.3, the image O1  SL.2; R/ is a co-compact Fuchsian subgroup; we denote it by O . An element  2 O of trace t > 2 is hyperbolic; write E0 .t; 1; O1 / for the number of conjugacy classes in O of primitive elements of trace t. Note that if  is hyperbolic, then for r > 1 we have f gO1 D fˇgO1 if and only if f r gO1 D fˇ r gO1 . We deduce that E.t; 1; O1 / D

X

0 0

E .s; 1; O1 /;

(8.10)

s6t

where the sum runs over the set of s such that if tr. / D s, there exists an integer r > 1 such that tr. r / D t. We can now reformulate for the group O the Selberg trace formula proved in Chap. 5. In what follows, h W C ! C is an even function, holomorphic in the vertical strip j Im rj < 12 C " for some " > 0, and satisfying the following decay property   jh.r/j D O .1 C Re.r//2ı for a constant ı > 0. Proposition 8.15 Denote by k D rk2 C 1=4, k 2 N, the set of eigenvalues of the hyperbolic Laplacian in L2 .O nH/, counted with multiplicities. Then C1 X

area.O / h.rk / D 4 kD0 C

Z

C1

rh.r/ tanh.r/ dr 1

Z C1 t 1 X E0 .t; 1; O1 / mX 1 e2kr=mt h.r/ dr 2mt sin .k=mt / 1 1 C e2r kD1

t2f0;1g

C

C1 X tD3

E0 .t; 1; O1 / arccosh.t=2/

C1 X g .2k arccosh.t=2// ; sinh .k arccosh.t=2// kD1

284

8 Jacquet-Langlands Correspondence

where area.O / is the area of a fundamental domain for the action of O on H, mt for t 2 f0; 1g is the order of the primitive element of trace t, and 1 g.u/ D 2

Z

C1

h.r/eiru dr

1

is the Fourier transform of h. Proof We have just rewritten the trace formula (Theorem 5.8) by replacing the sums over representatives of primitive hyperbolic and elliptic conjugacy classes in O by sums over the traces t 2 N, t ¤ 2, of these representatives. The number of primitive conjugacy classes of trace t is E0 .t; 1; O1 /. t u Note that the area area.O / can be explicitly calculated. Lemma 8.16 We have area.O / D

Y .p  1/: 3 pjdA

Idea of the proof Fix a Z-basis .e1 ; e2 ; e3 ; e4 / of A. Every element of A1 D A ˝ R can be written as x D x1 e1 C x2 e2 C x3 e3 C x4 e4

.xi 2 R/

and the group O 1 acts by left-multiplication. We obtain the lemma by calculating in two different ways the integral Z F

dx1    dx4 ;

(8.11)

where F is a fundamental domain for O 1 in fx 2 A1 j jNred .x/j 6 1g. The algebra A1 is isomorphic to the algebra of 2  2 matrices; let be an isomorphism. Denote by Eij ( j; k D 1; 2) the canonical basis of M2 .R/ and xij the corresponding coordinates of an element x 2 A1 , so that 

.x/ D

 x11 x12 : x21 x22

We then have Nred .x/ D det. .x// D x11 x22  x12 x21 . By definition of the discriminant we have dA dx1    dx4 D dx11 dx12 dx21 dx22 : A change of variables then implies that the integral (8.11) is equal to Z F

dx1    dx4 D

1 dA

D

2 dA

Z

1

0

Z

F2

Z tdt  F2

dx11 dx22 dx12 jx12 j

dx11 dx22 dx12 ; jx12 j

8.3 The Trace Formula

285

where F2 is a fundamental domain of O in the group SL˙ .2; R/ of 2  2 matrices of determinant ˙1. The quotient SL˙ .2; R/=O.2/ can be identified with the Poincaré upper half-plane H via the map g 7! z D g.i/ where the isotropy group of i is O.2/. Write dk for the Haar measure normalised in such a way as to give O.2/ total volume 1. We then have dx11 dx22 dx12 D d .z/dk: jx12 j Let FO be a fundamental domain for the action of O on H. Since an element of O acts trivially on H if and only if it is equal to ˙1, we find that Z FO

d .z/ D

2 

Z

dx11 dx22 dx12 : jx12 j

F2

In other words area.O / D

2dA 

Z F

dx1    dx4 :

(8.12)

Furthermore, one can show that the integral (8.11) is equal to the residue at s D 1 of the zeta function A .s/ of the algebra A. This function has an Euler product expansion and one has the identity A .s/ D .2s/.2s  1/

Y

.1  p12s /:

pjdA

Its residue at s D 1 is therefore equal to .2/ Y .p  1/: dA

(8.13)

pjdA

Finally, the lemma follows from (8.12), (8.13) and the well-known fact that .2/ D  2 =6. t u

Now consider case I. The identity (8.10) remains valid when one replaces O1 by 0 .N/. Using the fact that N is squarefree, the Selberg trace formula (Theorem 5.21) can be rewritten in the following way. Proposition 8.17 Let k D rk2 C 1=4, k 2 N, denote the set of eigenvalues of the hyperbolic Laplacian in L2 .0 .N/nH/, counted with multiplicity. Then C1 X

area.0 .N// h.rk / D 4 kD0 C

Z

C1

rh.r/ tanh.r/ dr 1

Z C1 t 1 X E0 .t; 1; 0 .N// mX 1 e2kr=mt h.r/ dr 2mt sin .k=mt / 1 1 C e2r kD1

t2f0;1g

286

8 Jacquet-Langlands Correspondence C1 X

C1 X

g .2k arccosh.t=2// sinh .k arccosh.t=2// tD3 kD1   

0 Z C1 0 1 1  !.N/ C ir C .1 C ir/ dr C2 g.0/ log.=2/  h.r/ 2 1  2  C1 C1 X X .n/ X log p C2 g.2k log p/ ; g.2 log n/  n pk nD1 kD0 C

E0 .t; 1; 0 .N// arccosh.t=2/

pjN; p prime

where area.0 .N// is the area of a fundamental domain for the action of 0 .N/ on H, mt for t 2 f0; 1g denotes the order of the primitive element of tracet, !.N/ is equal to the number of divisors of N and 1 g.u/ D 2

Z

C1

h.r/eiru dr

1

is the Fourier transform of h.

8.4 Jacquet-Langlands Correspondence and Applications The Jacquet-Langlands correspondence links the spectrum of the Laplacian of co-compact congruence hyperbolic surfaces to the spectrum of the Laplacian of congruence covers of the modular surface. The proof proceeds by a comparison of terms on the geometric side of the trace formula. We shall therefore consider the trace formula for different groups, and to avoid all confusion we shall write X

h.rk /

fk 2S

for the sum over an orthonormal basis of eigenfunctions in a subspace S, where k D rk2 C1=4 is the Laplacian eigenvalue corresponding to fk and h is a test function such as those arising in the trace formula of the preceding section. Theorem 8.18 Let O be a maximal order in a quaternion algebra A D Da;b .Q/ with a; b > 1 of discriminant dA > 1. Then the set of non-zero eigenvalues of the hyperbolic Laplacian in L2 .O nH/, counted with multiplicity, coincides with the set of eigenvalues associated with primitive Maaß forms for the group 0 .dA /. Proof We begin by showing that X fk 2L2 .O nH/

h.rk / D

X fk 2C˚Cprim .0 .dA /nH/

h.rk /

(8.14)

8.4 Jacquet-Langlands Correspondence and Applications

287

for every test function h. The two following lemmas will allow us to reduce the proof of (8.14) to a comparison of the right-hand sides of the trace formulas in Propositions 8.15 and 8.17. Define X

ˇ.n/ D

.k/ .n=k/;

(8.15)

kjn

where .n/ is the MÖbius function. In particular, if n is the product of r distinct primes, then ˇ.n/ D .2/r . In terms of Dirichlet convolutions the function ˇ is equal to  . Lemma 8.19 We have X

h.rk / D

fk 2Cprim .0 .dA /nH/

X mjdA

X

ˇ .dA =m/

h.rk /:

fk 2C.0 .m/nH/

Proof Let m be a divisor of dA . To a form f 2 Cprim .0 .m/nH/ there corresponds exactly  .dA =m/ oldforms in C.0 .dA /nH/, where .n/ is the number of positive divisors of n. Write ı.m; / for the dimension of the eigenspace of C.0 .m/nH/ corresponding to the eigenvalue , and write ı 0 .m; / for the dimension of the subspace of those which are primitive. Since C.0 .dA /nH/ is a direct sum of spaces of oldforms and newforms, we have X  .dA =m/ ı 0 .m; /: ı.dA ; / D mjdA

Inverting this formula yields ı 0 .dA ; / D

X

ˇ .dA =m/ ı.m; /:

mjdA

t u

The lemma now follows directly. Lemma 8.20 For a squarefree integer d > 1 we have X

ˇ .d=m/ D .1/!.d/ ;

mjd

where !.d/ denotes the number of prime divisors of d. Proof This follows immediately from the binomial theorem since X mjd

ˇ .d=m/ D

  !.d/ X !.d/ D .1  2/!.d/ D .1/!.d/ : .2/!.d/i i iD0

t u

288

8 Jacquet-Langlands Correspondence

Remark 8.21 The function is the inverse under convolution of the function 1, constantly equal to 1 on N. Lemma 8.20 is the formula ˇ1D which is the same as saying ˇ  1 D  .  1/ D . In the same way the function ˇ is the inverse under convolution of 1  1 D d, where d is the divisor function: d.n/ D 2!.n/ . We deduce that ˇ  d D ı;

(8.16)

where ı – the unit for convolution product – is the function which is everywhere zero except at 1, where it equals 1. Lemmas 8.19 and 8.20 imply X mjdA

X

ˇ .dA =m/

h.rk /

fk 2C˚C.0 .m/nH/

D h.r0 / C

X

h.rk /

fk 2C.0 .m/nH/

mjdA

X

D

X

ˇ .dA =m/ h.rk /:

fk 2C˚Cprim .0 .dA /nH/

We are thus led to establish the following identity X

h.rk / D

fk 2L2 .O nH/

X

ˇ .dA =m/

X

h.rk /;

(8.17)

fk 2C˚C.0 .m/nH/

mjdA

since it is equivalent to (8.14). To this end we compare the right-hand sides of the trace formulas given in Propositions 8.15 and 8.17. We do so with the help of the following lemmas, which treat the contributions of each of the conjugacy classes (identity, elliptic, hyperbolic and parabolic) appearing in these formulas. Lemma 8.22 We have area.O / D

X

ˇ .dA =m/ area.0 .m//:

mjdA

Proof According to Lemma 8.16 and Exercise 2.18, we must show that dA

Y Y 1 X 1 1 1C D : ˇ .dA =m/ m p p

pjdA

mjdA

pjm

8.4 Jacquet-Langlands Correspondence and Applications

289

This is equivalent to the identity Y Y 1 X 1 1C 1 D ;  .dA =m/ m p p

dA

pjdA

mjdA

pjm

which is clearly dA

Y 1  X X 0 Y 1 D : 1C 1 d p p 0 0 mjdA d jm

pjdA

(8.18)

pjd

Since both sides of this equation are multiplicative arithmetic functions in the integer dA , it suffices to verify (8.18) for dA prime. This is a trivial verification. u t In the elliptic and hyperbolic contributions to the trace formulas, only the numbers E0 .t; 1;  / depend on the group  . The following lemma identifies these terms. Lemma 8.23 For every trace t, we have E0 .t; 1; O1 / D

X

ˇ .dA =m/ E0 .t; 1; 0 .m//:

mjdA

Proof The elliptic and hyperbolic cases can be treated in the same way. We may therefore assume that t > 2. Since E.3; 1; O1 / D E0 .3; 1; O1 /, a simple induction on t starting with (8.10) shows that it suffices to verify E.t; 1; O1 / D

X

ˇ .dA =m/ E.t; 1; 0 .m//:

(8.19)

mjdA

Let us show more generally that if ˝ is an order in a quadratic field extension of Q, then X E.˝; O1 / D ˇ .dA =m/ E.˝; 0 .m//: (8.20) mjdA

According to Lemma 8.14 the expression (8.20) indeed implies (8.19); moreover, from Theorem 8.8 we see that the above expression is equivalent to Y pjdA

1

 ˝  p

D

X mjdA

ˇ .dA =m/

 ˝  Y : 1C p

(8.21)

pjm

The discriminant dA is the product of an even number, say 2r, of primes. Let us put ˇ ˇ ˇ ˇ ˝  ˝  ˇ ˇ ˇ ˇ ˇ ˇ ˇ k D ˇ p j pjdA ; D 1 ˇ and e D ˇ p j pjdA ; D 0 ˇˇ : p p

290

8 Jacquet-Langlands Correspondence

It is clear that (  ˝  Y 0 1 D p 22re pjdA

if k > 0; otherwise.

Furthermore, all of the non-zero terms of the right-hand sideof  (8.21) correspond to ˝ the various possible products of the k C e primes such that p ¤ 1, which then leads to the expression X mjdA

   k e X  ˝  X Y e 2rij i j k 1C D ˇ .dA =m/ .2/ 2 1 j i p jD0 iD0 pjm

D2

2re

e X

.1/ 2

j ej

jD0

 X   k e i k : .1/ j i iD0

The expression (8.21) now follows from the binomial theorem, and more precisely from e X

.1/ 2

j ej

jD0

  e D .2  1/e D 1 j

and k X iD0

  ( .1  1/k D 0 k D .1/ i 1 i

if k > 0; if k D 0: t u

The parabolic terms only arise in the context of the groups 0 .m/; their contribution to (8.17) is of the form X

ˇ .dA =m/ 2

mjdA

!.m/

  X CC f . p/ ; pjm

where C is a constant (independent of m) and f depends only on p. The following lemma shows that this contribution vanishes. Lemma 8.24 If dA > 1, we have X mjdA

ˇ .dA =m/ 2!.m/ D 0:

8.4 Jacquet-Langlands Correspondence and Applications

291

Moreover, if f is an arbitrary function and if dA has at least two distinct prime divisors, then X X ˇ .dA =m/ 2!.m/ f . p/ D 0; mjdA

pjm

where the outer sum runs over all the divisors of dA and the inner sum runs over all prime divisors of m. Proof The first part follows from (8.16) or from the following computation: X

ˇ .dA =m/ 2!.m/ D

!.dA /

X

.2/!.dA /i  2i

iD0

mjdA

  !.dA / D .2  2/!.dA / D 0: i

We deduce from this that X X X X ˇ .dA =m/ 2!.m/ f . p/ D f . p/ ˇ .dA =mp/ 2!.mp/ mjdA

pjm

D2

mjdA =p

pjdA

X pjdA

f . p/

X mjdA =p

 dA =p !.m/ 2 ˇ D 0: m 

t u

We have thus proved formula (8.14); let us now show how to deduce the theorem from this. For this we fix r 2 iŒ1=2; 0 [ Œ0; C1Œ and write  D r2 C 1=4. Let ı > 0 be such that for every eigenvalue  ¤  of the hyperbolic Laplacian in L2 .O nH/ or in Cprim .0 .dA /nH/, we have j   j > ı. Given a constant C > 0, we put

hC .r/ D exp C.r2  r2 /2 :

(8.22)

The function hC is a test function to which formula (8.14) applies; it satisfies 0 < hC .r/ 6 1 for r 2 iR [ R and .j   j > ı and C > 2/ H) hC .r/ < eCı

2 =2

 h1 .r/:

(8.23)

Now recall that each of the sums which arise in (8.14) is absolutely convergent for every function h D hC ; in particular this is the case for h D h1 . Property (8.23) then implies, by letting C tend to infinity, that the multiplicity with which  appears in the spectrum of the hyperbolic Laplacian in L2 .O nH/ is equal to the multiplicity with which  appears in the spectrum of the hyperbolic Laplacian in Cprim .0 .dA /nH/. This completes the proof of the theorem. t u As a corollary of Theorems 7.1 and 8.18, we get the following result

292

8 Jacquet-Langlands Correspondence

Theorem 8.25 Let O be a maximal order in a quaternion algebra A D Da;b .Q/ with a; b > 0 of discriminant dA > 1. Then 1 .O nH/ >

5 : 36

Remark 8.26 The theorem remains true when O is replaced by a congruence subgroup. In this case as well, we expect to be able to replace 5=36 by 1=4 and this lower bound would then be optimal: Theorem 4.31 and the Jacquet-Langlands correspondence – extended to non-maximal subgroups – allows one to construct a quaternion algebra A of discriminant dA > 1 and a congruence subgroup   O such that 1 . nH/ D

1 : 4

8.5 Commentary and References For more on the Jacquet-Langlands correspondence, the reader can consult [43, 55, 65]. The first two of these references are written in the adelic language but the proof that we have given here is nevertheless based on the same approach: a comparison of two trace formulas.

§ 8.1 The reader is encouraged to look at [92, 132] for a more comprehensive study of the arithmetic of quaternion algebras.   Here are a few (non-trivial) examples – taken from Hejhal [55] – of maximal orders (  denotes the Legendre symbol). Example 8.27 Let a and b be two positive relatively prime integers. Assume that a and b are squarefree and such that 8 ˆ ab > 1 ˆ ˆ ˆ ˆ ˆ a  1 .mod 4/; b odd ˆ 0;

where B."; / D a..; //u .."; "//uC .."; "//;   t     e 0 10 1x : ; u .x/ D and a.t/ D uC .x/ D x1 01 0 et

(9.29)

Henceforth the set B."; / will denote the above open neighborhood of the identity in PSL.2; R/. Theorem 9.25 Let  be a probability measure on X D C   nG=Kp which is invariant under the action of the diagonal group A  feg D

˚  0  0 

 feg  G:

Assume that  is 1. of strongly positive entropy; 2. Tp -recurrent. Then  is GL.2; R/-invariant. The proof of Theorem 9.25 is difficult and technical (it forms the heart of the proof of Theorem 9.24) but has nothing to do with spectral theory or arithmetic; it is purely ergodic theoretic along the lines of Theorem 9.16. We shall content ourselves here with the more modest goal of showing how the underlying arithmetic, realized by the action of the Hecke operators, allows one to verify entropy hypotheses 1 and 2, which are otherwise hard to get a hold of.

9.6 Use of Hecke Operators In this section we explain why any arithmetic quantum limit satisfies the hypotheses of Theorem 9.25. We begin by checking the strongly positive entropy hypothesis. We must show that the density j j2 d associated with a Hecke eigenfunction is sufficiently “diffuse”. We begin by showing that on the tree Tp , the distributional density defined by a Hecke eigenfunction cannot give too much mass to a vertex or a geodesic axis. In what follows we shall express this property by saying that an eigenfunction cannot be too concentrated on a vertex or along a geodesic axis. The idea is to find enough Hecke translates of the balls B."; / which are pairwise “almost” disjoint.

328

9 Arithmetic Quantum Unique Ergodicity

9.6.1 Local Contributions Let p be a prime belonging to P (see § 9.5). Denote by e the base point in the tree Tp D PGL.2; Qp /=PGL.2; Zp / corresponding to the class of the lattice L0 . The Hecke operators arise in the following way: an eigenfunction f W Tp ! C of the Hecke operator ıp cannot be too concentrated on a vertex (Proposition 9.26); moreover, if f is the restriction of a Maaß form, we show that it cannot be too concentrated along a geodesic axis of Tp passing through e (Proposition 9.27).

Local Contribution I In this first subsection we prove that an eigenfunction for the Hecke operator ıp on the tree Tp cannot be too concentrated on a vertex. This property is true more generally for eigenfunctions of the Laplace operator on a regular tree. Let f W Tp ! C. Define ıpk f .x/ D

X

f .y/;

dTp .x;y/Dk T

where dTp is the combinatorial distance in the tree Tp . Finally, write Bn p for the ball of radius n centered at e in the tree Tp equipped with the distance function dTp . Proposition 9.26 Assume that ıp f D f for some  2 R. Then there exists a constant c independent of  and p such that for all n > 0, X

j f .y/j2 > cnj f .e/j2 :

(9.30)

Tp y2Bn

Proof It is clear that ıpk ı ıp D ıpkC1 C pıpk1 if k > 2 and ıp ı ıp D ıp2 C . p C 1/ Id : Thus f is an eigenfunction for all the ıpk . Write pk for the corresponding eigenvalue. The sequence .pk / satisfies the linear recurrence relation associated with the degree 2 equation X 2  X C p. We then distinguish two cases, according to whether p p jj > 2 p or jj 6 2 p.

9.6 Use of Hecke Operators

329

ˇ p ˇ p We begin by assuming that jj > 2 p. We then write cosh ˛ D ˇ=2 pˇ. The p p two roots of the polynomial X 2  X C p are then sgn./ p e˛ and sgn./ p e˛ and a simple calculation gives  1  pe2˛ k˛ 1  pe2˛ k˛ e C e sgn./pk=2 : pk D p.1  e2˛ / p.1  e2˛ /

With the convention that p0 D 1 we thus obtain n X

p2k D pn

kD0

sinh.2n C 1/˛ > .2n C 1/pn : sinh ˛

In other words, ˇ ˇ ˇ ˇ

X dTp .e;y/2f0;2;:::;2ng

ˇ ˇ f .y/ˇˇ > .2n C 1/pn f .e/:

The Cauchy-Schwarz inequality then implies X

j f .y/j2 > n2 j f .e/j2 ;

dTp .e;y/2f0;2;:::;2ng

yielding (9.30). p p Now assume that jj 6 2 p and write cos D =2 p. In the same way as above one may show that n X

p2k D pn

kD0

sin.2n C 1/ : sin

On the other hand, the Cauchy-Schwarz inequality implies X

j f .y/j2 >

dTp .e;y/D2k

ˇP ˇ2 ˇ ˇ ˇ d.e;y/D2k f .y/ˇ . p C 1/p2k1 D

jp2k j2 j f .e/j2 : . p C 1/p2k1

Subtracting (9.31) with n D k  1 from (9.31) with n D k we get

p2k D p

k

 sin.2k  1/ sin.2k C 1/  : sin p sin

(9.31)

330

9 Arithmetic Quantum Unique Ergodicity

Thus X

j f .y/j2 > cj f .e/j2

d.x;y/D2k

if .2k C 1/ 2 Œ2=5; 3=5 .mod /. If n is sufficiently large (n > c1 = ), we can assume that jfk j 1 6 k 6 n; .2k C 1/ 2 Œ2=5; 3=5 .mod /gj > c2 n so that (9.30) is satisfied for some constant c. On the other hand, if n is sufficiently small (n 6 c3 = ) we have sin.2nC1/  n so that (9.31) implies sin ˇ ˇX ˇ ˇ n ˇ p2k ˇˇ > npn : ˇ kD0

This gives X

j f .y/j2 > cn2 j f .e/j2 > cnj f .e/j2 :

y2BT2n

We can thus choose a constant c so that (9.30) always holds.

t u

In order to determine the local contribution away from a geodesic in Tp we shall use the hypothesis that f is the restriction of a Maaß form. In this way we can appeal to the approximation to the Ramanujan-Petersson conjecture established in § 7.6.

Ramanujan-Petersson Conjecture Suppose firstly that O1 D SL.2; Z/ and write  for this group. Let  W  nH ! C, .z/ D

X

p ar y K .2jrjy/e.rx/;

r2Z

be a non-constant Maaß form, which we assume to be an eigenfunction for all Hecke operators and normalized so that ar D 1. Then  is primitive and cuspidal, and we have Sp  D

p

p ap :

p Let  D p ap . Fix a “Tp -leaf” embedded in  nH. The restriction of  to this leaf defines a function f W Tp ! C which is a ıp -eigenfunction with eigenvalue . The

9.6 Use of Hecke Operators

331

Ramanujan-Petersson conjecture implies that for any " > 0,   jj D O" p1=2C" ; where the implied constant is independent of p. The Ramanujan-Petersson conjecture is still open. Nevertheless, Theorem 7.42 implies that there exists a universal constant ı > 0 such that   jj D O p1ı : (9.32) Neither ı nor the implied constant depends on p. This result remains true for more general O1 and p … P. Just as for Theorem 8.25, the Jacquet-Langlands correspondence (sufficiently refined) allows us to reduce to the case of a Hecke-Maaß newform for a certain group 0 .N/, where p does not divide N.

Local Contribution II Let Ap be the geodesic axis in the metric space Tp of the element   p0 2 PGL.2; Qp /: 01 Then Ap is the line in Tp passing through the vertices Œ. pn Zp /  Zp  2 Tp , n 2 Z. Figure 9.1 – realized by U. Goertz – represents an axis in the tree T2 . Fig. 9.1 An axis in the tree T2

332

9 Arithmetic Quantum Unique Ergodicity

We introduce two subsets of “transversal” vertices to the axis Ap . First let S1 be the set of p  1 neighbors of e in Tp not belonging to Ap . Next let S2 be the union over all the x 2 S1 of the p neighbors of x different from e. We have jS1 j D p  1 S1 D

and jS2 j D p2  p;

  ˇ pb ˇ eˇ0 0 such that for any 2 C1 .X/ of L2 -norm 1 which is an eigenfunction for all Hecke operators Sp , for all x 2 X and for all " > 0, we have .xB."; 0 // 6 c" ; where d .x/ D j .x/j2 dm.x/ and dm is a fixed Haar measure on SL.2; R/. Proof Fix x 2 X. It follows from Proposition 9.30 that the function X

1ˇxB.";0 /

ˇ2R"

is everywhere less than or equal to c0 . Multiplying this inequality by j j2 and integrating over a fundamental domain for the action of O1 on SL.2; R/, we obtain X

.ˇxB."; 0 // 6 c0 :

ˇ2R"

Now recall that the set R" is a union of sets Rp for p running through P" . For every p 2 P" , we have Z

X

.ˇxB."; 0 // D

ˇ2Rp

X B.";0 / ˇ2R p

j .ˇxb/j2 dm.b/:

But it follows from Proposition 9.27 that there is a universal constant c1 such that for every b, X

j .ˇxb/j2 >

ˇ2Rp

We deduce from this that X .ˇxB."; 0 // > ˇ2Rp

c1 j .xb/j2 : p12ı

c1 12ı p

.xB."; 0 //:

338

9 Arithmetic Quantum Unique Ergodicity

Summing over the set of p 2 P" , we find X

.ˇxB."; 0 // > c1

ˇ2R"

X

 .xB."; 0 //; 12ı 1

p

p2P"

whence .xB."; 0 // 6

c0 c1

X p2P"

1

1

p12ı

:

(9.35)

Note that the set of primes p … P is finite and that the sum X pjD"

1 p12ı

;

being less than or equal to log D" , is uniformly bounded from above by log c0 C log "1 , where c0 is given by Lemma 9.28. Moreover, from the prime number theorem we see that X

1

p4 6c"1

p12ı

Z 6

c1=4 "1=4 1

d .x= log x/ D " x12ı

for a certain constant  > 0 depending only on c and ı. Inserting these estimates into (9.35) we deduce the theorem. All of the above constants depend a priori on x but, by the compactness of X, we can choose them uniformly. t u Corollary 9.33 The microlocal lift of any quantum arithmetic limit is of strongly positive entropy. Proof Let  be a quantum arithmetic weak limit of a sequence ji .x/j2 d .x/, where each i 2 C1 .O1 nH/ is of L2 -norm 1, is a Laplacian eigenfunction i D i i (i D 1=4 C ri2 ! C1) and an eigenfunction of all Hecke operators Sp for p 2 P. From Theorem 9.3 (and its proof), each measure ji .x/j2 d .x/ lifts to a measure e  i D ji j2 dm on X D O1 nSL.2; R/, where 1

1=2

Œri

X

.i /n : i D q 1=2 2Œri  C 1 nDŒri1=2   of . The Moreover, the sequence e  i converges weakly to the microlocal lift e function i being itself an eigenfunction of all Hecke operators, we can apply Theorem 9.32 to it. Passing to the weak limit we find that the microlocal lift of  is of strongly positive entropy. t u

9.7 Commentary and References

339

In particular, Corollary 9.33 implies the following result. Corollary 9.34 Let S D O1 nH be as above. Let  be a quantum arithmetic limit on S and the singular support of  with respect to Lebesgue measure . Then if is contained in a finite union of points and closed geodesics, we have D ¿.

9.6.4 Tp -Recurrence To conclude we fix a prime p 2 P and note that the proofs of Corollary 9.33 and Proposition 9.26 imply the following proposition, where c is the constant given in Proposition 9.26.  its Proposition 9.35 Let  be a quantum arithmetic limit on S D O1 nH and e microlocal lift to X. Let n 2 N, x 2 X. Let r > 0 be sufficiently small. Then X

e .Br .y// > cn .Br .x//;

(9.36)

y2t.x;BTn /

where t.x; BTn / denotes the set of points in the Tp -leaf passing through x and at distance 6 n from x in Tp . Similarly to § 9.2 we can “disintegrate” the measure e  along Tp -leaves. Proposition 9.35 then implies that for e  -almost all x 2 X, e 1 x;T .T/ D 1: . This in turn is equivalent – again as in § 9.2 – to the Tp -recurrence of e From Corollary 9.33 we can thus apply Theorem 9.25 to the measure e  on X, concluding the proof of Theorem 9.24.

9.7 Commentary and References The quantum unique ergodicity conjecture is due to Rudnick and Sarnak [108]. It is a remarkable conjecture: it posits that the chaotic aspect of the geodesic flow on S manifests itself at the quantum level. Indeed, we can think of the equation  D  as the quantization of the Hamiltonian equation generating the geodesic flow, here  D 1=„2 with „ equal to the Planck constant (or quantization level). Conjecture 9.1 followed the work of Šnirel0 man [125], Zelditch [144] and Colin de Verdière [29], who proved a weak version: Corollary 9.8. The proof of their theorem is based on the notion of microlocal lift. The quantum unique ergodicity conjecture is surely out of reach for the time being. Nevertheless, spectacular progress has recently been made by Nalini Anantharaman [2] and, in the case of arithmetic surfaces, by Elon Lindenstrauss [82] by ergodic theoretic methods. In this chapter we have described a part of Lindenstrauss’s work by making the link with

340

9 Arithmetic Quantum Unique Ergodicity

general ergodic theoretic results, for the proofs of which we refer the reader to the original articles.

§ 9.1 The fact that the geodesic flow on a finite volume hyperbolic surface is mixing is a theorem of E. Hopf; see for example [7]. This gives a precise sense in which the geodesic flow is chaotic. The Weyl calculus and microlocal analysis are the subject of the books [90, 104]. The microlocal lift is usually defined via microlocal analysis. This is not the approach we have followed here. Our “by hand” construction of this lift then leaves in the background the term “microlocal”.

§ 9.2 The presentation that we give here of the microlocal lift is due to Zelditch [144], Wolpert [142] and Lindenstrauss [82]. We have primarily followed the article [82] of Lindenstrauss.

§ 9.4 This subsection is based on a text written by Sébastien Gouëzel for a study group on the work of Lindenstrauss. Theorem 9.13 is due to Rokhlin; for a proof, see for example [48]. We give four equivalent definitions of Tp -recurrence. The fourth is the definition that one can find in the article of Lindenstrauss [82], the first is the classical definition. For the construction of invariant measures supported on Cantor sets on the circle the reader can consult [50, Th. 4.1.1]. Historically, Theorem 9.16 is the first result in the style of Einsiedler, Katok, Lindenstrauss. It is due to Rudolph [109] and is motivated by an important work of Furstenberg [40]. The proof that we reproduce here is due to Host [58]. We have not defined the entropy of a measure. For that the reader can consult, for example, [50]. The entropy of fp is equal to Z 

S1

log '.x/ d .x/:

In particular, this integral must be strictly negative. An invariant measure can be decomposed into ergodic components (see [50, Th. 4.1.12]). We can then verify that if satisfies (9.22), then almost all ergodic components of are of entropy > . For the Birkhoff ergodicity theorem we refer the reader to [50, Cor. 4.1.9]. More generally than Proposition 9.19, we can show that a measure is Tp -recurrent if and only if all of its ergodic components are of positive entropy (see [82]).

9.7 Commentary and References

341

Taking p D 3, Theorem 9.16 and Proposition 9.19 imply Rudolph’s theorem in its original form: Theorem 9.36 Let be a measure on S1 which is invariant under multiplication by 2 and 3. Assume that every ergodic component of with respect to multiplication by 3 is of positive entropy (with respect to 3). Then is Lebesgue measure. The relation between Theorem 9.16 and the theorem of Einsiedler, Katok and Lindenstrauss is then obvious: one can think of the action of A1 as multiplication by 3 and the action of A2 as multiplication by 2.

§ 9.5 We have followed Serre’s book [117] to describe the tree associated with PGL.2; Qp /. Theorem 9.24 is proved in the case of compact hyperbolic surfaces by Lindenstrauss in [82]. The case of the modular surface presents an additional difficulty: the mass can escape to infinity. Soundararajan showed that this is not the case in [126]. One then finally obtains the general Theorem 9.24 which confirms the arithmetic version of the conjecture of Rudnick and Sarnak. It is conjectured that the spectrum of the Laplacian on an arithmetic hyperbolic surface has bounded multiplicities. Under this hypothesis it is not difficult to verify – using the quantum ergodicity theorem – that the quantum unique ergodicity conjecture for arithmetic surfaces follows from Theorem 9.24. Nevertheless, bounding the multiplicity of the spectrum seems, at present, to be completely out of reach. Compared with Theorems 9.11 or 9.16, we make an additional hypothesis on the measure  in the statement of Theorem 9.25. Indeed, in Theorem 9.11 we do not ask that  be Tp recurrent and in Theorem 9.16 we do not ask that  have strongly positive entropy. This difference comes from the fact that, in the case of Theorem 9.11, if is an ergodic A1  A2 invariant measure with positive entropy with respect to A1 , then is automatically recurrent with respect to the foliation induced by the second factor of SL.2; R/. Similarly, in the case of Theorem 9.16, if  is ergodic and Tp -recurrent then  is of positive entropy with respect to multiplication by 2.

§ 9.6 The idea of the proof that arithmetic quantum limits are of positive entropy goes back to the article of Rudnick and Sarnak [108]. The general result is due to Bourgain and Lindenstrauss [17] and the proof that we give is taken from Silberman’s thesis (issuing from joint work with Venkatesh, see [124] for a third approach). Lemma 9.28 reproduces [17, Lem. 3.3] and Lemma 9.29 reproduces [17, Lem. 3.2]. Theorem 9.32 is the main result of [17]. It directly implies the strongly positive entropy of arithmetic quantum limits. In the case of a general negatively curved Riemannian manifold, Nalini Anantharaman [2] showed that every quantum limit (not necessarily arithmetic!) is of positive entropy. This remarkable theorem implies that if  is a quantum limit,  has at least one ergodic component of positive entropy. Nevertheless, this does not allow one to deduce the statement of Corollary 9.33, for it does not rule out the possibility that other ergodic components of  are of zero entropy. Corollary 9.34 is due to Rudnick and Sarnak [108]. It is historically the first result in the direction of their conjecture.

342

9 Arithmetic Quantum Unique Ergodicity

9.8 Exercises Exercise 9.37 (Sensitivity to initial conditions) 1. Consider two geodesics 1 ; 2 W RC ! R2 in the Euclidean plane. Let " > 0 and T > 0. Assume that 1 and 2 coincide at t D 0 and are close to each other at time T, i.e., distEuc .1 .T/; 2 .T// 6 ". Show that distEuc .1 .2T/; 2 .2T// 6 2". 2. Now consider two geodesics 1 ; 2 W RC ! H in the hyperbolic plane. Let " > 0 and T > 0. Assume that 1 and 2 coincide at t D 0 and are close to each other at time T, i.e., .1 .T/; 2 .T// 6 ". Show that if " > 0 is fixed and T goes to infinity, we have9 :

.1 .T/; 2 .T//  2T: Exercise 9.38 Show that the space of K-finite functions is dense in C1 . nG/ and that it corresponds to the restriction to the unit tangent bundle (viewed as the level hypersurface jwj2 D 1=2 in H  R2 ) of functions in C1 .H/Œw; w. Here C1 .H/Œw; w is the space of polynomials in w and w with coefficients in C1 .H/. Exercise 9.39 Let .˛; ˇ/ be the geodesic in H with endpoints ˛; ˇ 2 P1 .R/ D R[f1g. We send the reader to § 4.4 for the notion of the size of a geodesic excursion with respect to the Farey triangulation of H. 1. Let ˛ 2 Œ1; 0/ and ˇ > 1. Show that the n-th excursion (beginning with that at infinity) of  D .˛; ˇ/ is associated with the rational number pn1 =qn1 D Œb0 ; b1 ; : : : ; bn1 ; 0; : : :  obtained by truncating the continued fraction expansion of ˇ to order n  1. 2. Show that a geodesic of S D PSL.2; Z/nH is compact if and only if the associated sequence is periodic, and that a geodesic in S is relatively compact if and only if its sequence of excursion sizes is bounded. 3. Deduce that there exist A-orbits in PSL.2; Z/nPSL.2; R/ which are relatively compact but not compact.

Thus, even a precise control on the geodesic flow over an interval Œ0; T cannot lead to an estimation on .1 .2T/; 2 .2T// (other than that implied by the triangle inequality). The future is independent of the past; this is a very simple manifestation of deterministic chaos.

9

Appendix A

Three Coordinate Systems for H

In this appendix we shall describe three systems of coordinates for the half-plane H that we use at various places in the text.

Polar Coordinates Let p0 2 H be an arbitrary base point. For any point p 2 H  fp0 g, there exists a unique geodesic parametrized by arc length  W Œ0; C1/ ! H with .0/ D p0 and passing through p. Let r D r. p/ be the distance from p0 to p, so that .r. p// D p. Furthermore, fix an arbitrary unit tangent vector v at p0 and let D . p/ 2 Œ; / be the directly oriented angle from v to the tangent vector along  at p0 . Then .r; / D .r. p/; . p// are the polar coordinates of p relative to the choice of the base point p0 and the vector v. In polar coordinates, the hyperbolic metric admits the following expression: ds2 D dr2 C sinh2 rd 2 :

(A.1)

Rather than taking . p/ in the interval Œ; /, we can view . p/ as an element of the unit circle S1 D R=Œs 7! s C 2:

Fermi Coordinates By replacing the reference to the base point p0 by a reference to a fixed geodesic, we obtain the Fermi coordinates. They are defined as follows. Let 0 W t 7! 0 .t/ 2 H be © Springer International Publishing Switzerland 2016 N. Bergeron, The Spectrum of Hyperbolic Surfaces, Universitext, DOI 10.1007/978-3-319-27666-3

343

A Three Coordinate Systems for H

344

a fixed geodesic line in the hyperbolic plane, parametrized by arc length dt. Then 0 separates H into two half-planes, one to the right and one to the left of 0 . For every point p 2 H we thus have a signed distance r from p to 0 : positive on one side, negative on the other, and zero if p 2 0 . In these coordinates the hyperbolic metric becomes ds2 D dr2 C cosh2 rdt2 :

(A.2)

We follow the following sign convention: if 0 is oriented, the signed distance r from a point p to 0 is negative if p is to the left of 0 and positive otherwise.

Horocyclic Coordinates The last coordinates we introduce are useful in the study of cusps. Let p0 be a point on the boundary of H, i.e., a point of the extended complex plane b C lying in b RD R [ f1g, the union of the real axis and the point at infinity. Euclidean circles (or lines if p0 D 1) contained in H [ fp0 g and passing through p0 are called horocycles with “center” p0 . The elements in SL.2; R/ send horocycles to horocycles. In the special case where p0 D 1, the horocycles with 1 as their center are horizontal Euclidean lines, and the vertical lines are geodesics having 1 as an endpoint. These vertical and horizontal lines intersect each other orthogonally. More generally, regardless of the position of the point p0 , the geodesics having p0 as an endpoint and the horocycles having p0 as their center form an orthogonal family. Now let h W t 7! h.t/ 2 H be a horocycle parametrized by arc length. The hyperbolic plane is again separated into two half-planes by h. We choose the parametrization of h for which p0 is to the left of h. By convention the signed distance r from a point p to h is negative if p is to the left of h, positive if p is to the right of h, and zero if p 2 h. There exists a unique real number t such that the geodesic passing through p and perpendicular to h meets h at h.t/. By definition, the horocyclic coordinates are .r; t/. In horocyclic coordinates the hyperbolic metric is expressed as ds2 D dr2 C e2r dt2 :

(A.3)

Appendix B

The Gamma Function and Bessel Functions

The Gamma Function The Euler Gamma function is defined for Re.s/ > 0 as the Mellin transform of the exponential function, see (1.26). Integration by parts produces the relation  .s C 1/ D s .s/: From this it follows, via a recurrence argument, that  .s/ admits a meromorphic continuation to all of C, whose only poles are simple and located at s D k for integer k > 0. Moreover, we have  .n/ D .n  1/Š for n > 1. We refer the reader to the book of Titchmarsh [129, particularly §4.41 and 4.42] for the proofs of the following properties (the last three are referred to collectively as Stirling’s formula): 1. We have  .s/ ¤ 0 for all s 2 C. 2. We have j . C it/j D .2/1=2 jtj 1=2 ejtj=2 .1 C Oa;b .t1 //; uniformly for all a 6 6 b and jtj > 1. 3. We have r  . / 

2 

e

(B.1)

(B.2)

as 2 R tends toward C1.

© Springer International Publishing Switzerland 2016 N. Bergeron, The Spectrum of Hyperbolic Surfaces, Universitext, DOI 10.1007/978-3-319-27666-3

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346

B The Gamma Function and Bessel Functions

4. We have 0 .s/ D log jsj C O" .s1 /; 

(B.3)

uniformly in every angle j arg.s/j 6   " The book by Whittaker and Watson [138] contains a mine of information on special functions. The Gamma function forms the subject of Chapter XII. In the following paragraph we shall state a few properties of Bessel functions used in the text. One can find proofs of these properties in [138, Chap. XVII] or [135].

Bessel Functions These are the solutions to the differential equation z2 f 00 C zf 0  .z2 C  2 /f D 0;

(B.4)

where  is a complex number. Equation (B.4) is singular at z D 0; we can eliminate this singularity by cutting the complex z-plane along the interval .1; 0. Equation (B.4) then admits two linearly independent solutions which are holomorphic in z 2 C  .1; 0. One of them is easily obtained as the sum of the series C1 X kD0

 z C2k 1 kŠ .k C 1 C / 2

(B.5)

which converges absolutely on the entire complex plane. We denote it by I .z/ and  is called the order of I . As a function of  the function I is entire. Replacing  with  leaves equation (B.4) invariant, so that the function I .z/ is another solution. The solutions I .z/, I .z/ are linearly independent if and only if the Wronskian W.I .z/; I .z// D 2.z/1 sin  is not identically zero, which is to say that  is not an integer. If  D n is an integer we have the relation In .z/ D In .z/:

(B.6)

To obtain a suitable pair of linearly independent solutions for arbitrary  we form the linear combination K .z/ D

 .sin /1 .I .z/  I .z//; 2

(B.7)

B The Gamma Function and Bessel Functions

347

while passing to the limit  ! n to define Kn . The functions I .z/ and K .z/ are always linearly independent since the Wronskian is W.I .z/; K .z// D z1 : For  D n integral, the function In .z/ is entire. If  is not an integer there is a discontinuity along the negative real axis. More precisely, we have I .x C "i/  I .x  "i/  2i sin./I .x/ for x > 0 when " tends toward 0. The Bessel functions of different orders are linked by the following recurrence relations: I1 .z/  IC1 .z/ D 2z1 I .z/; I1 .z/ C IC1 .z/ D 2I0 .z/; .z I .z//0 D z I1 .z/; .z I .z//0 D z IC1 : (For the K-Bessel functions the above formulae remain valid as long as we change the sign of the right-hand side.) The Bessel functions of order 1=2 are elementary functions: I1=2 .z/ D .2=z/1=2 sinh z;

K1=2 .z/ D .=2z/1=2 ez :

With the help of the above recurrence formulae we can find elementary expressions for the Bessel functions of order half an odd integer. For the K-Bessel functions we have the following asymptotic behavior: K .x/ 

 ./  x  as x ! 0: 2 2

(B.8)

21  ./ C O.x2 /; x

(B.9)

For 0 < x < 1, K .x/ D

where the constant in the O is uniform for  2 .0 ; 1=2/ and x 2 .0; 1=2/. For x > 1 C  2 K .x/ D

  1=2 2x

  1 C  2  ; ex 1 C O x

(B.10)

348

B The Gamma Function and Bessel Functions

where the constant in the O is uniform for  2 .0 ; 1=2/ and x > 2. More precisely, with D 4 2 r K .x/ 

 .  1/.  9/ 1  x C ::: : e C 1C 2x 8x 2Š.8x/2

There are various integral representations for Bessel functions. Here are a few for the K-Bessel functions:   Z 1 1  z  C1 2 C .t  1/1=2 etz dt 2 2 1   Z 1  z  C1 2 .t C 1/1=2 cos.tz/dt D  1=2   C 2 2 0   1=2  Z 1 1 z 1 t   C e e .t .1 C t=2z//1=2 dt D 2z 2 0 Z C1   z 1 exp  .t C 1=t/ t1 dt D 2 0 2 Z C1 D ez cosh t cosh.t/dt;

K .z/ D  1=2 

0

where Re.z/ > 0 and Re./ > 1=2. From the last integral representation we easily deduce the two following properties: 1. For fixed  and x > 0 the function K .x/ is positive and decreasing. 2. For fixed x > 0 and  > 0 the function K .x/ is positive and increasing.

Appendix C

Elementary Bounds on Hyper-Kloosterman Sums by Valentin Blomer and Farrell Brumley

Let q and n be integers such that q; n > 2, and let r be an integer prime to q. Define Sn .r; q/ D

X

.r/./n ;

(C.1)

 .mod q/

the star indicating that the sum is taken over primitive characters. For a character  mod q we denote by X

./ D

.x/e.x=q/

(C.2)

x .mod q/

the Gauß sum, where e.x/ D e2ix . The aim of this appendix is to prove the following elementary bound. Theorem C.1 Let p be a prime not dividing n and q D p2˛ with ˛ > 2. If r admits no n-th root r1=n in .Z=p˛ Z/ then Sn .r; q/ D 0; otherwise Sn .r; q/ D .q/q.n1/=2

X

  e nr1=n =q ;

(C.3)

 n 1 .mod p˛ /

where  is the Euler totient function.

(V. Blomer): Universiät Göttingen, Mathematisches Institut, Bunsenstr. 3-5, 37073 Göttingen, Allemagne. e-mail: [email protected] (F. Brumley): Institut Galilée, Université Paris 13, 99 avenue J.-B. Clément, 93430 Villetaneuse, France. e-mail: [email protected] © Springer International Publishing Switzerland 2016 N. Bergeron, The Spectrum of Hyperbolic Surfaces, Universitext, DOI 10.1007/978-3-319-27666-3

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Elementary Bounds on Hyper-Kloosterman Sums

Analogous formulae can be obtained for q D p2˛C1 with ˛ > 1 and p fixed. For simplicity, we only treat the case of even powers here. Note that the condition p − n can be eliminated if one is content with the upper bound Sn .r; q/ 6 nq.n1/=2 . We begin by relating Sn .r; q/ to the hyper-Kloosterman sum X

Kn .r; q/ D

e

x1 ;:::;xn .mod q/ x1 xn r .mod q/

x C    C x  1 n : q

(C.4)

Since r is prime to q, the above sum runs over all invertible residue classes mod q. Lemma C.2 For all q > 2 we have .q/1 Sn .r; q/ D Kn .r; q/. Proof The idea is to use Fourier inversion on the group G D .Z=qZ/ . For primitive characters , we interpret the Gauß sum ./ as the Fourier transform, evaluated at , of the function x 7! e.x=q/ defined on G. On the other hand, .q/1 Sn .r; q/ can be seen as the inverse Fourier transform, evaluated at r, of the b of all characters of G. The convolution function  7! ./n defined on the group G theorem (or just a simple change of summation order) then states that .q/1 Sn .r; q/ is the convolution of the function e.x=q/ iterated n times with itself. This multiconvolution of exponentials is nothing other than Kn .r; q/. For the above argument to go through, one must check that (C.1) can be extended to all characters. This is possible since ./ D 0 unless  is primitive. Indeed, if the conductor of  is pˇ with 0 6 ˇ < ˛, then writing x D y C pˇ z with y .mod pˇ / and z .mod p˛ˇ / we find X

./ D

X

.y C pˇ z/e

y (mod pˇ / z (mod p˛ˇ /

X

D

.y/e .y=p˛ /

y (mod pˇ /

 y C pˇ z 

X

p˛   e z=p˛ˇ D 0;

z (mod p˛ˇ /

t u

since the inner sum vanishes.

To prove Theorem C.1 it therefore suffices to show that if r has an n-th root r1=n in .Z=p˛ Z/ then Kn .r; q/ D q.n1/=2

X

  e nr1=n =q

.q D p2˛ /I

(C.5)

 n 1 .mod p˛ /

otherwise Kn .r; q/ D 0. Remark C.3 There are n  1 free variables in (C.4), each running through the q elements of Z=qZ. As each term is of size 1 in absolute value, a trivial bound on Kn .r; q/ is given by qn1 . Thus (C.5) expresses square-root cancellation in the number of free variables.

C Elementary Bounds on Hyper-Kloosterman Sums

351

When q D p is prime, the optimal bound jKn .r; p/j 6 np.n1/=2 is a deep result of Deligne [35]. Other the other hand, when q D p˛ with ˛ > 1, the exponential sums can be explicitly calculated, and in an elementary way. This is a very general principle that applies in a large variety of situations (see [64, Section 12]). Finally, note that we can easily obtain a bound on Kn .r; q/ which is intermediate between the trivial bound qn1 and the optimal bound q.n1/=2 . Indeed, the Gauß sum p ./ of a primitive character  mod q satisfies j./j D q. The expression (C.1) n=2 of Sn .r; q/ then implies jSn .r; q/j 6 .q/q , and from Lemma C.2 we see that jKn .r; q/j 6 qn=2 : To simplify the notation we write f .x/ D f .x1 ; : : : ; xn1 / D x1 C    C xn1 C

r : x1    xn1

Then Kn .r; q/ D

X

e. f .x/=q/;

(C.6)

x .mod q/

where the star indicates that the sum is taken over invertible residue classes. The gradient of f is given by the column vector of rational functions 0 B rf .x/ D B @

1 1

r .x1 xn1 /x1

:: :

r

1 C C: A

(C.7)

.x1 xn1 /xn1

Lemma C.4 (Taylor expansion) For j D 1; : : : ; n  1, let xj be an invertible class .mod p2˛ /. Let xj D yj C p˛ zj where yj and zj are determined mod p˛ , and yj is invertible. We put x D .xj /j and similarly for y and z. Then f .x/ D f .y C p˛ z/  f .y/ C p˛ rf .y/  z .mod p2˛ /;

(C.8)

where a  b denotes the usual scalar product of vectors. Proof The congruence (C.8) is valid for any rational function f as long as x; y are distinct from the poles of f , but we can also verify it directly for the particular function f by a direct computation. Indeed, by definition of f , ˛

f .y C p z/ D

n1 X jD1

˛

yj C p zj C r

n1 Y

.yj C p˛ zj /1 :

jD1

(C.9)

352

C

Elementary Bounds on Hyper-Kloosterman Sums

Now ˛ 2 2˛ 2 2˛ .yj C p˛ zj /.y1 j  p zj yj / D 1  p .zj =yj /  1 .mod p /;

so that ˛ 2 .mod p2˛ / .yj C p˛ zj /1  y1 j  p zj yj

and n1 Y

˛

.yj C p zj /

1



jD1

n1 Y

˛ 2 .y1 j  p zj yj /

jD1

 .y1    yn1 /

1

˛

 p .y1    yn1 /

1

n1 X

2˛ y1 j zj .mod p /:

jD1

t u

Inserting this last expression in (C.9) we obtain (C.8).

We now return to the proof of (C.5). We shall see that only the critical points of f contribute to the final expression for Kn .r; q/. These are the points where the exponential e. f .x/=q/ becomes “stationary”. This phenomenon is similar to that for oscillatory integrals over the reals, where only a small neighborhood around the stationary points contribute to the asymptotic behavior of the integral. The p-adic case which is the subject of this appendix is much simpler, and we obtain here not just an asymptotic formula but an exact one. Proof of Theorem C.1 We write Kn .r; p2˛ / D

X

X

 e

y .mod p˛ / z .mod p˛ /

D

X



e f .y/=p

y .mod p˛ /

2˛

f .y/ C p˛ rf .y/  z p2˛  X z .mod p˛ /



 rf .y/  z : e p˛ 

The inner sum vanishes unless rf .y/ is identically 0 mod p˛ , in which case it evaluates to p˛.n1/ . In other words, Kn .r; p2˛ / D p˛.n1/

X

e. f .y/=p2˛ /:

(C.10)

p˛ /

y .mod rf .y/0 .mod p˛ /

At this point, the proof already shows that this expression is independent of the choice of representatives y .mod p˛ /.

C Elementary Bounds on Hyper-Kloosterman Sums

353

We now look more closely at the solutions to rf .y/ D 0 .mod p˛ /. According to Definition (C.7) these are precisely the invertible classes y such that .y1    yn1 /yj  r .mod p˛ /

j D 1; : : : ; n  1:

This implies y1      yn1 .mod p˛ /, which gives ynj  r .mod p˛ / for 1 6 j 6 n1. If r admits no n-th root r1=n in .Z=p˛ Z/ , then f has no critical points; the sum in (C.10) is therefore empty and Kn .r; q/ vanishes. Otherwise the solutions to yn  r .mod p˛ / are given by r1=n , where  runs over a complete set of representatives of solutions to the equation yn D 1 .mod p˛ /. We are free to make a suitable choice of such representatives, and as p − n, Hensel’s lemma [47, Proposition 3.4.2] assures us that we can take  to satisfy  n D 1 .mod p2˛ /. Thus f .y; : : : ; y/ D f .r1=n .; : : : ; //  nr1=n  .mod p2˛ /: Inserting this into (C.10) we obtain (C.5).

t u

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Index of notation

j  j, 2 h;  iG , 206   , 271 p

b C, 5

C . nH/, 76 Cc1 .H/, 54 D . nH/, 72 E . nH/, 112 "D , 129 e.z/, 1 Qf , 215 f , 232 Qf , 233 fzrad , 58 0 .N/, 213 grad, 72 r, 55 h./, 62

h.r/, 64 HomG .; /, 205 J.1 ; 2 /, 231 k. /, 60 k.z; w/, 60 ./, 162 .n/, 235 N , 162 PW.C/, 67 b R, 6  ./, 230 Tn , 219 TnN , 219 Tp -recurrent, 315 tr.T/, 82 U.t/, 64 U.z; w/, 64 triv , 230

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Index

Abel transform, 65 Adjoint, 54 Admissible, 164 Arithmetic hyperbolic surface, 16 Arithmetic surface, 45 Automorphic kernel, 73 Axis, 160

Bessel functions, 346 Boundary of H, 344 Bounded operator, 54

Casimir operator, 303 Character, 206 Closed operator, 54 Cocompact group, 12 Collar, 202 Commensurable, 36 Commensurator, 218 Compact operator, 81 Compact part, 78 Completely integrable system, 298 Congruence subgroup, 17, 49 Conjugate, 35 Contour shift, 111 Cuspidal function, 57, 76 Cusps, 12, 14

Degree formula, 207 Density, 310 Dirichlet character, 230

Dirichlet convolution, 261 Dirichlet domain, 12 Discrete, 11 Discriminant, 268 of an algebra, 270 of a number field, 130 reduced, 269 Division algebra, 49

Eigenspace, 55 Eigenvalue, 55 Eigenvector, 55 Eisenstein series, 100 complete, 101, 235 incomplete, 107, 112 Eisenstein transform, 118 Elliptic matrix, 10 Equivalent quadratic forms, 36 representations, 205 Ergodic, 299 measure, 319 Euler totient function, 51 Even (Galois representation), 148 Exceptional Isomorphism, 37 Excursion, 133

Faithful representation, 210 Farey triangulation, 132 Fermi coordinates, 343 Fuchsian group, 11 of the first kind, 11

© Springer International Publishing Switzerland 2016 N. Bergeron, The Spectrum of Hyperbolic Surfaces, Universitext, DOI 10.1007/978-3-319-27666-3

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366 Fundamental discriminant, 270 domain, 12 solution to the heat equation, 67

Gamma function, 345 Geodesic flow, 296 Geodesics, 7 Geometric side of the trace formula, 156 Green’s formula, 55

Hadamard-De la Vallée Poussin theorem, 116 Hamiltonian equation, 297 flow, 296 Hecke character, 138, 139 Hecke operators, 125, 218, 325 Height, 31 Hermite-Mahler criterion, 32 Hilbert-Schmidt operator, 82 Horocycle, 344 Horocyclic coordinates, 344 Hyperbolic lines, 4 matrix, 10 metric, 5 surface, 11

Ideles, 189 Ikehara theorem, 117 Invariant integral operator, 61 measure, 317 operator, 53 subspace, 205 Irreducible representation, 205 Iwasawa decomposition, 32

Jacobi sums, 231

Index Maaß-Selberg relations, 113 Maximal order, 269 Measurable partition, 313 Mellin inversion, 107 transform, 107 Mercer’s theorem, 82 Microlocal lift, 305 Möbius function, 235 Modular surface, 16

Narrow class number, 131 Narrowly equivalent, 130 Newform, 221 Norm, 10, 127, 162 Normalized newform, 226

Observables, 298 Oldform, 220 Operator, 54 Optimal embedding, 273 Orbital integral, 156 Order, 109, 267 of index n, 271

Parabolic matrix, 10 Phragmén-Lindelöf principle, 108 Poincaré model, 4 Point-pair invariant, 60 Poisson summation formula, 2 Polar coordinates, 343 Pretrace formula, 159 Prime geodesic, 160 Prime number theorem, 116 Primitive character, 230 element, 21, 127, 160 ideal, 131 quadratic form, 129 Principal congruence subgroup, 44 of level N, 45 Principal parts, 78 Properly discontinuously, 11

K-finite, 302

Laplacian, 1, 14, 72 Lattice, 31 Left regular representation, 206 Legendre symbol, 271 Logarithmic integral, 184

Quantum limit, 295 Quaternion algebra, 35

Radial function, 58 Radialization, 58

Index Ramanujan-Petersson conjecture, 260, 330 Rankin-Selberg L-function, 239 Rapid decay at infinity, 240 Rational representation, 39 Reduced norm, 35 Representation, 205 Residual spectrum, 146 Riemann hypothesis, 146 Riemann zeta function, 25 Schrödinger equation, 300 Schwartz class, 2 Selberg kernel, 154 Selberg trace formula, 155, 162 Selberg transform, 64 Separable, 54 Siegel domain, 32 Size of an excursion, 133 Spectral side of the trace formula, 156 State, 299 Stirling’s formula, 109, 345 Strong approximation theorems, 272 Strong positive entropy, 319, 326 Symmetric kernel, 74 Symmetric operator, 54

367 Total principal part, 78 Trace, 35, 82 Trace class operator, 82 Transitive, 6 Twisted Eisenstein series, 234 Twisted Maaß form, 232

Unbounded operator, 54 Unimodular, 153 Unitary dual, 156

Volume, 31

Wave equation, 123 Weak- topology, 295 Weil explicit formula, 189 Weyl law, 182 Whittaker function, 57

Zariski topology, 43

Index of names

Abel, 65, 68, 162, 178, 190 Anantharaman, vii, 339, 341 Arthur, 188, 189 Atkin, 220, 221, 263 Bolte, vi, 23, 293 Bourgain, 341 Bump, 95, 145, 147 Buser, v, 19, 94, 96 Cheeger, 96 Cohen, 189 Colin de Verdière, 310, 339 Deligne, 217, 264 Dirichlet, 3, 12, 17, 76, 148, 182, 230, 232, 235, 239, 252–254, 257, 259–261, 264, 271, 287 Eisenstein, 99–101, 103, 105–107, 110, 112–114, 116, 118, 120–123, 126, 127, 135, 138, 146, 166, 168, 171, 234, 235, 239, 240, 245 Fermat, 50, 131 Furstenberg, 312, 340 Gamburd, 23, 211 Gauß, 190, 230–232, 263 Ghys, 9 Goldfeld, 147, 189 Hecke, 125–127, 138–140, 147, 149, 214, 217–221, 227, 229, 230, 251, 253, 260, 262, 322, 325–328, 330, 333, 337, 338 Heisenberg, 299 Hejhal, 21, 292 Hermite, 32, 41 Host, vii, 312, 340 Huber, 21 Iwaniec, v, 19, 22, 94–96, 146, 165, 189

Jacobi, 231, 263 Jacquet, vi, 23, 263, 267, 286, 331 Johansson, vi, 23, 293 Kloosterman, 264 Kuznetsov, 21 Landau, 261, 264 Langlands, v, vi, 23, 147, 148, 188, 217, 252, 267, 286, 331 Lax, 146 Lehner, 220, 221, 263 Li, 263 Lindenstrauss, vi, 19, 96, 125, 145, 146, 190, 295, 311, 317, 322, 326, 339–341 Luo, vi, 23, 96, 253, 262, 264 Maaß, 19, 20, 23, 24, 99, 113, 127, 138, 140, 141, 145, 147–149, 190, 213–216, 219–224, 227–230, 232, 239, 240, 252, 253, 260, 263, 286, 328, 330–332 Mahler, 32, 41 Margulis, 262, 311, 312 Miyake, 263, 272 Pell, 50, 129, 131, 136 Petersson, 260, 330, 331 Phillips, 96, 146, 190 Poincaré, 4, 9, 14, 16, 285, 298, 300 Ramanujan, 260, 330, 331 Randol, 190 Rankin, 214, 239, 240, 252, 260, 263, 264 Riemann, v, 3, 5, 6, 21, 22, 25, 101, 103, 106, 113, 116, 122, 127, 138, 146, 175, 190, 214, 217, 228, 252, 253, 274 Roelcke, 22, 96 Rudnick, vi, 23, 253, 262, 264, 311, 339, 341 Rudolph, vii, 340

© Springer International Publishing Switzerland 2016 N. Bergeron, The Spectrum of Hyperbolic Surfaces, Universitext, DOI 10.1007/978-3-319-27666-3

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370 Sarnak, vi, vii, 22–24, 96, 189, 190, 211, 253, 262, 264, 311, 339, 341 Schnirelman, 310, 339 Schrödinger, 295, 300 Selberg, v, 19–24, 64, 65, 67, 68, 93, 96, 113, 123, 124, 146, 148, 149, 153–157, 162, 164–166, 172, 181, 183, 185, 186, 188–191, 193, 196, 201, 213, 214, 239, 240, 251, 252, 260, 262–264, 267, 282, 283, 285, 308 Serre, 188, 211, 264 Shimura, 50 Siegel, 12, 14, 32, 50, 147, 190

Index of names Silberman, 341 Soundararajan, 341 Taylor, 148 Tunnell, 148 Venkatesh, vi, 19, 96, 125, 145, 146, 190, 341 Venkov, 21 Vignéras, 19 Weil, 147, 189, 217, 264 Weyl, 181, 182, 187, 189, 301 Wielonsky, 147 Wolpert, 96, 340 Zagier, 147 Zelditch, 308, 310, 339, 340