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Library of Congress Cataloging-in-Publication Data Iwaniec, Henryk. Spectral methods of automorphic forms / Henryk Iwaniec.-2nd ed. p. cm.-(Graduate studies in mathematics, ISSN 1065-7339 ; v. 53) First ed. published in Revista matemática iberoamericana in 1995. Includes bibliographical references and index. ISBN 0-8218-3160-7 (acid-free paper) 1. Automorphic functions. 2. Automorphic forms. 3. Spectral theory (Mathematics) I. Title. II. Series. QA353.A9 I88 2002 511.3 3-dc21 2002027749 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given.

Preface xi Chapter 0. Introduction 1 Chapter 1. The Classical Modular Forms 3 1.1. Periodic functions 3 1.2. Elliptic functions 6 1.3. Modular functions 10 1.4. The Fourier expansion of Eisenstein series 13 1.5. The modular group 15 1.6. The linear space of modular forms 18 Chapter 2. Automorphic Forms in General 23 2.1. The hyperbolic plane 23 2.2. The classification of motions 27 2.3. Discrete groups-Fuchsian groups 29 2.4. Congruence groups 34 2.5. Double coset decomposition 37 2.6. Multiplier systems 40 2.7. Automorphic forms 42 2.8. The eta-function and the theta-function 44 vu viii Contents Chapter 3. The Eisenstein and the Poincare Series 47 3.1. General Poincare series 47 3.2. Fourier expansion of Poincare series 49 3.3. The Hilbert space of cusp forms 52 Chapter 4. Kloosterman Sums 57 4.1. General Kloosterman sums 57 4.2. Kloosterman sums for congruence groups 58 4.3. The classical Kloosterman sums 59 4.4. Power-moments of Kloosterman sums 61 4.5. Sums of Kloosterman sums 65 4.6. The Salie sums 66 Chapter 5. Bounds for the Fourier Coefficients of Cusp Forms 69 5.1. General estimates 69 5.2. Estimates by Kloosterman sums 72 5.3. Coefficients of cusp forms with theta multiplier

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