Spectral Methods of Automorphic Forms

Abstract

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.

Estimating Additive Character Sums for Fuchsian Groups

“…See [Iw1] section 3.4 for these results. As already mentioned, E a (σ b z, s) has a simple pole at s = 1 with residue Vol(Γ\H) −1 .…”

“…We begin with a brief review of some background material (see [Iw1], [Sh], for example, for more details). Let Γ denote a Fuchsian group of the first kind, i.e., Γ ≤ P SL 2 (R) acts discretely on the upper half plane H = {z ∈ C | Im(z) > 0}, and every point on ∂H is the limit of some orbit.…”

“…If χ is trivial, then it has a simple pole at s = 1 with residue Vol(Γ\H) −1 . The importance of this series comes from, among other things, its role in the spectral theory of automorphic functions [Iw1] and its usefulness in deriving the analytic continuation of various L-functions.…”

On largeness and multiplicity of the first eigenvalue of finite area hyperbolic surfaces

“…Thus any λ 1 (S)-eigenfunction φ is a linear combination of residues of Eisenstein series (see [I]). It follows from [I,Thorem 6.9] that the y s term can not occur in the Fourier development (see (2.1)) of these residues in P t . Hence φ has a Fourier development in P t of the form (see §1):…”

“…M. N. Huxley [Hu] proved this conjecture for Γ n with n ≤ 6. Several attempts have been made to prove it (see [I,Chapter 11] for details) in the general case. The best known bound is 975 4096 due to Kim and Sarnak [K-S].…”