Local Analysis of Selberg's Trace Formula
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1040
Anton Good
Local Analysis of Selberg's Trace Formula
S pringer-Verlag Berlin Heidelberg New York Tokyo 1983
Author Anton Good Forschungsinstitut fur Mathematik, ETH ZOrich 8092 ZOrich, Switzerland
AMS Subject Classifications (1980): 10015, 10G10, 22E40, 30F35, 33A 75, 43A80 ISBN 3-540-12713-5 Springer-Verlag Berlin Heidelberg New York Tokyo ISBN 0-387-12713-5 Springer-Verlag New York Heidelberg Berlin Tokyo This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to ·Verwertungsgesellschaft Wort", Munich. @) by Springer-Verlag Berlin Heidelberg 1983 Printed in Germany
Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2146/3140-543210
Table of contents 1.
Introduction
1
2.
Preliminaries
3.
Decompositions of
4.
Integral representation of eigenfunctions
26
5.
Fourier coefficients and Kloosterman sums
39
6.
Computation of some integrals I
47
7.
Poincare series and their Fourier series expansions
73
8.
computation of some integrals II
80
9.
Analytic continuations and functional equations
86
10.
Sum formulae (first form)
98
11.
Sum formulae (second form)
113
Literature
123
Index of notations
125
Index of terminology
128
14 G
17
1. Introduction In number theory identities and functional equations have played an eminent role for a long time. They often come in pairs as e.g. the 1-dimensional Poisson summation formula and the functional equation for the Riemann zeta-function. With their help many arithmetical functions have been studied. In these notes we derive a large number of identities from our investigations of certain meromorphic functions which satisfy functional equations relating their values at
sand
1-s • In very special cases these functions are essentially Heeke's zeta-functions with Gr6ssencharacters
of quadratic fields. In general.
the functional equations do no longer follow from a Poisson summation formula but basically stem from the fact that selfadjoint operators have only
£!!!
eigenvalues. Our results have links with problems from
different branches of mathematics. We mention just one example from the following fields: algebra: structure of discrete groups. analysis: spectral theory of Laplacians. geometry: distribution of geodesics on Riemann surfaces (cf. Example 3 in sect. 11). number theory: Mean-value theorems for zeta-functions (cf. [4J ,[8],[11]). automorphic forms: Rankin-Selberg convolution method for non-arithmetic groups (cf. [7J). The identities we are going to prove connect algebraic data of discrete groups with spectral data of differential operators. The general set up can be described as follows. Let which there is given a differential operator
S
be a manifold on
D and a measure
ill
2
A
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