An Approach to the Selberg Trace Formula via the Selberg Zeta-Function
The Notes give a direct approach to the Selberg zeta-function for cofinite discrete subgroups of SL (2,#3) acting on the upper half-plane. The basic idea is to compute the trace of the iterated resolvent kernel of the hyperbolic Laplacian in order to arri
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1253 Jurgen Fischer
An Approach to the Selberg Trace Formula via the Selberg Zeta-Function
Springer-Verlag Berlin Heidelberg New York London Paris Tokyo
Author
Jurqen Fischer Mathematisches Institut EinsteinstraBe 62, 4400 Munster, Federal Republic of Germany
Mathematics Subject Classification (1980): 10D 12, 10D40, 10H 10, 58G25 ISBN 3-540-15208-3 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-15208-3 Springer-Verlag New York Berlin Heidelberg
This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. Duplication of this publication or parts thereof is only permitted under the provisions of the German Copyright Law of September 9, 1965, in its version of June 24, 1985, and a copyright fee must always be paid. Violations fall under the prosecution act of the German Copyright Law. © Springer-Verlag Berlin Heidelberg 1987 Printed in Germany Printing and binding: Druckhaus Beltz, Hemsbach/Bergstr. 2146/3140-543210
TABLE OF CONTENTS page Introduction The mathematical background 7
The contents of this volume 1.
Basic facts
14
1•1
Notations
14
15
1.2
Cofinite groups, fundamental domains
1.3
The linear operators
1 .4
The eigenvalue problem of automorphic forms, the resolvent of the differential operator -Zk
22
1 .5
Eisenstein series
28
1 .6
Spectral decomposition of expansion in k
2.
The trace of the iterated resolvent kernel
.
I [S,k],
D
-b k
multiplier systems
,
17
orthogonal series
35
40
-b k
41
2.1
Representation of the trace by the eigenvalues of
2.2
The contribution of the hyperbolic elements
2.3
The contribution of the elliptic elements
57
2.4
The
69
2.5
The resolvent trace formula
105
3.
The entire function selberg zeta-function
113
3.1
Definition and functional equation of the Selberg zeta-function
3.2
The growth of
3.3
The distribution of the eigenvalues of the Weyl-Selberg asymptotic formula
3.4
WeierstraB factorization of the entire function an analogue of the Euler-Mascheroni constant
4.
The general Selberg trace formula
of the parabolic elements
associated with the
=,
47
11 3
_
117
-Zk'
127
=,
145 162
Index
176
Index of notations
177
References
180
INTRODUCTION THE MATHEMATICAL BACKGROUND
In 1949 H. MaaB [Ma 1] extended the classical Riemann-Heeke correspondence between Dirichlet series with functional equation and automorphic forms. For that purpose he introduced a new class of automorphic functions which are real-analytic on the upper half-plane with respect to a certain
subgroup
r < PSL(2,IR) ,
ill,
automorphic
and satisfy the
wave equation for the Laplacian for the hyperbolic metric on
ill
(1)
with some parameter
A.
These MaaB wave forms turned out to be of key
importance for the subsequent development of the theory of modular forms and its applications to number theory (
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