Fourier Coefficients of Automorphic Forms
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865 Roelof W. Bruggeman
Fourier Coefficients of Automorphic Forms
Springer-Verlag Berlin Heidelberg New York 1981
Author
Roelof W. Bruggeman Mathematisch Instituut, Rijksuniversiteit Utrecht Budapestlaan 6, Utrecht, The Netherlands
AMS Subject Classifications (1980): 10D15, 10D40, 22 E40, 42A 16, 43A65, 44A 15 ISBN 3-540-10839-4 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-10839-4 Springer-Verlag New York Heidelberg Berlin
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© by Springer-Verlag Berlin Heidelberg 1981 Printed in Germany
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Table of contents Introduction 2
The universal covering group G
19
3
Representations
24
4
Automorphic models
33
5
Standard and Whittaker models
41
6
Intertwining operators
49
7
Fourier coefficients of automorphic models
61
8
The operators G a Eisenstein models
72
9
84
2
10
L ( I'\ G, X)
11
The analytic continuation of the Eisenstein model
12
Spectral decomposition of
93 e 2
L (I'\G,X)
T
103 J 25
13
The space H
133
14
Convolution operators
148
15
Scalar product of Poincare series
166
16
Sum formula
177
Literature
196
o
Index of notations
198
Index of terminology
201
I. Introduction.
1.1. A sum formula for the modular case
To give an idea of the subject of these notes I discuss in this subsection a simple case. Holomorphic modular cusp forms are well known; of many possible references I mention [23], [39], [42]. These cusp forms are holomorphic functions on the upper half plane h ; {z E Cllm z
> a},
(1.1.1)
satisfying the transformation property
Ct,S,y,o Ell, CtO-Sy; I, yz+u for some positive even number b, the weight, and having a Fourier series expansion fez) ; a e2rrinz. (1.1.2) The numbers a
; (yz+o)bf(z),
n
are
n
the Fourier coefficients of f. For a fixed weight b the
space of such cusp forms has finite dimension; it may be provided with the Petersson scalar product. For fixed n the map f be described as f
(f,p)
n p
an is linear on such a space, hence it may
for some cusp form p . By (.,.) n
p
I denote the Petersson
scalar product. Up to a multiple this Pn is given by a Poincare series. Petersson has given a formula for the Fourier coefficients of these Poincare series, see [ 28] or [23], p.298. The n-th Fourier coefficient of the m-th Poincare series of weight b equals
Ib Ib- I 00 -I 1 ro + 2rr(-l) 2 (n/m) 2 2 L S(n,m;c)c J (4rrc- "mn). m,n b_ 1 J _ is a Bessel function, and is a Kloosterman sum: b 1 (1.1.4) S(n,m;c); L e2rric-l(nx+mx-; x mod c (x,c);1 with xX=' I(c). (1.1.3)
Let f
I'" .,f be an orthonormal basis for the space of cusp fo
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