Stabile Modulformen und Eisensteinreihen
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1219 Rainer Weissauer
Stabile Modulformen und Eisensteinreihen
Springer-Verlag Berlin Heidelberg New York London Paris Tokyo
Lecture Notes in Mathematics Edited by A. Oold and B. Eckmann
1219 Rainer Weissauer
Stabile Modulformen und Eisensteinreihen
Springer-Verlag Berlin Heidelberg New York London Paris Tokyo
Autor
Rainer Weissauer Mathematisches lnstitut, Universitat Heidelberg 1m Neuenheimer Feld 288, 6900 Heidelberg, Federal Republic of Germany
Mathematics Subject Classification (1980): 10-XX
ISBN 3-540-17181-9 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-17181-9·Springer-Verlag New York Berlin Heidelberg CIP-Kurztitelaufnahme der Deutschen Bibliothek. Weissauer, Rainer: Stabile Modulformen und Eisensteinreihen 1Rainer Weissauer. - Berlin; Heidelberg; New York; London; Paris; Tokyo: Springer, 1986. (Lecture notes in mathematics; 1219) ISBN 3-540-17181-9 (Berlin ...) ISBN 0-387-17181-9 (New York ...) NE:GT 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, broadcas1ing, 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.
© Springer-Verlag Berlin Heidelberg 1986 Printed in Germany Printing and binding: Druckhaus Beltz, Hemsbach/Bergstr. 2146/3140-543210
INHALTSVERZEICHNIS 1 Introduction
. . . .
1
2 Stabile Modulformen
7
3 Differentialoperatoren
23
4 Automorphe Formen
45
5 Hyperebenen . .
51
6 Eisensteinreihen
59
7 Eisensteinreihen vom Klingenschen Typ
68
8 Ableitungen der Klingenschen Eisensteinreihen
80
9 Polstellen der Eisensteinreihen
84
10 Der Grenzfall k -- n+2j +l 11 Das holomorphe diskrete Spektrum yon
.97
L 2 (r n \G)
108
12 Der Operator M(p, s)
112
13 Stabile Liftungen
123
14 Die Siegelschen Eisensteinreihen
131
Literaturverzeichnis
142
Symbolverzeichnis
144
Schlagwortindex
146
III
1 INTRODUCTION The central theme of this book is the so called Siegel -operator arising in the theory of Siegel modular forms. Is F a holomorphic modular form of weight k on Siegel's upper half space
a, = { Z = z(n) = Z'
: Im(Z) >
of degree n, then the -operator given by ( F) (Z)
=
o}
lim F(
defines another
modular form F of the same weight on Siegel's upper half space of degree one less. If the weight is large enough every modular form of even weight on H n -
1
can be obtained in
this way. This was shown first by Maaf using the theory of Poincare series [26] and then later by Klingen [16] using Eisenstein series. For this one usually has to define Eisenstein series of the following type
G(Z) = Lg(1I"(M(Z))) det(GZ + D)-k M
where 9 is a cuspform on Hi of weight k, Here 11" denotes the projection of H n on Hi, which maps a matrix Z to its upper j by j submatrix, Finally M(Z) = (AZ + B)(GZ + D)-l denotes the action of a symplectic matrix M
=
on
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