Singular Modular Forms and Theta Relations
This research monograph reports on recent work on the theory of singular Siegel modular forms of arbitrary level. Singular modular forms are represented as linear combinations of theta series. The reader is assumed toknow only the basic theory of Siegel m
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Lecture Notes in Mathematics Editors: A. Dold, Heidelberg B. Eckmann, ZUrich F. Takens, Groningen
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Eberhard Freitag
Singular Modular Forms and Theta Relations
Springer-Verlag Berlin Heidelberg New York London Paris Tokyo Hong Kong Barcelona Budapest
Author Eberhard Freitag Mathematisches Institut Universitat Heidelberg 1m Neuenheimer Feld 288 W-6900 Heidelberg, FRG
Mathematics Subject Classification (1991): II F, 10D
ISBN 3-540-54704-5 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-54704-5 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 any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. © Springer-Verlag Berlin Heidelberg 1991 Printed in Germany Typesetting: Camera ready by author Printing and binding: Druckhaus Beltz, Hemsbach/Bergstr, 46/3140-543210 - Printed on acid-free paper
Table of contents Introduction
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
Chapter I. Siegel modular forms Introduction
. 8 11 13 16
1 The symplectic group. . . 2
The Siegel upper half space
3
Multiplier systems . . . .
4
Siegel modular forms . . .
5 6
The theta multiplier system The Siegel -operator Appendix to sec.6: Highest weights .
21 29 33
Chapter II. Theta series with polynomial coefficients
38
1
Coefficient functions and the Gauss transformation
38
2 The general inversion formula . . . . . . . . .
41 44 48 51 57
3
The action of the theta group on coefficient functions .
4
The Eichler imbedding . . . . . . . . . . . . . .
5
Theta series with respect to positive definite quadratic forms
6
Theta series with harmonic forms as coefficients . Appendix to sec.6: The theory of harmonic forms
7
Some multiplier systems
.
Chapter III. Singular weights
62 66
70
1
Jacobi forms
2
Jacobi forms and theta series
70 75
3
Differential operators . . . .
83
4
The classification of singular weights
85
Chapter IV. Singular modular forms and theta series
89
1
The big singular space
2
Theta series contained in M .
89 94
. . .
VI
3
Fourier-Jacobi expansion
99
4
The hidden relations . .
5
A combinatorical problem
106 108
Chapter V. The fundamental lemma
111
1
Formulation of the lemma. . . . . .
111
2
Reduction to the case of a local ring R
116
3
The case, where the sum of isotropic vectors is isotropic
118
4
The case r = 2. . . . . . . . . . . . . . . . . . .
122
5
An exceptional isotropic structure The case of a ground field. . . . An exceptional isotropic structure The case of a ground ring .
6
7
The cas
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