Modular Forms on Half-Spaces of Quaternions
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1143 Aloys Krieg
Modular Forms on Half-Spaces of Quaternions
Springer-Verlag Berlin Heidelberg New York Tokyo
Author
Aloys Krieg Mathematisches Institut der Westfalischen Wilhelms-Universitat Einsteinstr. 62, 4400 Munster, Federal Republic of Germany
Mathematics Subject Classification (1980): 10020 ISBN 3-540-15679-8 Springer-Verlag Berlin Heidelberg New York Tokyo ISBN 0-387-15679-8 Springer-Verlag New York Heidelberg Berlin Tokyo
Library of Congress Cataloging-in-Publication Data. Krieg, Aloys, 1955 - Modular forms on halfspaces of quaternions. (Lecture notes in mathematics; 1143) Bibliography: p. Includes index. 1. Forms, Modular. 2. Quaternions. I. Title II. Series: Lecture notes in mathematics (SpringerVerlag; 1143. QA.L28 no. 1143 520 s 85-17284 [QA243] [512'.522] ISBN 0-387-15679-8 (U.S.) 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 1985 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2146/3140-543210
Table of contents v
Introduction Notations
XIII
Chapter I
Integral and Hermitian matrices
1
§1 §2 §3 §4 §5
Orderings
2
Hermitian matrices
14 20
MINKOWSKI's reduction theory
28
Applications of reduction theory
37
Chapter II
Modular group and fundamental domain
§1 §2 §3 §4
Symplectic group and half-space
42 43 54
Integral matrices
Modular group The fundamental domain
F(n;F)
58
Congruence subgroups
67
Chapter III
Modular forms
71
§1 §2 §3
Analytic class invariants
72
The vector space of modular forms
83
Modular forms with respect to congruence 92
subgroups
§4
Relations between SIEGEL modular forms, Hermitian modular forms and modular forms
96
of quaternions
99
Chapter IV
Theta-series
§1 §2 §3
Elementary properties of theta-series Theta-series as modular forms
100 110
Theta-series with respect to even quadratic forms
117
§4
Singular modular forms
126
Chapter V
EISENSTEIN- and POINCARE -series
136
§1 §2 §3 §4 §5
Integrals
137
EISENSTEIN-series
145
,
POINCARE -series
157
Metrization
166
Algebraical independence
174
IV
Chapter VI §1 §2 §3
Modular functions The field of modular functions Modular functions with respect to congruence subgroups and symmetric modular functions
182 183 189
Relations between SIEGEL modular functions, Hermitian modular functions and modular functions of quaternions
192
Bibliography
196
List of symbols
200
Index
202
Introduction The mathematical background
The theory of elliptic modular forms and functions developed from 19 t h century. One pro-
the investigation of elliptic functions in the ceeds from the upper half H:={z=x+iyElt;y>O}
(1)
in
It
By me
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