Siegel's Modular Forms and Dirichlet Series Course Given at the Univ
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Hans MaaB Universitat Heidelberg, Heidelberg/Deutschland
Siegel's Modular Forms and Dirichlet Series Course Given at the University of Maryland, 1969-1970
Springer-Verlag Berlin· Heidelberg· NewYork 1971
AMS Subject Classifications (1970): lOD20, 10HlO
ISBN 3-540-05563-0 Springer-Verlag Berlin . Heidelberg . New York ISBN 0-387-05563-0 Springer-Verlag New York . Heidelberg . Berlin 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 the publisher, the amount of the fee to be determined by agreement with the publisher.
© by Springer-Verlag Berlin - Heidelberg 1971. Library of Congress Catalog Card Number 73-171870. Printed in Germany. Offsetdruck:Julius Beltz, HemsbachiBergstr.
It is the large generalization, limited by a happy particularity, which is the fruitful conception. Alfred North Whitehead
Dedicated to the last great representative of a passing epoch
CARL LUDWIG SIEGEL on the occasion of his seventy-fifth birthday
PREFACE
These notes present the content of a course I delivered at the University of Maryland, College Park, between September 1969 and April 1970.
The choice of the subject was mainly determined by my
intention to show how Atle Selberg makes fascinating use of differential operators in order to prove certain functional equations.
Of
course one has to be somewhat familiar with his theory of weakly symmetric Riemannian spaces, but - as Selberg himself pointed out to me the main idea can be found already in Riemann's work.
Since Selberg
never published his idea, it might be of some value for the mathematical community to make available to a wider public the methods which were originally conceived by Selberg a long time ago. In November 1970 Mrs. Audrey A. Terras sent me two preprints [33, 34J showing that she obtained independently some of the results
of §17 by similar methods. The audience of my course, strongly interested in the subject, had an influence on these notes through many valuable comments; Professor John Horvath undertook the editorial task of preparing them for publication, whereas the typing was done with supreme accuracy by Mrs. John Vanderslice.
I enjoyed in gratitude the stimulating at-
mosphere of the Department of Mathematics of the University of Maryland, and felt encouraged to make more effort. Heidelberg, May 1971 H. MaaS
CONTENTS
H.
Preliminary remarks on topological groups
1
§2.
Automorphism groups of bilinear forms
8
§ 3.
Geometry in the representation space
22
§4.
Symplectic geometry
31
§ 5.
Weakly symmetric Riemannian spaces
51
§ 6.
The Riemannian space of all positive matrices
63
§ 7.
A generalization of J (X)
84
§ 8.
The Riemannian space
§9.
The reduction theory of positive quadrati
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