Cubic Metaplectic Forms and Theta Functions
The book is an introduction to the theory of cubic metaplectic forms on the 3-dimensional hyperbolic space and the author's research on cubic metaplectic forms on special linear and symplectic groups of rank 2. The topics include: Kubota and Bass-Milnor-S
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1677
Lecture Nates in Mathematics Editors: A. Dold, Heidelberg F. Takens, Groningen
1677
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Nikolai Proskurin
Cubic Metaplectic Forms and Theta Functions
Springer
Author Nikolai Proskurin St. Petersburg Branch of the Mathematical Institute Russian Academy of Sciences Fontanka 27 St. Petersburg 191011, Russia e-mail: [email protected] Library of Congress Cataloging-in-Publication Data
Proskur in. Ni ko 1a t , 1953Cubic metaplectic forms and theta functlons / Nikolai Proskurin. p. cm. -- (Lecture notes t n mathematics; 1677) Includes bibl iographical references and lndex. ISBN 3-540-63751-6 (softcover alk. paper) 1. Automorphic forms. 2. Discontinuous groups. 3. Functions, Theta. I. Title. II. Series: Lecture notes in mathematics (Springer-Verlag) ; 1677. QA3.L28 no. 1677 [QA243] 510 s--dc21 [512' .7J 97-45869 CIP
Mathematics Subject Classification (1991): Primary: 11F55, IIF60, I1F30 Secondary: 33C 10 ISSN 0075-8434 ISBN 3-540-63751-6 Springer-Verlag Berlin Heidelberg New York 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 1998 Printed in Germany
The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting: Camera-ready TEX output by the author SPIN: 10553411 46/3143-543210 - Printed on acid-free paper
Preface
The subject of these notes does not have a long history. It takes its origin from the very short paper published by Kubota in 1965. Let F be a totally imaginary algebraic number field containing the full group of m t h roots of 1, denoted by Ji-m(F) , and let be the principal congruence subgroup module ideal q in SL(n, OF)' OF being the integers ring of F. Kubota showed [51] that, under some conditions on q, the reciprocity low yields if c =I 0 if c = 0 th is a group homomorphism --+ Ji-m(F); here we write ("":"")m for the m degree residue symbol. This theorem has very far-reaching consequences. In a series of papers [52], [53], ... , [57] Kubota studied automorphic forms under the group with the homomorphism above as a multiplier system (= a factor of automorphy), the so called metaplectic forms of degree TTL That are real analytic forms, in the sence of Maaf and Selberg, defined on n'. where H :::
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