Mixed Automorphic Forms, Torus Bundles, and Jacobi Forms
This volume deals with various topics around equivariant holomorphic maps of Hermitian symmetric domains and is intended for specialists in number theory and algebraic geometry. In particular, it contains a comprehensive exposition of mixed auto
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Min Ho Lee
Mixed Automorphic Forms, Torus Bundles, and Jacobi Forms
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Author Min Ho LEE Department of Mathematics University of Northern Iowa Cedar Falls IA 50614, U.S.A. e-mail: [email protected]
Library of Congress Control Number: 2004104067
Mathematics Subject Classification (2000): 11F11, 11F12, 11F41, 11F46, 11F50, 11F55, 11F70, 14C30, 14D05, 14D07, 14G35 ISSN 0075-8434 ISBN 3-540-21922-6 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, specif ically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microf ilm 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 is a part of Springer Science + Business Media http://www.springeronline.com c Springer-Verlag Berlin Heidelberg 2004 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 specif ic 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 authors SPIN: 11006329
41/3142/du - 543210 - Printed on acid-free paper
To Virginia, Jenny, and Katie
Preface
This book is concerned with various topics that center around equivariant holomorphic maps of Hermitian symmetric domains and is intended for specialists in number theory and algebraic geometry. In particular, it contains a comprehensive exposition of mixed automorphic forms that has never appeared in book form. The period map ω : H → H of an elliptic surface E over a Riemann surface X is a holomorphic map of the Poincar´e upper half plane H into itself that is equivariant with respect to the monodromy representation χ : Γ → SL(2, R) of the discrete subgroup Γ ⊂ SL(2, R) determined by X. If ω is the identity map and χ is the inclusion map, then holomorphic 2-forms on E can be considered as an automorphic form for Γ of weight three. In general, however, such holomorphic forms are mixed automorphic forms of type (2, 1) that are defined by using the product of the usual weight two automorphy factor and a weight one automorphy factor involving ω and χ. Given a positive integer m, the elliptic variety E m can be constructed by essentially taking the fiber product of m copies of E over X, and holomorphic (m + 1)-forms on E m may be regarded as mixed automorphic forms of type (2, m). The generic fiber of E m is the product of m elliptic curves and is therefore an abelian variety, or a complex torus. Thus the elliptic variety E m is a complex torus bundle over t
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