The Geometry of some special Arithmetic Quotients
The book discusses a series of higher-dimensional moduli spaces, of abelian varieties, cubic and K3 surfaces, which have embeddings in projective spaces as very special algebraic varieties. Many of these were known classically, but in the last chapter a n
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1637
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Bruce Hunt
The Geometry of some special Arithmetic Quotients
Springer
Author Bruce Hunt MPI fur Mathematik in den Naturwissenschaften Inselstr. 22-26 D-04103 Leipzig, Germany
Library of Congress Cataloging-in-Publication Data
Hunt, Bruce, 1958The geometry of some special arithmetic quotients! Bruce Hunt. p. cm. -- (Lecture notes in mathematics; 1637) Includes bibliographical references and index. ISBN 3-540-61795-7 (softcover alk. paper) 1. Threefolds (Algebraic geometry) 2. Moduli theory. 3. Surfaces, Algebraic. I. Title. II. Series Lecture notes in mathematics (Springer-Verlag) ; 1637. OA3.L28 no. 1637 [OA573) 510 s--dc20 [516.3'52] 96-41832 CIP
Mathematics Subject Classification (199 I): 14130, 14KIO, 14K25, I IF55, 22E40, 32M15, 12EI2 ISSN 0075-8434 ISBN 3-540-6 I795-7 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 1996 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: 10479879 46/3144-543210 - Printed on acid-free paper
A space section of the invariant quintic L5
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Contents 0 Introduction
1
1 Moduli spaces of PEL structures 1.1 Shimura's construction . . . . . . . . . . . . . . . . . . 1.1.1 Endomorphism rings . . . . . ......... 1.1.2 Abelian varieties with given endomorphisms 1.1.3 Arithmetic groups 1.1.4 PEL structures 1.2 Moduli spaces . . . · .. 1.2.1 Moduli functors. 1.2.2 Examples · .. 1.2.3 Coarse and fine moduli spaces. 1.2.4 Moduli spaces of abelian varieties. 1.3 Hyperbolic planes. . . . . . 1.3.1 The rational groups 1.3.2 Arithmetic groups 1.3.3 Modular subvarieties
15 15 15 16 18 20 22 22 22 25 26 29 29 31 34
Arithmetic quotients 2.1 Siegel modular varieties 2.1.1 The groups · .. 2.1.2 Com pactifications 2.1.3 Modular subvarieties 2.1.4 Commensurable subgroups 2.2 Picard modular varieties 2.2.1 The groups · ..... 2.2.2 Com pactification . . . 2.2.3 Modular subvarieties 2.3 Domains of type IV n ' . ... 2.3.1 The groups · ...... 2.3.2 A four-dimens
