Diophantine Approximation and Abelian Varieties
The 13 chapters of this book centre around the proof of Theorem 1 of Faltings' paper "Diophantine approximation on abelian varieties", Ann. Math.133 (1991) and together give an approach to the proof that is accessible to Ph.D-level students in number theo
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B. Eckmann, ZUrich F. Takens, Groningen
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B. Edixhoven J.-H. Evertse (Eds.)
Diophantine Approximation and Abelian Varieties
Springer
Editors Bas Edixhoven Institut Mathematique Universite de Rennes I Campus de Beaulieu F-35042 Rennes Cedex France t
Jan-Hendrik Evertse Universiteit Leiden Wiskunde en Informatica Postbus 9512 NL-2300 RA Leiden The Netherlands t
1st edition 1993 2nd printing 1997 (with minor corrections) 3rd printing 2003 Cataloging-in-Publication Data applied for Die Deutsche Bibliothek - CIP-Einheitsaufhahme DiopbantiDe .pproDlDatioD and abeliaa varieties : introductory lectures lB. Edixhoven ; 1. -H. Evense (eel.). - 2. printing. - Berlin ~ Heidelberg; New York; Barcelona.; Budapest; HonsJeons ; London; Mailand ; Paris; Santa Clara; Sinppur ~ Tokio: Springer, 1997 (I...edme DOles in matbcmalics ; Vol. Ij66) ISBN 3-540-57528--6
Mathematics Subject Classification (1991): 14G05, 14G40, I1J99, I1G35 ISSN 0075-8434 ISBN 3-540-57528-6 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. t
Springer-Verlag Berlin Heidelberg New York a member of BertelsmannSpringer Science+Businiess Media GmbH © Springer-Verlag Berlin Heidelberg 1993 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/editor 46/3111-543 - Printed on acid-free paper
Preface
From April 12 to April 16, 1992, the instructional conference for Ph.D-students "Diophantine approximation and abelian varieties" was held in Soesterberg, The Netherlands. The intention of the conference was to give Ph.D-students in number theory and algebraic geometry (but anyone else interested was welcome) some acquaintance with each other's fields. In this conference a proof was presented of Theorem I of G. Faltings's paper "Diophantine approximation on abelian varieties", Ann. Math. 133 (1991),549-576, together with some background from diophantine approximation and algebraic geometry. These lecture notes consist of modified versions of the lectures given at the conference. We would like to thank F. Oort and R. Tijdeman for organizing the conference, the speakers for enabling us to pu
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