Mare Hindry Joseph H. Silverman Departement de Matbematiques Department of Mathematies Universire Denis Diderot Paris 7 Brown·University 75251 Paris Providenee, RI 02912 Franee USA [email protected] [email protected] Editorial Board S. Axler F.W. Gehring K.A. Ribet Mathematies Department Mathematies Department Mathematies Department San Franeiseo State East Hall University of Califomia University University of Miehigan at Berkeley San Franeiseo, CA 94132 Ann Arbor, MI 48109 Berkeley, CA 94720-3840 USA USA USA With 8 illustrations. Mathematics Subject Classifieation (1991): 11 Gxx, 14Gxx Library of Congress Cataloging-in-Publication Data Hindry, Marc. Diophantine geometry : an introduction / Marc Hindry, Joseph H. Silverman. p. cm. - (Graduate texts in mathematics ; 201) Includes bibliographical references and index. ISBN 978-0-387-98981-5 ISBN 978-1-4612-1210-2 (eBook) DOI 10.1007/978-1-4612-1210-2 l. Arithrnetical algebraic geometry. I. Silverman, Joseph H., 1955- 11. Title. ID. Series. QA242.5.H56 2000 5 12 '.7--dc2 1 99-057467 Printed on acid -free paper. © 2000 Springer Science+Business Media New York Originally published by Springer-Verlag New York, Inc. in 2000 Softcover reprint of the hardcover I st edition 2000 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher Springer Science+Business Media, LLC, except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrievaI, electronic adaptation, computer software, or by similar or dissirnilar methodology now known or hereafter developed is forbidden. The use of general descriptive names, trade names, trademarks, etc., in this publication, even if the former are not especially identified, is not to be taken as a sign that such names, as understood by the Trade Marks and Merchandise Marks Act, may accordingly be used freely by anyone. Production managed by MaryAnn Brickner; manufacturing supervised by Jeffrey Taub. Typeset by the authots using Textures. 9 8 7 6 5 432 1 ISBN 978-0-387-98981-5 .
viii Preface Dedekind domain! This view ofnumber theory, sometimes dubbed "arith meticgeometry," hasenjoyed considerable success, but to fully understand and exploit its power requires a substantial background in Grothendieck stylealgebraicgeometry, with thematerialina booksuchasHartshorne  providing a bare beginning.
Contents Preface vii Acknowledgments viii Contents x Detailed Contents for Part A xiii Introduction 1 PART A The Geometry ofCurves and Abelian Varieties 6 A.1 Algebraic Varieties 8 A.2 Divisors 34 A.3 Linear Systems 49 A.4 Algebraic Curves 67 A.5 Abelian Varieties over C 91 A.6 Jacobians over C 110 A.7 Abelian Varieties over Arbitrary Fields 119 A.8 Jacobians over Arbitrary Fields 134 A.9 Schemes 151 PARTB Height FUnctions 168 B.1 Absolute Values 170 B.2 Heights on Projective Space 174 B.3 Heights on Varieties 183 B.4 Canonical Height Functions 195 B.5 Canonical Heights on Abelian Varieties 199 B.6 Counting Rational Points on Varieties 210 B.7 Heights and Polynomials 224 B.8 Local Height Functions 237 B.9 Canonical Local Heights on Abelian Varieties 241 B.10 Introduction to Arakelov Theory 243 Exercises 251.