Lectures on the Mordell-Weil Theorem

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Jean-Pierre Serre

Lectures on the Mordell-Weil Theorem Third Edition

Jean-Pierre Serre

Lectures on the Mordell-Weil Theorem

Asped~f

Ivbthematia

Edited by Klas Diederich Vol. E 3:

G. Hector/U. Hirsch: Introduction to the Geometry of Foliations, Part B

Vol. E 5:

P. Stiller: Automorphic Forms and the Picard Number of on Elliptic Surface

Vol. E 6:

G. Faltings/G. Wustholz et 01.: Rational Points*

Vol. E 9:

A. Howard/P.-M. Wong (Eds.): Contribution to Several Complex Variables

Vol. E 10: A. J. Tromba (Ed.): Seminar of New Results in Nonlinear Partial Differential Equations * Vol. E 15: J.-P. Serre: lectures on the Mordell-Weil Theorem Vol. E 16: K. Iwasaki/H. Kimura/S. Shimomura/M. Yoshida: From Gauss to Painleve Vol. E 17: K. Diederich lEd.): Complex Analysis Vol. E 18: W. W. J. Hulsbergen: Conjectures in Arithmetic Algebraic Geometry Vol. E 19: R. Rocke: lectures on Nonlinear Evolution Equations Vol. E 20:

F. Hirzebruch, Th. Berger, R. Jung: Manifolds and Modular Forms*

Vol. E 21:

H. Fujimoto: Value Distribution Theory of the Gauss Mop of Minimal Surfaces in Rm

Vol. E 22:

D.

V. Anosov/A. A. Bolibruch: The Riemann-Hilbert Problem

Vol. E 23: A. P. Fordy/J. Vol. E 24:

C. Wood (Eds.): Harmonic Mops and Integrable Systems

D: S. Alexander: A History of Complex Dynamics

Vol. E 25: A. Tikhomirov/A. Tyurin (Eds.): Algebraic Geometry and its Applications Vol. E 26:

H. Skoda/J.-M. Trepreau IEds.): Contributions to Complex Analysis and AnalytiC Geometry

Vol. E 27:

D. N. Akhiezer: lie Group Actions in Complex Analysis

Vol. E 28:

R. Gerard, H. Tahara: Singular Nonlinear Partial Differential Equations

Vol. E 29:

R.-P. Holzapfel: Boll and Surface Arithmetics

Vol. E 30:

R. Huber: Etale Cohomology of Rigid AnalytiC Varieties and Adic Spaces

* A Publication af the Max-Planck-Institut fUr Mathematik,

Bonn

Jean-Pierre Serre

Lectures on the Mordell-Weil Theorem Translated and edited by Martin Brown from notes by Michel Waldschmidt 3rd edition

Springer Fachmedien Wiesbaden GmbH

II Vleweg

Prof. lean-Pierre Serre College de France Chaire d'Algebre et Geometrie 75005 Paris AMS Subject Classification: 14 G 13, 14 K 10, 14 K 15 1st edition 1989 2nd edition 1990 3rd edition 1997

All rights reserved © Springer Fachmedien Wiesbaden 1997 Originally published by Friedr. Vieweg & Sohn Verlagsgesellschaft mbH, BraunschweiglWiesbaden, 1997

No part of this publication may be reproduced, stored in a retrieval system or transmitted, mechanical, photocopying or otherwise, without prior permission of the copyright holder.

Cover design: Wolfgang Nieger, Wiesbaden Printed on acid-free paper ISSN 0179-2156 ISBN 978-3-663-10634-0 DOI 10.1007/978-3-663-10632-6

ISBN 978-3-663-10632-6 (eBook)

v

Foreword

This is a translation of "Autour du theoreme de Mordell-Well", a course given by J.-P. Serre at the College de France in 1980 and 1981. These notes were originally written weekly by Michel Waldschmidt and have been reproduced by Publications Mathematiques de l'Universite de Paris VI, by photocopying the handwritten