Lectures on the Mordell-Weil Theorem
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Jean-Pierre Serre
Lectures on the Mordell-Weil theorem
Jean-Pierre Serre
Lectures on the Mordell-Weil Theorem
Aspeds cl Mathematics Aspekte der Mathematik Editor: Klas Diederich
All volumes of the series are listed on pages 219-220
Jean-Pierre Serre
Lectures on the Mordell-Weil Theorem
Translated and edited by Martin Brown from notes by Michel Waldschmidt
Springer Fachmedien Wiesbaden GmbH
CIP-Titelaufnahme der Deutschen Bibliothek Serre, Jean-Pierre: Lectures on the Mordell-Weil theorem / Jean-Pierre Serre. Transl. and ed. by Martin Brown. From notes by Michel Waldschmidt. - Wiesbaden; Braunschweig: Vieweg, 1989 (Aspects of mathematics: E; Vol. 15)
NE: Waldschmidt, Michel [Bearb.]; Aspects of mathematics / E
Prof. Jean-Pierre Serre College de France Chaire d' Algebre et Geometrie 75005 Paris Ac\1S Subject Classification: 14 G 13, 14 K 10, 14 K 15 ISBN 978-3-528-08968-9 ISBN 978-3-663-14060-3 (eBook) DOI 10.1007/978-3-663-14060-3 All rights reserved © Springer Fachmedien Wiesbaden 1989
Originally published by Friedr. Vieweg & Sohn Verlagsgesellschaft mbR, Braunschweig in 1989.
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.
Produced by Wilhelm + Adam, Heusenstamm
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Foreword
This is a translation of "Auto ur du theoreme de Mordell-Weil", 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 manuscript. The present translation follows roughly the French text, with many modifications and rearrangements. We have not tried to give a detailed account of the new results due to Faltings, Raynaud, Gross-Zagier ... ; we have just mentioned them in notes at the appropriate places, and given bibliographical references.
Paris, Fall 1988
M.L.Brown J.-P. Serre
VII
CONTENTS
1. Summary. 1.1. Heights. 1.2. The Mordell-Weil theorem and Mordell's conjecture. 1.3. Integral points on algebraic curves. Siegel's theorem. 1.4. Balcer's method. 1.5. Hilbert's irreducibility theorem. Sieves.
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2. Heights. 2.1. The product formula. 2.2. Heights on Pm(K). 2.3. Properties of heights. 2.4. Northcott's finiteness theorem. 2.5. Quantitative form of Northcott's theorem. 2.6. Height associated to a morphism rj; : X - t P n . 2.7. The group Pic(X). 2.8. Heights and line bundles. 2.9. h c = 0(1) {:} c is of finite order (number fields). 2.10. Positivity of the height. 2.11. Divisors algebraically equivalent to zero. 2.12. Example-exercise:. projective plane blown up at a point.
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3. Normalised heights. 3.1. Neron-Tate normalisation. 3.2. Abelian varieties. 3.3. Quadraticity of hc on abelian varieties. 3.4. puality and Poincare divisors. 3.5. Example: elliptic curves. 3.6. Exercises on elliptic curves. 3.7. Applications to properties of heights. 3.8. Non-degeneracy. 3.9. Structure