Jean-Pierre Serre. Lectures on the. Mordell-Weil Theorem

Transcription

1 Jean-Pierre Serre Lectures on the Mordell-Weil Theorem

2 Aspeds cl Mathematics Aspekte der Mathematik Editor: Klas Diederich All volumes of the series are listed on pages

3 Jean-Pierre Serre Lectures on the Mordell-Weil Theorem Translated and edited by Martin Brown from notes by Michel Waldschmidt Springer Fachmedien Wiesbaden GmbH

4 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 Paris Ac\1S Subject Classification: 14 G 13, 14 K 10, 14 K 15 ISBN ISBN (ebook) DOI / All rights reserved Springer Fachmedien Wiesbaden 1989 Originally published by Friedr. Vieweg & Sohn Verlagsgesellschaft mbr, Braunschweig in 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

5 v 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 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

6 VII CONTENTS 1. Summary Heights The Mordell-Weil theorem and Mordell's conjecture Integral points on algebraic curves. Siegel's theorem Balcer's method Hilbert's irreducibility theorem. Sieves Heights The product formula Heights on Pm(K) Properties of heights Northcott's finiteness theorem Quantitative form of Northcott's theorem Height associated to a morphism rj; : X -t P n The group Pic(X) Heights and line bundles hc = 0(1) {:} c is of finite order (number fields) Positivity of the height Divisors algebraically equivalent to zero Example-exercise:. projective plane blown up at a point. 2E 3. Normalised heights. 2~ 3.1. Neron-Tate normalisation. 2~ 3.2. Abelian varieties. 3: 3.3. Quadraticity of hc on abelian varieties. 3~ 3.4. puality and Poincare divisors Example: elliptic curves. 3! 3.6. Exercises on elliptic curves Applications to properties of heights Non-degeneracy Structure of A(K): a preliminary result Back to 2.11 (c algebraically equivalent to zero) Back to 2.9 (torsion c). 4

7 VIII Contents 4. The Mordell-Weil theorem Hermite's finiteness theorem The Chevalley-Weil theorem The Mordell-Weil theorem The c1assical descent The number of points of bounded height on an abelian variety Explicit form of the weak Mordell-Weil theorem. 5. Mordell's conjecture Chabauty's theorem The Manin-Demjanenko theorem First application: Fermat quartics (Demjanenko) Second application: modular curves Xo(pn) (Manin) The generalised Mordell conjecture Mumford's theorem; preliminaries Application to heights: Mumford's inequality. 6. Local calculation of normalised heights Bounded sets Local heights Neron's theorem Relation with global heights Elliptic curves. 7. Siegel's method Quasi-integral sets Approximation of real numbers The approximation theorem on abelian varieties Application to curves of genus 2: Proof of Siegel's theorem Application to P(j(n)) Effectivity Bal.;:cr's method Reduction theorems Lower bounds for I:ßi log ai Application to P 1 - {O, 1, oo} Applications to other curves Applications to elliptic curves with good reduction outside 118 a given finite set of places.

8 Contents 9. Hilbert's irreducibility theorem Thin sets Specialisation of Galois groups Examples of degrees 2,3,4, Further properties of thin sets Hilbertian fields The irreducibility theorem: elernentary proof Thin sets in PI: upper bounds. 10. Construction of Galois extensions The method Extensions with Galois group Sn Extensions with Galois group An Further exarnples of Galois groups: use of elliptic curves Noether's method Infinite Galois extensions Recent results. 11. Construction of elliptic curves of large rank Neron's specialisation theorem Elliptic curves of rank ~ 9 over Q Elliptic curves of rank ~ 10 over Q Elliptic curves of rank ~ 11 over Q. 12. The large sieve Statement of the main theorem A lemma on finite groups The Davenport-Halberstam theorem Proof of the Davenport-Halberstarn theorem End of the proof of the main theorem. 13. Applications cf the large sieve to thin sets Statements cf results Proof of theorem l Proof of theorem Proof of theorem 3 from theorem 1. IX

9 x Appendix: The dass number 1 problem and integral points on modular curves. A.l. Historical remarks. A.2. Equivalent conditions for h( -p) = l. A.3. Orders of Rd. A.4. Elliptic curves with complex multiplication. A.5. Modular curves associated to normalisers of Cartan subgroups and their CM integral points. A.6. Examples. A.7. The Gel'fond-Linnik-Baker method. Bibliography. Index. Contents