Lectures on the Mordell-Weil Theorem (Aspects of Mathematics, 15)

Professor Jean-Pierre Serre ist ein renommierter französischer Mathematiker am Collège de France, Paris. 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 manuscript. The present translation follows roughly the French text, with many modi fications 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. M.L. Brown Paris, Fall 1988 I.-P. Serre VII CONTENTS 1. Summary. 1 1. 1. Heights. 3 1. 2. The Mordell-Weil theorem and Mordell's conjecture. 3 1. 3. Integral points on algebraic curves. Siegel's theorem. 4 1. 4. Baker's method. 5 1. 5. Hilbert's irreducibility theorem. Sieves. 5 2. Heights. 7 2. 1. The product formula. 7 2. 2. Heights on P m(K). 10 2. 3. Properties of heights. 13 2. 4. Northcott's finiteness theorem. 16 2. 5. Quantitative form of Northcott's theorem. 17 '2. 6. Height associated to a morphism Front Matter....Pages I-X Summary....Pages 1-6 Heights....Pages 7-28 Normalised Heights....Pages 29-48 The Mordell-Weil Theorem....Pages 49-57 Mordell’s Conjecture....Pages 58-80 Local Calculation of Normalised Heights....Pages 81-93 Siegel’s Method....Pages 94-107 Baker’s Method....Pages 108-120 Hilbert’s Irreducibility Theorem....Pages 121-136 Construction of Galois Extensions....Pages 137-151 Construction of Elliptic Curves of Large Rank....Pages 152-162 The Large Sieve....Pages 163-176 Application of the Large Sieve to Thin Sets....Pages 177-187 Back Matter....Pages 188-218 The book is based on a course given by J.-P. Serre at the Collège de France in 1980 and 1981. Basic techniques in Diophantine geometry are covered, such as heights, the Mordell-Weil theorem, Siegel's and Baker's theorems, Hilbert's irreducibility theorem, and the large sieve. Included are applications to, for example, Mordell's conjecture, the construction of Galois extensions, and the classical class number 1 problem. Comprehensive bibliographical references. Part of the "Aspects of Mathematics" series, this is an updated edition of the papers from the lectures on the Mordell-Weil theorem.