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.
Inhaltsverzeichnis
1. Summary. - 2. Heights. - 3. Normalised heights. - 4. The Mordell-Weil theorem. - 5. Mordell s conjecture. - 6. Local calculation of normalised heights. - 7. Siegel s method. - 8. Baker s method. - 9. Hilbert s irreducibility theorem. - 10. Construction of Galois extensions. - 11. Construction of elliptic curves of large rank. - 12. The large sieve. - 13. Applications of the large sieve to thin sets. - Appendix: The class number 1 problem and integral points on modular curves. - A. 1. Historical remarks. - 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.
