Lectures on the Mordell-Weil theorem. Transl. and ed. by Martin Brown from notes by Michel Waldschmidt. (English) Zbl 0676.14005

Serre, Jean-Pierre

Aspects of Mathematics, E 15. Braunschweig etc.: Friedr. Vieweg & Sohn. x, 218 p. DM 52.00 (1989).

These notes are based on a course given by J.-P. Serre at the Collège de France in 1980 and 1981. The notes provide an introduction to and summary of many of the major ideas and results in the theories of rational and integral points on algebraic varieties.
Beginning with a study of height functions, the book progresses to a proof of the weak Mordell-Weil theorem using the Chevalley-Weil theorem. After a brief discussion of descent arguments the full Mordell-Weil theorem is proved.
Following this is a description of Mordell’s conjecture, and a discussion of the major results (Chabauty’s theorem, the Manin-Demyanenko theorem, Mumford’s theorem) and applications that preceded Faltings’ proof. - Integral points and quasi-integral points on curves are studied via Siegel’s theorem and Baker’s method. The study includes a discussion of effectivity, and applications to the arithmetic of curves.
The problem of lifting rational points under morphisms is introduced with the notion of thin sets and Hilbert’s irreducibility theorem. Applications to the construction of Galois extensions with certain prescribed Galois groups, and to the construction of elliptic curves of large rank are given. There is also a discussion of the large sieve, and applications to the study of thin sets.
Finally, an appendix contains some remarks about the class number 1 problem, and connections with elliptic and modular curves.

Reviewer: S. Kamienny (Los Angeles)

MSC:

14G05	Rational points
14H52	Elliptic curves
11G05	Elliptic curves over global fields
11G50	Heights
14H25	Arithmetic ground fields for curves
14-02	Research exposition (monographs, survey articles) pertaining to algebraic geometry
11-02	Research exposition (monographs, survey articles) pertaining to number theory
14G25	Global ground fields in algebraic geometry
14H45	Special algebraic curves and curves of low genus
14K15	Arithmetic ground fields for abelian varieties
11R27	Units and factorization

Keywords:

rational points; integral points; height functions; Mordell-Weil theorem; Mordell’s conjecture; lifting rational points; Hilbert’s irreducibility theorem; class number one problem