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

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 CollĂ¨ge 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.

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 |