Diophantine geometry. An introduction.

*(English)*Zbl 0948.11023
Graduate Texts in Mathematics. 201. New York, NY: Springer. xiii, 558 p. (2000).

In 1922 Mordell conjectured that every algebraic curve of genus \(\geq 2\) has at most finitely many rational points. This conjecture was proved by Faltings in 1983. In 1991 Vojta gave a completely different proof, based on diophantine approximation. Vojta’s proof was then simplified by Bombieri. In the textbook under review, the authors work towards Bombieri’s proof, giving the reader the necessary background. Unlike several other textbooks in this field, the prerequisites are quite modest, so the book is very useful for instance for a graduate course on diophantine geometry. Each chapter goes along with many exercises. We give a brief overview of the contents of the book.

In part A, the authors, starting from scratch, give the necessary algebraic-geometric background on curves, surfaces, abelian varieties and Jacobians.

Part B is about height functions. The authors discuss heights on algebraic varieties, canonical heights, canonical heights on abelian varieties, local height functions, and canonical local height functions on abelian varieties. Further, a proof of Mumford’s gap principle for rational points on curves of genus \(\geq 2\) is included.

In part C about rational points on abelian varieties, the authors prove the Mordell-Weil theorem stating that for any number field \(k\) the group of \(k\)-rational points on an abelian variety is finitely generated.

In part D, the basics of diophantine approximation are introduced. The authors give a proof of Roth’s theorem on the approximation of algebraic numbers by rational numbers, and then deduce Siegel’s theorem that algebraic curves of genus \(\geq 1\) have only finitely many \(S\)-integral points.

In part E about rational points on curves of genus \(\geq 2\) the authors give a detailed account of Bombieri’s proof of Mordell’s conjecture.

Lastly, in part F, about further results and open problems, the authors give an overview of some of the recent developments (without proofs) and discuss some of the main conjectures. Among other things the authors discuss Faltings’ results on rational points on subvarieties of abelian varieties, the work of Szpiro, Ullmo and Zhang on Bogomolov’s conjecture, effectivity problems, the abc-conjecture, the Bombieri-Lang conjecture, Vojta’s conjecture and related conjectures.

In part A, the authors, starting from scratch, give the necessary algebraic-geometric background on curves, surfaces, abelian varieties and Jacobians.

Part B is about height functions. The authors discuss heights on algebraic varieties, canonical heights, canonical heights on abelian varieties, local height functions, and canonical local height functions on abelian varieties. Further, a proof of Mumford’s gap principle for rational points on curves of genus \(\geq 2\) is included.

In part C about rational points on abelian varieties, the authors prove the Mordell-Weil theorem stating that for any number field \(k\) the group of \(k\)-rational points on an abelian variety is finitely generated.

In part D, the basics of diophantine approximation are introduced. The authors give a proof of Roth’s theorem on the approximation of algebraic numbers by rational numbers, and then deduce Siegel’s theorem that algebraic curves of genus \(\geq 1\) have only finitely many \(S\)-integral points.

In part E about rational points on curves of genus \(\geq 2\) the authors give a detailed account of Bombieri’s proof of Mordell’s conjecture.

Lastly, in part F, about further results and open problems, the authors give an overview of some of the recent developments (without proofs) and discuss some of the main conjectures. Among other things the authors discuss Faltings’ results on rational points on subvarieties of abelian varieties, the work of Szpiro, Ullmo and Zhang on Bogomolov’s conjecture, effectivity problems, the abc-conjecture, the Bombieri-Lang conjecture, Vojta’s conjecture and related conjectures.

Reviewer: Jan-Hendrik Evertse (Leiden)

##### MSC:

11Gxx | Arithmetic algebraic geometry (Diophantine geometry) |

14-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to algebraic geometry |

11-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to number theory |

14Gxx | Arithmetic problems in algebraic geometry; Diophantine geometry |

11G10 | Abelian varieties of dimension \(> 1\) |

11J68 | Approximation to algebraic numbers |

11G30 | Curves of arbitrary genus or genus \(\ne 1\) over global fields |

11G50 | Heights |

14G05 | Rational points |

14G25 | Global ground fields in algebraic geometry |

14G40 | Arithmetic varieties and schemes; Arakelov theory; heights |

14K15 | Arithmetic ground fields for abelian varieties |