Heights and discreteness (after L. Szpiro, E. Ullmo and S. Zhang). (Hauteurs et discrétude (d’après L. Szpiro, E. Ullmo et S. Zhang).) (French) Zbl 1014.11042

Abbes, Ahmed

Séminaire Bourbaki. Volume 1996/97. Exposés 820-834. Paris: Société Mathématique de France, Astérisque. 245, 141-166, Exp. No. 825 (1997).

The author surveys the proofs of E. Ullmo [Ann. Math. (2) 147, No. 1, 167–179 (1998; Zbl 0934.14013)] and S. Zhang [Ann. Math. (2) 147, 159–165 (1998; Zbl 0991.11034)] concerning the Bogomolov conjecture. The main theorem (Zhang) states: If a subvariety \(X\) of an Abelian variety \(A\) over a number field is not a translate of a sub-Abelian variety by a torsion point, there exists a constant \(\varepsilon>0\) such that the algebraic points of \(X\) of Néron-Tate height smaller than \(\varepsilon\) are not Zariski-dense in \(X\).
Finally appendices on heights and on the Hilbert-Samuel arithmetic theorem are given.
For the entire collection see [Zbl 0910.00034].

Reviewer: O.Ninnemann (Berlin)

MSC:

11G35	Varieties over global fields
14G40	Arithmetic varieties and schemes; Arakelov theory; heights
14H25	Arithmetic ground fields for curves
14H40	Jacobians, Prym varieties

Keywords:

Bogomolov’s conjecture; Abelian variety; torsion point; Néron-Tate height

Citations:

Zbl 0934.14013; Zbl 0991.11034