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

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].


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