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].

MSC:

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

Citations:

Zbl 0934.14013; Zbl 0991.11034
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