zbMATH — the first resource for mathematics

Minorations des hauteurs normalisées des sous-variétés de variétés abeliennes. II. (Lower bounds for normalized heights of subvarieties of Abelian varieties. II.). (French) Zbl 1030.11026
The so-called Bogomolov problem asks whether, in an algebraic subvariety of an Abelian variety over a number field, the set of algebraic points of small height can be Zariski dense. The answer is known to be negative, apart from trivial degenerate cases, thanks to the work of E. Ullmo and S. Zhang. An alternative quantitative proof of this fact was given by the authors in the first part of this work [Contemp. Math. 210, 333-364 (1998; Zbl 0899.11027)]. The analogous question for tori has been also solved by the authors [Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 28, 489-543 (1999; Zbl 1002.11055)], who gave a completely explicit version, refining previous work of W. M. Schmidt [Lond. Math. Soc. Lect. Note Ser. 235, 157-187 (1996; Zbl 0917.11023)].
Here they produce a completely explicit version of their result for Abelian varieties. Moreover they also give explicit lower bounds for the successive minima of the normalized height for subvarieties of Abelian varieties over a number field. The results of this paper allowed Gaël Rémond to produce a completely explicit upper bound for the number of exceptional varieties in the theorems of Faltings and Vojta solving the conjectures of Mordell and Lang respectively [see G. Rémond, Invent. Math. 142, 513-545 (2000; Zbl 0972.11054)].

11G10 Abelian varieties of dimension \(> 1\)
14G40 Arithmetic varieties and schemes; Arakelov theory; heights
11J81 Transcendence (general theory)
Full Text: DOI