Minorations des hauteurs normalisées des sous-variétés de variétés abeliennes. II. (Lower bounds for normalized heights of subvarieties of Abelian varieties. II.)
type: Item
instance of: scholarly article
title: Minorations des hauteurs normalisées des sous-variétés de variétés abeliennes. II. (Lower bounds for normalized heights of subvarieties of Abelian varieties. II.)
zbMATH Open document ID: 1030.11026
DOI: 10.1007/PL00012437
author: Sinnou David
author: Patrice Philippon
published in: Commentarii Mathematici Helvetici
publication date: 2003-03-17T00:00:00Z
review text: 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 \textit{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 \textit{G. Rémond}, Invent. Math. 142, 513-545 (2000; Zbl 0972.11054)].
reviewed by: Michel Waldschmidt
Mathematics Subject Classification ID: 11G10
Mathematics Subject Classification ID: 14G40
Mathematics Subject Classification ID: 11J81
zbMATH DE Number: 1883362
zbMATH Keywords: heights
zbMATH Keywords: abelian varieties
zbMATH Keywords: Diophantine geometry
zbMATH Keywords: effectivity
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1883362