Vol. 4, No. 5, 2010

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ISSN: 1944-7833 (e-only)
ISSN: 1937-0652 (print)
Le problème de Bogomolov effectif sur les variétés abéliennes

Aurélien Galateau

Vol. 4 (2010), No. 5, 547–598
Abstract

On obtient une nouvelle minoration du minimum essentiel en petite codimension sur les variétés abéliennes, sous une conjecture concernant leurs idéaux premiers ordinaires. Cette minoration, déjà connue dans le cas torique depuis les travaux d’Amoroso et David, est optimale « à ε près » en le degré de la sous-variété. La preuve suit la méthode des pentes et est basée sur les propriétés p-adiques des points de torsion des variétés abéliennes.

We give a new lower bound for the essential minimum of subvarieties of abelian varieties with small codimension, under a conjecture about ordinary primes in abelian varieties. This lower bound, known in the toric case through the work of Amoroso and David, is best “up to an ε” in the degree of the subvariety. The proof follows the slope method and is based on the p-adic properties of torsion points in abelian varieties.

Keywords
Bogomolov, variété abélienne, minoration, hauteur, abelian variety, lower bound, height
Mathematical Subject Classification 2000
Primary: 11G10
Secondary: 11J81, 14G40
Milestones
Received: 25 June 2009
Revised: 16 November 2009
Accepted: 20 December 2009
Published: 10 July 2010
Authors
Aurélien Galateau
Mathematisches Institut
Universität Basel
Rheinsprung, 21
CH-4051 Basel
Switzerland
Bâtiment 425
Université Paris-Sud
91405 Orsay
France
http://www.math.u-psud.fr/~galateau