Vol. 2, No. 5, 2008

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Le défaut d'approximation forte pour les groupes algébriques commutatifs

David Harari

Vol. 2 (2008), No. 5, 595–611
Abstract

On établit une suite exacte décrivant l’adhérence des points rationnels d’un 1-motif dans ses points adéliques. On en déduit ensuite que le défaut d’approximation forte pour un groupe algébrique commutatif G est essentiellement mesuré par son groupe de Brauer algébrique via l’obstruction de Brauer-Manin entière.

We give an exact sequence describing the closure of the set of rational points of a 1-motive in its adelic points. From this we deduce that for a commutative algebraic group, the defect of strong approximation is essentially controlled by its algebraic Brauer group, by means of the integral Brauer-Manin obstruction.

Keywords
approximation forte, groupe de Brauer, $1$-motif, strong approximation, Brauer group, $1$-motive
Mathematical Subject Classification 2000
Primary: 14L15
Secondary: 12G05, 11G09
Milestones
Received: 22 April 2008
Revised: 26 May 2008
Accepted: 26 June 2008
Published: 4 July 2008
Authors
David Harari
Université Paris-Sud
Laboratoire de Mathématiques d’Orsay
F-91405 Orsay Cedex
France