Vol. 3, No. 2, 2009

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Obstruction de descente et obstruction de Brauer–Manin étale

Cyril Demarche

Vol. 3 (2009), No. 2, 237–254
Abstract

Soit X une variété projective lisse géométriquement intègre sur un corps de nombres. On considère deux obstructions au principe de Hasse sur X : l’obstruction de Brauer–Manin appliquée aux revêtements étales de X et l’obstruction de descente sur X. On démontre que la première est plus forte que la seconde. On en déduit, grâce à un exemple récent de Poonen, que l’obstruction de descente est insuffisante pour expliquer tous les contrexemples au principe de Hasse.

Let X be a smooth, projective and geometrically integral variety over a number field. We consider two obstructions to the Hasse principle on X: the Brauer–Manin obstruction applied to étale covers of X and the descent obstruction on X. We prove that the first one is at least as strong as the second. Combining this with a recent example of Poonen shows that the descent obstruction is not sufficient to explain all counterexamples to the Hasse principle.

Keywords
principe de Hasse, obstruction de Brauer–Manin, obstruction de descente, cohomologie galoisienne, torseurs, Hasse principle, Brauer–Manin obstruction, descent obstruction, Galois cohomology, torsors
Mathematical Subject Classification 2000
Primary: 11G35
Secondary: 14G05, 11E72
Milestones
Received: 11 July 2008
Revised: 21 October 2008
Accepted: 18 November 2008
Published: 15 March 2009
Authors
Cyril Demarche
Université Paris-Sud
Laboratoire de Mathématiques d’Orsay
91405 Orsay Cedex
France
http://www.math.u-psud.fr/~demarche/