Obstruction de descente et obstruction de Brauer–Manin étale

Abstract

Soit X une variete projective lisse geometriquement integre sur un corps de nombres. On considere deux obstructions au principe de Hasse sur X : l’obstruction de Brauer–Manin appliquee aux revetements etales de X et l’obstruction de descente sur X. On demontre que la premiere est plus forte que la seconde. On en deduit, grâce a un exemple recent 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 etale 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.