Vol. 14, No. 8, 2020

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Sous-groupe de Brauer invariant et obstruction de descente itérée

Yang Cao

Vol. 14 (2020), No. 8, 2151–2183
DOI: 10.2140/ant.2020.14.2151
Abstract

Pour une variété quasi-projective, lisse, géométriquement intègre sur un corps de nombres k, on montre que l’obstruction de descente itérée est équivalente à l’obstruction de descente. Ceci généralise un résultat de Skorobogatov, et ceci répond à une question ouverte de Poonen. Les outils principaux sont la notion de sous-groupe de Brauer invariant et la notion d’obstruction de Brauer–Manin étale invariante pour une k-variété munie d’une action d’un groupe linéaire connexe.

For a quasi-projective smooth geometrically integral variety over a number field k, we prove that the iterated descent obstruction is equivalent to the descent obstruction. This generalizes a result of Skorobogatov, and this answers an open question of Poonen. Our main tools are the notion of invariant Brauer subgroup and the notion of invariant étale Brauer–Manin obstruction for a k-variety equipped with an action of a connected linear algebraic group.

Keywords
Hasse principle, Brauer–Manin obstruction, algebraic group
Mathematical Subject Classification 2010
Primary: 11G35
Secondary: 14G05, 20G35
Milestones
Received: 23 July 2019
Revised: 10 February 2020
Accepted: 12 April 2020
Published: 18 September 2020
Authors
Yang Cao
IAZD, Leibniz Universität Hannover
Germany