Sous-groupe de Brauer invariant et obstruction de descente itérée

Cao, Yang

doi:10.2140/ant.2020.14.2151

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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.

Correction: 10.2140/ant.2023.17.261

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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

Correction: 24 March 2023

Authors

Yang Cao
IAZD, Leibniz Universität Hannover Germany

Vol. 14, No. 8, 2020