Descent obstruction is equivalent to étale Brauer-Manin obstruction
type: Item
instance of: scholarly article
title: Descent obstruction is equivalent to étale Brauer-Manin obstruction
zbMATH Open document ID: 1180.14017
published in: Mathematische Annalen
publication date: 2009-06-16T00:00:00Z
review text: The author completes the proof of the result stated in the title. More precisely, for a smooth projective geometrically integral \(k\)-variety \(X\) defined over a number field \(k\) and a torsor \(f: Y\to X\) under a linear \(k\)-group \(G\) denote \[ X(\mathbb A_k)^f = \bigcup_{[\sigma ]\in H^1(k,G)} f^{\sigma }(Y^{\sigma }(\mathbb A_k)), \] where \(f^{\sigma}: Y^{\sigma }\to X\) is the twist of \(f\) by the 1-cocycle \(\sigma\) and \(\mathbb A_k\) stands for the ring of adèles of \(k\). Then the descent obstruction \(X(\mathbb A_k)^{\text{{desc}}}\) is defined as the intersection of \(X(\mathbb A_k)^f\) where \(f\) ranges over all torsors under all linear \(k\)-groups. The author's goal is to compare \(X(\mathbb A_k)^{\text{{desc}}}\) with the étale Brauer--Manin obstruction, defined as \[ X(\mathbb A_k)^{\text{{ét, Br}}} = \bigcap_f\bigcup_{[\sigma ]\in H^1(k,F)} f^{\sigma }\left(Y^{\sigma }(\mathbb A_k)^{\text{{Br}}}\right), \] where \(f\) ranges over all torsors under all finite \(k\)-groups. The question is motivated by the fact that the étale Brauer--Manin obstruction explains the counter-example to the Hasse principle constructed by the author [Invent. Math. 135, No. 2, 399--424 (1999; Zbl 0951.14013)] but does not explain the counter-example constructed by \textit{B.~Poonen} [Insufficiency of the Brauer--Manin obstruction applied to étale covers, to appear in Ann. Math., cf. \url{arXiv:0806.1312}]. In the above cited paper Poonen asked whether in his counter-example the descent obstruction is empty or not, and, more generally, whether one always has the inclusion \[ X(\mathbb A_k)^{\text{ét, Br}} \subset X(\mathbb A_k)^{\text{desc}}. \] The latter inclusion was proved by \textit{C. Demarche} [Algebra and Number Theory 3, No. 2, 237--254 (2009; Zbl 1247.11090)]. In the paper under review the opposite inclusion is proved. The key result is the following formula for the descent obstruction (Theorem 1.1): if \(f: Y\to X\) is a torsor under a finite \(k\)-group…
Mathematics Subject Classification ID: 14G05
Mathematics Subject Classification ID: 14G25
Mathematics Subject Classification ID: 14J27
Mathematics Subject Classification ID: 14J28
Mathematics Subject Classification ID: 14L30
zbMATH DE Number: 5565277
zbMATH Keywords: Brauer-Manin obstruction
zbMATH Keywords: Hasse principle
zbMATH Keywords: weak approximation
Wikidata QID: Q57395844
author: Alexei N. Skorobogatov
reviewed by: Boris Kunyavskiĭ
MaRDI profile type: Publication
full work available at URL: s00208-008-0314-4
OpenAlex ID: W2151652950
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DOI: 10.1007/S00208-008-0314-4
description: scientific article; zbMATH DE number 5565277