La descente sur les variétés rationnelles. II. (The descent on rational varieties. II)
type: Item
instance of: scholarly article
title: La descente sur les variétés rationnelles. II. (The descent on rational varieties. II)
zbMATH Open document ID: 0659.14028
DOI: 10.1215/S0012-7094-87-05420-2
author: Jean-Louis Colliot-Thélène
author: Jean-Jacques Sansuc
published in: Duke Mathematical Journal
publication date: 1987-01-01T00:00:00Z
review text: [For part I see Journées de géometrie algébrique, Angers/France 1979, 223-237 (1980; Zbl 0451.14018).] The article under review gives a general framework for the study of rational points on varieties X over number fields k. For such a variety the Hasse-principle holds if it has a k-rational point, provided that it has rational points over all local fields \(k_ v\) arising from completions). Yu. I. Manin has found an obstruction to the Hasse- principle, involving Brauer-groups. The authors observe that one can give a geometric interpretation by forming torsors \(E\to X\) (under tori), such that the Manin-obstruction vanishes for E. Any rational point on X lifts to one such torsor E, and it is hoped that E satisfies the Hasse- principle. This is true for many important examples. However these assertions have been shown elsewhere. The bulk of this paper consists of general and very detailed cohomological constructions and computations.
Mathematics Subject Classification ID: 14M20
Mathematics Subject Classification ID: 14G05
Mathematics Subject Classification ID: 14G25
Mathematics Subject Classification ID: 14J25
zbMATH DE Number: 4077431
zbMATH Keywords: rational points on varieties
zbMATH Keywords: obstruction to the Hasse-principle
zbMATH Keywords: torsor
MaRDI profile type: Publication
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full work available at URL: s0012-7094-87-05420-2
OpenAlex ID: W2048504073

La descente sur les variétés rationnelles. II. (The descent on rational varieties. II) (reviewed by)
reviewed by: Gerd Faltings