The reciprocity obstruction for rational points on compactifications of torsors under tori over fields with global duality
type: Item
instance of: scholarly article
title: The reciprocity obstruction for rational points on compactifications of torsors under tori over fields with global duality
zbMATH Open document ID: 1025.14003
author: J. van Hamel
published in: Documenta Mathematica
publication date: 2003-11-04T00:00:00Z
full work available at URL: https://eudml.org/doc/123022
full work available at URL: 08.html
review text: Summary: This paper studies the reciprocity obstruction to the local-global principle for compactifications of torsors under tori over a generalised global field of characteristic zero. Under a non-divisibility condition on the second Tate-Shafarevich group for tori, it is shown that the reciprocity obstruction is the only obstruction to the local-global principle. This gives in particular an elegant unified proof of \textit{J.-J. Sansuc}'s result [J. Reine Angew. Math. 327, 12-80 (1981; Zbl 0468.14007)] on the Brauer-Manin obstruction for compactifications of torsors under tori over number fields, and Scheiderer's result on the reciprocity obstruction for compactifications of torsors under tori over \(p\)-adic function fields [see \textit{C. Scheiderer} and \textit{J. van Hamel}, Math. Ann. 326, 155-183 (2003; Zbl 1050.14016)].
Mathematics Subject Classification ID: 14G25
Mathematics Subject Classification ID: 14M17
Mathematics Subject Classification ID: 14F20
Mathematics Subject Classification ID: 11G35
Mathematics Subject Classification ID: 14G40
zbMATH DE Number: 2000815
zbMATH Keywords: arithmetic algebraic geometry
zbMATH Keywords: Diophantine geometry
zbMATH Keywords: varieties over global fields
zbMATH Keywords: Brauer-Manin obstruction
zbMATH Keywords: homogeneous spaces over global fields
zbMATH Keywords: obstruction to the local-global principle
zbMATH Keywords: compactifications of torsors
zbMATH Keywords: Tate-Shafarevich group
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 2000815