Weak approximation, Brauer and \(R\)-equivalence in algebraic groups over arithmetical fields. II.
type: Item
instance of: scholarly article
title: Weak approximation, Brauer and \(R\)-equivalence in algebraic groups over arithmetical fields. II.
zbMATH Open document ID: 1040.20038
author: Nguyêñ Quôć Thǎńg
published in: Journal of Mathematics of Kyoto University
publication date: 2002-01-01T00:00:00Z
review text: The main object of the paper under review is a connected linear algebraic group \(G\) defined over a number field \(k\). The author shows that the finite abelian groups \(G(k)/\text{Br}\) (the group of Brauer equivalence classes) and \(A(S,G)\) (the defect of weak approximation with respect to a finite set \(S\) of valuations of \(k\)) are birational invariants of \(G\) {as groups}. He also presents some formulas for \(G(k)/\text{Br}\) and proves the following bound for the defect of weak approximation: if \(H\) runs over connected \(k\)-subgroups of \(G\), then the order of \(A(S,H)\) is bounded by a constant depending only on \(G\), \(k\), \(S\). A similar result holds for the order of the Tate-Shafarevich group of \(H\). For Part I, cf. ibid. 40, 247--291 (2000; Zbl 1014.20019).
Mathematics Subject Classification ID: 20G25
Mathematics Subject Classification ID: 20G30
Mathematics Subject Classification ID: 20G10
Mathematics Subject Classification ID: 11E72
Mathematics Subject Classification ID: 18G50
zbMATH DE Number: 1998393
zbMATH Keywords: algebraic group
zbMATH Keywords: weak approximation
zbMATH Keywords: \(R\)-equivalence
reviewed by: Boris Kunyavskiĭ
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1215/kjm/1250283872
OpenAlex ID: W4240006696
DOI: 10.1215/KJM/1250283872
Recommended article: Weak approximation, Brauer and \(R\)-equivalence in algebraic groups over arithmetical fields
Recommended article: Q4016455
Recommended article: A note on obstructions to weak approximation and Brauer and R-equivalence relations for homogeneous spaces over global fields
Recommended article: Q3425772
Recommended article: Q3980268
Recommended article: Q4026919
Recommended article: On weak approximation in algebraic groups and related varieties defined by systems of forms
Recommended article: Q3979371
Recommended article: On Galois cohomology and weak approximation of connected reductive groups over fields of positive characteristic
Recommended article: Obstructions to weak approximation for reductive groups over \(p\)-adic function fields
description: scientific article; zbMATH DE number 1998393