On weak approximation in algebraic groups and related varieties defined by systems of forms
type: Item
instance of: scholarly article
title: On weak approximation in algebraic groups and related varieties defined by systems of forms
zbMATH Open document ID: 0859.20040
DOI: 10.1016/0022-4049(95)00141-7
published in: Journal of Pure and Applied Algebra
publication date: 1996-12-11T00:00:00Z
review text: The main objects of the reviewed paper are semisimple linear algebraic groups \(G\) of the adjoint type defined over an arbitrary field \(k\). The author is interested whether the weak approximation property with respect to an arbitrary finite set of valuations holds for \(G\). It turns out that the answer is positive for many classical and exceptional groups as well as for some related varieties defined by a system of forms. For groups defined over a global field \(k\), the author considers another approximation property (integral approximation) which is in a sense intermediate between weak and strong approximation. He proves (under mild restrictions on a connected \(k\)-group \(G\)) that this property holds outside some finite set of valuations of \(k\). Some of these results were announced in an earlier paper [the author, Contemp. Math. 131, Pt. 1, 423-426 (1992; Zbl 0763.11021)].
Mathematics Subject Classification ID: 20G15
Mathematics Subject Classification ID: 20G30
Mathematics Subject Classification ID: 11E57
zbMATH DE Number: 943212
zbMATH Keywords: integral approximation
zbMATH Keywords: semisimple linear algebraic groups
zbMATH Keywords: weak approximation
zbMATH Keywords: finite set of valuations
zbMATH Keywords: system of forms
zbMATH Keywords: global fields
zbMATH Keywords: strong approximation
author: Nguyêñ Quôć Thǎńg
reviewed by: Boris Kunyavskiĭ
MaRDI profile type: Publication
full work available at URL: 0022-4049(95)00141-7
OpenAlex ID: W2131438841
cites work: Groupes reductifs
cites work: Complements à l'article: ''Groupes reductifs''
cites work: On algebraic group varieties
cites work: Sur le principe de Hasse et l'approximation faible, et sur une hypothèse de Schinzel
cites work: La descente sur les variétés rationnelles. II. (The descent on rational varieties. II)
cites work: Principal homogeneous spaces under flasque tori; applications
cites work: Q3228640
cites work: Conjugacy Classes of Maximal Tori in Simple Real Algebraic Groups and Applications
cites work: Halbeinfache Gruppenschemata über Dedekindringen
cites work: Eine Bemerkung zum schwachen Approximationssatz
cites work: Q5557997
cites work: Starke Approximation in algebraischen Gruppen. I.
cites work: Q4026919
cites work: Q5675788
cites work: New improved exact sequences of Witt groups
cites work: Round quadratic forms
cites work: Congruence Properties of Zariski-Dense Subgroups I
cites work: Q3473279
cites work: On the relative theory of Tamagawa numbers
cites work: Reduced K-theory and approximation in algebraic groups
cites work: A.D. Alexandrov spaces with curvature bounded below
cites work: Strong approximation for semi-simple groups over function fields
cites work: Questions of rationality for solvable algebraic groups over non-perfect fields
cites work: Groupe de Brauer et arithmétique des groupes algébriques linéaires sur un corps de nombres.
cites work: Q3708857
cites work: Q4173562
cites work: Q3469169
cites work: Q4016455
cites work: Q4016232
cites work: Q5648549
cites work: Q4170852
cites work: Q3964707
cites work: TOROIDAL FANO VARIETIES AND ROOT SYSTEMS
description: scientific article; zbMATH DE number 943212