On almost strong approximation for algebraic groups
type: Item
instance of: scholarly article
title: On almost strong approximation for algebraic groups
zbMATH Open document ID: 1027.20024
DOI: 10.1016/S0021-8693(02)00100-X
author: J. S. Hsia
author: Wai Kiu Chan
published in: Journal of Algebra
publication date: 2003-02-12T00:00:00Z
review text: If \(G\) is a simply connected reductive group defined over a number field \(k\) and \(\infty\) is the set of all infinite places of \(k\), then \(G\) has strong approximation with respect to \(\infty\) if and only if the Archimedean part of any \(k\)-simple component of the adèle group \(G_\mathbb{A}\) is non-compact. Using affine Bruhat-Tits buildings the authors formulate an almost strong approximation (ASAP) for groups of compact type. The validity of ASAP for \(G(k)\) is proved for all classical groups of compact type whose Tits indices over \(k\) are not \(^2A_n^{(d)}\) with \(d\geq 3\). Some consequences of ASAP in the study of the genera of integral forms of algebraic groups of compact type in the sense of \textit{B. H. Gross} [Invent. Math. 124, No. 1-3, 263-279 (1996; Zbl 0846.20049)] are presented. When specialized to the groups SU and Spin for quadratic or skew-quadratic forms, the authors obtain results on integral representation of positive definite quadratic and Hermitian or skew-Hermitian forms.
Mathematics Subject Classification ID: 20G30
Mathematics Subject Classification ID: 11E12
Mathematics Subject Classification ID: 11E57
zbMATH DE Number: 1867989
zbMATH Keywords: almost strong approximation
zbMATH Keywords: algebraic groups
zbMATH Keywords: quadratic forms
zbMATH Keywords: simply connected reductive groups
zbMATH Keywords: integral representations
zbMATH Keywords: skew Hermitian forms
reviewed by: Kanat Abdukhalikov
MaRDI profile type: Publication
cites work: Arithmetic subgroups of algebraic groups
cites work: Q4693148
cites work: Reductive groups on a local field. II. Groups schemes. Existence of valuated root datum
cites work: Q4177654
cites work: Maximal integral forms of the algebraic group \(G_2\) defined by finite subgroups
cites work: Die Ähnlichkeitsklassen indefiniter Gitter
cites work: Q4266494
cites work: Groups over \(\mathbb{Z}\)
cites work: Almost strong approximations for definite quadratic spaces
cites work: Klassenzahlen indefiniter quadratischer Formen in drei oder mehr Veränderlichen
cites work: Klassenzahlen definiter quadratischer Formen
cites work: Starke Approximation in algebraischen Gruppen. I.
cites work: Lectures on Galois cohomology of classical groups. With an appendix by T. Springer
cites work: Q3997794
cites work: Q3293461
cites work: A.D. Alexandrov spaces with curvature bounded below
cites work: Q3708857
cites work: Arithmetic of unitary groups
cites work: Q5648549
cites work: Q3050630
description: scientific article; zbMATH DE number 1867989