On almost strong approximation for some exceptional groups.
type: Item
instance of: scholarly article
title: On almost strong approximation for some exceptional groups.
zbMATH Open document ID: 1075.20016
published in: Journal of Algebra
publication date: 2004-09-23T00:00:00Z
review text: The main object of the paper under review is a simply connected semisimple linear algebraic group \(G\) defined over a number field \(k\). \(G\) satisfies the strong approximation property with respect to a finite set \(S\) of places of \(k\) if and only if \(G_S=\prod_{{\mathfrak p}\in S}G(k_{\mathfrak p})\) is not compact. Thus if, say, \(G_\infty\) is not compact, one has to use a place \(\mathfrak q\) such that strong approximation holds for \(G\) with respect to \(\{\mathfrak q\}\cup\infty\). In a situation like that, inspired by the paper of \textit{J.~S.~Hsia} and \textit{M.~Jöchner} [Invent. Math. 129, No. 3, 471-487 (1997; Zbl 0894.11013)], where the case of quadratic forms was treated, \textit{W.~K.~Chan} and \textit{J.~S.~Hsia} [J. Algebra 254, No. 2, 441-461 (2002; Zbl 1027.20024)] introduced the notion of almost strong approximation property (ASAP) which allows one to partially recover the loss of the arithmetical information due to throwing away the place \(\mathfrak q\), proved ASAP for all classical groups of compact type whose Tits index over \(k\) is not \(^2A^{(d)}_n\) (\(d\geq 3\)), and applied this property to the study of integral forms of \(G\). In the paper under review, ASAP is established for all groups of compact type defined over a totally real number field \(k\) for the following types: \(^{3,6}D_4\), \(G_2\), \(F_4\), \(E_7\), and \(E_8\). Note that the proof is not case-by-case, as for classical groups, but relies upon the fact that all the above groups are quasi-split over a quadratic extension of \(k\).
Mathematics Subject Classification ID: 20G30
Mathematics Subject Classification ID: 11E12
Mathematics Subject Classification ID: 11E57
zbMATH DE Number: 2102467
zbMATH Keywords: almost strong approximations
zbMATH Keywords: exceptional groups
zbMATH Keywords: simply connected semisimple groups over number fields
zbMATH Keywords: affine Bruhat-Tits buildings
zbMATH Keywords: classical groups of compact type
zbMATH Keywords: Tits indices
author: Wai Kiu Chan
reviewed by: Boris Kunyavskiĭ
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2002.11.007
OpenAlex ID: W1999459332
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DOI: 10.1016/J.JALGEBRA.2002.11.007
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description: scientific article; zbMATH DE number 2102467