On the congruence subgroup problem. II
type: Item
instance of: scholarly article
title: On the congruence subgroup problem. II
zbMATH Open document ID: 0603.20044
DOI: 10.1007/BF01388793
author: M. S. Raghunathan
published in: Inventiones Mathematicae
publication date: 1986-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/143360
review text: This is a sequel of the author's earlier paper of the same title [Inst. Hautes Etud. Sci., Publ. Math. 46, 107-161 (1976; Zbl 0347.20027)]. Let k be a global field and G be a connected simply connected absolutely simple k-algebraic group and G(k) the k-rational points of G. For a finite set of valuations of k containing all the archimedean valuations, let A be the ring of S-integers in k. The family of S-arithmetic (resp. S- congruence) groups forms a fundamental system of neighbourhoods of the identity for the structure of a topological group, whose completion is denoted by \(\hat G(a)\) (resp. \(\hat G(c)\)). Then \(\hat G(a)\) and \(\hat G(c)\) are locally compact topological groups and there is a natural map \(\pi: \hat G(a)\to \hat G(c)\). The congruence subgroup problem is the determination of \(C(S,G)=\ker \pi\) which is the main aim of this paper. When k-rank \(G\geq 2\), this is solved by the author (cf. op. cit.). Therefore, in this paper the author treats the case when k-rank G\(=1.\) Let \(G(k)^+\) be the subgroup of \(G(k)\) generated by unipotent elements contained in unipotent radicals of k-parabolic subgroups. Note that \(G(k)=G(k)^+\) except in the case when G is an outer k-form of \(E_ 6\) of k-rank 1. Assume that G is isotropic over k and \(\sum_{v\in S}k_ v\)-rank \(G\geq 2\). Let \(\hat G(a)^+\) be the closure of \(G(k)^+\) in \(\hat G(a)\). Then, the author proves that \(\pi^+: G(a)^+\to \hat G(c)\) is a surjective central extension and \(C(S,G)\) is a 2-step solvable group with \(C(S,G)/C^+(S,G)=G(k)/G(k)^+\) where \(C^+(S,G)\) is the kernel of \(\pi^+\). From the result, it follows that \(C^+(S,G)\) is isomorphic to a quotient of \(\mu\) (k), the group of roots of unity in k; further it is trivial if S contains a non-archimedean valuation or there is \(v\in S\) such that \(k_ v\) is real and the relative \(k_ v\) root space of G corresponding to any long root is of dimension 1. To prove the theorem, the author introduces another family of sub…
Mathematics Subject Classification ID: 20G30
Mathematics Subject Classification ID: 20G35
Mathematics Subject Classification ID: 22E46
Mathematics Subject Classification ID: 20H05
zbMATH DE Number: 3973168
zbMATH Keywords: arithmetic subgroup
zbMATH Keywords: global field
zbMATH Keywords: connected simply connected absolutely simple k-algebraic group
zbMATH Keywords: rational points
zbMATH Keywords: archimedean valuations
zbMATH Keywords: fundamental system of neighbourhoods
zbMATH Keywords: completion
zbMATH Keywords: locally compact topological groups
zbMATH Keywords: congruence subgroup problem
zbMATH Keywords: unipotent elements
zbMATH Keywords: unipotent radicals
zbMATH Keywords: k-parabolic subgroups
zbMATH Keywords: universal central extension
zbMATH Keywords: adèle group
reviewed by: Eiichi Abe
MaRDI profile type: Publication
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OpenAlex ID: W3159709446
description: scientific article; zbMATH DE number 3973168