On the group of norm 1 elements in a division algebra
type: Item
instance of: scholarly article
title: On the group of norm 1 elements in a division algebra
zbMATH Open document ID: 0639.20026
DOI: 10.1007/BF01456282
author: M. S. Raghunathan
published in: Mathematische Annalen
publication date: 1988-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/164335
review text: Let k be a global field of characteristic p, V its set of valuations. For \(v\in V\) and v non-archimedean let \(k_ v\) denote the completion of k at v. Let D be a finite dimensional central division algebra of degree d over k. Let \(D_ v=D\otimes_ kk_ v\), \(S_ 0=\{v\in V|\) \(D_ v\) is a division algebra\}. For \(v\in S_ 0\) we let \(p_ v\) denote the maximal ideal of the maximal compact subring \(R_ v\) of \(D_ v\). Let \(Nred_ D\) (respectively \(Nred_{D_ v})\) be the reduced norm homomorphism in D (respectively in \(D_ v).\) The author is interested in obtaining information on normal subgroups of \(D^ 1=\{x\in D^*|\) \(Nred(x)=1\}\). According to Margulis any non-central normal subgroup of \(D^ 1\) has finite index in \(D^ 1\). On the other hand one has the following class of non-central normal subgroups of \(D^ 1\). Let I denote the collection of all maps \(r: S_ 0\to {\mathbb{Z}}^+\). For each \(v\in I\), let \(D^ 1(r)=\{x\in D^ 1|\) \(x- 1\in p_ v^{r(v)}\) for each \(v\in S_ 0\}\). Then \(D^ 1(r)\) is a non-central normal subgroup of \(D^ 1\). All subgroups \(\{D^ 1(r)|\) \(r\in I\}\) constitute a fundamental system of neighbourhoods of 1 in \(D^ 1\) for the \(S_ 0\)-adic topology on \(D^ 1\), namely the topology induced on \(D^ 1\) from the product \(\prod_{v\in S_ 0}D_ v^ 1\) (where \(D^ 1_ v=\{x\in D^*_ v|\) \(Nred(x)=1\}\) through the diagonal imbedding of \(D^ 1\) in that group. It is clear that a subgroup of finite index is closed in the \(S_ 0\)-adic topology if and only if it contains \(D^ 1(r)\) for some \(r\in I\). In this situation the natural question arises: (Q) Is every normal subgroup of \(D^ 1\) \(S_ 0\)-adically closed? Margulis has shown that when \(d=2\) and \(p\neq 2\) (Q) has an affirmative answer. The main result of the paper under review is the following Theorem. Let \(N\subset D^ 1\) be a subgroup normalized by \(D^*\) and not central in \(D^ 1\). Let \(\bar N\) be the closure of N in \(D^ 1\) for the \(S_ 0\)-adic topology on \(D^ 1\…
Mathematics Subject Classification ID: 20G30
Mathematics Subject Classification ID: 16Kxx
Mathematics Subject Classification ID: 16U60
Mathematics Subject Classification ID: 20E07
zbMATH DE Number: 4042154
zbMATH Keywords: global field
zbMATH Keywords: valuations
zbMATH Keywords: completion
zbMATH Keywords: finite dimensional central division algebra
zbMATH Keywords: reduced norm
zbMATH Keywords: non-central normal subgroups
zbMATH Keywords: fundamental system of neighbourhoods
zbMATH Keywords: subgroup of finite index
zbMATH Keywords: \(S_ 0\)-adically closed subgroup
reviewed by: V. I. Yanchevskiĭ
MaRDI profile type: Publication
cites work: Q5302599
cites work: Q3856973
cites work: Q3969027
cites work: The Norm 1 Group of -Adic Division Algebra
description: scientific article; zbMATH DE number 4042154