On the group of norm 1 elements in a division algebra
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 \(D^ 1\). Then \(\bar N/N\) is perfect. The main result above is equivalent to the following theorems. Theorem 1. Let \(N\subset D^ 1\) be a normal subgroup and \(\bar N\) its closure in \(D^ 1\) for the \(S_ 0\)-adic topology. Then \(\bar N/N\) is perfect. Theorem 2. If \(N\subset D^ 1\) is an \(S_ 0\)-adically closed subgroup of finite index in \(D^ 1\), \([N,N]\) is \(S_ 0\)-adically closed. Taking \(N=D\) 1 in theorem 2 the author gets Theorem 3. \([D^ 1,D^ 1]=D^ 1({\underline 1})\) where \({\underline 1}\in I\) is defined by \({\underline 1}(v)=1\) for all \(v\in S_ 0\) \((D^ 1({\underline 1})=\{x\in D^*|\) \(x\in [D^ 1_ v,D^ 1_ v]\) for all \(v\in S_0\}\). Consequently, \(D^ 1/[D^ 1,D^ 1]\cong \prod_{v\in S_ 0}F^ 1_ v\) where \(F_ v=R_ v/p_ v\) and \(F^ 1_ v=\{x\in F_ v|\) \(Norm_{F_ v/f_ v}(x)=1\}\). When \(p=0\) and \(v((d,2))=0\) theorem 3 is due to Platonov and Rapinchuk. The main result above also enables to complete Margulis' theorem for quaternion algebras: Theorem 4. (Q) has an affirmative answer when \(p=2\), \(d=2\).
finite dimensional central division algebra
scientific article; zbMATH DE number 4042154