On the congruence subgroup problem. II
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 subgroups of G(k). Fix a faithful k-representation \(\rho\) once for all and for any ideal \({\mathfrak a}\neq 0\) in A, let G(\({\mathfrak a})=\{x\in G(k)\); \(\rho(x)\in GL(n)\), \(\rho(x)\equiv 1 (\bmod {\mathfrak a})\}\); E(\({\mathfrak a})\) the subgroup of \(G({\mathfrak a})\) generated by unipotents in \(G({\mathfrak a})\) which are contained in unipotent radicals of k-parabolic subgroups. The family \(\{E({\mathfrak a})\); \({\mathfrak a}\) non-zero ideal\(\}\) defines another topological structure on G whose completion is denoted by \(\hat G(e)\). Let \(\hat G(e)^+\) be the closure of \(G(k)^+\) in \(\hat G(e)\). Giving a universal central extension of the S-adèle group of G in terms of generators and relations, the author shows that if k-rank G\(=1\) and \(\sum_{v\in S}k_ v\)-rank \(G\geq 2\), \(\hat G(e)\to \hat G(c)\) is a central extension of \(\hat G(c)\). The main result can be shown from this fact. In appendix, the author gives a general result (over arbitrary field k) on k-rank 1 algebraic groups which seems to be of independent interest.
connected simply connected absolutely simple k-algebraic group
scientific article; zbMATH DE number 3973168
scientific article