Strong Approximation for Semi-Simple Groups Over Function Fields

Abstract

Let K be a field and A be a K-algebra. For a variety V, defined over K, we shall let V(A) denote the set of A-rational points of V. In case A is a locally compact topological ring, V(A) has a natural locally compact Hausdorff topology induced by the topology on A (see Weil [25: App. III]); in the sequel we shall assume V(A) endowed with this topology. If K is a local field (i.e., a non-discrete locally compact field) and V is a smooth K-variety, then V(K) is a K-analytic manifold. Moreover, when V is a K-group, V(K) is a K-analytic group. Let G be a connected semi-simple affine algebraic group defined, isotropic and almost simple over a local field K of arbitrary characteristic. Let G = G(K). Let G+ be the normal subgroup of G generated by the K-rational points of the unipotent radicals of parabolic K-subgroups of G. The object of this paper is to prove the following: