Solution of the congruence subgroup problem for solvable algebraic groups

Abstract

Let k be an algebraic number field of finite degree over the field Q of rational numbers. We denote by o the ring of integers in k. In general, for a subring A, containing 1, of a universal domain Ω we denote by GL(n, A) the subgroup of GL(n, Ω) consisting of matrices x = (xij) with xij ∈ A and det x ∈ A×, the group of units of A. Now, we consider an algebraic group G in GL(n, Ω) defined over k.