Non-standard automorphisms and non-congruence subgroups of [formula omitted] over Dedekind domains contained in function fields

Abstract

Let K be an algebraic function field of one variable with constant field k and let C be the Dedekind domain consisting of all those elements of K which are integral outside a fixed place ∞ of K. We introduce “non-standard” automorphisms of the group SL 2 ( C ) , generalizing a result of Reiner for the special case SL 2 ( k [ t ] ) . For the (arithmetic) case where k is finite, we use these to transform congruence subgroups into non-congruence subgroups of almost any level. This enables us to investigate the existence, number, and minimal index of non-congruence subgroups of prescribed level. We provide also a group-theoretic characterization of those SL 2 ( C ) where C is a principal ideal domain.