Non-standard automorphisms and non-congruence subgroups of SL2 over Dedekind domains contained in function fields

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 SL2(C), generalizing a result of Reiner for the special case SL2(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 SL2(C) where C is a principal ideal domain.