Non-standard automorphisms and non-congruence subgroups of SL2 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.

Elliptic points of the Drinfeld modular groups

“…Clearly it suffices to prove this for the case s = ∞. The proof is based on an approach introduced in [MS1]. Let T (a) = I 2 + E 12 (a ∈ A) and for each A-ideal q let Γ (q) = {X ∈ SL 2 (A) : X ≡ I 2 (mod q)}.…”

“…We call them Reiner automorphisms because the original idea for the construction, for the ring A = F q [t], is in [Re]. In [MS1] we had generalized these automorphisms to any A but still only for the cusp ∞ (and used them to map congruence subgroups to non-congruence subgroups).…”