Genuine non-congruence subgroups of Drinfeld modular groups

Abstract

Let A be the ring of elements in an algebraic function field K over a finite field F-q which are integral outside a fixed place infinity. In an earlier paper we have shown that the Drinfeld modular group G = GL(2)(A) has automorphisms which map congruence subgroups to non-congruence subgroups. Here we prove the existence of (uncountably many) normal genuine non-congruence subgroups, defined to be those which remain non-congruence under the action of every automorphism of G. In addition, for all but finitely many cases we evaluate ngncs(G), the smallest index of a normal genuine non-congruence subgroup of G, and compare it to the minimal index of an arbitrary normal non-congruence subgroup. (C) 2016 Elsevier B.V. All rights reserved