Genuine non-congruence subgroups of Drinfeld modular groups

TL;DR

This paper proves the existence of uncountably many normal genuine non-congruence subgroups in Drinfeld modular groups and evaluates their minimal indices, highlighting their distinct automorphism-invariant properties.

Contribution

It establishes the existence of uncountably many normal genuine non-congruence subgroups and computes their minimal indices, advancing understanding of subgroup structures in Drinfeld modular groups.

Findings

Existence of uncountably many normal genuine non-congruence subgroups.

Evaluation of the minimal index of such subgroups.

Comparison with minimal index of arbitrary normal non-congruence subgroups.

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 $\infty$. 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.