Genuine non-congruence subgroups of Drinfeld modular groups

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.

Elliptic points of the Drinfeld modular groups

Let $K$ be an algebraic function field with constant field ${mathbb F}_q$. Fix a place $infty$ of $K$ of degree $delta$ and let $A$ be the ring of elements of $K$ that are integral outside $infty$. We give an explicit description of the elliptic points for the action of the Drinfeld modular group $G=GL_2(A)$ on the Drinfelds upper half-plane $Omega$ and on the Drinfeld modular curve $G!setminus!Omega$. It is known that under the {it building map} elliptic points are mapped onto vertices of the {it Bruhat-Tits tree} of $G$. We show how such vertices can be determined by a simple condition on their stabilizers. Finally for the special case $delta=1$ we obtain from this a surprising free product decomposition for $PGL_2(A)$.

Nonrational genus zero function fields and the Bruhat-Tits tree

Let K be a function field with constant field k and let infinity be a fixed place of K. Let C be the Dedekind domain consisting of all those elements of K which are integral outside infinity. The group G=GL_2(C) is important for a number of reasons. For example, when k is finite, it plays a central role in the theory of Drinfeld modular curves. Many properties follow from the action of G on its associated Bruhat-Tits tree, T. Classical Bass-Serre theory shows how a presentation for G can be derived from the structure of the quotient graph (or fundamental domain) GT. The shape of this quotient graph (for any G) is described in a fundamental result of Serre. However there are very few known examples for which a detailed description of GT is known. (One such is the rational case, C=k[t], i.e. when K has genus zero and infinity has degree one.) In this paper we give a precise description of GT for the case where the genus of K is zero, K has no places of degree one and infinity has degree two. Among the known examples a new feature here is the appearance of vertex stabilizer subgroups (of G) which are of quaternionic type.