Elliptic points of the Drinfeld modular groups

Abstract

Let $K$ be an algebraic function field with constant field ${\mathbb F}_q$. Fix a place $\infty$ of $K$ of degree $δ$ 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 Drinfeld's upper half-plane $Ω$ and on the Drinfeld modular curve $G\!\setminus\!Ω$. 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 $δ=1$ we obtain from this a surprising free product decomposition for $PGL_2(A)$.