Elliptic points of the Drinfeld modular groups

TL;DR

This paper explicitly describes elliptic points for Drinfeld modular groups acting on Drinfeld's upper half-plane and modular curves, revealing how to identify vertices via stabilizers and deriving a free product decomposition for PGL2(A) when the degree is one.

Contribution

It provides an explicit characterization of elliptic points and a method to determine vertices in the Bruhat-Tits tree, including a new free product decomposition for PGL2(A) when δ=1.

Findings

Elliptic points correspond to vertices with specific stabilizer conditions.

Vertices can be identified using a simple stabilizer-based criterion.

For δ=1, PGL2(A) admits a free product decomposition.

Abstract

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 Drinfeld's 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 $PGL_2(A)$.