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 $\delta$ 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 $G=G{L}_2(A)$ 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 $PG{L}_2(A)$.

30 pages

References in corpus (1)

• The stabilizers in a Drinfeld modular group of the vertices of its Bruhat-Tits tree: an elementary approach

Cited by in corpus (2)

• Quasi-inner automorphisms of Drinfeld modular groups
• On the automorphisms of the Drinfeld modular groups