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

TL;DR

This paper provides an elementary matrix-based method to determine the structure of vertex stabilizers in the Bruhat-Tits tree associated with Drinfeld modular groups, extending previous results and analyzing vertex valencies for specific cases.

Contribution

It introduces an elementary approach to explicitly compute all vertex stabilizers in the Bruhat-Tits tree for Drinfeld modular groups, expanding on prior theoretical work.

Findings

Explicit structure of all vertex stabilizers determined.

All possible vertex valencies identified for degree 1 place.

Extends previous results by Serre, Takahashi, and authors.

Abstract

Let $K$ be an algebraic function field of one variable with constant field $k$ and let $C$ be the Dedekind domain consisting of all those elements of $K$ which are integral outside a fixed place $\infty$ of $K$. When $k$ is finite the group $GL_2(C)$ plays a central role in the theory of Drinfeld modular curves analagous to that played by $SL_2(Z)$ in the classical theory of modular forms. When $k$ is finite (resp. infinite) we refer to a group $GL_2(C)$ as an arithmetic (resp. non-arithmetic) Drinfeld modular group. Associated with $GL_2(C)$ is its Bruhat-Tits tree, $T$. The structure of the group is derived from that of the quotient graph $GL_2(C)\backslash T$. Using an elementary approach which refers explicitly to matrices we determine the structure of all the vertex stabilizers of $T$. This extends results of Serre, Takahashi and the authors. We also determine all possible valencies of the vertices of $GL_2(C)\backslash T$ for the important special case where $\infty$ has degree 1.