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

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.