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 $G{L}_2(C)$ plays a central role in the theory of Drinfeld modular curves analagous to that played by $S{L}_2(Z)$ in the classical theory of modular forms. When $k$ is finite (resp. infinite) we refer to a group $G{L}_2(C)$ as an arithmetic (resp. non-arithmetic) Drinfeld modular group. Associated with $G{L}_2(C)$ is its Bruhat-Tits tree, $T$. The structure of the group is derived from that of the quotient graph $G{L}_2(C)\backslash T$. 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 $G{L}_2(C)\backslash T$ for the important special case where $\infty$ has degree 1.

v2: 33 pages, some improvements, title changed, second author's address updated

References in corpus (1)

• Nonrational genus zero function fields and the Bruhat-Tits tree

Cited by in corpus (5)

• Elliptic points of the Drinfeld modular groups
• Quasi-inner automorphisms of Drinfeld modular groups
• Genuine non-congruence subgroups of Drinfeld modular groups
• On the automorphisms of the Drinfeld modular groups
• Simultaneous diagonalization of vector bundles