The vertex stabilizers of the Bruhat-Tits tree of a Drinfeld modular group

Details The vertex stabilizers of the Bruhat-Tits tree of a Drinfeld modular group The vertex stabilizers of the Bruhat-Tits tree of a Drinfeld modular group

2012-03-16

arxiv.org/abs/1203.3617v1

Mathematics

Group Theory

31 pages

Scientific paper

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 Drinfeld modular curves analagous to that played by $SL_2({\mathbb Z})$ 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 $GL_2(C)\backslash T$. In this paper 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.

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