On computing quaternion quotient graphs for function fields

Abstract

Let $\Lambda$ be a maximal ${\mathbb{F}}_q[T]$-order in a division quaternion algebra over ${\mathbb{F}}_q(T)$ which is split at the place $\infty$. The present article gives an algorithm to compute a fundamental domain for the action of the group of units ${\Lambda }^{\ast }$ on the Bruhat-Tits tree $\mathcal{T}$ associated to $PG{L}_2({\mathbb{F}}_q((1/T)))$. This action is a function field analog of the action of a co-compact Fuchsian group on the upper half plane. The algorithm also yields an explicit presentation of the group ${\Lambda }^{\ast }$ in terms of generators and relations. Moreover we determine an upper bound for its running time using that ${\Lambda }^{\ast }\backslash \mathcal{T}$ is {\em almost} Ramanujan.

30 pages, 2 figures