On computing quaternion quotient graphs for function fields

Details On computing quaternion quotient graphs for function fields On computing quaternion quotient graphs for function fields

2010-10-22

arxiv.org/abs/1010.4826v2

Journal de Th\'{e}orie des Nombres de Bordeaux 24 (2012), 73--99

Mathematics

Number Theory

30 pages, 2 figures

Scientific paper

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 $\Lambda^*$ on the Bruhat-Tits tree $\mathcal{T}$ associated to $PGL_2(\mathbb{F}_q((1/T)))$. 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 $\Lambda^*$ in terms of generators and relations. Moreover we determine an upper bound for its running time using that $\Lambda^*\backslash\mathcal{T}$ is {\em almost} Ramanujan.

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