Fundamental domains for quaternionic S-arithmetic groups over totally real fields

Abstract

Let $B$ be a totally-definite quaternion algebra over a totally real field $F$, let $\mathfrak{p}$ be a prime ideal of $F$, and let $\Gamma$ be the group of reduced norm-$1$ elements of an Eichler $\mathcal{O}_F[1/\mathfrak{p}]$-order $R$ inside $B$. We give an algorithm to compute the fundamental domain for the action of $\Gamma$ on the Bruhat-Tits tree of $\operatorname{GL}_2(F_\mathfrak{p})$. Using this, we tabulate Shimura curves of genus up to $3$ over any totally real field which can be $\mathfrak{p}$-adically uniformized for some prime $\mathfrak{p}$.