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 $\mathfrak{ p}$.
Categories: math.NT