Computing fundamental domains for the Bruhat–Tits tree for ${\rm GL}_2(\mathbf{Q}_p)$, $p$-adic automorphic forms, and the canonical embedding of Shimura curves

Abstract We describe algorithms that allow the computation of fundamental domains in the Bruhat–Tits tree for the action of discrete groups arising from quaternion algebras. These algorithms are used to compute spaces of rigid modular forms of arbitrary even weight, and we explain how to evaluate such forms to high precision using overconvergent methods. Finally, these algorithms are applied to the calculation of conjectural equations for the canonical embedding of p -adically uniformizable rational Shimura curves. We conclude with an example in the case of a genus 4 Shimura curve.