Mumford curves covering p-adic Shimura curves and their fundamental domains

We give an explicit description of fundamental domains associated with the p-adic uniformisation of families of Shimura curves of discriminant Dp and level N ≥ 1, for which the one-sided ideal class number h(D, N) is 1. The results obtained generalise those in Schottky groups and Mumford curves, Springer, Berlin, 1980 for Shimura curves of discriminant 2p and level N = 1. The method we present here enables us to find Mumford curves covering Shimura curves, together with a free system of generators for the associated Schottky groups, p-adic good fundamental domains, and their stable reduction-graphs. The method is based on a detailed study of the modular arithmetic of an Eichler order of level N inside the definite quaternion algebra of discriminant D, for which we generalise the classical results of Hurwitz. As an application, we prove general formulas for the reduction-graphs with lengths at p of the families of Shimura curves considered.