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

TL;DR

This paper explicitly describes fundamental domains for p-adic uniformizations of certain Shimura curves, generalizing previous results, and provides methods to find Mumford curves covering these Shimura curves along with their fundamental groups and reduction-graphs.

Contribution

It introduces a new method to explicitly determine fundamental domains and Mumford curves covering Shimura curves with specific discriminant and level, extending classical results.

Findings

Explicit descriptions of fundamental domains for p-adic Shimura curves.

Construction of Mumford curves covering Shimura curves with generators for Schottky groups.

Formulas for reduction-graphs and their lengths at p.

Abstract

We give an explicit description of fundamental domains associated to the $p$-adic uniformisation of families of Shimura curves of discriminant $Dp$ and level $N\geq 1$, for which the one-sided ideal class number $h(D,N)$ is $1$. The obtained results generalise those in \cite[Ch. IX]{Gerritzen_vanderPut1980} 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. This 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 classical results of Hurwitz \cite{Hurwitz1896}. As an application, we prove general formulas for the reduction-graphs with lengths at $p$ of the considered families of Shimura curves.