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

Amorós, Laia; Milione, Piermarco

doi:10.1090/tran/7397

Mumford curves covering $p$-adic Shimura curves and their fundamental domains
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by Laia Amorós and Piermarco Milione PDF

Trans. Amer. Math. Soc. 371 (2019), 1119-1149 Request permission

Abstract:

We give an explicit description of fundamental domains associated with the $p$-adic uniformisation of families of Shimura curves of discriminant $Dp$ and level $N\geq 1$, 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.

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