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

Laia Amoros Carafi, Piermarco Milione

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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 ≥ 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.

Original languageEnglish
Pages (from-to)1119-1149
Number of pages31
JournalTransactions of the American Mathematical Society
Volume371
Issue number2
DOIs
Publication statusPublished - 1 Jan 2019
MoE publication typeA1 Journal article-refereed

Keywords

  • Shimura curves
  • Mumford curves
  • p-adic fundamental domains
  • UNIFORMIZATION
  • CONSTRUCTION
  • POINTS

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