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

2016/08/17 by Laia Amorós, Amorós, Laia, Piermarco Milione +1
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Analytic Number Theory Research

paper · pdf · doi:10.48550/arxiv.1608.04891

Abstract

We give an explicit description of fundamental domains associated to the\np-adic uniformisation of families of Shimura curves of discriminant Dp and\nlevel N\≥ 1, for which the one-sided ideal class number h(D,N) is 1.\nThe obtained results generalise those in cite[Ch. IX]Gerritzen_vanderPut1980\nfor Shimura curves of discriminant 2p and level N=1. The method we present\nhere enables us to find Mumford curves covering Shimura curves, together with a\nfree system of generators for the associated Schottky groups, p-adic good\nfundamental domains and their stable reduction-graphs. This is based on a\ndetailed study of the modular arithmetic of an Eichler order of level N\ninside the definite quaternion algebra of discriminant D, for which we\ngeneralise classical results of Hurwitz citeHurwitz1896. As an application,\nwe prove general formulas for the reduction-graphs with lengths at p of the\nconsidered families of Shimura curves.\n

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