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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



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

Authors: Laia Amorós and Piermarco Milione
Journal: Trans. Amer. Math. Soc. 371 (2019), 1119-1149
MSC (2010): Primary 11G18, 11R52, 14G35, 14G22
Published electronically: July 31, 2018
MathSciNet review: 3885173
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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\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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Additional Information

Laia Amorós
Affiliation: Faculté des Sciences, de la Technologie et de la Communication, 6 rue Richard Coudenhove-Kalergi, L-1359 Luxembourg

Piermarco Milione
Affiliation: Department of Mathematics and Systems Analysis, School of Science, Aalto University, FI-00076 Aalto, Finland

Keywords: Shimura curves, Mumford curves, $p$-adic fundamental domains
Received by editor(s): September 29, 2016
Received by editor(s) in revised form: April 11, 2017
Published electronically: July 31, 2018
Article copyright: © Copyright 2018 American Mathematical Society