Mumford curves covering $p$-adic Shimura curves and their fundamental domains
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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.References
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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
- Email: laia.amoros@aalto.fi
- Piermarco Milione
- Affiliation: Department of Mathematics and Systems Analysis, School of Science, Aalto University, FI-00076 Aalto, Finland
- MR Author ID: 1189750
- Email: piermarcomilione@gmail.com
- Received by editor(s): September 29, 2016
- Received by editor(s) in revised form: April 11, 2017
- Published electronically: July 31, 2018
- © Copyright 2018 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 371 (2019), 1119-1149
- MSC (2010): Primary 11G18, 11R52, 14G35, 14G22
- DOI: https://doi.org/10.1090/tran/7397
- MathSciNet review: 3885173