Reference : Mumford curves covering p-adic Shimura curves and their fundamental domains
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Physical, chemical, mathematical & earth Sciences : Mathematics
Mumford curves covering p-adic Shimura curves and their fundamental domains
Amoros Carafi, Laia mailto [University of Luxembourg > Faculty of Science, Technology and Communication (FSTC) > Mathematics Research Unit >]
Milione, Piermarco [Aalto University]
[en] Shimura curves ; Mumford curves ; p-adic fundamental domains
[en] We give an explicit description of fundamental domains associated to the p-adic uniformisa- tion 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 obtained results generalise those in [19, Ch. IX] 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. This 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 classical results of Hurwitz [20]. As an application, we prove general formulas for the reduction-graphs with lengths at p of the considered families of Shimura curves.

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