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

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 [GvdP80, 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 reductiongraphs. 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 [Hur96]. As an application, we prove general formulas for the reduction-graphs with lengths at p of the families of Shimura curves considered.

CM points on Shimura curves and $p$-adic binary quadratic forms

“…The classes of p-adic binary quadratic forms are obtained by the action of a discrete and cocompact subgroup of PGL 2 (Q p ) arising from the p-adic uniformization of the Shimura curve. We finally compute families of p-adic binary quadratic forms associated to an infinite family of Shimura curves studied in [2]. This extends results of Alsina-Bayer [1] to the p-adic context.…”

“…Recall that Q p 2 Q p is the set of the Q p 2 -points of the p-adic upper-half plane H p (cf. [2] for a brief overview). We wish to underline the difference between Proposition 1.3 and its archimedean analog.…”

“…In the following example we compute families of binary quadratic forms arising from the p-adic uniformization of Shimura curves of discriminant 2p and level N = 1, when p ≡ 1 (mod 4). We refer the reader to [2] for an explicit description of the p-adic uniformization this and other families of Shimura curves.…”

“…Anyway, this is not an easy problem as there is no general method for computing p-adic fundamental domains with respect to the cocompact groups Γ p,+ . In [2] we computed fundamental domains for certain infinite families of Shimura curves in relation to Mumford curves covering them. It will be the object of a future work to making explicit a reduction point algorithm for those fundamental domains.…”