BCN Spring 2016 Workshop: Number Theory & K-theory

$p$-adic uniformization of Shimura curves through Mumford curves


Piermarco Milione


Let $X(D,N)$ be the Shimura curve associated to an Eichler order of level $N$ in an indefinite quaternion $\mathbb{Q}$-algebra of discriminant $D>1$, and let us fix a prime $p\vert D$.


Thanks to the works of \v{C}erednik and Drinfel'd, it is well-known that $X(D,N)$ admits a $p$-adic uniformization which can be expressed, \emph{inter alia}, as a rigid analytic isomorphism


\Gamma_{p}\backslash \mathcal{H}_{p}\otimes_{\mathbb{Q}_{p}}\mathbb{Q}_{p^{2}}\simeq (X(D,N)\otimes_{\mathbb{Q}}\mathbb{Q}_{p})^{rig},


where $\mathcal{H}_{p}$ denotes the $p$-adic upper half-plane and $\Gamma_{p}$ is a discrete cocompact subgroup of $\mathrm{PGL}_{2}(\mathbb{Q}_{p})$. As a consequence, the $p$-adic Shimura curve $X(D,N)\otimes_{\mathbb{Q}}\mathbb{Q}_{p}$ is the twist over $\mathbb{Q}_{p^{2}}$ of a finite quotient of some Mumford curve associated to a cocompact Schottky group $\Gamma_{p}^{Sch}\subseteq\Gamma_{p}$. Moreover the group $\Gamma_{p}$ arises as the units group of an Eichler order of level $N$ over $\mathbb{Z}[1/p]$ inside the definite quaternion algebra of discriminant $Dp^{-1}$.


In this talk we want to present the method we developed in order to find the Schottky group $\Gamma_{p}^{Sch}\subseteq\Gamma_{p}$ together with a free system of generators for it, at least in the cases of those definite Eichler orders having ideal class number equal to $1$. This generalizes some nice results of Gerritzen and van der Put on Mumford curves arising from the quaternion algebra of discriminant $2$. Consequently, we give an explicit description of the rigid analytic structure of the Mumford curve associate to the group $\Gamma_{p}^{Sch}$ (such as a good fundamental domain in $\mathcal{H}_{p}$ and its stable reduction graph), as well as of the the rigidification of the $p$-adic Shimura curve $X(D,N)\otimes_{\mathbb{Q}}\mathbb{Q}_{p}$.


As an application of our results, we can easily and clearly obtain formulas describing the reduction graphs with lengths of the Shimura curves in question (generalizing some famous formulas of Kurihara), and also formulas for the genera of the reduction of these curves.


This is a joint work with Laia Amor\'{o}s.


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