On the p-adic uniformization of quaternionic Shimura curves

TL;DR

This paper proves a p-adic uniformization theorem for quaternionic Shimura curves associated with a division algebra over a totally real field, using integral models and formal completions.

Contribution

It establishes a new uniformization theorem for Shimura curves with non-split p-adic places, extending p-adic uniformization theory to integral models.

Findings

Proves Cherednik uniformization for certain Shimura curves.

Defines integral models over p-adic integers.

Establishes uniformization for the formal completion of the model.

Abstract

Let D be a quaternion division algebra over a totally real number field F which splits exactly at one infinite place. We assume that there is a p-adic place where D doesn't split. Then the associated Shimura curve has a Cherednik unifomization by the p-adic upper half plane. We define an integral model of the Shimura curve an over the integers O of this p-adic place. We prove an uniformization theorem for the completion of the model over the formal spectrum Spf O.