On optimal $p$-adic uniformization of unitary Shimura curves

TL;DR

This paper extends previous work on $p$-adic uniformization of Shimura curves by exploring new variants, including maximal level at a special place and explicit local models for anisotropic unitary groups.

Contribution

It introduces two variants of $p$-adic uniformization for Shimura curves, expanding the scope to maximal levels and explicit local Shimura varieties for anisotropic unitary groups.

Findings

Allows maximal level at the special $p$-adic place in the RSZ variant.

Provides explicit determination of local Shimura varieties for anisotropic unitary groups.

Extends the framework of $p$-adic uniformization to new settings.

Abstract

The paper is a continuation of the paper of Kudla-Rapoport-Zink on $p$-adic uniformization of Shimura curves associated to a group of binary unitary similitudes. Here we consider two variants: first, the RSZ variant, for which we can allow any level which is maximal at the chosen special $p$-adic place where the group is anisotropic; second, the unitary group variant. The latter is based on an explicit determination of the integral local Shimura variety associated to an anisotropic unitary group over a $p$-adic local field.