Regular models of ramified unitary Shimura varieties at maximal parahoric level

TL;DR

This paper constructs a semi-stable model for ramified unitary Shimura varieties with maximal parahoric level, overcoming flatness issues of naive models, and demonstrating semi-stable reduction.

Contribution

It introduces a new flat semi-stable splitting model for ramified unitary Shimura varieties at maximal parahoric level, improving upon previous non-flat models.

Findings

The genuine splitting model is flat with semi-stable reduction.

The naive splitting model fails to be flat in this setting.

The approach clarifies the structure of Shimura varieties at ramified primes.

Abstract

We use the idea of splitting models to define and study a semi-stable model for unitary Shimura varieties of signature $(n-1,1)$ with maximal parahoric level structure at ramified primes. In this case, the ``naive'' splitting model defined by Pappas and Rapoport fails to be flat in a crucial way. We prove that the genuine splitting model in this case is flat with semi-stable reduction.