Integral models of certain PEL Shimura varieties with $${\Gamma_1(p)}$$ Γ 1 ( p ) -type level structure

Abstract

We study p-adic integral models of certain PEL Shimura varieties with level subgroup at p related to the \({\Gamma_1(p)}\)-level subgroup in the case of modular curves. We will consider two cases: the case of Shimura varieties associated with unitary groups that split over an unramified extension of \({\mathbb{Q}_p}\) and the case of Siegel modular varieties. We construct local models, i.e. simpler schemes which are etale locally isomorphic to the integral models. Our integral models are defined by a moduli scheme using the notion of an Oort–Tate generator of a group scheme. We use these local models to find a resolution of the integral model in the case of the Siegel modular variety of genus 2. The resolution is regular with special fiber a nonreduced divisor with normal crossings.