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

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