  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 \'{e}tale 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.
