On the geometry of integral models of Shimura varieties with ${\mathrm{\Gamma }}_1(p)$-level structure

We study integral models of some Shimura varieties with bad reduction at a prime $p$, namely the Siegel modular variety and Shimura varieties associated with some unitary groups. We focus on the case where the level structure at $p$ is given by the pro-unipotent radical of an Iwahori subgroup, and we analyze the geometry of the integral models that have been proposed until now: we show that they are almost never normal and in some cases not flat over ${\mathbb{Z}}_p$. We do so by showing the failure of these geometric properties on the corresponding local models, and we explain how the local model diagrams can be interpreted using the root stack construction.