On the geometry of integral models of Shimura varieties with $Γ_1(p)$-level structure
Giulio Marazza
TL;DR
This work analyzes the geometry of integral models for Siegel and unitary Shimura varieties at a prime $p$ under pro-$p$ Iwahori level structure ($\Gamma_1(p)$). It develops local-model diagrams via root stacks and Oort–Tate data to study flatness and normality, showing that many natural constructions fail to be normal or flat in general, with precise results depending on signature and case (unitary vs Siegel). It also computes a normalization for a Drinfeld-type specialization of a pro-$p$ unitary local model in the $(1,n-1)$ case, illustrating how root-stack atlases can resolve certain singularities. The findings highlight subtle geometric pathologies in bad-reduction settings and offer a framework for analyzing them through local models and OT theory, with implications for arithmetic applications and the use of local models in Shimura-variety reductions.
Abstract
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.