# Local model of Hilbert-Siegel moduli schemes in $\Gamma_1(p)$-level

## TL;DR

This paper constructs a local model for Hilbert-Siegel moduli schemes with specific level structure and bad reduction, using advanced Lie complex techniques to relate integral and local models over a number field.

## Contribution

It introduces a novel approach to local models of Hilbert-Siegel schemes with $ ext{Gamma}_1(p)$-level, employing a variant of the Lie complex over the small Zariski site.

## Key findings

- Explicit local model construction for Hilbert-Siegel schemes with $ ext{Gamma}_1(p)$-level.
- Calculation of the $ extbf{F}_q$-equivariant Lie complex of Raynaud group schemes.
- Establishment of a relation between integral and local models using Lie complex techniques.

## Abstract

We construct a local model for Hilbert-Siegel moduli schemes with $\Gamma_1(p)$-level bad reduction over $\text{Spec }\mathbb{Z}_{q}$, where $p$ is a prime unramified in the totally real field and $q$ is the residue cardinality over $p$. Our main tool is a variant over the small Zariski site of the ring-equivariant Lie complex $_A\underline{\ell}_G^{\vee}$ defined by Illusie in his thesis, where $A$ is a commutative ring and $G$ is a scheme of $A$-modules. We use it to calculate the $\mathbb{F}_{q}$-equivariant Lie complex of a Raynaud group scheme, then relate the integral model and the local model.