Local model of Hilbert-Siegel moduli schemes in ${\Gamma }_1(p)$-level

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.

40 pages; in French. Comments are welcome!