The prismatic realization functor for Shimura varieties of abelian type

TL;DR

This paper constructs a prismatic $F$-gauge model for Shimura varieties of abelian type, leading to new insights into their $p$-adic geometry, including an analogue of the Serre--Tate deformation theorem.

Contribution

It introduces a prismatic realization functor for Shimura varieties of abelian type, providing new tools for understanding their $p$-adic properties and deformations.

Findings

Developed a prismatic $F$-gauge model for universal local systems.

Established an abelian-type analogue of the Serre--Tate deformation theorem.

Provided a prismatic characterization of integral canonical models.

Abstract

For the integral canonical model $\mathscr{S}_{\mathsf{K}^p}$ of a Shimura variety $\mathrm{Sh}_{\mathsf{K}_0\mathsf{K}^p}(\mathbf{G},\mathbf{X})$ of abelian type at hyperspecial level $K_0=\mathcal{G}(\mathbb{Z}_p)$, we construct a prismatic $F$-gauge model for the `universal' $\mathcal{G}(\mathbb{Z}_p)$-local system on $\mathrm{Sh}_{\mathsf{K}_0\mathsf{K}^p}(\mathbf{G},\mathbf{X})$. We use this to obtain several new results about the $p$-adic geometry of Shimura varieties, notably an abelian-type analogue of the Serre--Tate deformation theorem (realizing an expectation of Drinfeld in the abelian-type case) and a prismatic characterization of these models at individual level.