The prismatic realization functor for Shimura varieties of abelian type

Abstract

For the integral canonical model ${\mathcal{S}}_{{\mathsf{K}}^p}$ 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.

arXiv comment59 pages. The portion concerning an integral analogue of Fontaine's crystalline functor has been separated, and is now included in arXiv:2504.16282