The prismatic realization functor for Shimura varieties of abelian type

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 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 new \(p\)-adic Hodge theoretic information about these Shimura varieties, and to provide a prismatic characterization of these models. To do this, we make several advances in integral \(p\)-adic Hodge theory, notably the development of an integral analogue of the functor \(D_{\mathrm{crys}}\).