Canonical integral models for Shimura varieties of toral type

Abstract

We prove the Pappas–Rapoport conjecture on the existence of canonical integral models of Shimura varieties with parahoric level structure in the case where the Shimura variety is defined by a torus. As an important ingredient, we show, using the Bhatt–Scholze theory of prismatic $F$-crystals, that there is a fully faithful functor from $\mathcal{𝒢}$-valued crystalline representations of $\mathrm{Gal}<!--FUNCTION\; APPLICATION-->(\overline{K}∕K)$ to $\mathcal{𝒢}$-shtukas over $\mathrm{Spd}<!--FUNCTION\; APPLICATION-->({\mathcal{𝒪}}_K)$, where $\mathcal{𝒢}$ is a parahoric group scheme over ${\mathbb{Z}}_p$ and ${\mathcal{𝒪}}_K$ is the ring of integers in a $p$-adic field $K$.