Canonical integral models for Shimura varieties of toral type

TL;DR

This paper proves the existence of canonical integral models for Shimura varieties of toral type, utilizing prismatic F-crystals and establishing a functor between crystalline representations and shtukas.

Contribution

It confirms the Pappas-Rapoport conjecture for toral Shimura varieties and introduces a functor linking crystalline Galois representations to shtukas via prismatic theory.

Findings

Proved existence of canonical integral models for toral Shimura varieties.

Established a fully faithful functor from crystalline Galois representations to shtukas.

Applied prismatic F-crystals to relate Galois representations and shtukas.

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 Gal$(\bar{K}/K)$ to $\mathcal{G}$-shtukas over Spd$(\mathcal{O}_K)$, 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 $K$.