On certain integral Frobenius period maps for Shimura varieties and their reductions

TL;DR

This paper develops an integral Frobenius period map for Shimura varieties, linking their crystalline and mod p structures, and introduces a framework for base reduction diagrams to analyze their reductions.

Contribution

It formulates a new integral Frobenius period map for Shimura varieties and relates it to local G-shtukas and double G-zips, advancing the understanding of their reductions.

Findings

Derived a period map from the mod p fiber to the moduli stack of local G-shtukas

Established the association of the pair (S, S) with a double G-zip

Introduced a framework of base reduction diagrams for analyzing reductions

Abstract

We formulate an integral Frobenius period map for the framed crystalline prismatization of the $p$-integral model $\mathcal{S}$ of a Shimura variety with good reduction. By analyzing reductions of this map, we derive a period map from the mod $p$ fiber $S$ of $\mathcal{S}$ to the moduli stack of 1-1 truncated local $G$-shtukas in the prismatic topology, which refines the zip period map of $S$ within this topology. Furthermore, we show that the pair $(\mathcal{S}, S)$ is associated with a double $G$-zip. Additionally, we introduce a framework of base reduction diagrams.