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

TL;DR

The paper constructs an integral Frobenius period map for the framed crystalline prismatization of a $p$-integral Shimura model with good reduction, connecting arithmetic and function-field period data via prismatic topology. It defines the Breuil–Kisin prismatic and crystalline Frobenius period maps, and analyzes their reductions to the mod $p$ fiber, establishing a reduction framework that recovers and refines the classical zip period map. A key outcome is the emergence of a prismatic zip gauge and a double $G$-zip over the Shimura pair $(\mathcal{S}, S)$, tying together prismatic torsors, loop groups, and zips in a unified base-reduction diagram framework. These results provide a geometric bridge between the integral $p$-adic and function-field perspectives, with explicit constructions of torsors, period domains, and reduction morphisms that preserve tensor data and Frobenius structures. The framework offers tools for translating arithmetic Frobenius data into function-field period maps, enabling refined local- and global-zip analyses in the prismatic setting.

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.

Theorems & Definitions (78)

• Definition 1.1
• Definition 1.2
• Definition 1.3
• Definition 1.4
• Definition 1.5: Base Reduction Diagram
• Definition 1.6
• Definition 1.7
• Remark 1.8
• Theorem 1.9
• Remark 1.10