Log prismatic $F$-crystals and purity

TL;DR

This paper establishes a criterion for the semistability of p-adic local systems on rigid-analytic varieties by analyzing prismatic F-crystals and proving a purity theorem, linking local and global properties.

Contribution

It introduces a new approach using analytic prismatic F-crystals and proves a prismatic purity theorem that characterizes semistability of local systems.

Findings

Semistability of local systems is equivalent to semistability on irreducible components.

Proves a prismatic purity theorem for Breuil-Kisin log prisms.

Establishes a criterion connecting local and global semistability.

Abstract

Our goal is to study $p$-adic local systems on a rigid-analytic variety with semistable formal model. We prove that such a local system is semistable if and only if so are its restrictions to the points corresponding to the irreducible components of the special fiber. For this, the main body of the paper concerns analytic prismatic $F$-crystals on the absolute logarithmic prismatic site of a semistable $p$-adic log formal scheme. Analyzing Breuil-Kisin log prisms, we obtain a prismatic purity theorem and deduce the above purity theorem for semistable local systems.