$\mathbb{Z}_p$-lattices in semistable Galois representations

TL;DR

This paper establishes an equivalence between certain prismatic F-crystals and $Z_p$-lattices in semistable Galois representations, providing a new linear algebraic framework for understanding these representations.

Contribution

It introduces a novel equivalence between logarithmic prismatic F-crystals and lattices in semistable Galois representations, extending Breuil--Kisin module techniques.

Findings

Equivalence between prismatic F-crystals and Galois lattices.

Description of Galois representations via logarithmic Breuil--Kisin modules.

Framework for linear algebraic analysis of semistable Galois representations.

Abstract

We show that the category of logarithmic prismatic F-crystals on $(\mathcal{O}_K, \varpi^{\mathbb{N}})$ is equivalent to the category of $\mathbb{Z}_p$-lattices in semistable $\text{Gal}_K$-representations. We then apply our method to describe such Galois representations using linear algebraic data via various "logarithmic" versions of Breuil--Kisin modules.