$\mathbb{Z}_p$-lattices in 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.