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