Integral $p$-adic Hodge theory in the imperfect residue field case

TL;DR

This paper develops an integral $p$-adic Hodge theory framework for fields with imperfect residue fields, classifying semi-stable Galois representations and $p$-divisible groups via new categorical equivalences.

Contribution

It introduces a fully faithful functor linking integral semi-stable representations to Breuil-Kisin modules in the imperfect residue field case.

Findings

Classification of semi-stable representations via filtered $(, )$-modules

Construction of a functor to Breuil-Kisin $G_K$-modules

Classification of $p$-divisible groups using minuscule Breuil-Kisin modules

Abstract

Let $K$ be a mixed characteristic complete discrete valuation field with residue field admitting a finite $p$-basis, and let $G_K$ be the Galois group. We first classify semi-stable representations of $G_K$ by weakly admissible filtered $(\varphi,N)$-modules with connections. We then construct a fully faithful functor from the category of \emph{integral} semi-stable representations of $G_K$ to the category of Breuil-Kisin $G_K$-modules. Using the integral theory, we classify $p$-divisible groups over the ring of integers of $K$ by minuscule Breuil-Kisin modules with connections.