The $p$-adic monodromy theorem in the imperfect residue field case

Abstract

Let $K$ be a complete discrete valuation field of mixed characteristic $\left(0,p\right)$ and $G_K$ the absolute Galois group of $K$. In this paper, we will prove the $p$-adic monodromy theorem for $p$-adic representations of $G_K$ without any assumption on the residue field of $K$, for example the finiteness of a $p$-basis of the residue field of $K$. The main point of the proof is a construction of $\left(\phi ,G_K\right)$-module ${\tilde{\mathbb{N}}}_{rig}^{\nabla +}\left(V\right)$ for a de Rham representation $V$, which is a generalization of Pierre Colmez’s ${\tilde{\mathbb{N}}}_{rig}^{+}\left(V\right)$. In particular, our proof is essentially different from Kazuma Morita’s proof in the case when the residue field admits a finite $p$-basis.

We also give a few applications of the $p$-adic monodromy theorem, which are not mentioned in the literature. First, we prove a horizontal analogue of the $p$-adic monodromy theorem. Secondly, we prove an equivalence of categories between the category of horizontal de Rham representations of $G_K$ and the category of de Rham representations of an absolute Galois group of the canonical subfield of $K$. Finally, we compute $H^1$ of some $p$-adic representations of $G_K$, which is a generalization of Osamu Hyodo’s results.