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

TL;DR

This paper proves the p-adic monodromy theorem for p-adic Galois representations over fields with arbitrary residue fields, extending previous results and introducing new constructions and applications in p-adic Hodge theory.

Contribution

It generalizes the p-adic monodromy theorem to cases with imperfect residue fields by constructing a new (phi,G_K)-module and providing novel applications.

Findings

Established the p-adic monodromy theorem without residue field assumptions

Constructed a generalized (phi,G_K)-module for de Rham representations

Computed H^1 for certain p-adic Galois representations

Abstract

Let K be a complete discrete valuation field of mixed characteristic (0,p) 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 (phi,G_K)-module Nrig^+(V) for a de Rham representation V, which is a generalization of Pierre Colmez' Nrig^+(V). 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.