Crystalline and semi-stable representations in the imperfect residue field case

TL;DR

This paper extends the theory of p-adic representations to imperfect residue fields, establishing conditions for potential crystallinity and semi-stability, and proving Fontaine's p-adic monodromy theorem in this broader context.

Contribution

It introduces a new criterion linking representations over imperfect residue fields to those over perfect closures, and proves Fontaine's p-adic monodromy theorem in this setting.

Findings

Characterization of potentially crystalline and semi-stable representations over imperfect residue fields.

Establishment of the p-adic monodromy theorem for imperfect residue fields.

Connection between representations over K and its perfect closure K^{pf}.

Abstract

Let K be a p-adic local field with residue field k such that [k:k^p]=p^e<\infty and V be a p-adic representation of Gal(\bar{K}/K). Then, by using the theory of p-adic differential modules, we show that V is a potentially crystalline (resp. potentially semi-stable) representation of Gal(\bar{K}/K) if and only if V is a potentially crystalline (resp. potentially semi-stable) representation of Gal(\bar{K^{pf}}/K^{pf}) where K^{pf}/K is a certain p-adic local field whose residue field is the smallest perfect field k^{pf} containing k. As an application, we prove the p-adic monodromy theorem of Fontaine in the imperfect residue field case.