Repr\'esentations galoisiennes p-adiques et (phi,tau)-modules

TL;DR

This paper develops a new classification of p-adic Galois representations using (phi, tau)-modules, extending Fontaine's theory by replacing the cyclotomic extension with a different p-adic extension, and connects it to Kisin's modules.

Contribution

It introduces (phi, tau)-modules as an analogue of Fontaine's (phi,Gamma)-modules for a new p-adic extension, providing a novel classification method.

Findings

Classifies p-adic Galois representations via (phi, tau)-modules.

Links (phi, tau)-modules to Kisin's (phi, N_nabla)-modules.

Proves representations of finite height are potentially semi-stable.

Abstract

Let p be an odd prime number and K be a p-adic field. In this paper, we develop an analogue of Fontaine's theory of (phi,Gamma)-modules replacing the p-cyclotomic extension by the extension K_infty obtained by adding to K a compatible system of p^n-th roots of a fixed uniformizer pi of K. As a result, we obtain a new classification of p-adic representations of G_K = Gal(Kbar/K) by some (phi, \tau)-modules. We then make a link between the theory of (phi,tau)-modules discussed above and the so-called theory of (phi,N_nabla)$-modules developped by Kisin. As a corollary, we answer a question of Tong Liu: we prove that, if K is a finite extension of Q_p, every representation of G_K of E(u)-finite height is potentially semi-stable.