Représentations galoisiennes p-adiques et (phi,tau)-modules

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, τ)-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.

51 pages serious problem in Section 3 fixed; 2010-65

References in corpus (1)

• Some Bounds for ramification of $p^n$-torsion semi-stable representations

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• Crystalline condition for $A_{\inf }$-cohomology and ramification bounds
• Cohomology of $(\phi ,\tau )$-modules