LOCALLY ANALYTIC VECTORS AND OVERCONVERGENT (phi, tau)-MODULES

Abstract

Let p be a prime, let K be a complete discrete valuation field of characteristic 0 with a perfect residue field of characteristic p, and let G(K) be the Galois group. Let pi be a fixed uniformizer of K, let K-infinity be the extension by adjoining to K a system of compatible p(n) th roots of pi for all n, and let L be the Galois closure of K-infinity. Using these field extensions, Caruso constructs the (phi, tau)-modules, which classify p-adic Galois representations of G(K). In this paper, we study locally analytic vectors in some period rings with respect to the p -adic Lie group Gal(L/K), in the spirit of the work by Berger and Colmez. Using these locally analytic vectors, and using the classical overconvergent (phi, Gamma)-modules, we can establish the overconvergence property of the (phi, tau)-modules.