Locally analytic vectors and overconvergent (φ,τ)-modules

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 GK be the Galois group. Let π be a fixed uniformizer of K, let K∞ be the extension by adjoining to K a system of compatible pnth roots of π for all n, and let L be the Galois closure of K∞. Using these field extensions, Caruso constructs the (φ,τ)-modules, which classify p-adic Galois representations of GK. 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 (φ,Γ)-modules, we can establish the overconvergence property of the (φ,τ)-modules.