LOCALLY ANALYTIC VECTORS AND OVERCONVERGENT -MODULES

Abstract

AbstractLet$p$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$\unicode[STIX]{x1D70B}$be a fixed uniformizer of$K$, let$K_{\infty }$be the extension by adjoining to$K$a system of compatible$p^{n}$th roots of$\unicode[STIX]{x1D70B}$for all$n$, and let$L$be the Galois closure of$K_{\infty }$. Using these field extensions, Caruso constructs the$(\unicode[STIX]{x1D711},\unicode[STIX]{x1D70F})$-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$\operatorname{Gal}(L/K)$, in the spirit of the work by Berger and Colmez. Using these locally analytic vectors, and using the classical overconvergent$(\unicode[STIX]{x1D711},\unicode[STIX]{x1D6E4})$-modules, we can establish the overconvergence property of the$(\unicode[STIX]{x1D711},\unicode[STIX]{x1D70F})$-modules.