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 $K_{\infty }$ 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 $K_{\infty }$. 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 $\mathrm{Gal}(L/K)$, 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.

Some minor corrections (materials in Subsection 2.1 are substantially re-arranged.) Final version, to appear, J. Inst. Math. Jussieu