Locally analytic vectors and overconvergent $(\varphi, \tau)$-modules

TL;DR

This paper investigates locally analytic vectors in period rings related to $p$-adic Galois representations, demonstrating the overconvergence of $(, au)$-modules by building on classical overconvergent $(, )$-modules and the work of Berger and Colmez.

Contribution

It establishes the overconvergence property of $(, au)$-modules using locally analytic vectors and classical overconvergent $(, )$-modules, extending previous frameworks.

Findings

Proves overconvergence of $(, au)$-modules.

Analyzes locally analytic vectors in period rings.

Connects $(, au)$-modules with classical overconvergent 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 $(\varphi, \tau)$-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 $(\varphi, \Gamma)$-modules, we can establish the overconvergence property of the $(\varphi, \tau)$-modules.