# Wach models and overconvergence of \'etale $(\varphi, \Gamma)$-modules

## TL;DR

This paper provides a new proof that all étale $(,g)$-modules over a finite extension of $Q_p$ are overconvergent, and establishes an explicit lower bound for the overconvergence radius using Wach models.

## Contribution

It offers an alternative proof of overconvergence for étale $(,g)$-modules and introduces a uniform lower bound for the overconvergence radius based on Wach models.

## Key findings

- All étale $(,g)$-modules are overconvergent over finite extensions of $Q_p$
- Explicit lower bound for the overconvergence radius established
- Method uses Wach models in modulo $p^n$ Galois representations

## Abstract

A classical result of Cherbonnier and Colmez says that all \'etale $(\varphi, \Gamma)$-modules are overconvergent. In this paper, we give another proof of this fact when the base field $K$ is a finite extension of $\mathbb Q_p$. Furthermore, we obtain an explicit ("uniform") lower bound for the overconvergence radius, which was previously not known. The method is similar to that in a previous joint paper with Tong Liu. Namely, we study Wach models (when $K$ is unramified) in modulo $p^n$ Galois representations, and use them to build an overconvergence basis.