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

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.