Wach models and overconvergence of étale (φ,Γ)-modules

Abstract

A classical result of Cherbonnier and Colmez says that all étale (φ,Γ)-modules are overconvergent. In this paper, we give another proof of this fact when the base field K is a finite extension of Qp. 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 pn Galois representations, and use them to build an overconvergence basis.