Wach models and overconvergence of étale $(\phi ,\Gamma )$-modules

Abstract

A classical result of Cherbonnier and Colmez says that all étale $(\phi ,\Gamma )$-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.

Final version. To appear, J. Number Theory. arXiv admin note: text overlap with arXiv:1606.07216