On families of $\varphi,\Gamma$-modules

Abstract

Berger and Colmez (2008) formulated a theory of families of overconvergent étale $\left(\phi ,\Gamma \right)$-modules associated to families of $p$-adic Galois representations over $p$-adic Banach algebras. In contrast with the classical theory of $\left(\phi ,\Gamma \right)$-modules, the functor they obtain is not an equivalence of categories. In this paper, we prove that when the base is an affinoid space, every family of (overconvergent) étale $\left(\phi ,\Gamma \right)$-modules can locally be converted into a family of $p$-adic representations in a unique manner, providing the “local” equivalence. There is a global mod $p$ obstruction related to the moduli of residual representations.