On families ofφ,Γ-modules

Berger and Colmez (2008) formulated a theory of families of overconvergent étale .';/-modules associated to families of p-adic Galois representations over p-adic Banach algebras.In contrast with the classical theory of .';/-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 .';/-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.