Berger and Colmez (2008) formulated a theory of families of overconvergent étale -modules associated to families of -adic Galois representations over -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 -adic representations in a unique manner, providing the “local” equivalence. There is a global mod obstruction related to the moduli of residual representations.
"On families of $\varphi,\Gamma$-modules." Algebra Number Theory 4 (7) 943 - 967, 2010. https://doi.org/10.2140/ant.2010.4.943