On Families of (Phi,Gamma)-modules

TL;DR

This paper establishes a local equivalence between families of overconvergent étale (Phi,Gamma)-modules and p-adic Galois representations over affinoid spaces, clarifying the relationship in p-adic Hodge theory.

Contribution

It proves that over affinoid spaces, families of overconvergent étale (Phi,Gamma)-modules can be locally converted into p-adic representations uniquely, extending the classical theory.

Findings

Local equivalence of (Phi,Gamma)-modules and p-adic representations over affinoid spaces

Existence of a global mod p obstruction related to residual representations

Clarification of the relationship between (Phi,Gamma)-modules and Galois representations

Abstract

Berger and Colmez introduced a theory of families of overconvergent \'etale (Phi,Gamma)-modules associated to families of p-adic Galois representations over p-adic Banach algebras. However, in contrast with the classical theory of (Phi,Gamma)-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) \'etale (Phi,Gamma)-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.