Relative p-adic Hodge theory, II: (φ,Γ)-modules

Building on foundations introduced in a previous paper, we give several p-adic analytic descriptions of the categories of étale Zp-local systems and étale Qp-local systems on an affinoid algebra over a finite extension of Qp (or more generally, over the fraction field of the Witt vectors of a perfect field of characteristic p). These include generalizations of Fontaine’s theory of (φ, Γ)-modules, the refinement of Fontaine’s construction introduced by Cherbonnier and Colmez, and a recent description by Fargues and Fontaine in terms of semistable vector bundles on a certain scheme. Our descriptions depend on the embedding of the associated affinoid space into an affine toric variety, but there are natural functoriality maps relating the constructions for different choices of the embedding; these may be used to give analogous descriptions over more general rigid or Berkovich analytic spaces.