Multivariable (φ,Γ)-modules and locally analytic vectors

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Let K be a finite extension of Qₚ and let Gₖ = Gal(Qₚ/K). There is a very useful classification of p-adic representations of Gₖ in terms of cyclotomic (φ,Γ)-modules (cyclotomic means that Γ= Gal(K∞/K) where K∞ is the cyclotomic extension of K). One particularly convenient feature of the cyclotomic theory is the fact that any (φ,Γ)-module is overconvergent. Questions pertaining to the p-adic local Langlands correspondence lead us to ask for a generalization of the theory of (φ,Γ)-modules, with the cyclotomic extension replaced by an infinitely ramified p-adic Lie extension K∞ / K. It is not clear what shape such a generalization should have in general. Even in the case where we have such a generalization, namely the case of a Lubin-Tate extension, most (φ,Γ)-modules fail to be overconvergent. In this article, we develop an approach that gives a solution to both problems at the same time, by considering the locally analytic vectors for the action of Γ inside some big modules defined using Fontaine's rings of periods. We show that, in the cyclotomic case, we recover the ususal overconvergent (φ,Γ)-modules. In the Lubin-Tate case, we can prove, as an application of our theory, a folklore conjecture in the field stating that (φ,Γ)-modules attached to F-analytic representations are overconvergent.