Rigid character groups, lubin-tate theory, and (ϕ,γ)-modules

The construction of the p-adic local Langlands correspondence for GL₂(Q_p) uses in an essential way Fontaine’s theory of cyclotomic (ϕ,Γ)-modules. Here cyclotomic means that Γ = Gal(Q_p(μ_p∞)/Q_p) is the Galois group of the cyclotomic extension of Q_p. In order to generalize the p-adic local Langlands correspondence to GL₂(L), where L is a finite extension of Q_p, it seems necessary to have at our disposal a theory of Lubin-Tate (ϕ,Γ)-modules. Such a generalization has been carried out to some extent, by working over the p-adic open unit disk, endowed with the action of the endomorphisms of a Lubin-Tate group. The main idea of our article is to carry out a Lubin-Tate generalization of the theory of cyclotomic (ϕ,Γ)-modules in a different fashion. Instead of the p-adic open unit disk, we work over a character variety, that parameterizes the locally L-analytic characters on o_L. We study (ϕ,Γ)-modules in this setting, and relate some of them to what was known previously.