Lubin-Tate Deformation Spaces and (φ,Γ)-Modules

Jean-Marc Fontaine has shown that there exists an equivalence of categories between the category of continuous Zp-representations of a given Galois group and the category of etale (φ,Γ)-modules over a certain ring. This work attempts to answer the question of whether there exists a theory of (φ,Γ)-modules for the Lubin-Tate tower. We construct this tower via the rings Rn which parametrize deformations of level n of a given formal module. One can choose prime elements πn in each ring Rn in a compatible way, and consider the tower of fields (K′ n)n obtained by localizing at πn, completing, and passing to fraction fields. By taking the compositum Kn = K0K ′ n of each field with a certain unramified extension K0 of the base field K ′ 0 one obtains a tower of fields (Kn)n which is strictly deeply ramified in the sense of Anthony Scholl. This is a first step towards showing that there exists a theory of (φ,Γ)-modules for this tower.