Lubin-Tate Deformation Spaces and $(\phi,\Gamma)$-Modules

Jean-Marc Fontaine has shown that there exists an equivalence of categories between the category of continuous $\mathbb{Z}_p$-representations of a given Galois group and the category of \'{e}tale $(\phi,\Gamma)$-modules over a certain ring. This work attempts to answer the question of whether there exists a theory of $(\phi,\Gamma)$-modules for the Lubin-Tate tower. We construct this tower via the rings $R_n$ which parametrize deformations of level $n$ of a given formal module. One can choose prime elements $\pi_n$ in each ring $R_n$ in a compatible way, and consider the tower of fields $(K'_n)_n$ obtained by localizing at $\pi_n$, completing, and passing to fraction fields. By taking the compositum $K_n = K_0 K'_n$ of each field with a certain unramified extension $K_0$ of the base field $K'_0$ one obtains a tower of fields $(K_n)_n$ which is strictly deeply ramified in the sense of Anthony Scholl. This is a first step towards showing that there exists a theory of $(\phi,\Gamma)$-modules for this tower.