Lubin-Tate Deformation Spaces and Fields of Norms

Abstract

We construct a tower of fields from the rings $R_n$ which parametrize pairs $(X,\lambda)$, where $X$ is a deformation of a fixed one-dimensional formal group $\mathbb{X}$ of finite height $h$, together with a Drinfeld level-$n$ structure $\lambda$. We choose principal prime ideals $\mathfrak{p}_n \mid (p)$ in each ring $R_n$ in a compatible way and consider the field $K'_n$ obtained by localizing $R_n$ at $\mathfrak{p}_n$, completing, and passing to the fraction field. By taking the compositum $K_n = K'_n K_0$ of each field with the completion $K_0$ of a certain unramified extension of $K'_0$, we obtain a tower of fields $(K_n)_n$ which we prove to be strictly deeply ramified in the sense of Anthony Scholl. When $h=2$ we also investigate the question of whether this is a Kummer tower.