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 $K_n^{\prime }$ obtained by localizing $R_n$ at ${\mathfrak{p}}_n$, completing, and passing to the fraction field. By taking the compositum $K_n=K_n^{\prime }{K}_0$ of each field with the completion $K_0$ of a certain unramified extension of $K_0^{\prime }$, 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.

arXiv comment23 pages; title changed (from "Lubin-Tate Deformation Spaces and $(\varphi ,\Gamma )$-Modules"); paper significantly reorganized; some expository material removed; many arguments re-written