Lubin-Tate Deformation Spaces and Fields of Norms

TL;DR

This paper constructs a tower of fields from deformation spaces of formal groups, analyzes their ramification properties, and explores their potential Kummer tower structure in the case of height two.

Contribution

It introduces a new tower of fields from deformation spaces and proves its strict deep ramification, also examining Kummer properties for height two.

Findings

The tower is strictly deeply ramified.

The tower's structure is compatible with deformation parameters.

For height two, the tower may have Kummer properties.

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.