Galois actions on torsion points of universal one-dimensional formal modules

TL;DR

This paper investigates the Galois actions on torsion points of universal one-dimensional formal modules over local non-Archimedean fields, demonstrating the surjectivity of certain Galois representations associated with these modules.

Contribution

It establishes the surjectivity of the Galois representation on the Tate module for the etale quotient of the universal deformation of formal modules with Drinfeld level structures.

Findings

Galois representation on Tate modules is surjective

Universal deformation spaces are characterized by Drinfeld level structures

Results apply to formal modules over local non-Archimedean fields

Abstract

Let $F$ be a local non-Archimedean field with ring of integers $o$. Let $\bf X$ be a one-dimensional formal $o$-module of $F$-height $n$ over the algebraic closure of the residue field of $o$. By the work of Drinfeld, the universal deformation $X$ of $\bf X$ is a formal group over a power series ring $R_0$ in $n-1$ variables over the completion of the maximal unramified extension of $o$. For $h \in \{0,...,n-1\}$ let $U_h$ be the subscheme of $\Spec(R_0)$ where the connected part of the associated divisible module of $X$ has height $h$. Using the theory of Drinfeld level structures we show that the representation of the fundamental group of $U_h$ on the Tate module of the etale quotient is surjective.