Galois actions on torsion points of one-dimensional formal modules

Let F be a local non-Archimedean field with ring of integers o and uniformizer ϖ, and fix an algebraically closed extension k of the residue field of o. Let X be a one-dimensional formal o-module of F-height n over k. By the work of Drinfeld, the universal deformation X of X is a formal group over a power series ring R0 in n−1 variables over Wo(k). For h∈{0,…,n−1} let Uh⊂Spec(R0) be the locus where the connected part of the associated ϖ-divisible module X[ϖ∞] has height h. Using the theory of Drinfeld level structures we show that the representation of π1(Uh) on the Tate module of the étale quotient is surjective.