Galois actions on torsion points of one-dimensional formal modules

Abstract

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 R 0 in n − 1 variables over W o ( k ) . For h ∈ { 0 , … , n − 1 } let U h ⊂ Spec ( R 0 ) 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 π 1 ( U h ) on the Tate module of the étale quotient is surjective.