Iterated extensions and relative Lubin-Tate groups

TL;DR

This paper proves that certain Galois extensions generated by iterated roots of a polynomial over a local field are abelian and can be described using torsion points of a generalized Lubin-Tate group, extending classical local class field theory.

Contribution

It generalizes the theory of Lubin-Tate groups to relative settings and characterizes Galois extensions generated by iterated polynomial roots as torsion points of these groups.

Findings

Galois extensions are abelian under the given conditions

Extensions are generated by torsion points of relative Lubin-Tate groups

The approach involves p-adic periods and local class field theory.

Abstract

Let K be a finite extension of Q_p with residue field F_q and let P(T) = T^d + a_{d-1}T^{d-1} + ... +a_1 T, where d is a power of q and a_i is in the maximal ideal of K for all i. Let u_0 be a uniformizer of O_K and let {u_n}_{n \geq 0} be a sequence of elements of Q_p^alg such that P(u_{n+1}) = u_n for all n \geq 0. Let K_infty be the field generated over K by all the u_n. If K_infty / K is a Galois extension, then it is abelian, and our main result is that it is generated by the torsion points of a relative Lubin-Tate group (a generalization of the usual Lubin-Tate groups). The proof of this involves generalizing the construction of Coleman power series, constructing some p-adic periods in Fontaine's rings, and using local class field theory.