Galois Extensions of Height-One Commuting Dynamical Systems

TL;DR

This paper proves that certain p-adic dynamical systems generated by commuting power series produce abelian Galois extensions and must include torsion series, linking them to formal group endomorphisms.

Contribution

It establishes that height-one commuting power series with Galois-generating points are necessarily formal group endomorphisms, revealing their algebraic structure.

Findings

Galois extensions are abelian

Presence of a maximal order torsion series

Series are endomorphisms of height-one formal groups

Abstract

We consider a dynamical system consisting of a pair of commuting power series, one noninvertible and another nontorsion invertible, of height one with coefficients in the $p$-adic integers. Assuming that each point of the dynamical system generates a Galois extension over the base field, we show that these extensions are in fact abelian, and, using results and considerations from the theory of the field of norms, we also show that the dynamical system must include a torsion series of maximal order. From an earlier result, this shows that the series must in fact be endomorphisms of some height-one formal group.