Ramification Filtrations of Certain Abelian Lie Extensions of Local Fields

TL;DR

This paper characterizes when certain abelian Lie extensions of local fields are APF extensions and explores related invertible power series, advancing understanding of ramification in local field extensions.

Contribution

It provides a necessary and sufficient condition for APF extensions arising from specific abelian Lie groups of power series, linking formal group theory with local field extensions.

Findings

Characterization of APF extensions via power series conditions

Application to invertible power series commuting with a fixed series

Insights into ramification filtrations of abelian Lie extensions

Abstract

Let $G\subset x{\mathbb F}_q[\![x]\!]$ ($q$ is a power of the prime $p$) be a subset of formal power series over a finite field such that it forms a compact abelian $p$-adic Lie group of dimension $d\ge 1$. We establish a necessary and sufficient condition for the APF extension of local field corresponding to $\left({\mathbb F}_q(\!(x)\!), G\right)$ under the field of norms functor to be an extension of $p$-adic fields. We then apply this result to study family of invertible power series with coefficients in a $p$-adic integers ring and commute with a fixed noninvertible power series under the composition of power series.