Wintenberger's Functor for Abelian Extensions

TL;DR

This paper extends Wintenberger's functor, which relates totally ramified abelian p-adic Lie extensions over local fields to automorphism groups, to include more general abelian pro-p groups.

Contribution

The paper generalizes Wintenberger's equivalence to encompass arbitrary abelian pro-p groups, broadening its applicability in local field extension theory.

Findings

Extended the functor to arbitrary abelian pro-p groups.

Maintained the equivalence between field extensions and automorphism groups.

Broadened the scope of Wintenberger's original construction.

Abstract

Let $k$ be a finite field. Wintenberger used the field of norms to give an equivalence between a category whose objects are totally ramified abelian $p$-adic Lie extensions $E/F$, where $F$ is a local field with residue field $k$, and a category whose objects are pairs $(K,A)$, where $K\cong k((T))$ and $A$ is an abelian $p$-adic Lie subgroup of $\Aut_k(K)$. In this paper we extend this equivalence to allow $\Gal(E/F)$ and $A$ to be arbitrary abelian pro-$p$ groups.