Fiercely ramified cyclic extensions of p-adic fields with imperfect residue field

TL;DR

This paper characterizes the ramification behavior of fierce cyclic Galois extensions of p-adic fields with imperfect residue fields, using Kato's refined Swan conductor to establish necessary and sufficient conditions for possible ramification data.

Contribution

It introduces a complete characterization of ramification data for fierce cyclic extensions of local fields with imperfect residue fields, expanding understanding of ramification in such contexts.

Findings

Provides necessary and sufficient conditions for ramification data

Associates ramification data with differential forms on residue fields

Utilizes Kato's refined Swan conductor to analyze ramification

Abstract

We study the ramification of fierce cyclic Galois extensions of a local field $K$ of characteristic zero with a one-dimensional residue field of characteristic $p>0$. Using Kato's theory of the refined Swan conductor, we associate to such an extension a ramification datum, consisting of a sequence of pairs $(\delta_i,\omega_i)$, where $\delta_i$ is a positive rational number and $\omega_i$ a differential form on the residue field of $K$. Our main result gives necessary and sufficient conditions on such sequences to occur as a ramification datum of a fierce cyclic extension of $K$.