Hasse-Arf property and abelian extensions for local fields with imperfect residue fields

TL;DR

The paper extends ramification theory for local fields by generalizing the Hasse-Arf criterion to fields with imperfect residue fields. It leverages non-log upper ramification groups in the Abbes-Saito framework and uses tangentially dominant base changes to reduce to the perfect-residue-field setting, proving that L/K is abelian when L/M is abelian and all non-log ramification breaks are integers. Furthermore, it constructs converses to the Hasse-Arf theorem in the imperfect case, showing that non-abelian inertia can yield extensions with non-integral upper ramification breaks, both in classical and non-log contexts. These results broaden the applicability of ramification theory to a wider class of local fields, providing new tools for understanding abelianity and ramification in the imperfect setting.

Abstract

For a finite totally ramified extension $L$ of a complete discrete valuation field $K$ with the perfect residue field of characteristic $p>0$, it is known that $L/K$ is an abelian extension if the upper ramification breaks are integers and if the wild inertia group is abelian. We prove a similar result without the assumption that the residue field is perfect. As an application, we prove a converse to the Hasse-Arf theorem for a complete discrete valuation field with the imperfect residue field. More precisely, for a complete discrete valuation field $K$ with the residue field $\overline{K}$ of residue characteristic $p>2$ and a finite non-abelian Galois extension $L/K$ such that the Galois group of $L/K$ is equal to the inertia group $I$ of $L/K$, we construct a complete discrete valuation field $K'$ with the residue field $\overline{K}$ and a finite Galois extension $L'/K'$ which has at least one non-integral upper ramification break and whose Galois group and inertia group are isomorphic to $I$.

Theorems & Definitions (46)

• Proposition 1.1: feshas
• Theorem 1.2: Theorem \ref{['imperfect']}
• Theorem 1.3: converse
• Theorem 1.4: Theorem \ref{['converse2']}
• Theorem 1.5: Theorem \ref{['converse3']}
• Proposition 2.1: feshas
• Lemma 2.2
• proof
• Lemma 2.3
• proof