On the Hasse-Arf property of local fields

TL;DR

The paper investigates when a finite Galois totally ramified extension $F/K$ of complete DVRs has the Hasse-Arf property, i.e., integer upper-numbering ramification jumps. It develops necessary defining equations for $F$ in terms of ramification jumps and proves that the Hasse-Arf property is equivalent to these equations being extremely rigid. Using Lucas's theorem and ramification-group theory, it shows that, for jumps prime to $p$, the condition reduces to the minimal-polynomial relation $f_i^{[F_{i+1}:F_i]}+ \ast = \kappa_i f_{i-1} + \ast$ with $\kappa_i \in K^\times$ for all $i>1$. The paper illustrates the theory with Witt-vector/Artin-Schreier-type constructions (e.g., Obus–Pries) to exhibit explicit ramification data and minimal-term behavior, bridging abstract criteria with concrete examples.

Abstract

Let $F/K$ be a finite Galois totally & wildly ramified extension of complete discrete valuation fields. We say that the extension has the Hasse-Arf property if the ramification jumps in upper numbering are integers. We give necessary defining equations for $F$ in terms of the ramification jumps. In order for the Hasse-Arf property to hold, these equations become very strict. We prove that the last assertion is an equivalence condition, thus in terms of these defining equations, the Hasse-Arf property becomes an equivalence condition.

Theorems & Definitions (28)

• Lemma 1
• proof
• Remark 2
• Corollary 3
• proof
• Lemma 4
• proof
• Lemma 5
• proof
• Lemma 6