Generalized Hasse-Herbrand functions in positive characteristic

TL;DR

This paper extends properties of the classical Hasse-Herbrand function to a broader setting involving extensions of complete discrete valuation fields in positive characteristic, even when the residue field of the extension is not perfect.

Contribution

It introduces and analyzes the generalized Hasse-Herbrand function $ ext{psi}_{L/K}^{ ext{ab}}$ in positive characteristic, establishing its key properties similar to the classical case.

Findings

$ ext{psi}_{L/K}^{ ext{ab}}$ is continuous and piecewise linear.

It is increasing and convex.

It satisfies certain integrality properties.

Abstract

Let $L/K$ be an extension of complete discrete valuation fields of positive characteristic, and assume that the residue field of $K$ is perfect. The residue field of $L$ is not assumed to be perfect. In this paper, we show that the generalized Hasse-Herbrand function $\psi_{L/K}^{\mathrm{ab}}$ has properties similar to those of its classical counterpart. In particular, we prove that $\psi_{L/K}^{\mathrm{ab}}$ is continuous, piecewise linear, increasing, convex, and satisfies certain integrality properties.