On ramification filtrations and $p$-adic differential modules, I: equal characteristic case

TL;DR

This paper establishes the equivalence of conductors from arithmetic ramification filtrations and differential modules for Galois groups over equal characteristic fields, leading to a Hasse-Arf theorem and related applications.

Contribution

It proves the equality of conductors from different approaches in equal characteristic, extending the Hasse-Arf theorem and linking differential and arithmetic ramification theories.

Findings

Conductors from ramification filtrations match differential Artin and Swan conductors.

Hasse-Arf theorem is valid for arithmetic ramification filtrations in equal characteristic.

Comparison between differential Artin conductors and Borger's conductors is established.

Abstract

Let $k$ be a complete discretely valued field of equal characteristic $p > 0$ with possibly imperfect residue field and let $G_k$ be its Galois group. We prove that the conductors computed by the arithmetic ramification filtrations on $G_k$ coincide with the differential Artin conductors and Swan conductors of Galois representations of $G_k$. As a consequence, we give a Hasse-Arf theorem for arithmetic ramification filtrations in this case. As applications, we obtain a Hasse-Arf theorem for finite flat group schemes; we also give a comparison theorem between the differential Artin conductors and Borger's conductors.