On refined ramification filtrations in the equal characteristic case

TL;DR

This paper introduces refined ramification invariants for Galois representations over equal characteristic p fields using p-adic differential modules, linking them to existing cohomological definitions and studying their variation.

Contribution

It defines refined Artin and Swan conductors in the equal characteristic setting using differential modules, aligning with Saito's cohomological approach and exploring their behavior under toroidal variation.

Findings

Refined Swan conductors coincide with Saito's cohomological definition.

Description of ramification filtration subquotients using these invariants.

Analysis of how Swan conductors vary toroidally.

Abstract

Let k be a complete discrete valuation field of equal characteristic p>0. Using the tools of p-adic differential modules, we define refined Artin and Swan conductors for a representation of the absolute Galois group $G_k$ with finite local monodromy; this leads to a description of the subquotients of the ramification filtration on $G_k$. We prove that our definition of the refined Swan conductors coincide with that is given by Saito, which uses etale cohomology. We also study its relation with the toroidal variation of the Swan conductors.