Arithmetic and Differential Swan Conductors of rank one representations with finite local monodromy

TL;DR

This paper proves the equivalence of arithmetic and differential Swan conductors for rank one Galois representations with finite local monodromy over fields with possibly non perfect residue fields, extending Fontaine's formalism.

Contribution

It establishes the equality of arithmetic and differential Swan conductors in a non perfect residue field setting, generalizing previous theories and providing new insights into the Refined Swan Conductor.

Findings

Arithmetic and differential Swan conductors coincide for rank one representations.

Extension of Fontaine's formalism to non perfect residue fields.

New interpretation of the Refined Swan Conductor.

Abstract

We consider a complete discrete valuation field of characteristic p, with possibly non perfect residue field. Let V be a rank one continuous representation with finite local monodromy of its absolute Galois group. We will prove that the Arithmetic Swan conductor of V (defined after Kato in [Kat89] which fits in the more general theory of [AS02] and [AS06]) coincides with the Differential Swan conductor of the associated differential module $D^{\dag}(V)$ defined by Kedlaya in [Ked]. This construction is a generalization to the non perfect residue case of the Fontaine's formalism as presented in [Tsu98a]. Our method of proof will allow us to give a new interpretation of the Refined Swan Conductor.