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

We consider a complete discrete valuation field of characteristic p, with possibly nonperfect residue field. Let V be a rank one continuous representation of its absolute Galois group with finite local monodromy. we will prove that the arithmetic Swan conductor of V (defined after K. Kato, which fits in the more general theory of Abbes-Saito) coincides with the differential Swan conductor of the associated differential module D(dagger)(V) defined by K. Kedlaya. This construction is a generalization to the nonperfect residue case of the Fontaine's formalism as presented in the work of N. Tsuzuki. Our method of proof will allow us to give a new interpretation of the refined Swan conductor.