Swan conductors for p-adic differential modules, I: A local construction

Abstract

We define a numerical invariant, the differential Swan conductor, for certain differential modules on a rigid analytic annulus over a $p$-adic field. This gives a definition of a conductor for $p$-adic Galois representations with finite local monodromy over an equal characteristic discretely valued field, which agrees with the usual Swan conductor when the residue field is perfect. We also establish analogues of some key properties of the usual Swan conductor, such as integrality (the Hasse–Arf theorem), and the fact that the graded pieces of the associated ramification filtration on Galois groups are abelian and killed by $p$.