  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.
