Vol. 1, No. 3, 2007

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ISSN: 1944-7833 (e-only)
ISSN: 1937-0652 (print)
Swan conductors for p-adic differential modules, I: A local construction

Kiran S. Kedlaya

Vol. 1 (2007), No. 3, 269–300
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.

Keywords
$p$-adic differential modules, Swan conductors, wild ramification, Hasse–Arf theorem, imperfect residue fields
Mathematical Subject Classification 2000
Primary: 11S15
Secondary: 14F30
Milestones
Received: 12 February 2007
Revised: 13 August 2007
Accepted: 17 September 2007
Published: 1 August 2007
Authors
Kiran S. Kedlaya
Department of Mathematics
Massachusetts Institute of Technology
77 Massachusetts Avenue
Cambridge, MA 02139
http://math.mit.edu/~kedlaya