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

Kedlaya, Kiran

doi:10.2140/ant.2007.1.269

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

Kiran S. Kedlaya

Vol. 1 (2007), No. 3, 269–300

DOI: 10.2140/ant.2007.1.269

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

Vol. 1, No. 3, 2007