Vol. 4, No. 8, 2010

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On ramification filtrations and $p$-adic differential modules, I: the equal characteristic case

Liang Xiao

Vol. 4 (2010), No. 8, 969–1027

Let k be a complete discretely valued field of equal characteristic p > 0 with possibly imperfect residue field, and let Gk be its Galois group. We prove that the conductors computed by the arithmetic ramification filtrations on Gk defined by Abbes and Saito (Amer. J. Math 124:5, 879–920) coincide with the differential Artin conductors and Swan conductors of Galois representations of Gk defined by Kedlaya (Algebra Number Theory 1:3, 269–300). As a consequence, we obtain a Hasse–Arf theorem for arithmetic ramification filtrations in this case. As applications, we obtain a Hasse–Arf theorem for finite flat group schemes; we also give a comparison theorem between the differential Artin conductors and Borger’s conductors (Math. Ann. 329:1, 1–30).

ramification, $p$-adic differential equation, Swan conductors, Artin conductors, Hasse–Arf theorem
Mathematical Subject Classification 2000
Primary: 11S15
Secondary: 14G22, 12H25
Received: 14 May 2009
Revised: 25 May 2010
Accepted: 22 June 2010
Published: 24 February 2011
Liang Xiao
University of Chicago
321 Eckhart Hall
5734 S University Avenue
Chicago, IL 60637
United States