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
Abstract

Let $k$ be a complete discretely valued field of equal characteristic $p>0$ with possibly imperfect residue field, and let ${G}_{k}$ be its Galois group. We prove that the conductors computed by the arithmetic ramification filtrations on ${G}_{k}$ 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 ${G}_{k}$ 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).

Keywords
ramification, $p$-adic differential equation, Swan conductors, Artin conductors, Hasse–Arf theorem
Mathematical Subject Classification 2000
Primary: 11S15
Secondary: 14G22, 12H25