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On ramification filtrations and p-adic differential equations, II: mixed characteristic case

Part of: Algebraic number theory: local and $p$-adic fields Differential and difference algebra

Published online by Cambridge University Press:  30 November 2011

Liang Xiao
Liang Xiao*
Affiliation:
Department of Mathematics, University of Chicago, 5734 S. University Avenue, Chicago, IL 60637, USA (email: lxiao@math.uchicago.edu)
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Abstract

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Let K be a complete discrete valuation field of mixed characteristic (0,p), with possibly imperfect residue field. We prove a Hasse–Arf theorem for the arithmetic ramification filtrations on GK, except possibly in the absolutely unramified and non-logarithmic case, or the p=2 and logarithmic case. As an application, we obtain a Hasse–Arf theorem for filtrations on finite flat group schemes over 𝒪K.

Keywords

ramificationp-adic differential equationsimperfect residue fields

MSC classification

Secondary: 11S15: Ramification and extension theory 12H25: $p$-adic differential equations

Information

Type
Research Article
Information
Compositio Mathematica , Volume 148 , Issue 2 , March 2012 , pp. 415 - 463
Copyright
Copyright © Foundation Compositio Mathematica 2011

References

[AM04]Abbes, A. and Mokrane, A., Sous-groupes canoniques et cycles évanescents p-adiques pour les variétés abéliennes, Publ. Math. Inst. Hautes Études Sci. 99 (2004), 117–162.CrossRefGoogle Scholar
[AS02]Abbes, A. and Saito, T., Ramification of local fields with imperfect residue fields, Amer. J. Math. 124 (2002), 879–920.CrossRefGoogle Scholar
[AS03]Abbes, A. and Saito, T., Ramification of local fields with imperfect residue fields, II, Doc. Math. Extra Vol. (2003), 5–72 (Kazuya Kato’s fiftieth birthday).Google Scholar
[And07]André, Y., Structure des connexions méromorphes formelles de plusieurs variables et semi-continuité de l’irrégularité, Invent. Math. 170 (2007), 147–198.CrossRefGoogle Scholar
[Ber90]Berkovich, V. G., Spectral theory and analytic geometry over non-Archimedean fields, Mathematical Surveys and Monographs, vol. 33 (American Mathematical Society, Providence, RI, 1990).Google Scholar
[Bor04]Borger, J. M., Conductors and the moduli of residual perfection, Math. Ann. 329 (2004), 1–30.CrossRefGoogle Scholar
[CD94]Christol, G. and Dwork, B., Modules différentiels sur des couronnes, Ann. Inst. Fourier (Grenoble) 44 (1994), 663–701.CrossRefGoogle Scholar
[CM00]Christol, G. and Mebkhout, Z., Sur le théorème de l’indice des équations différentielles p-adiques. III, Ann. of Math. (2) 151 (2000), 385–457.CrossRefGoogle Scholar
[CR94]Christol, G. and Robba, P., Équations différentielles p-adiques. Applications aux sommes exponentielles, Actualités Mathématiques (Hermann, Paris, 1994).Google Scholar
[Hat06]Hattori, S., Ramification of a finite flat group scheme over a local field, J. Number Theory 118 (2006), 145–154.CrossRefGoogle Scholar
[Hat08]Hattori, S., Tame characters and ramification of finite flat group schemes, J. Number Theory 128 (2008), 1091–1108.CrossRefGoogle Scholar
[Kat89]Kato, K., Logarithmic structures of Fontaine–Illusie, in Algebraic analysis, geometry, and number theory (Baltimore, MD, 1988) (Johns Hopkins University Press, Baltimore, MD, 1989), 191–224.Google Scholar
[KS04]Kato, K. and Saito, T., On the conductor formula of Bloch, Publ. Math. Inst. Hautes Études Sci. 100 (2004), 5–151.CrossRefGoogle Scholar
[Ked05]Kedlaya, K. S., Local monodromy of p-adic differential equations: an overview, Int. J. Number Theory 1 (2005), 109–154.CrossRefGoogle Scholar
[Ked07]Kedlaya, K. S., Swan conductors for p-adic differential modules. I. A local construction, Algebra Number Theory 1 (2007), 269–300.CrossRefGoogle Scholar
[Ked10]Kedlaya, K. S., p-adic differential equations, Cambridge Studies in Advanced Mathematics, vol. 125 (Cambridge University Press, Cambridge, 2010).CrossRefGoogle Scholar
[Ked11]Kedlaya, K. S., Swan conductors for p-adic differential modules, II: Global variation, J. Inst. Math. Jussieu 10 (2011), 191–224.CrossRefGoogle Scholar
[KX10]Kedlaya, K. S. and Xiao, L., Differential modules on p-adic polyannuli, J. Inst. Math. Jussieu 9 (2010), 155–201; Erratum, J. Inst. Math. Jussieu 9 (2010), 669–671.CrossRefGoogle Scholar
[Sai]Saito, T., Ramification of local fields with imperfect residue fields, III, Math. Ann., to appear, doi:10.1007/s00208-011-0652-5.CrossRefGoogle Scholar
[Ser79]Serre, J.-P., Local fields, Graduate Texts in Mathematics, vol. 67 (Springer, New York, 1979).CrossRefGoogle Scholar
[Swe68]Sweedler, M. E., Structure of inseparable extensions, Ann. of Math. (2) 87 (1968), 401–410.CrossRefGoogle Scholar
[Xia10]Xiao, L., On ramification filtrations and p-adic differential equations, I: equal characteristic case, Algebraic Number Theory 4 (2010), 969–1027.CrossRefGoogle Scholar