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Fiercely ramified cyclic extensions of p-adic fields with imperfect residue field

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Abstract

We study the ramification of fierce cyclic Galois extensions of a local field K of characteristic zero with a one-dimensional residue field of characteristic p > 0. Using Kato’s theory of the refined Swan conductor, we associate to such an extension a ramification datum, consisting of a sequence of pairs (δ i , ω i ), where δ i is a positive rational number and ω i a differential form on the residue field of K. Our main result gives necessary and sufficient conditions on such sequences to occur as a ramification datum of a fierce cyclic extension of K.

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References

  1. Abbes A., Saito T.: Ramification of local fields with imperfect residue field. Am. J. Math. 124, 879–920 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  2. Cartier P.: Une nouvelles opération sur les formes différentielles. C. R. Acad. Sci. Paris 244, 426–428 (1957)

    MATH  MathSciNet  Google Scholar 

  3. Epp H.P.: Eliminating wild ramification. Invent. math. 19, 235–249 (1973)

    Article  MATH  MathSciNet  Google Scholar 

  4. Garuti, M.A.: Linear systems attached to cyclic inertia. In: Arithmetic Fundamental Groups and Noncommutative Algebra (Berkeley, CA, 1999), volume 70 of Proceedings of Symposium in Pure Mathematics, pp. 377–386. American Mathematical Society (2002)

  5. Hyodo, O.: Wild ramification in the imperfect residue field case. In: Galois Representations and Arithmetic Algebraic Geometry, Number 12 in Advanced Studies in Pure Mathematics, pp. 287–314 (1987)

  6. Kato K.: A generalization of local class field theory by using K-groups, I. J. Fac. Sci. Univ. Tokyo Sec. IA 26, 303–376 (1979)

    MATH  Google Scholar 

  7. Kato K.: A generalization of local class field theory by using K-groups, II. J. Fac. Sci. Univ. Tokyo Sec. IA 27, 603–683 (1980)

    MATH  Google Scholar 

  8. Kato, K.: Galois cohomology of complete discrete valuation fields. In: Algebraic K-theory II, Number 967 in LNM, pp. 215–236. Springer (1982)

  9. Kato, K.: Swan conductors with differential values. In: Galois Representations and Arithmetic Algebraic Geometry, Number 12 in Advanced Studies in Pure Mathematics, pp. 315–342 (1987)

  10. Kato K.: Vanishing cycles, ramification of valuations, and class field theory. Duke Math. J. 55(3), 629–659 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  11. Obus A., Pries R.: Wild tame-by-cyclic extensions. J. Pure Appl. Algebra 214(5), 565–573 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  12. Obus, A., Wewers, S.: Cyclic extensions and the local lifting problem. Ann. Math. (to appear) arXiv:1203.5057 (2012)

    Google Scholar 

  13. Schmid H.L.: Zur Arithmetik der zyklischen p-Körper. J. Reine Angew. Math. 176, 161–167 (1937)

    Google Scholar 

  14. Serre J.-P.: Corps locaux. Hermann, Paris (1968)

    Google Scholar 

  15. Thomas L.: Ramification groups in Artin–Schreier–Witt extensions. J. Théorie des Nombres Bordeaux 17, 689–720 (2005)

    Article  MATH  Google Scholar 

  16. Zhukov I.: On ramification theory in the imperfect residue field case. Sbornik Math. 194(12), 1747–1774 (2003)

    Article  MATH  MathSciNet  Google Scholar 

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  1. Institut für Reine Mathematik, Universität Ulm, Helmholtzstr. 18, 89081, Ulm, Germany

    Stefan Wewers

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  1. Stefan Wewers
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Correspondence to Stefan Wewers.

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Wewers, S. Fiercely ramified cyclic extensions of p-adic fields with imperfect residue field. manuscripta math. 143, 445–472 (2014). https://doi.org/10.1007/s00229-013-0630-1

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  • DOI: https://doi.org/10.1007/s00229-013-0630-1

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