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St. Petersburg Mathematical Journal

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Fesenko reciprocity map


Authors: K. I. Ikeda and E. Serbest
Original publication: Algebra i Analiz, tom 20 (2008), nomer 3.
Journal: St. Petersburg Math. J. 20 (2009), 407-445
MSC (2000): Primary 11S20, 11S31
DOI: https://doi.org/10.1090/S1061-0022-09-01054-1
Published electronically: April 7, 2009
MathSciNet review: 2454454
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Abstract: In recent papers, Fesenko has defined the non-Abelian local reciprocity map for every totally ramified arithmetically profinite (APF) Galois extension of a given local field $ K$, by extending the work of Hazewinkel and Neukirch-Iwasawa. The theory of Fesenko extends the previous non-Abelian generalizations of local class field theory given by Koch-de Shalit and by A. Gurevich. In this paper, which is research-expository in nature, we give a detailed account of Fesenko's work, including all the skipped proofs.


References [Enhancements On Off] (What's this?)

  • 1. I. B. Fesenko, Local reciprocity cycles, Invitation to Higher Local Fields (Münster, 1999) (I. B. Fesenko and M. Kurihara, eds.), Geom. Topol. Monogr., vol. 3, Geom. Topol. Publ., Coventry, 2000, pp. 293-298. MR 1804942 (2001k:11239)
  • 2. -, Nonabelian local reciprocity maps, Class Field Theory -- Its Centenary and Prospect (Tokyo, 1998) (K. Miyake, ed.), Adv. Stud. Pure Math., vol. 30, Math. Soc. Japan, Tokyo, 2001, pp. 63-78. MR 1846451 (2002f:11177)
  • 3. -, On the image of noncommutative local reciprocity map, Homology, Homotopy Appl. 7 (2005), 53-62. MR 2200206 (2006m:11171)
  • 4. I. B. Fesenko and S. V. Vostokov, Local fields and their extensions. A constructive approach, Transl. Math. Monogr., vol. 121, Amer. Math. Soc., Providence, RI, 1993. MR 1218392 (94d:11095)
  • 5. J.-M. Fontaine and J.-P. Wintenberger, Le ``corps des normes'' de certaines extensions algébriques de corps locaux, C. R. Acad. Sci. Paris Sér. A-B 288 (1979), A367-A370. MR 0526137 (80b:12015)
  • 6. -, Extensions algébriques et corps des normes des extensions APF des corps locaux, C. R. Acad. Sci. Paris Sér. A-B 288 (1979), A441-A444. MR 0527692 (80h:12014)
  • 7. A. Gurevich, Description of Galois groups of local fields with the aid of power series, Ph. D. Thesis, Humboldt Univ., Berlin, 1998.
  • 8. M. Hazewinkel, Local class field theory is easy, Adv. Math. 18 (1975), 148-181. MR 0389858 (52:10688)
  • 9. K. I. Ikeda and M. Ikeda, Two lemmas on formal power series, Turkish J. Math. 23 (1999), 435-440. MR 1773548 (2001f:13032)
  • 10. K. I. Ikeda and E. Serbest, Non-abelian local class field theory, Preprint.
  • 11. -, A remark on Sen's theory on Galois representations (in preparation).
  • 12. K. Iwasawa, Local class field theory, Oxford Univ. Press, Clarendon Press, New York, 1986. MR 0863740 (88b:11080)
  • 13. H. Koch and E. de Shalit, Metabelian local class field theory, J. Reine Angew. Math. 478 (1996), 85-106. MR 1409054 (97f:11095)
  • 14. F. Laubie, Une théorie du corps de classes local non abélien, Compositio Math. 143 (2007), 339-362. MR 2309990 (2008b:11124)
  • 15. J. Neukirch, Class field theory, Grundlehren Math. Wiss., Bd. 280, Springer-Verlag, Berlin, 1986. MR 0819231 (87i:11005)
  • 16. J.-P. Serre, Corps locaux, 2-me éd., Publ. Inst. Math. Univ. Nancago, No. 8, Actualités Sci. Indust., No. 1296, Hermann, Paris, 1968. MR 0354618 (50:7096)
  • 17. J.-P. Wintenberger, Le corps des normes de certaines extensions infinies de corps locaux; applications, Ann. Sci. École Norm. Sup. (4) 16 (1983), 59-89. MR 0719763 (85e:11098)

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Additional Information

K. I. Ikeda
Affiliation: Department of Mathematics, Istanbul Bilgi University, Kurtuluş Deresi Cad. No. 47, Dolapdere, 34440 Beyoǧlu, Istanbul, Turkey
Address at time of publication: Department of Mathematics, Yeditepe University, 26 Aǧustos Yerleşimi, İnönü Mah., Kayışdaǧı Cad., 34755 Kadıköy, Istanbul, Turkey
Email: ilhan@bilgi.edu.tr, ilhan.ikeda@yeditepe.edu.tr

E. Serbest
Affiliation: Department of Mathematics, Atilim University, Kizilcaşar Köyü, Incek, 06836 Gölbaşı, Ankara, Turkey
Email: eserbest@atilim.edu.tr

DOI: https://doi.org/10.1090/S1061-0022-09-01054-1
Keywords: Local fields, higher-ramification theory, APF extensions, Fontaine--Wintenberger field of norms, Fesenko reciprocity map, non-Abelian local class field theory, $p$-adic local Langlands correspondence
Received by editor(s): August 20, 2007
Published electronically: April 7, 2009
Dedicated: Dedicated to our teacher Mehpare Bilhan
Article copyright: © Copyright 2009 American Mathematical Society

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