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
Full-text PDF
Abstract | References | Similar Articles | Additional Information
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 , 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.
- 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)
Retrieve articles in St. Petersburg Mathematical Journal with MSC (2000): 11S20, 11S31
Retrieve articles in all journals with MSC (2000): 11S20, 11S31
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