Research Article
Year 2025,
Volume: 54 Issue: 2, 542 - 561, 28.04.2025
Abstract
The ``local class field theory'', which can be defined as the description of the extensions of a given local field $K$ with finite residue field of $q=p^f$ elements in terms of the algebraic and analytic objects depending only on the base $K$ is one of the central problems of modern number theory. The theory developed for the abelian extensions, around the fundamental works of Artin and Hasse in the first quarter of the 20th century.
It is natural to ask if one could construct this theory including the non-abelian extensions of the base field. There are two approches to this problem. One approach is based on the ideas of Langlands, and the other on Koch. Koch's method was later generalized by Fesenko and Koch-de Shalit for specific type of non-abelian extensions of the base field. Laubie extended Koch-de Shalit's work and constructed a local non-abelian class field theory for $K$. On the other hand, İkeda and Serbest extended Fesenko's works to construct a non-abelian local class field theory for $K$, containing a $p^{th}$ root of unity.
In this study, we extended İkeda-Serbest's construction of the local reciprocity map for $K$ containing a $p^{th}$ root of unity to any local field. Also we have shown that the extended map satisfies the certain functoriality and ramification theoretic properties.
References
- [1] S. Bedikyan, Abelyen olmayan yerel sınıf cisim kuramı üzerine, PhD thesis, Mimar Sinan Fine Arts University, 2013.
- [2] K. S. Brown, Cohomology of Groups, Springer Verlag New York, 1982.
- [3] I. B. Fesenko, Local reciprocity cycles, Geometry & Topology Monographs 3, 293-298, 2000.
- [4] I. B. Fesenko, Noncommutative local reciprocity maps, Advanced Studies in Pure Math. 30, 63-78, 2001.
- [5] I. B. Fesenko, On the image of noncommutative local reciprocity map, Homology, Homotopy and Appl. 7, 53-62, 2005.
- [6] I. B. Fesenko and S. V. Vostokov, Local Fields and Their Extensions 2nd ed, AMS Translations of Mathematical Monographs 121, Amer. Math. Soc. Providence, RI, 2002.
- [7] 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 Math. 288, 367-370, 1979.
- [8] J.M. Fontaine and J.P. Wintenberger, Extensions algébriques et corps des normes des extensions APF des corps locaux, C. R. Acad. Sci. Paris Sér. A Math. 288, 441-444, 1979.
- [9] A. Gurevich, Description of Galois groups of local fields with the aid of power series, PhD thesis, Humboldt University, 1997.
- [10] M. Hazewinkel, Local class field theory is easy, Adv. Math. 18, 148-181, 1975.
- [11] K. Iwasawa, Local Class Field Theory, Oxford Mathematical Monographs, Oxford Univ. Press, Clarendon, 1986.
- [12] K. İ. İkeda, On the metabelian local Artin map I: Galois conjugation law, Turkish J. Math. 24, 25-58, 2000.
- [13] K. İ. İkeda and E. Serbest, Fesenko reciprocity map, Algebra i Analiz 20 (3), 112-162, 2008.
- [14] K. İ. İkeda and E. Serbest Generalized Fesenko reciprocity map, Algebra i Analiz, 20 (4), 118-159, 2008.
- [15] K. İ. İkeda and E. Serbest, Non-abelian local reciprocity law, Manuscripta Math. 132, 19-49, 2010.
- [16] K. İ. İkeda and E. Serbest, Ramification theory in non-abelian local class field theory, Acta Arith. 144, 373-393, 2010.
- [17] A. S. Kazancıoğlu, Laubie ve genelleştirilmiş Fesenko karşılıklılık ilkelerinin ilişkisi üzerine, PhD thesis, Istanbul Technical University, 2012.
- [18] H. Koch, Local class field theory for metabelian extensions, Proceed. 2nd Gauss Symposium. Conf. A: Mathematics and Theor. Physics (Munich, 1993), de Gruyter, Berlin, 287-300, 1995.
- [19] H. Koch and E. de Shalit, Metabelian local class field theory, J. reine angew. Math., 478, 85-106, 1996.
- [20] S. Lang, Algebra, Springer-Verlag New York, 2002.
- [21] F. Laubie, Une théorie du corps de classes local non abélien, Composito Math. 143, 339-362, 2007.
- [22] J. Neukirch, Class Field Theory, Springer-Verlag, Berlin, 1986.
- [23] J. P. Serre, Local Fields, Springer-Verlag New York, 1979.
- [24] J. S. Wilson, Profinite Groups, Oxford University Press New York, 1998.
Year 2025,
Volume: 54 Issue: 2, 542 - 561, 28.04.2025
Abstract
References
- [1] S. Bedikyan, Abelyen olmayan yerel sınıf cisim kuramı üzerine, PhD thesis, Mimar Sinan Fine Arts University, 2013.
- [2] K. S. Brown, Cohomology of Groups, Springer Verlag New York, 1982.
- [3] I. B. Fesenko, Local reciprocity cycles, Geometry & Topology Monographs 3, 293-298, 2000.
- [4] I. B. Fesenko, Noncommutative local reciprocity maps, Advanced Studies in Pure Math. 30, 63-78, 2001.
- [5] I. B. Fesenko, On the image of noncommutative local reciprocity map, Homology, Homotopy and Appl. 7, 53-62, 2005.
- [6] I. B. Fesenko and S. V. Vostokov, Local Fields and Their Extensions 2nd ed, AMS Translations of Mathematical Monographs 121, Amer. Math. Soc. Providence, RI, 2002.
- [7] 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 Math. 288, 367-370, 1979.
- [8] J.M. Fontaine and J.P. Wintenberger, Extensions algébriques et corps des normes des extensions APF des corps locaux, C. R. Acad. Sci. Paris Sér. A Math. 288, 441-444, 1979.
- [9] A. Gurevich, Description of Galois groups of local fields with the aid of power series, PhD thesis, Humboldt University, 1997.
- [10] M. Hazewinkel, Local class field theory is easy, Adv. Math. 18, 148-181, 1975.
- [11] K. Iwasawa, Local Class Field Theory, Oxford Mathematical Monographs, Oxford Univ. Press, Clarendon, 1986.
- [12] K. İ. İkeda, On the metabelian local Artin map I: Galois conjugation law, Turkish J. Math. 24, 25-58, 2000.
- [13] K. İ. İkeda and E. Serbest, Fesenko reciprocity map, Algebra i Analiz 20 (3), 112-162, 2008.
- [14] K. İ. İkeda and E. Serbest Generalized Fesenko reciprocity map, Algebra i Analiz, 20 (4), 118-159, 2008.
- [15] K. İ. İkeda and E. Serbest, Non-abelian local reciprocity law, Manuscripta Math. 132, 19-49, 2010.
- [16] K. İ. İkeda and E. Serbest, Ramification theory in non-abelian local class field theory, Acta Arith. 144, 373-393, 2010.
- [17] A. S. Kazancıoğlu, Laubie ve genelleştirilmiş Fesenko karşılıklılık ilkelerinin ilişkisi üzerine, PhD thesis, Istanbul Technical University, 2012.
- [18] H. Koch, Local class field theory for metabelian extensions, Proceed. 2nd Gauss Symposium. Conf. A: Mathematics and Theor. Physics (Munich, 1993), de Gruyter, Berlin, 287-300, 1995.
- [19] H. Koch and E. de Shalit, Metabelian local class field theory, J. reine angew. Math., 478, 85-106, 1996.
- [20] S. Lang, Algebra, Springer-Verlag New York, 2002.
- [21] F. Laubie, Une théorie du corps de classes local non abélien, Composito Math. 143, 339-362, 2007.
- [22] J. Neukirch, Class Field Theory, Springer-Verlag, Berlin, 1986.
- [23] J. P. Serre, Local Fields, Springer-Verlag New York, 1979.
- [24] J. S. Wilson, Profinite Groups, Oxford University Press New York, 1998.
There are 24 citations in total.
Details
| Primary Language | English |
|---|---|
| Subjects | Algebra and Number Theory |
| Journal Section | Mathematics |
| Authors | |
| Early Pub Date | August 27, 2024 |
| Publication Date | April 28, 2025 |
| Submission Date | November 15, 2023 |
| Acceptance Date | June 3, 2024 |
| Published in Issue | Year 2025 Volume: 54 Issue: 2 |