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Year 2021, Volume: 50 Issue: 5, 1225 - 1250, 15.10.2021

Kazim İlhan Ikeda , Erol Serbest

https://doi.org/10.15672/hujms.834042

Abstract

References

  • [1] A. Abbes and T. Saito, Ramification of local fields with imperfect residue fields, Amer. J. Math. 124 (5), 879–920, 2002.
  • [2] E. Artin and J. Tate, Class Field Theory, AMS Chelsea Publishing, Vol.366, American Mathematical Society, Providence, Rhode Island, 2008.
  • [3] C. Barwick and P. Heine, Pyknotic objects I. Basic notions, arXiv:1904.09966 [math.AG], 2019. Retrieved November 30, 2020, from the arXiv database.
  • [4] S. Bloch, Algebraic K-theory and class field theory for arithmetic surfaces, Ann. Math. 114, 229–266, 1981.
  • [5] O. Braunling, M. Groechenig and J. Wolfson, Geometric and analytic structures on higher adèles, Res. Math. Sci. 3, Paper No. 22, 56 pages, 2016.
  • [6] A. Cámara, Topology on rational points over n-local fields, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Math. RACSAM 110 (2), 417-432, 2016.
  • [7] I.B. Fesenko, Class field theory of multi-dimensional local fields of characteristic zero, with residue field of positive characteristic, Algebra i Analiz 3 (3), 165-196, 1991.
  • [8] I.B. Fesenko, On class field theory of multi-dimensional local fields of positive characteristic, Algebraic K-theory, Adv. Soviet Math., Vol. 4, Amer. Math. Soc., Providence, RI, 1991, 103-127.
  • [9] I.B. Fesenko, Multi-dimensional local class field theory, Dokl. Akad. Nauk SSSR 318 (1), 47-50, 1991.
  • [10] I.B. Fesenko, Abelian local p-class field theory, Math. Anal. 301, 561-586, 1995.
  • [11] I.B. Fesenko, Abelian extensions of complete discrete valuation fields, Number Theory: Séminaire de Théorie des Nombres de Paris 1993-94 (Sinnou David ed.), Cambridge Univ. Press, Cambridge, 47-74, 1996.
  • [12] I.B. Fesenko, Topological Milnor K-groups of higher local fields, Invitation to Higher Local Fields (Ed. I. B. Fesenko, M. Kurihara), Geometry & Topology Monographs 3, Warwick, 61-74, 2000.
  • [13] I.B. Fesenko, Explicit higher local class field theory, Invitation to Higher Local Fields (Ed. I. B. Fesenko, M. Kurihara), Geometry & Topology Monographs 3, Warwick, 95-101, 2000.
  • [14] I.B. Fesenko, Sequential topologies and quotients of the Milnor K-groups of higher local fields, Algebra i Analiz 13 (3), 198–221, 2001.
  • [15] I.B. Fesenko and S.V. Vostokov, Local Fields and Their Extensions (2nd ed.), AMS Translations of Mathematical Monographs 121, AMS, Providence, Rhode Island, 2002.
  • [16] H. Hasse, Die Normenresttheorie relative-Abelscher Zahlkörper als Klassenkörper im Kleinen, J. Reine Angew. Math. (Crelle) 162, 145-154, 1930.
  • [17] M. Hazewinkel, Local class field theory is easy, Advances in Math. 18 (2), 148-181, 1975.
  • [18] A. Huber, On the Parshin-Be˘ılinson adèles for schemes, Abh. Math. Sem. Univ. Hamburg 61, 249–273, 1991.
  • [19] O. Hyodo, Wild ramification in the imperfect residue field case, Galois Groups and Their Representations (Nagoya, 1981), Adv. Stud. Pure Math. 2, North-Holland, Amsterdam, 287–314, 1983.
  • [20] K.I. Ikeda and E. Serbest, Local non-abelian Kato-Parshin reciprocity law, in preparation.
  • [21] K. Iwasawa, Local Class Field Theory, Oxford Mathematical Monographs, Oxford Univ. Press., Clarendon, 1986.
  • [22] K. Kato, A generalization of local class field theory by using K-groups I, II, III, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 26, 303-376, 1979; 27, 603-683, 1980; 29, 31-43, 1982.
  • [23] K. Kato, Vanishing cycles, ramification of valuations, and class field theory, Duke Math. J. 55 (3), 629-659, 1987.
  • [24] K. Kato, Swan conductors for characters of degree one in the imperfect residue field case, Algebraic K-theory and Algebraic Number Theory (Honolulu, HI, 1987), Contemp. Math. 83, Amer. Math. Soc., Providence, RI, 101-131, 1989.
  • [25] K. Kato, Existence theorem for higher local fields, Invitation to Higher Local Fields (Ed. I. B. Fesenko, M. Kurihara), Geometry & Topology Monographs 3, Warwick, 165-195, 2000.
  • [26] K. Kato and S. Saito, Two-dimensional class field theory, Galois Groups and Their Representations (Nagoya, 1981), Adv. Stud. Pure Math. 2, North-Holland, Amsterdam, 103–152, 1983.
  • [27] K. Kato and S. Saito, Global class field theory of arithmetic schemes, Applications of Algebraic K-theory to Algebraic Geometry and Number Theory, Part I, II (Boulder, Colorado, 1983), Contemp. Math., 55, Amer. Math. Soc., Providence, RI, 255–331, 1986.
  • [28] K. Kato and T. Saito Coincidence of two Swan conductors of abelian characters, Épijournal Géom. Algébrique 3, Art. 15, 16 pp, 2019.
  • [29] Y. Kawada and I. Satake, Class formations II, J. Fac. Sci. Univ. Tokyo 7, 453-490, 1955.
  • [30] Y. Koya, A generalization of class formation by using hypercohomology, Invent. Math. 101, 705-715, 1990.
  • [31] Y. Koya, A generalization of Tate-Nakayama theorem by using hypercohomology, Proc. Japan Acad., Ser. A 69 (3), 53-57, 1993.
  • [32] Y. Koya, Class field theory without theorem 90, Algebra Colloq. 1 (4), 347-358, 1994.
  • [33] K. Kurano and K. Shimomoto, An elementary proof of the Cohen-Gabber theorem in the equal characteristic p > 0 case, Tohoku Math. J. 70, 377-389, 2018.
  • [34] S. Lichtenbaum, The construction of weight-two arithmetic cohomology, Invent. Math. 88, 183-215, 1987.
  • [35] V.G. Lomadze, On the ramification theory of two-dimensional local fields, Math. USSR Sbornik 37, 349–365, 1980.
  • [36] A.I. Madunts and I.B. Zhukov, Multi-dimensional complete fields : Topology and other basic constructions, Trudy S.-Peterb. Mat. Obshch. 1995, English translation in Amer. Math. Soc. Transl. (Ser. 2)165, 1-34, 1995.
  • [37] M. Morrow, An introduction to higher dimensional local fields and adèles, arXiv:1204.0586v2 [math.AG], 2012. Retrieved October 25, 2020, from the arXiv database.
  • [38] M. Morrow, Continuity of the norm map on Milnor K-theory, J. K-Theory 9 (3), 565– 577, 2012.
  • [39] J. Neukirch, Neubegründung der Klassenkörpertheorie, Math. Z. 186, 557–574, 1984.
  • [40] J. Neukirch, Class Field Theory, Springer-Verlag, Berlin, 1986.
  • [41] D.V. Osipov, n-dimensional local fields and adèles on n-dimensional schemes, Surveys in Contemporary Mathematics, London Math. Soc. Lecture Note Ser., 347, Cambridge Univ. Press, Cambridge, 131–164, 2008.
  • [42] A.N. Parshin, Class fields and algebraic K-theory, Uspekhi Mat. Nauk 30 (1), 253– 254, 1975.
  • [43] A.N. Parshin, On the arithmetic of two-dimensional schemes. I. Distributions and residues, Izv. Akad. Nauk SSSR Ser. Mat. 40 (4), 736–773, 1976.
  • [44] A.N. Parshin, Abelian coverings of arithmetic schemes, Sov. Math., Dokl. 19, 1438-1442, 1978.
  • [45] A.N. Parshin, Local class field theory, Trudy Mat. Inst. Steklov 165, 143-170, 1985.
  • [46] P. Scholze, Lectures on Condensed Mathematics–Joint work with D. Clausen, Bonn Lectures, 2019.
  • [47] M. Spiess, Class formations and higher dimensional local class field theory, Journal of Number Theory 62, 273–283, 1997.
  • [48] J. Tate, The higher dimensional cohomology groups of class field theory, Ann. of Math. (2nd Series) 56, 294-297, 1952.
  • [49] L. Xiao and I. B. Zhukov, Ramification of higher local fields, approaches and questions, Algebra i Analiz 26 (5), 1-63, 2014.
  • [50] I.B. Zhukov, Higher dimensional local fields, Invitation to Higher Local Fields (Münster, 1999) (Ed. I. B. Fesenko, M. Kurihara), Geom. Topol. Monogr. 3, Geom. Topol. Publ., Coventry, 5-18, 2000.
  • [51] I.B. Zhukov, An approach to higher ramification theory, Invitation to Higher Local Fields (Münster, 1999) (Ed. I. B. Fesenko, M. Kurihara), Geom. Topol. Monogr., 3, Geom. Topol. Publ., Coventry, 143-150, 2000.

Local abelian Kato-Parshin reciprocity law: A survey

Year 2021, Volume: 50 Issue: 5, 1225 - 1250, 15.10.2021

Kazim İlhan Ikeda , Erol Serbest

https://doi.org/10.15672/hujms.834042

Abstract

Let $K$ denote an $n$-dimensional local field. The aim of this expository paper is to survey the basic arithmetic theory of the $n$-dimensional local field $K$ together with its Milnor $K$-theory and Parshin topological $K$-theory; to review Kato's ramification theory for finite abelian extensions of the $n$-dimensional local field $K$, and to state the local abelian higher-dimensional $K$-theoretic generalization of local abelian class field theory of Hasse, which is developed by Kato and Parshin. The paper is geared toward non-abelian generalization of this theory.

Keywords

higher-dimensional local fields Milnor K-theory local class field theory

References

  • [1] A. Abbes and T. Saito, Ramification of local fields with imperfect residue fields, Amer. J. Math. 124 (5), 879–920, 2002.
  • [2] E. Artin and J. Tate, Class Field Theory, AMS Chelsea Publishing, Vol.366, American Mathematical Society, Providence, Rhode Island, 2008.
  • [3] C. Barwick and P. Heine, Pyknotic objects I. Basic notions, arXiv:1904.09966 [math.AG], 2019. Retrieved November 30, 2020, from the arXiv database.
  • [4] S. Bloch, Algebraic K-theory and class field theory for arithmetic surfaces, Ann. Math. 114, 229–266, 1981.
  • [5] O. Braunling, M. Groechenig and J. Wolfson, Geometric and analytic structures on higher adèles, Res. Math. Sci. 3, Paper No. 22, 56 pages, 2016.
  • [6] A. Cámara, Topology on rational points over n-local fields, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Math. RACSAM 110 (2), 417-432, 2016.
  • [7] I.B. Fesenko, Class field theory of multi-dimensional local fields of characteristic zero, with residue field of positive characteristic, Algebra i Analiz 3 (3), 165-196, 1991.
  • [8] I.B. Fesenko, On class field theory of multi-dimensional local fields of positive characteristic, Algebraic K-theory, Adv. Soviet Math., Vol. 4, Amer. Math. Soc., Providence, RI, 1991, 103-127.
  • [9] I.B. Fesenko, Multi-dimensional local class field theory, Dokl. Akad. Nauk SSSR 318 (1), 47-50, 1991.
  • [10] I.B. Fesenko, Abelian local p-class field theory, Math. Anal. 301, 561-586, 1995.
  • [11] I.B. Fesenko, Abelian extensions of complete discrete valuation fields, Number Theory: Séminaire de Théorie des Nombres de Paris 1993-94 (Sinnou David ed.), Cambridge Univ. Press, Cambridge, 47-74, 1996.
  • [12] I.B. Fesenko, Topological Milnor K-groups of higher local fields, Invitation to Higher Local Fields (Ed. I. B. Fesenko, M. Kurihara), Geometry & Topology Monographs 3, Warwick, 61-74, 2000.
  • [13] I.B. Fesenko, Explicit higher local class field theory, Invitation to Higher Local Fields (Ed. I. B. Fesenko, M. Kurihara), Geometry & Topology Monographs 3, Warwick, 95-101, 2000.
  • [14] I.B. Fesenko, Sequential topologies and quotients of the Milnor K-groups of higher local fields, Algebra i Analiz 13 (3), 198–221, 2001.
  • [15] I.B. Fesenko and S.V. Vostokov, Local Fields and Their Extensions (2nd ed.), AMS Translations of Mathematical Monographs 121, AMS, Providence, Rhode Island, 2002.
  • [16] H. Hasse, Die Normenresttheorie relative-Abelscher Zahlkörper als Klassenkörper im Kleinen, J. Reine Angew. Math. (Crelle) 162, 145-154, 1930.
  • [17] M. Hazewinkel, Local class field theory is easy, Advances in Math. 18 (2), 148-181, 1975.
  • [18] A. Huber, On the Parshin-Be˘ılinson adèles for schemes, Abh. Math. Sem. Univ. Hamburg 61, 249–273, 1991.
  • [19] O. Hyodo, Wild ramification in the imperfect residue field case, Galois Groups and Their Representations (Nagoya, 1981), Adv. Stud. Pure Math. 2, North-Holland, Amsterdam, 287–314, 1983.
  • [20] K.I. Ikeda and E. Serbest, Local non-abelian Kato-Parshin reciprocity law, in preparation.
  • [21] K. Iwasawa, Local Class Field Theory, Oxford Mathematical Monographs, Oxford Univ. Press., Clarendon, 1986.
  • [22] K. Kato, A generalization of local class field theory by using K-groups I, II, III, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 26, 303-376, 1979; 27, 603-683, 1980; 29, 31-43, 1982.
  • [23] K. Kato, Vanishing cycles, ramification of valuations, and class field theory, Duke Math. J. 55 (3), 629-659, 1987.
  • [24] K. Kato, Swan conductors for characters of degree one in the imperfect residue field case, Algebraic K-theory and Algebraic Number Theory (Honolulu, HI, 1987), Contemp. Math. 83, Amer. Math. Soc., Providence, RI, 101-131, 1989.
  • [25] K. Kato, Existence theorem for higher local fields, Invitation to Higher Local Fields (Ed. I. B. Fesenko, M. Kurihara), Geometry & Topology Monographs 3, Warwick, 165-195, 2000.
  • [26] K. Kato and S. Saito, Two-dimensional class field theory, Galois Groups and Their Representations (Nagoya, 1981), Adv. Stud. Pure Math. 2, North-Holland, Amsterdam, 103–152, 1983.
  • [27] K. Kato and S. Saito, Global class field theory of arithmetic schemes, Applications of Algebraic K-theory to Algebraic Geometry and Number Theory, Part I, II (Boulder, Colorado, 1983), Contemp. Math., 55, Amer. Math. Soc., Providence, RI, 255–331, 1986.
  • [28] K. Kato and T. Saito Coincidence of two Swan conductors of abelian characters, Épijournal Géom. Algébrique 3, Art. 15, 16 pp, 2019.
  • [29] Y. Kawada and I. Satake, Class formations II, J. Fac. Sci. Univ. Tokyo 7, 453-490, 1955.
  • [30] Y. Koya, A generalization of class formation by using hypercohomology, Invent. Math. 101, 705-715, 1990.
  • [31] Y. Koya, A generalization of Tate-Nakayama theorem by using hypercohomology, Proc. Japan Acad., Ser. A 69 (3), 53-57, 1993.
  • [32] Y. Koya, Class field theory without theorem 90, Algebra Colloq. 1 (4), 347-358, 1994.
  • [33] K. Kurano and K. Shimomoto, An elementary proof of the Cohen-Gabber theorem in the equal characteristic p > 0 case, Tohoku Math. J. 70, 377-389, 2018.
  • [34] S. Lichtenbaum, The construction of weight-two arithmetic cohomology, Invent. Math. 88, 183-215, 1987.
  • [35] V.G. Lomadze, On the ramification theory of two-dimensional local fields, Math. USSR Sbornik 37, 349–365, 1980.
  • [36] A.I. Madunts and I.B. Zhukov, Multi-dimensional complete fields : Topology and other basic constructions, Trudy S.-Peterb. Mat. Obshch. 1995, English translation in Amer. Math. Soc. Transl. (Ser. 2)165, 1-34, 1995.
  • [37] M. Morrow, An introduction to higher dimensional local fields and adèles, arXiv:1204.0586v2 [math.AG], 2012. Retrieved October 25, 2020, from the arXiv database.
  • [38] M. Morrow, Continuity of the norm map on Milnor K-theory, J. K-Theory 9 (3), 565– 577, 2012.
  • [39] J. Neukirch, Neubegründung der Klassenkörpertheorie, Math. Z. 186, 557–574, 1984.
  • [40] J. Neukirch, Class Field Theory, Springer-Verlag, Berlin, 1986.
  • [41] D.V. Osipov, n-dimensional local fields and adèles on n-dimensional schemes, Surveys in Contemporary Mathematics, London Math. Soc. Lecture Note Ser., 347, Cambridge Univ. Press, Cambridge, 131–164, 2008.
  • [42] A.N. Parshin, Class fields and algebraic K-theory, Uspekhi Mat. Nauk 30 (1), 253– 254, 1975.
  • [43] A.N. Parshin, On the arithmetic of two-dimensional schemes. I. Distributions and residues, Izv. Akad. Nauk SSSR Ser. Mat. 40 (4), 736–773, 1976.
  • [44] A.N. Parshin, Abelian coverings of arithmetic schemes, Sov. Math., Dokl. 19, 1438-1442, 1978.
  • [45] A.N. Parshin, Local class field theory, Trudy Mat. Inst. Steklov 165, 143-170, 1985.
  • [46] P. Scholze, Lectures on Condensed Mathematics–Joint work with D. Clausen, Bonn Lectures, 2019.
  • [47] M. Spiess, Class formations and higher dimensional local class field theory, Journal of Number Theory 62, 273–283, 1997.
  • [48] J. Tate, The higher dimensional cohomology groups of class field theory, Ann. of Math. (2nd Series) 56, 294-297, 1952.
  • [49] L. Xiao and I. B. Zhukov, Ramification of higher local fields, approaches and questions, Algebra i Analiz 26 (5), 1-63, 2014.
  • [50] I.B. Zhukov, Higher dimensional local fields, Invitation to Higher Local Fields (Münster, 1999) (Ed. I. B. Fesenko, M. Kurihara), Geom. Topol. Monogr. 3, Geom. Topol. Publ., Coventry, 5-18, 2000.
  • [51] I.B. Zhukov, An approach to higher ramification theory, Invitation to Higher Local Fields (Münster, 1999) (Ed. I. B. Fesenko, M. Kurihara), Geom. Topol. Monogr., 3, Geom. Topol. Publ., Coventry, 143-150, 2000.
There are 51 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Mathematics
Authors

Kazim İlhan Ikeda 0000-0001-6349-3541

Erol Serbest 0000-0002-3047-0826

Publication Date October 15, 2021
Published in Issue Year 2021 Volume: 50 Issue: 5

Cite

APA Ikeda, K. İ., & Serbest, E. (2021). Local abelian Kato-Parshin reciprocity law: A survey. Hacettepe Journal of Mathematics and Statistics, 50(5), 1225-1250. https://doi.org/10.15672/hujms.834042
AMA Ikeda Kİ, Serbest E. Local abelian Kato-Parshin reciprocity law: A survey. Hacettepe Journal of Mathematics and Statistics. October 2021;50(5):1225-1250. doi:10.15672/hujms.834042
Chicago Ikeda, Kazim İlhan, and Erol Serbest. “Local Abelian Kato-Parshin Reciprocity Law: A Survey”. Hacettepe Journal of Mathematics and Statistics 50, no. 5 (October 2021): 1225-50. https://doi.org/10.15672/hujms.834042.
EndNote Ikeda Kİ, Serbest E (October 1, 2021) Local abelian Kato-Parshin reciprocity law: A survey. Hacettepe Journal of Mathematics and Statistics 50 5 1225–1250.
IEEE K. İ. Ikeda and E. Serbest, “Local abelian Kato-Parshin reciprocity law: A survey”, Hacettepe Journal of Mathematics and Statistics, vol. 50, no. 5, pp. 1225–1250, 2021, doi: 10.15672/hujms.834042.
ISNAD Ikeda, Kazim İlhan - Serbest, Erol. “Local Abelian Kato-Parshin Reciprocity Law: A Survey”. Hacettepe Journal of Mathematics and Statistics 50/5 (October 2021), 1225-1250. https://doi.org/10.15672/hujms.834042.
JAMA Ikeda Kİ, Serbest E. Local abelian Kato-Parshin reciprocity law: A survey. Hacettepe Journal of Mathematics and Statistics. 2021;50:1225–1250.
MLA Ikeda, Kazim İlhan and Erol Serbest. “Local Abelian Kato-Parshin Reciprocity Law: A Survey”. Hacettepe Journal of Mathematics and Statistics, vol. 50, no. 5, 2021, pp. 1225-50, doi:10.15672/hujms.834042.
Vancouver Ikeda Kİ, Serbest E. Local abelian Kato-Parshin reciprocity law: A survey. Hacettepe Journal of Mathematics and Statistics. 2021;50(5):1225-50.

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