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ON A GROUP CLOSELY RELATED WITH THE AUTOMORPHIC LANGLANDS GROUP

  • Received : 2018.07.16
  • Accepted : 2019.09.25
  • Published : 2019.12.30

Abstract

Let LK denote the hypothetical automorphic Langlands group of a number field K. In our recent study, we briefly introduced a certain unconditional non-commutative topological group ${\mathfrak{W}}{\mathfrak{A}}{\frac{\varphi}{K}}$, called the Weil-Arthur idèle group of K, which, assuming the existence of LK, comes equipped with a natural topological group homomorphism $NR{\frac{\varphi}{K}^{Langlands}}$ : ${\mathfrak{W}}{\mathfrak{A}}{\frac{\varphi}{K}}$ → LK that we called the "Langlands form" of the global nonabelian norm-residue symbol of K. In this work, we present a detailed construction of ${\mathfrak{W}}{\mathfrak{A}}{\frac{\varphi}{K}}$ and $NR{\frac{\varphi}{K}^{Langlands}}$ : ${\mathfrak{W}}{\mathfrak{A}}{\frac{\varphi}{K}}$ → LK, and discuss their basic properties.

Acknowledgement

Supported by : Yeditepe University

References

  1. J. Arthur, A note on the automorphic Langlands group, Canad. Math. Bull. 45 (2002), no. 4, 466-482. https://doi.org/10.4153/CMB-2002-049-1 https://doi.org/10.4153/CMB-2002-049-1
  2. I. Barnea and S. Shelah, The abelianization of inverse limits of groups, Israel J. Math. 227 (2018), no. 1, 455-483. https://doi.org/10.1007/s11856-018-1741-x https://doi.org/10.1007/s11856-018-1741-x
  3. N. Bourbaki, General topology. Chapters 1-4, translated from the French, reprint of the 1966 edition, Elements of Mathematics (Berlin), Springer-Verlag, Berlin, 1989. https://doi.org/10.1007/978-3-642-61701-0
  4. V. Drinfeld, On the pro-semisimple completion of the fundamental group of a smooth variety over a finite field, Adv. Math. 327 (2018), 708-788. https://doi.org/10.1016/j.aim.2017.06.029 https://doi.org/10.1016/j.aim.2017.06.029
  5. E. S. T. Fan, A note on the cohomology of the Langlands group, Trans. Amer. Math. Soc. 367 (2015), no. 4, 2905-2920. https://doi.org/10.1090/S0002-9947-2014-06230-2 https://doi.org/10.1090/s0002-9947-2014-06230-2
  6. I. Fesenko, Nonabelian local reciprocity maps, in Class eld theory-its centenary and prospect (Tokyo, 1998), 63-78, Adv. Stud. Pure Math., 30, Math. Soc. Japan, Tokyo, 2001. https://doi.org/10.2969/aspm/03010063
  7. J.-M. Fontaine and J.-P. Wintenberger, Le "corps des normes" de certaines extensions algebriques de corps locaux, C. R. Acad. Sci. Paris Ser. A-B 288 (1979), no. 6, A367-A370.
  8. J.-M. Fontaine and J.-P. Wintenberger, Extensions algebrique et corps des normes des extensions APF des corps locaux, C. R. Acad. Sci. Paris Ser. A-B 288 (1979), no. 8, A441-A444.
  9. S. P. Franklin and B. V. S. Thomas, A survey of $k_{\omega}$-spaces, Topology Proc. 2 (1977), no. 1, 111-124 (1978).
  10. H. Glockner, R. Gramlich, and T. Hartnick, Final group topologies, Kac-Moody groups and Pontryagin duality, Israel J. Math. 177 (2010), 49-101. https://doi.org/10.1007/ s11856-010-0038-5 https://doi.org/10.1007/s11856-010-0038-5
  11. M. I. Graev, On free products of topological groups, Izvestiya Akad. Nauk SSSR. Ser. Mat. 14 (1950), 343-354.
  12. K. Ikeda, On the non-abelian global class field theory, Ann. Math. Que. 37 (2013), no. 2, 129-172. https://doi.org/10.1007/s40316-013-0004-9 https://doi.org/10.1007/s40316-013-0004-9
  13. K. Ikeda, Basic properties of the non-Abelian global reciprocity map, in Mathematics in the 21st century, 45-92, Springer Proc. Math. Stat., 98, Springer, Basel, 2015. https://doi.org/10.1007/978-3-0348-0859-0_5
  14. K. Ikeda and E. Serbest, Generalized Fesenko reciprocity map, St. Petersburg Math. J.20 (2009), no. 4, 593-624; translated from Algebra i Analiz 20 (2008), no. 4, 118-159.https://doi.org/10.1090/S1061-0022-09-01063-2 https://doi.org/10.1090/S1061-0022-09-01063-2
  15. K. Ikeda and E. Serbest, Non-abelian local reciprocity law, Manuscripta Math. 132 (2010), no. 1-2, 19-49. https://doi.org/10.1007/s00229-010-0336-6 https://doi.org/10.1007/s00229-010-0336-6
  16. K. Ikeda and E. Serbest, Ramification theory in non-abelian local class field theory, Acta Arith. 144 (2010), no. 4, 373-393. https://doi.org/10.4064/aa144-4-4 https://doi.org/10.4064/aa144-4-4
  17. H. Koch and E. de Shalit, Metabelian local class field theory, J. Reine Angew. Math. 478 (1996), 85-106. https://doi.org/10.1515/crll.1996.478.85 https://doi.org/10.1515/crll.1996.478.85
  18. R. E. Kottwitz, Stable trace formula: cuspidal tempered terms, Duke Math. J. 51 (1984), no. 3, 611-650. https://doi.org/10.1215/S0012-7094-84-05129-9 https://doi.org/10.1215/S0012-7094-84-05129-9
  19. R. P. Langlands, Automorphic representations, Shimura varieties, and motives. EinMarchen, in Automorphic forms, representations and L-functions (Proc. Sympos. PureMath., Oregon State Univ., Corvallis, Ore., 1977), Part 2, 205-246, Proc. Sympos. PureMath., XXXIII, Amer. Math. Soc., Providence, RI, 1979. https://doi.org/10.1090/pspum/033.2
  20. R. P. Langlands, Functoriality and Reciprocity, Two Lectures at the Institute for Advanced Studies, Seminar Notes, March 2011. http://publications.ias.edu/sites/default/les/functoriality.pdf
  21. F. Laubie, Une theorie du corps de classes local non abelien, Compos. Math. 143 (2007),no. 2, 339-362. https://doi.org/10.1112/S0010437X06002600 https://doi.org/10.1112/S0010437X06002600
  22. K. Miyake, Galois-theoretic local-global relations in nilpotent extensions of algebraicnumber fields, in Seminaire de Theorie des Nombres, Paris, 1989-90, 191-207, Progr.Math., 102, Birkhauser Boston, Boston, MA, 1992. https://doi.org/10.1007/978-1-4757-4269-5_14 https://doi.org/10.1007/978-1-4757-4269-5_14
  23. S. A. Morris, Free products of topological groups, Bull. Austral. Math. Soc. 4 (1971),17-29. https://doi.org/10.1017/S0004972700046219 https://doi.org/10.1017/S0004972700046219
  24. J. Neukirch, A. Schmidt, and K. Wingberg, Cohomology of number fields, second edition,Grundlehren der Mathematischen Wissenschaften, 323, Springer-Verlag, Berlin, 2008.https://doi.org/10.1007/978-3-540-37889-1 https://doi.org/10.1007/978-3-540-37889-1
  25. E. T. Ordman, Free products of topological groups which are $k_{\omega}$-spaces, Trans. Amer.Math. Soc. 191 (1974), 61-73. https://doi.org/10.2307/1996981 https://doi.org/10.2307/1996981
  26. J. Tate, Number theoretic background, in Automorphic forms, representations and L-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Part2, 3-26, Proc. Sympos. Pure Math., XXXIII, Amer. Math. Soc., Providence, RI, 1979.https://doi.org/10.1090/pspum/033.2 https://doi.org/10.1090/pspum/033.2