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


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.



Supported by : Yeditepe University


  1. J. Arthur, A note on the automorphic Langlands group, Canad. Math. Bull. 45 (2002), no. 4, 466-482.
  2. I. Barnea and S. Shelah, The abelianization of inverse limits of groups, Israel J. Math. 227 (2018), no. 1, 455-483.
  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.
  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.
  5. E. S. T. Fan, A note on the cohomology of the Langlands group, Trans. Amer. Math. Soc. 367 (2015), no. 4, 2905-2920.
  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.
  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. 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.
  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.
  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.
  15. K. Ikeda and E. Serbest, Non-abelian local reciprocity law, Manuscripta Math. 132 (2010), no. 1-2, 19-49.
  16. K. Ikeda and E. Serbest, Ramification theory in non-abelian local class field theory, Acta Arith. 144 (2010), no. 4, 373-393.
  17. H. Koch and E. de Shalit, Metabelian local class field theory, J. Reine Angew. Math. 478 (1996), 85-106.
  18. R. E. Kottwitz, Stable trace formula: cuspidal tempered terms, Duke Math. J. 51 (1984), no. 3, 611-650.
  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.
  20. R. P. Langlands, Functoriality and Reciprocity, Two Lectures at the Institute for Advanced Studies, Seminar Notes, March 2011.
  21. F. Laubie, Une theorie du corps de classes local non abelien, Compos. Math. 143 (2007),no. 2, 339-362.
  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.
  23. S. A. Morris, Free products of topological groups, Bull. Austral. Math. Soc. 4 (1971),17-29.
  24. J. Neukirch, A. Schmidt, and K. Wingberg, Cohomology of number fields, second edition,Grundlehren der Mathematischen Wissenschaften, 323, Springer-Verlag, Berlin, 2008.
  25. E. T. Ordman, Free products of topological groups which are $k_{\omega}$-spaces, Trans. Amer.Math. Soc. 191 (1974), 61-73.
  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.