Fesenko, Ivan Class field theory, its three main generalisations, and applications. (English) Zbl 1482.11152 EMS Surv. Math. Sci. 8, No. 1-2, 107-133 (2021). In this paper, the author presents branches of class field theory. He gives us special and general approaches to class field theory, and discusses their roles. In particular, the three main generalisations of class field theory: Higher-class field theory, Langlands correspondences and anabelian geometry, and their further developments are discussed. He suggests several directions of unification of the generalisations of class field theory. Reviewer: Abdelmalek Azizi (Oujda) Cited in 1 Document MSC: 11R37 Class field theory 11S37 Langlands-Weil conjectures, nonabelian class field theory 12-03 History of field theory 19F05 Generalized class field theory (\(K\)-theoretic aspects) Keywords:class field theory; general class field theory; special class field theory; higher class field theory; Langlands correspondences; anabelian geometry; elliptic curves over global fields; zeta integrals; higher adelic geometry and analysis; IUT theory PDFBibTeX XMLCite \textit{I. Fesenko}, EMS Surv. Math. Sci. 8, No. 1--2, 107--133 (2021; Zbl 1482.11152) Full Text: DOI References: [1] J. Arthur, Automorphic representations and number theory. In1980 Seminar on Harmonic Analysis, McGill University (Montréal, 1980), pp. 3-54, Canad. Math. Soc. Conf. Proceedings 1. Amer. Math. Soc., Providence, RI, 1981. Available fromhttpW//www.math.toronto.edu/ arthur/pdf/11.pdf [2] E. Artin and J. Tate,Class field theory. W. A. Benjamin, Inc., New York-Amsterdam, 1968 Zbl0176.33504MR0223335 · Zbl 0176.33504 [3] P. Etingof, E. Frenkel, and D. Kazhdan, An analytic version of the Langlands correspondence for complex curves, 2020arXiv:1908.09677v3 · Zbl 1475.14021 [4] I. Fesenko, Class field theory of multidimensional local fields of characteristic0with residue field of positive characteristic. Part I.Algebra i Analiz3(1991), no. 3, 165-196; English transl. inSt. Petersburg Math.3(1992), no. 3, 649-678 Zbl0791.11063MR1150559; Part II.Algebra i Analiz3(1991), no. 5, 168-189; English transl. inSt. Petersburg Math. J.3 (1992), no. 5, 1103-1126 Zbl0770.11052MR1186241 · Zbl 0791.11063 [5] I. Fesenko, Local class field theory: the perfect residue field case.Izv. Ross. Akad. Nauk Ser. Mat.57(1993), no. 4, 72-91; English transl. inRuss. Acad. Scienc. Izvest. Math.43(1994), no. 1, 65-81 Zbl0826.11056MR1243352 · Zbl 0826.11056 [6] I. Fesenko, On just infinite pro-p-groups and arithmetically profinite extensions of local fields. J. Reine Angew. Math.517(1999), 61-80 Zbl0997.11107MR1728547 · Zbl 0997.11107 [7] I. Fesenko, Nonabelian local reciprocity maps. InClass field theory—its centenary and prospect (Tokyo, 1998), pp. 63-78, Adv. Stud. Pure Math. 30, Math. Soc. Japan, Tokyo, 2001 Zbl1039.11085MR1846451 [8] I. Fesenko, Analysis on arithmetic schemes. Part I.Docum. Math.Kazuya Kato’s fiftieth birthday (2003), Extra. Vol., 261-284 Zbl1130.11335MR2046602; Part II.J. K-Theory5 (2010), no. 3, 437-557 Zbl1225.14019MR2658047; Part III. Available fromhttpsW//www. maths.nottingham.ac.uk/personal/ibf/a3.pdf [9] I. Fesenko, Adelic approach to the zeta function of arithmetic schemes in dimension two. Mosc. Math. J.8(2008), no. 2, 273-317 Zbl1158.14023MR2462437 · Zbl 1158.14023 [10] I. Fesenko, Arithmetic deformation theory via arithmetic fundamental groups and nonarchimedean theta-functions, notes on the work of Shinichi Mochizuki.Eur. J. Math.1(2015), no. 3, 405-440 Zbl1416.11119MR3401899 · Zbl 1416.11119 [11] I. Fesenko, G. Ricotta, and M. Suzuki, Mean-periodicity and zeta functions.Ann. Inst. Fourier (Grenoble)62(2012), no. 5, 1819-1887 Zbl1305.11079MR3025155 · Zbl 1305.11079 [12] I. Fesenko and S. V. Vostokov,Local fields and their extensions. With a foreword by I. R. Shafarevich. Second edn., Transl. Math. Monogr. 121. Amer. Math. Soc., Providence, RI, 2002 Zbl1156.11046MR1915966 [13] I. Fesenko, S. V. Vostokov, and S. H. Yoon, Generalised Kawada-Satake method for Mackey functors in class field theory.Eur. J. Math.4(2018), no. 3, 953-987 Zbl1426.19004 MR3851125 · Zbl 1426.19004 [14] E. Frenkel, Langlands program, trace formulas, and their geometrization.Bull. Amer. Math. Soc. (N.S.)50(2013), no. 1, 1-55 Zbl1329.11121MR2994994 · Zbl 1329.11121 [15] E. Frenkel, Is there an analytic theory of automorphic functions for complex algebraic curves? SIGMA Symmetry Integrability Geom. Methods Appl.16(2020), Paper No. 042, 31pp Zbl1446.14005MR4098670 · Zbl 1446.14005 [16] D. Gaitsgory, Outline of the proof of the geometric Langlands conjecture forGL2.Astérisque (2015), no. 370, 1-112 Zbl1406.14008MR3364744 · Zbl 1406.14008 [17] C. Haesemeyer and C. A. Weibel,The norm residue theorem in motivic cohomology. Ann. of Math. Stud. 200, Princeton Univ. Press, Princeton, NJ, 2019 Zbl1433.14001MR3931681 · Zbl 1433.14001 [18] G. Harder and D. A. Kazhdan, Automorphic forms on GL2over function fields (after V. G. Drinfel’d). InAutomorphic forms, representations andL-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Part 2, pp. 357-379, Proc. Sympos. Pure Math., XXXIII, Amer. Math. Soc., Providence, R.I., 1979 Zbl0442.12011 MR546624 · Zbl 0442.12011 [19] H. Hasse, History of class field theory. InAlgebraic Number Theory (Proc. Instructional Conf., Brighton, 1965), pp. 266-279, Thompson, Washington, D.C., 1967 Zbl0153.07403 MR0218330 [20] Yu. Hoshi, A. Minamide, and Sh. Mochizuki, Group-theoreticity of numerical invariants and distinguished subgroups of configuration space groups, preprint, 2017. Available fromhttpW// www.kurims.kyoto-u.ac.jp/ motizuki/Numerical [21] Y. Ihara,On congruence monodromy problems. MSJ Mem. 18, Math. Soc. Japan, Tokyo, 2008 Zbl0228.14010MR2499876 · Zbl 1166.14301 [22] K. I. Ikeda, On the non-abelian global class field theory.Ann. Math. Qué.37(2013), no. 2, 129-172 Zbl1318.11145MR3117741 · Zbl 1318.11145 [23] K. I. Ikeda and E. Serbest, Fesenko reciprocity map.Algebra i Analiz20(2008), no. 3, 112- 162; reprinted inSt. Petersburg Math. J.20(2009), no. 3, 407-445 Zbl1206.11138 MR2454454 · Zbl 1206.11138 [24] K. I. Ikeda and E. Serbest, Generalized Fesenko reciprocity map.Algebra i Analiz20(2008), no. 4, 118-159; reprinted inSt. Petersburg Math. J.20(2009), no. 4, 593-624 Zbl1206.11140MR2473746 · Zbl 1206.11140 [25] K. I. Ikeda and E. Serbest, Non-abelian local reciprocity law.Manuscripta Math.132(2010), no. 1-2, 19-49 Zbl1205.11128MR2609287 · Zbl 1205.11128 [26] K. I. Ikeda and E. Serbest, Ramification theory in non-abelian local class field theory.Acta Arith.144(2010), no. 4, 373-393 Zbl1237.11050MR2684288 · Zbl 1237.11050 [27] E. Inaba, On matrix equations for Galois extensions of fields with characteristicp.Natur. Sci. Rep. Ochanomizu Univ.12(1961), no. 2, 26-36 Zbl0106.25802MR140512 · Zbl 0106.25802 [28] E. Inaba, On generalized Artin-Schreier equations.Natur. Sci. Rep. Ochanomizu Univ.13 (1962), no. 2, 1-13 Zbl0115.24903MR155817 · Zbl 0115.24903 [29] K. Iwasawa, A note on functions. InProceedings of the International Congress of Mathematicians (Cambridge, Mass., 1950). Vol. 1, p. 322, Amer. Math. Soc., Providence, R.I., 1952 [30] K. Iwasawa, Letter to J. Dieudonné. InZeta functions in geometry (Tokyo, 1990), pp. 445-450, Adv. Stud. Pure Math., 21, Kinokuniya, Tokyo, 1992 Zbl0835.11002MR1210798 [31] K. Iwasawa,Hecke’sL-functions. Lectures at Princeton University (Spring 1964). With a foreword by John Coates and Masato Kurihara. SpringerBriefs Math., Springer, Singapore, 2019 Zbl1455.11007MR3969976 [32] K. Kato, A generalization of local class field theory by usingK-groups. Part I.J. Fac. Sci. Univ. Tokyo Sect. IA Math.26(1979), no. 2, 303-376 Zbl0428.12013MR550688; Part II. J. Fac. Sci. Univ. Tokyo Sect. IA Math.27(1980), no. 3, 603-683 Zbl0463.12006 MR603953 · Zbl 0428.12013 [33] K. Kato, Galois cohomology of complete discrete valuation fields. InAlgebraicK-theory, Part II (Oberwolfach, 1980), pp. 215-238, Lecture Notes in Math. 967, Springer, Berlin-New York, 1982 Zbl0506.12022MR689394 [34] K. Kato and S. Saito, Two-dimensional class field theory. InGalois groups and their representations (Nagoya, 1981), pp. 103-152, Adv. Stud. Pure Math. 2, North-Holland, Amsterdam, 1983 Zbl0544.12011MR732466 [35] K. Kato and S. Saito, Global class field theory of arithmetic schemes. InApplications of algebraicK-theory to algebraic geometry and number theory, Part I, II (Boulder, Colo., 1983), pp. 255-331, Contemp. Math. 55, Amer. Math. Soc., Providence, RI, 1986 Zbl0614.14001 MR862639 [36] Y. Kawada, Class formations. III.J. Math. Soc. Japan7(1955), 453-490 Zbl0101.02903 MR79612 · Zbl 0101.02903 [37] M. Kerz, Higher class field theory and the connected component.Manuscripta Math.135 (2011), no. 1-2, 63-89 Zbl1221.19004MR2783387 · Zbl 1221.19004 [38] M. Kerz and A. Schmidt, Covering data and higher dimensional global class field theory.J. Number Theory129(2009), no. 10, 2569-2599 Zbl1187.14028MR2541032 · Zbl 1187.14028 [39] N. Kurokawa, Special values of Selberg zeta functions. InAlgebraicK-theory and algebraic number theory (Honolulu, HI, 1987), pp. 133-150, Contemp. Math. 83, Amer. Math. Soc., Providence, RI, 1989 Zbl0684.10038MR991979 [40] L. Lafforgue, Chtoucas de Drinfeld et correspondance de Langlands.Invent. Math.147(2002), no. 1, 1-241 Zbl1038.11075MR1875184 · Zbl 1038.11075 [41] L. Lafforgue, Le principe de fonctorialité de Langlands comme un problème de généralisation de la loi d’addition, IHES preprint, 2016. Available fromhttpW//preprints.ihes.fr/2016/M/ M-16-27.pdf [42] L. Lafforgue, Du transfert automorphe de Langlands aux formules de Poisson non linéaires. Ann. Inst. Fourier (Grenoble)66(2016), no. 3, 899-1012 Zbl1417.11153MR3494164 · Zbl 1417.11153 [43] V. Lafforgue, Chtoucas pour les groupes réductifs et paramétrisation de Langlands globale.J. Amer. Math. Soc.31(2018), no. 3, 719-891 Zbl1395.14017MR3787407 · Zbl 1395.14017 [44] S. Lang, Unramified class field theory over function fields in several variables.Ann. of Math. (2)64(1956), 285-325 Zbl0089.26201MR83174 · Zbl 0089.26201 [45] S. Lang, Sur les sériesLd’une variété algébrique.Bull. Soc. Math. France84(1956), 385-407 Zbl0089.26301MR88777 · Zbl 0089.26301 [46] R. P. Langlands, Where stands functoriality today? InRepresentation theory and automorphic forms (Edinburgh, 1996), pp. 457-471, Proc. Sympos. Pure Math. 61, Amer. Math. Soc., Providence, RI, 1997 Zbl0901.11032MR1476510 · Zbl 0901.11032 [47] S. Lysenko, On the automorphic sheaves forGSp4, 2019arXiv:1901.04447 I. Fesenko132 [48] S. Lysenko, Moduli of metaplectic bundles on curves and theta-sheaves.Ann. Sci. Éc. Norm. Supér. (4)39(2006), no. 3, 415-466 Zbl1111.14029MR2265675 · Zbl 1111.14029 [49] S. Lysenko, Geometric theta-lifting for the dual pairSO2m;Sp2n.Ann. Sci. Éc. Norm. Supér. (4)44(2011), no. 3, 427-493 Zbl1229.22015MR2839456 · Zbl 1229.22015 [50] A. Minamide and H. Nakamura, The automorphism groups of the profinite braid groups. To appear inAmer. J. Math.arXiv:1904.06749 [51] S. Mochizuki, The profinite Grothendieck conjecture for closed hyperbolic curves over number fields.J. Math. Sci. Univ. Tokyo3(1996), no. 3, 571-627 Zbl0889.11020MR1432110 · Zbl 0889.11020 [52] S. Mochizuki, The local pro-panabelian geometry of curves.Invent. Math.138(1999), no. 2, 319-423 Zbl0935.14019MR1720187 · Zbl 0935.14019 [53] S. Mochizuki, Topics surrounding the anabelian geometry of hyperbolic curves. InGalois groups and fundamental groups, pp. 119-165, Math. Sci. Res. Inst. Publ., 41, Cambridge Univ. Press, Cambridge, 2003 Zbl1053.14029MR2012215 · Zbl 1053.14029 [54] S. Mochizuki, Topics in absolute anabelian geometry. Part I: Generalities.J. Math. Sci. Univ. Tokyo19(2012), no. 2, 139-242 Zbl1267.14039MR2987306; Part II: Decomposition groups and endomorphisms.J. Math. Sci. Univ. Tokyo20(2013), no. 2, 171-269 Zbl1367.14011MR3154380; Part III: Global reconstruction algorithms.J. Math. Sci. Univ. Tokyo22(2015), no. 4, 939-1156 Zbl1358.14024MR3445958 · Zbl 1358.14024 [55] S. Mochizuki, Alien copies, Gaussians and inter-universal Teichmüller theory. InInteruniversal Teichmüller Theory Summit 2016, pp. 23-192, RIMS Kôkyûroku Bessatsu B84, Res. Inst. Math. Sci. (RIMS), Kyoto, 2021 [56] S. Mochizuki, Inter-universal Teichmüller theory. Part I: Construction of Hodge theaters.Publ. Res. Inst. Math. Sci.57(2021), no. 1, 3-207 Zbl7317908MR4225473; Part II: Hodge- Arakelov-theoretic evaluation.Publ. Res. Inst. Math. Sci.57(2021), no. 1, 209-401 Zbl7317909MR4225474; Part III: Canonical splittings of the log-theta-lattice.Publ. Res. Inst. Math. Sci.57(2021), no. 1, 403-626 Zbl7317910MR4225475; Part IV: Log-volume computations and set-theoretic foundations.Publ. Res. Inst. Math. Sci.57(2021), no. 1, 627- 723 Zbl7317911MR4225476 · Zbl 1465.14002 [57] F. Morel,A1-algebraic topology over a field. Lecture Notes in Math. 2052, Springer, Heidelberg, 2012 Zbl1263.14003MR2934577 · Zbl 1263.14003 [58] M. Morrow, Integration on product spaces and GLnof a valuation field over a local field. Commun. Number Theory Phys.2(2008), no. 3, 563-592 Zbl1171.28303MR2482943 · Zbl 1171.28303 [59] M. Morrow, Integration on valuation fields over local fields.Tokyo J. Math.33(2010), no. 1, 235-281 Zbl1203.11081MR2682892 · Zbl 1203.11081 [60] J. Neukirch,Algebraic number theory. Grundlehren Math. Wiss. 322, Springer, Berlin, 1999 Zbl0956.11021MR1697859 · Zbl 0956.11021 [61] J. Neukirch,Class field theory. Grundlehren Math. Wiss. 280, Springer, Berlin, 1986 Zbl0587.12001MR819231 · Zbl 0587.12001 [62] A. N. Parshin, Local class field theory.Proc. Steklov Inst. Math.165(1985), no. 3, 157-185 Zbl0579.12012MR752939 · Zbl 0579.12012 [63] A. N. Parshin, Galois cohomology and the Brauer group of local fields.Proc. Steklov Inst. Math.(1991), no. 4, 191-201 Zbl0731.11064MR1092028 · Zbl 0731.11064 [64] N. Schappacher, On the history of Hilbert’s twelfth problem: a comedy of errors. InMatériaux pour l’histoire des mathématiques au XXesiècle (Nice, 1996), pp. 243-273, Sémin. Congr. 3, Soc. Math. France, Paris, 1998 Zbl1044.01530MR1640262 · Zbl 1044.01530 [65] M. Suzuki, Positivity of certain functions associated with analysis on elliptic surfaces.J. Number Theory131(2011), no. 10, 1770-1796 Zbl1237.11028MR2811546 · Zbl 1237.11028 [66] M. Suzuki, Two-dimensional adelic analysis and cuspidal automorphic representations of GL.2/. InMultiple Dirichlet series, L-functions and automorphic forms, pp. 339-361, Progr. Math. 300, Birkhäuser/Springer, New York, 2012 Zbl1277.11074MR2952583 · Zbl 1277.11074 [67] A. Tamagawa, The Grothendieck conjecture for affine curves.Compositio Math.109(1997), no. 2, 135-194 Zbl0899.14007MR1478817 · Zbl 0899.14007 [68] J. T. Tate, Fourier analysis in number fields, and Hecke’s zeta-functions. InAlgebraic Number Theory (Proc. Instructional Conf., Brighton, 1965), pp. 305-347, Thompson, Washington, D.C., 1967 MR0217026 [69] S. Tsujimura, Combinatorial Belyi cuspidalization and arithmetic subquotients of the Grothendieck-Teichmüller group.Publ. Res. Inst. Math. Sci.56(2020), no. 4, 779-829 Zbl1460.14068MR4162297 · Zbl 1460.14068 [70] S. V. Vostokov, Explicit formulas for the Hilbert symbol. InInvitation to higher local fields (Münster, 1999), pp. 81-90, Geometry and Topology Monographs 3, Geometry and Topology Publications, Coventry, 2000. Available fromhttpsW//msp.org/gtm/2000/03/p008.xhtml · Zbl 1008.11053 [71] R. Waller, Measure and integration on GL2over a two-dimensional local field.New York J. Math.25(2019), 396-422 Zbl1457.11167MR3982247 · Zbl 1457.11167 [72] A. Weil,Basic number theory. Russian edn, Osnovy teorii chisel, Mir, 1972 · Zbl 0238.12001 [73] G. Wiesend, A construction of covers of arithmetic schemes.J. Number Theory121(2006), no. 1, 118-131 Zbl1120.14014MR2268759 · Zbl 1120.14014 [74] G. Wiesend, Class field theory for arithmetic schemes.Math. Z.256(2007), no. 4, 717-729 Zbl1115 · Zbl 1115.14016 This reference list is based on information provided by the publisher or from digital mathematics libraries. 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