The field-of-norms functor and the Hilbert symbol for higher local fields
Journal de Théorie des Nombres de Bordeaux, Volume 24 (2012) no. 1, pp. 1-39.

The field-of-norms functor is applied to deduce an explicit formula for the Hilbert symbol in the mixed characteristic case from the explicit formula for the Witt symbol in characteristic p>2 in the context of higher local fields. Is is shown that a “very special case” of this construction gives Vostokov’s explicit formula.

Dans cet article nous appliquons le foncteur corps des normes pour déduire, dans le cas de caractéristique mixte, une formule explicite pour le symbole de Hilbert de la formule explicite pour le symbole de Witt en caractéristique p>2 dans le contexte des corps locaux multidimensionnels. On montre que la formule explicite de Vostokov est un cas très particulier de notre construction.

Received:
Published online:
DOI: 10.5802/jtnb.786
Classification: 11S20,  11S31,  11S70
Keywords: higher local fields, field-of-norms, Hilbert Symbol, Vostokov’s pairing
Victor Abrashkin 1; Ruth Jenni 

1 Department of Mathematical Sciences Durham University Science Laboratories South Rd, Durham DH1 3LE United Kingdom
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Victor Abrashkin; Ruth Jenni. The field-of-norms functor and the Hilbert symbol for higher local fields. Journal de Théorie des Nombres de Bordeaux, Volume 24 (2012) no. 1, pp. 1-39. doi : 10.5802/jtnb.786. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.786/

[1] V. Abrashkin The field of norms functor and the Brueckner-Vostokov formula. Math. Annalen 308 (1997), 5–19. | MR: 1446195 | Zbl: 0895.11049

[2] V. Abrashkin An analogue of the field-of-norms functor and of the Grothendieck Conjecture. J. Algebraic Geom. 16 (2007), no.  4, 671–730. | MR: 2357687

[3] T. V. Belyaeva, S. V.  Vostokov The Hilbert symbol in a complete multidimensional field.I. J. Math. Sci (N.Y.) 120 (2004), no.  4, 1483–1500. | MR: 1875716

[4] S. Bloch, K. Kato p-adic étale cohomology. Publ.  Math.  IHES 63 (1986), 107–152. | Numdam | MR: 849653 | Zbl: 0613.14017

[5] H.  Bass, J. Tate, The Milnor ring of a global field. Lect. Notes Math. 342, Springer-Verlag, Berlin, 1973, 474–486. | MR: 442061 | Zbl: 0299.12013

[6] I. B. Fesenko, Sequential topologies and quotients of the Milnor K-groups of higher local fields. Algebra i Analiz 13 (2001), no.3, 198–221; English translation in: St. Petersburg Math. J. 13 (2002), issue 3, 485–501. | MR: 1850194

[7] I. Fesenko, S. Vostokov, Local Fields and their Extensions. Translations of Mathematical Monographs, vol. 121, Amer. Math. Soc., Providence, Rhode Island, 2002. | MR: 1915966 | Zbl: 0781.11042

[8] J.-M.Fontaine, Representations p-adiques des corps locaux (1-ere partie). In: The Grothendieck Festschrift, A Collection of Articles in Honor of the 60th Birthday of Alexander Grothendieck, vol.  II, 1990, 249–309. | MR: 1106901 | Zbl: 0743.11066

[9] T. Fukaya, The theory of Coleman power series for K 2 . J. Algebraic Geom. 12 (2003), No. 1, 1–80. | MR: 1948685

[10] A. I. Madunts, I. B. Zhukov, Multidimensional 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. | MR: 1363290 | Zbl: 0874.11078

[11] K. Kato, The explicit reciprocity law and the cohomology of Fontaine-Messing. Bull. Soc. Math. France 109 (1991), no.4, 397–441. | Numdam | MR: 1136845 | Zbl: 0752.14015

[12] F.  Laubie, Extensions de Lie et groupes d’automorphismes de corps locaux. Comp. Math. 67 (1988), 165–189. | Numdam | MR: 951749 | Zbl: 0649.12012

[13] J. Neukirch, Class Field theory. Springer-Verlag, Berlin and New York, 1986. | MR: 819231 | Zbl: 0587.12001

[14] A. N. Parshin, Class fields and algebraic K-theory. (Russian). Uspekhi Mat. Nauk 30 (1975), 253–254. | MR: 401710 | Zbl: 0302.14005

[15] A. N. Parshin, Local class field theory. (Russian). Algebraic Geometry and its applications, Trudy Mat. Inst. Steklov 165 (1985), 143–170; English translation in: Proc. Steklov Inst. Math., 1985, issue 3, 157–185. | MR: 752939 | Zbl: 0579.12012

[16] A. N. Parshin, Galois cohomology and Brauer group of local fields. Trudy Mat. Inst. Steklov (1990); English translation in: Proc. Steklov Inst. Math., 1991, issue 4, 191–201. | MR: 1092028 | Zbl: 0731.11064

[17] T. Scholl, Higher fields of norms and (ϕ,Γ)-modules. Doc. Math., Extra vol. (2006), 685–709 (electronic). | MR: 2290602

[18] I. R. Shafarevich A general reciprocity law(Russian), Mat. Sb. 26(68) (1950), 113–146 | MR: 31944 | Zbl: 0036.15901

[19] S. Vostokov, An explicit form of the reciprocity law. Izv. Akad. Nauk SSSR, Ser. Mat. (1978); English translation in: Math. USSR Izv. 13 (1979), 557–588. | MR: 522940 | Zbl: 0467.12018

[20] S. Vostokov, Explicit construction of the theory of class fields of a multidimensional local field. Izv. Akad. Nauk SSSR Ser. Mat., 49 (1985), 283–308. | MR: 791304 | Zbl: 0608.12017

[21] S.Zerbes, The higher Hilbert pairing via (φ,G)-modules. ArXiv:0705.4269

[22] I. Zhukov, Higher dimensional local fields. (Münster, 1999), Geom. Topol. Monogr. (2000), no.3, 5–18. | MR: 1804916

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