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Izv. Akad. Nauk SSSR Ser. Mat., 1978, Volume 42, Issue 6, Pages 1288–1321 (Mi izv1967)  

This article is cited in 29 scientific papers (total in 29 papers)

Explicit form of the law of reciprocity

S. V. Vostokov

Abstract: A pairing in the multiplicative group of a local field (a finite extension of a $p$-adic number field) is defined in terms of the expansion of elements into series in a local uniformizing parameter. The main properties of this pairing are proved: bilinearity, skew-symmetry, invariance with respect to choice of local uniformizing parameter, and independence of the choice of expansion of elements into series in this uniformizing parameter. In addition, the norm property of this pairing is examined. The main result of this paper is that this pairing coincides with Hilbert's norm residue symbol. This yields an explicit formula for the latter.
Bibliography: 9 titles.

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English version:
Mathematics of the USSR-Izvestiya, 1979, 13:3, 557–588

Bibliographic databases:

UDC: 519.48
MSC: Primary 12B10; Secondary 12B25
Received: 06.06.1978

Citation: S. V. Vostokov, “Explicit form of the law of reciprocity”, Izv. Akad. Nauk SSSR Ser. Mat., 42:6 (1978), 1288–1321; Math. USSR-Izv., 13:3 (1979), 557–588

Citation in format AMSBIB
\by S.~V.~Vostokov
\paper Explicit form of the law of reciprocity
\jour Izv. Akad. Nauk SSSR Ser. Mat.
\yr 1978
\vol 42
\issue 6
\pages 1288--1321
\jour Math. USSR-Izv.
\yr 1979
\vol 13
\issue 3
\pages 557--588

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    This publication is cited in the following articles:
    1. S. V. Vostokov, “A norm pairing in formal modules”, Math. USSR-Izv., 15:1 (1980), 25–51  mathnet  crossref  mathscinet  zmath  isi
    2. V. A. Kolyvagin, “Formal groups and the norm residue symbol”, Math. USSR-Izv., 15:2 (1980), 289–348  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    3. S. V. Vostokov, “Symbols on formal groups”, Math. USSR-Izv., 19:2 (1982), 261–284  mathnet  crossref  mathscinet  zmath
    4. S. V. Vostokov, “Explicit construction of class field theory for a multidimensional local field”, Math. USSR-Izv., 26:2 (1986), 263–287  mathnet  crossref  mathscinet  zmath
    5. Michel Gros, Masato Kurihara, “Régulateurs syntomiques et valeurs de fonctionsL p-adiques I”, Invent math, 99:1 (1990), 293  crossref  mathscinet  zmath  isi
    6. Robert A. Bell, “On properties of the n-dimensional norm residue symbol in higher local class field theory”, Journal of Pure and Applied Algebra, 108:1 (1996), 1  crossref
    7. Robert A. Bell, “On the properties of the Vostokov and Parshin pairing in higher local class field theory”, Journal of Pure and Applied Algebra, 108:1 (1996), 7  crossref
    8. V. A. Abrashkin, “Explicit formulae for the Hilbert symbol of a formal group over the Witt vectors”, Izv. Math., 61:3 (1997), 463–515  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    9. S. V. Vostokov, G. K. Pak, “Norm series in multidimensional local fields”, J. Math. Sci. (N. Y.), 130:3 (2005), 4675–4688  mathnet  crossref  mathscinet  zmath
    10. Proc. Steklov Inst. Math., 241 (2003), 98–109  mathnet  mathscinet  zmath
    11. M. V. Bondarko, S. V. Vostokov, F. Lorenz, “The Hilbert pairing for formal groups over $\sigma$-rings”, J. Math. Sci. (N. Y.), 134:6 (2006), 2445–2476  mathnet  crossref  mathscinet  zmath  elib  elib
    12. K. F. Lai, S. V. Vostokov, “The Kneser relation and the Hilbert pairing in multidimensional local field”, Math Nachr, 280:16 (2007), 1780  crossref  mathscinet  zmath  isi  elib
    13. S. V. Vostokov, “The classical reciprocity law for power residues as an analog of the Abelian integral theorem”, St. Petersburg Math. J., 20:6 (2009), 929–936  mathnet  crossref  mathscinet  zmath  isi  elib
    14. S. V. Vostokov, M. A. Ivanov, “Eisenstein's reciprocity law for Lubin–Tate formal groups”, J. Math. Sci. (N. Y.), 180:3 (2012), 269–277  mathnet  crossref
    15. M. A. Ivanov, “Product of $p^n$-power residues as an Abelian integral”, St. Petersburg Math. J., 24:2 (2013), 275–281  mathnet  crossref  mathscinet  zmath  isi  elib  elib
    16. S. V. Vostokov, M. A. Ivanov, “Integralnaya teorema Koshi i klassicheskii zakon vzaimnosti”, Uchen. zap. Kazan. un-ta. Ser. Fiz.-matem. nauki, 154, no. 2, Izd-vo Kazanskogo un-ta, Kazan, 2012, 73–82  mathnet
    17. S. S. Afanas'eva, B. M. Bekker, S. V. Vostokov, “The Hilbert symbol in multi-dimensional local fields for Lubin–Tate formal groups”, J. Math. Sci. (N. Y.), 192:2 (2013), 137–153  mathnet  crossref  mathscinet
    18. S. V. Vostokov, V. V. Volkov, G. K. Pak, “The Hilbert symbol of a polynomial formal group”, J. Math. Sci. (N. Y.), 192:2 (2013), 196–199  mathnet  crossref
    19. S. V. Vostokov, I. L. Klimovitskii, “Primary Elements in Formal Modules”, Proc. Steklov Inst. Math., 282, suppl. 1 (2013), S140–S149  mathnet  crossref  crossref  isi  elib
    20. S. V. Vostokov, “Shafarevich's paper “A general reciprocity law””, Sb. Math., 204:6 (2013), 781–800  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    21. S. S. Afanas'eva, “The Hilbert symbol in multidimensional local fields for Lubin–Tate formal groups. 2”, J. Math. Sci. (N. Y.), 202:3 (2014), 346–359  mathnet  crossref  mathscinet
    22. S. V. Vostokov, V. V. Volkov, “Explicit formula for Hilbert pairing on polynomial formal modules”, St. Petersburg Math. J., 26:5 (2015), 785–796  mathnet  crossref  mathscinet  isi  elib  elib
    23. St. Petersburg Math. J., 26:6 (2015), 859–865  mathnet  crossref  mathscinet  isi  elib  elib
    24. S. V. Vostokov, V. V. Volkov, M. V. Bondarko, “Explicit form of Hilbert symbol for polynomial formal groups over multidimensional local field. I”, J. Math. Sci. (N. Y.), 219:3 (2016), 370–374  mathnet  crossref  mathscinet
    25. A. I. Madunts, R. P. Vostokova, “Formal modules for generalized Lubin–Tate groups”, J. Math. Sci. (N. Y.), 219:4 (2016), 553–564  mathnet  crossref  mathscinet
    26. S. V. Vostokov, V. V. Volkov, “Explicit form of the Hilbert symbol on polynomial formal module for multidimensional local field. II”, J. Math. Sci. (N. Y.), 222:4 (2017), 394–403  mathnet  crossref  mathscinet
    27. S. V. Vostokov, R. P. Vostokova, I. A. Budanaev, “Asymmetric ID-based encryption system, using an explicit pairing function of the reciprocity law”, Bul. Acad. Ştiinţe Repub. Mold. Mat., 2016, no. 2, 40–44  mathnet
    28. A. I. Madunts, “Formal modules for relative Lubin–Tate formal groups”, J. Math. Sci. (N. Y.), 232:5 (2018), 704–716  mathnet  crossref  mathscinet
    29. S. O. Gorchinskiy, Vik. S. Kulikov, A. N. Parshin, V. L. Popov, “Igor Rostislavovich Shafarevich and His Mathematical Heritage”, Proc. Steklov Inst. Math., 307 (2019), 1–21  mathnet  crossref  crossref  isi  elib
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