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Izv. Akad. Nauk SSSR Ser. Mat., 1979, Volume 43, Issue 5, Pages 1054–1120 (Mi izv1748)  

This article is cited in 13 scientific papers (total in 13 papers)

Formal groups and the norm residue symbol

V. A. Kolyvagin

Abstract: This paper investigates the canonical pairing associated with a one-dimensional formal group law $F$ over the ring of integers of a finite extension of $\mathbf Q_p$ and an isogeny $f\colon F\to F$, just as the Hilbert symbol is associated with the multiplicative law and the isogeny “raising to the $p$th power”. Formulas are obtained which generalize the formulas of Artin–Hasse, Iwasawa, and Wiles. The formulas describe the values of the symbol in terms of $p$-adic differentiation, the logarithm of the formal group law, the norm, and the trace.
Bibliography: 8 titles.

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English version:
Mathematics of the USSR-Izvestiya, 1980, 15:2, 289–348

Bibliographic databases:

UDC: 519.4
MSC: Primary 12B10; Secondary 12B15
Received: 25.12.1978

Citation: V. A. Kolyvagin, “Formal groups and the norm residue symbol”, Izv. Akad. Nauk SSSR Ser. Mat., 43:5 (1979), 1054–1120; Math. USSR-Izv., 15:2 (1980), 289–348

Citation in format AMSBIB
\by V.~A.~Kolyvagin
\paper Formal groups and the norm residue symbol
\jour Izv. Akad. Nauk SSSR Ser. Mat.
\yr 1979
\vol 43
\issue 5
\pages 1054--1120
\jour Math. USSR-Izv.
\yr 1980
\vol 15
\issue 2
\pages 289--348

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    This publication is cited in the following articles:
    1. S. V. Vostokov, “Symbols on formal groups”, Math. USSR-Izv., 19:2 (1982), 261–284  mathnet  crossref  mathscinet  zmath
    2. 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
    3. T. Fukaya, “Explicit reciprocity laws for $p$-divisible groups over higher dimensional local fields”, crll, 2001:531 (2001), 61  crossref  mathscinet  zmath  elib
    4. V. A. Kolyvagin, “Fermat's equation over the tower of cyclotomic fields”, Izv. Math., 65:3 (2001), 503–541  mathnet  crossref  crossref  mathscinet  zmath
    5. S. V. Vostokov, F. Lorenz, “An explicit formula for the Hilbert symbol for Honda groups in a multidimensional local field”, Sb. Math., 194:2 (2003), 165–197  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    6. 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
    7. S. V. Vostokov, E. V. Ferens-Sorotskiy, “Hilbert pairing for the polynomial formal groups”, Vestnik St Petersb Univ Math, 43:1 (2010), 18  crossref
    8. S. S. Afanas'eva, G. K. Pak, “Norm series for Honda formal groups”, J. Math. Sci. (N. Y.), 183:5 (2012), 577–583  mathnet  crossref
    9. S. S. Afanas'eva, “Norm series for multi-dimensional Honda formal groups”, J. Math. Sci. (N. Y.), 192:2 (2013), 127–136  mathnet  crossref  mathscinet
    10. St. Petersburg Math. J., 26:6 (2015), 859–865  mathnet  crossref  mathscinet  isi  elib  elib
    11. S. V. Vostokov, I. I. Nekrasov, “Lubin–Tate formal module in a cyclic unramified $p$-extension as Galois module”, J. Math. Sci. (N. Y.), 219:3 (2016), 375–379  mathnet  crossref  mathscinet
    12. D. A. Malinin, “One construction of integral representations of $p$-groups and some applications”, Chebyshevskii sb., 16:3 (2015), 322–338  mathnet  elib
    13. S. V. Vostokov, I. L. Klimovitskii, P. N. Pital, “Universal Approach to the Arithmetics of Formal Groups”, Math. Notes, 104:2 (2018), 204–209  mathnet  crossref  crossref  mathscinet  isi  elib
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