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Izv. Akad. Nauk SSSR Ser. Mat., 1979, Volume 43, Issue 4, Pages 765–794 (Mi izv1730)  

This article is cited in 15 scientific papers (total in 15 papers)

A norm pairing in formal modules

S. V. Vostokov

Abstract: A pairing of the multiplicative group of a local field (a finite extension of the field of $p$-adic numbers $\mathbf Q_p$) with the group of points of a Lubin–Tate formal group is defined explicitly. The values of the pairing are roots of an isogeny of the formal group. The main properties of this pairing are established: bilinearity, invariance under the choice of a local uniformizing element, and independence of the method of expanding elements into series with respect to this uniformizing element.
These properties of the pairing are used to prove that it agrees with the generalized Hilbert norm residue symbol when the field over whose ring of integers the formal group is defined is totally ramified over $\mathbf Q_p$. This yields an explicit expression for the generalized Hilbert symbol on the group of points of the formal group.
Bibliography: 12 titles.

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English version:
Mathematics of the USSR-Izvestiya, 1980, 15:1, 25–51

Bibliographic databases:

UDC: 519.48
MSC: 12B10, 12B25
Received: 03.01.1979

Citation: S. V. Vostokov, “A norm pairing in formal modules”, Izv. Akad. Nauk SSSR Ser. Mat., 43:4 (1979), 765–794; Math. USSR-Izv., 15:1 (1980), 25–51

Citation in format AMSBIB
\by S.~V.~Vostokov
\paper A~norm pairing in formal modules
\jour Izv. Akad. Nauk SSSR Ser. Mat.
\yr 1979
\vol 43
\issue 4
\pages 765--794
\jour Math. USSR-Izv.
\yr 1980
\vol 15
\issue 1
\pages 25--51

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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. 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
    4. S. V. Vostokov, A. N. Zinoviev, “Arithmetic of the module of roots of the isogeny of a formal group in the case of small ramification”, J. Math. Sci. (N. Y.), 145:1 (2007), 4765–4772  mathnet  crossref  mathscinet  zmath  elib  elib
    5. S. V. Vostokov, E. V. Ferens-Sorotskiy, “Hilbert pairing for the polynomial formal groups”, Vestnik St Petersb Univ Math, 43:1 (2010), 18  crossref
    6. 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
    7. 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
    8. 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
    9. 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
    10. 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
    11. 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
    12. St. Petersburg Math. J., 26:6 (2015), 859–865  mathnet  crossref  mathscinet  isi  elib  elib
    13. E. V. Ikonnikova, “Hensel–Shafarevich canonical basis in Lubin–Tate formal modules”, J. Math. Sci. (N. Y.), 219:3 (2016), 462–472  mathnet  crossref  mathscinet
    14. 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
    15. Vostokov S., “Skew-symmetric pairing on polynomial formal modules”, Lobachevskii J. Math., 38:1 (2017), 170–176  crossref  isi  scopus
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