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Zap. Nauchn. Sem. POMI, 2014, Volume 430, Pages 53–60 (Mi znsl6082)  

This article is cited in 3 scientific papers (total in 3 papers)

Explicit form of Hilbert symbol for polynomial formal groups over multidimensional local field. I

S. V. Vostokov, V. V. Volkov, M. V. Bondarko

St. Petersburg State University, St. Petersburg, Russia

Abstract: Let $K$ be a multidimensional local field with characteristic different from characteristic of its residue field, $c$ be a unit of $K$ and $F_c(X,Y)=X+Y+cXY$ be a polynomial formal group, which defines formal module $F_c(\mathfrak M)$ over maximal ideal of ring of integers in $K$. Assume that $K$ contains group of the roots of isogeny $[p^m]_c(X)$, which we denote by $\mu_{F_c,m}$. Let $\mathcal H$ be the multiplicative group of Cartier curves and $\mathcal H_c$ be a formal analogue of the module $F_c(\mathfrak M)$. In the current work we construct formal symbol $\{\cdot,\cdot\}_c\colon K_n(\mathcal H)\times\mathcal H_c\to\mu_{F_c,m}$ and check its basic properties. This is the first step in construction of the explicit formula for the Hilbert symbol.

Key words and phrases: Hilbert symbol, multidimensional local field, formal groups, polynomial formal groups.

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English version:
Journal of Mathematical Sciences (New York), 2016, 219:3, 370–374

Bibliographic databases:

UDC: 512.741
Received: 30.09.2014

Citation: S. V. Vostokov, V. V. Volkov, M. V. Bondarko, “Explicit form of Hilbert symbol for polynomial formal groups over multidimensional local field. I”, Problems in the theory of representations of algebras and groups. Part 27, Zap. Nauchn. Sem. POMI, 430, POMI, St. Petersburg, 2014, 53–60; J. Math. Sci. (N. Y.), 219:3 (2016), 370–374

Citation in format AMSBIB
\Bibitem{VosVolBon14}
\by S.~V.~Vostokov, V.~V.~Volkov, M.~V.~Bondarko
\paper Explicit form of Hilbert symbol for polynomial formal groups over multidimensional local field.~I
\inbook Problems in the theory of representations of algebras and groups. Part~27
\serial Zap. Nauchn. Sem. POMI
\yr 2014
\vol 430
\pages 53--60
\publ POMI
\publaddr St.~Petersburg
\mathnet{http://mi.mathnet.ru/znsl6082}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=3486761}
\transl
\jour J. Math. Sci. (N. Y.)
\yr 2016
\vol 219
\issue 3
\pages 370--374
\crossref{https://doi.org/10.1007/s10958-016-3112-7}


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    This publication is cited in the following articles:
    1. 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
    2. Volkov V.V., “On the Norm Property of the Hilbert Symbol For Polynomial Formal Modules in a Multidimensional Local Field”, Vestnik St. Petersburg Univ. Math., 49:4 (2016), 320–324  crossref  mathscinet  zmath  isi  scopus
    3. S. V. Vostokov, S. S. Afanas'eva, M. V. Bondarko, V. V. Volkov, O. V. Demchenko, E. V. Ikonnikova, I. B. Zhukov, I. I. Nekrasov, P. N. Pital, “Explicit constructions and the arithmetic of local number fields”, Vestnik St. Petersburg Univ. Math., 50:3 (2017), 242–264  crossref  mathscinet  isi  scopus
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