Advertisement

Vestnik St. Petersburg University: Mathematics

, Volume 49, Issue 4, pp 320–324 | Cite as

On the norm property of the Hilbert symbol for polynomial formal modules in a multidimensional local field

  • V. V. VolkovEmail author
Mathematics

Abstract

In a two-dimensional local field K containing the pth root of unity, a polynomial formal group F c (X, Y) = X + Y + cXY acting on the maximal ideal M of the ring of integers б K and a constructive Hilbert pairing {·, ·} c : K 2(K) × F c (M) → <ξ> c , where <ξ> c is the module of roots of [p] c (pth degree isogeny of F c ) with respect to formal summation are considered. For the extension of two-dimensional local fields L/K, a norm map of Milnor groups Norm: K 2(L) → K 2(K) is considered. Its images are called norms in K 2(L). The main finding of this study is that the norm property of pairing {·, ·}c: {x,β} c : = 0 ⇔ x is a norm in K 2(K([p] c -1 (β))), where [p] c -1 (β) are the roots of the equation [p] c = β, is checked constructively.

Keywords

the Hilbert symbol multidimensional local field formal groups polynomial formal groups norm property 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    S. V. Vostokov, V. V. Volkov, and M. V. Bondarko, “An explicit form of Hilbert symbol for polynomial formal modules over a multidimensional local field. I,” Zap. Nauchn. Semin. POMI 430, 53–60 (2014).MathSciNetGoogle Scholar
  2. 2.
    S. V. Vostokov and V. V. Volkov, “An explicit form of Hilbert symbol for polynomial formal modules over a multidimensional local field. II,” Zap. Nauchn. Semin. POMI 443, 46–60 (2016).MathSciNetGoogle Scholar
  3. 3.
    I. B. Fesenko and S. V. Vostokov, Local Fields and Their Extensions, 2nd ed. (Am. Math. Soc., Providence, RI, 2002).zbMATHGoogle Scholar
  4. 4.
    S. V. Vostokov, “Explicit construction of class field theory for a multidimensional local field,” Math. USSRIzv., 26, 283–308 (1985).MathSciNetGoogle Scholar
  5. 5.
    A. N. Parshin, “Local class field theory,” Proc. Steklov Inst. Math. 165, 157–185 (1985).MathSciNetzbMATHGoogle Scholar
  6. 6.
    A. A. Suslin, “Algebraic K-theory and the norm residue homomorphism,” Itogi Nauki Tekh., Ser.: Sovrem. Probl. Mat. Nov. Dostizh. 25, 115–208 (1984).MathSciNetzbMATHGoogle Scholar

Copyright information

© Allerton Press, Inc. 2016

Authors and Affiliations

  1. 1.St. Petersburg State UniversitySt. PetersburgRussia

Personalised recommendations