[go: up one dir, main page]
More Web Proxy on the site http://driver.im/ Skip to main content
Springer Nature Link
Log in

Formal Groups over Sub-Rings of the Ring of Integers of a Multidimensional Local Field

  • Mathematics
  • Published:
Vestnik St. Petersburg University, Mathematics Aims and scope Submit manuscript

Abstract

We constructed so-called convergence rings for the ring of integers of a multidimensional local field. The convergence ring is a sub-ring of the ring of integers with the property that any power series with coefficients from the sub-ring converges when replacing a variable by an arbitrary element of the maximal ideal. The properties of convergence rings and an explicit formula for their construction are derived. Note that the multidimensional case is fundamentally different from the case of the classical (one-dimensional) local field, where the whole ring of integers is the convergence ring. Next, we consider a multidimensional local field with zero characteristics of the penultimate residue field. For each convergence ring of such a field, we introduce a homomorphism that allows us to construct a formal group over the same ring with a logarithm having coefficients from the field for a power series with coefficients from the ring, and we give an explicit formula for the coefficients. In addition, by isogeny with coefficients from this ring, we construct a generalization of the formal Lubin—Tate group over this ring, study the endomorphisms of these formal groups, and derive a criterion for their isomorphism. We prove a one-to-one correspondence between formal groups created by ring homomorphism and by isogeny. Also, for any finite extension of a multidimensional local field with zero characteristic of the penultimate residue field, we consider the point group generated by the corresponding Lubin—Tate formal group.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
£29.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price includes VAT (United Kingdom)

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. J. Lubin and J. Tate, “Formal complex multiplication in local fields,” Ann. Math. 81, 380–387 (1985).

    Article  MathSciNet  MATH  Google Scholar 

  2. A. I. Madunts, “Formal Modules for Relative Formal Lubin-Tate Groups,” J. Math. Sci. (N. Y.) 232, 704–716 (2018).

    Article  MathSciNet  MATH  Google Scholar 

  3. E. de Shalite, “Relative Lubin—Tate groups,” in Proc. Am. Math. Soc. 95, 1–4 (1985).

    Article  MathSciNet  MATH  Google Scholar 

  4. A. I. Madunts, “Classification of generalized formal Lubin-Tate groups over multidimensional local fields,” J. Math. Sci. (N. Y.) 234, 175–179 (2018).

    Article  MathSciNet  MATH  Google Scholar 

  5. S. S. Afanas’eva, “The Hilbert symbol in higher-dimensional local fields for formal Lubin-Tate groups. II,” J. Math. Sci. (N. Y.) 202, 346–359 (2013).

    Article  MATH  Google Scholar 

  6. A. I. Madunts, “Lubin-Tate formal groups over the ring of integers of a multidimensional local field,” J. Math. Sci. (N. Y.) 281, 221–226 (2001).

    MathSciNet  MATH  Google Scholar 

  7. A. I. Madunts, Convergence of Sequences and Series in Multidimensional Complete Fields, Candidate’s Dissertation in Mathematics and Physics (St. Petersburg State Univ., St. Petersburg, 1995).

    MATH  Google Scholar 

  8. I. Zhukov, Abelian Extensions and Topological K-groups of Multidimensional Local Fields, Candidate’s Dissertation in Mathematics and Physics (St. Petersburg State Univ., St. Petersburg, 1991).

    Google Scholar 

  9. A. N. Paršin, “On the arithmetic of two-dimensional schemes. I. Distributions and residues,” Math. USSR-Izv. 40, 695–729 (1976).

    MathSciNet  Google Scholar 

  10. I. B. Fesenko, “Class field theory of multidimensional local fields of characteristic 0 with residue field of positive characteristic,” Algebra Anal. 3, 165–196 (1991).

    MathSciNet  MATH  Google Scholar 

  11. I. B. Fesenko, “The theory of local fields. Local class field theory. Multidimensional local class field theory,” Algebra Anal. 4, 403–438 (1992).

    MathSciNet  MATH  Google Scholar 

  12. I. Zhukov, “Structural theorem for complete fields,” Tr. S.-Peterb. Mat. O-va. 3, 215–234 (1994).

    MATH  Google Scholar 

  13. I. Zhukov and A. I. Madunts, “Additive and multiplicative expansions in multidimensional local fields,” J. Math. Sci. (N. Y.) 272, 186–196 (2000).

    MATH  Google Scholar 

  14. M. Hazewinkel, Formal Groups and Applications (Academic, New York, 1978).

    MATH  Google Scholar 

  15. K. Iwasawa, Local Class Field Theory (Oxford Univ. Press, Oxford, 1986; Mir, Moscow, 1983).

    MATH  Google Scholar 

  16. I. R. Shafarevich, “The general reciprocity law,” Mat. Sb. (N. S.) 26(68) (1), 113–146 (1950).

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. St. Petersburg State University, St. Petersburg, 199034, Russia

    A. I. Madunts & S. V. Vostokov

  2. Ustinov Baltic State Technical University, St. Petersburg, 190005, Russia

    R. P. Vostokova

Authors
  1. A. I. Madunts
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. S. V. Vostokov
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. R. P. Vostokova
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding authors

Correspondence to A. I. Madunts, S. V. Vostokov or R. P. Vostokova.

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Madunts, A.I., Vostokov, S.V. & Vostokova, R.P. Formal Groups over Sub-Rings of the Ring of Integers of a Multidimensional Local Field. Vestnik St.Petersb. Univ.Math. 52, 59–65 (2019). https://doi.org/10.3103/S1063454119010084

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.3103/S1063454119010084

Keywords