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Coefficient rings of Tate formal groups determining Krichever genera

  • E. Yu. Bunkova
  • V. M. Buchstaber
  • A. V. Ustinov
Article
  • 23 Downloads

Abstract

The paper is devoted to problems at the intersection of formal group theory, the theory of Hirzebruch genera, and the theory of elliptic functions. In the focus of our interest are Tate formal groups corresponding to the general five-parametric model of the elliptic curve as well as formal groups corresponding to the general four-parametric Krichever genus. We describe coefficient rings of formal groups whose exponentials are determined by elliptic functions of levels 2 and 3.

Keywords

Formal Group STEKLOV Institute Elliptic Curve Complex Manifold Elliptic Function 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    V. M. Buchstaber, “Functional equations associated with addition theorems for elliptic functions and two-valued algebraic groups,” Usp. Mat. Nauk 45 (3), 185–186 (1990) [Russ. Math. Surv. 45 (3), 213–215 (1990)].MathSciNetGoogle Scholar
  2. 2.
    V. M. Buchstaber, “Complex cobordism and formal groups,” Usp. Mat. Nauk 67 (5), 111–174 (2012) [Russ. Math. Surv. 67, 891–950 (2012)].MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    V. M. Buchstaber and E. Yu. Bunkova, “Krichever formal groups,” Funkts. Anal. Prilozh. 45 (2), 23–44 (2011) [Funct. Anal. Appl. 45, 99–116 (2011)].MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    V. M. Buchstaber and E. Yu. Bunkova, “The universal formal group that defines the elliptic function of level 3,” Chebyshev. Sb. 16 (2), 66–78 (2015).Google Scholar
  5. 5.
    V. M. Buchstaber, A. S. Mishchenko, and S. P. Novikov, “Formal groups and their role in the apparatus of algebraic topology,” Usp. Mat. Nauk 26 (2), 131–154 (1971) [Russ. Math. Surv. 26 (2), 63–90 (1971)].MathSciNetzbMATHGoogle Scholar
  6. 6.
    V. M. Buchstaber and T. E. Panov, Toric Topology (Am. Math. Soc., Providence, RI, 2015), Math. Surv. Monogr. 204.Google Scholar
  7. 7.
    V. M. Buchstaber and A. V. Ustinov, “Coefficient rings of formal groups,” Mat. Sb. 206 (11), 19–60 (2015) [Sb. Math. 206, 1524–1563 (2015)].MathSciNetCrossRefGoogle Scholar
  8. 8.
    A. Hattori, “Integral characteristic numbers for weakly almost complex manifolds,” Topology 5 (3), 259–280 (1966).MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    M. Hazewinkel, Formal Groups and Applications (Academic, New York, 1978).zbMATHGoogle Scholar
  10. 10.
    F. Hirzebruch, “Elliptic genera of level N for complex manifolds,” Preprint 88–24 (Max-Planck-Inst. Math., Bonn, 1988).Google Scholar
  11. 11.
    F. Hirzebruch, T. Berger, and R. Jung, Manifolds and Modular Forms (Friedr. Vieweg, Wiesbaden, 1992), Aspects Math. E20.CrossRefzbMATHGoogle Scholar
  12. 12.
    T. Honda, “Formal groups and zeta-functions,” Osaka J. Math. 5, 199–213 (1968).MathSciNetzbMATHGoogle Scholar
  13. 13.
    I. M. Krichever, “Generalized elliptic genera and Baker–Akhiezer functions,” Mat. Zametki 47 (2), 34–45 (1990) [Math. Notes 47, 132–142 (1990)].MathSciNetzbMATHGoogle Scholar
  14. 14.
    M. Lazard, “Sur les groupes de Lie formels un paramètre,” Bull. Soc. Math. France 83, 251–274 (1955).MathSciNetzbMATHGoogle Scholar
  15. 15.
    S. P. Novikov, “The methods of algebraic topology from the viewpoint of cobordism theory,” Izv. Akad. Nauk SSSR, Ser. Mat. 31 (4), 855–951 (1967) [Math. USSR, Izv. 1, 827–913 (1967)].MathSciNetGoogle Scholar
  16. 16.
    S. P. Novikov, “Adams operators and fixed points,” Izv. Akad. Nauk SSSR, Ser. Mat. 32 (6), 1245–1263 (1968) [Math. USSR, Izv. 2, 1193–1211 (1968)].MathSciNetGoogle Scholar
  17. 17.
    S. Ochanine, “Sur les genres multiplicatifs définis par des intégrales elliptiques,” Topology 26 (2), 143–151 (1987).MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    J. B. Von Oehsen, “Elliptic genera of level N and Jacobi polynomials,” Proc. Am. Math. Soc. 122 (1), 303–312 (1994).MathSciNetzbMATHGoogle Scholar
  19. 19.
    D. Quillen, “On the formal group laws of unoriented and complex cobordism theory,” Bull. Am. Math. Soc. 75 (6), 1293–1298 (1969).MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    R. E. Stong, “Relations among characteristic numbers. I, II,” Topology 4 (3), 267–281 (1965); 5 (2), 133–148 (1966).MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    J. T. Tate, “The arithmetic of elliptic curves,” Invent. Math. 23 (3–4), 179–206 (1974).MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    E. Witten, “Elliptic genera and quantum field theory,” Commun. Math. Phys. 109, 525–536 (1987).MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2016

Authors and Affiliations

  • E. Yu. Bunkova
    • 1
  • V. M. Buchstaber
    • 1
    • 2
  • A. V. Ustinov
    • 3
    • 4
  1. 1.Steklov Mathematical Institute of Russian Academy of SciencesMoscowRussia
  2. 2.Institute for Information Transmission Problems (Kharkevich Institute)Russian Academy of SciencesMoscowRussia
  3. 3.Khabarovsk Division of the Institute of Applied MathematicsFar Eastern Branch of the Russian Academy of SciencesKhabarovskRussia
  4. 4.Pacific National UniversityKhabarovskRussia

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