Functional Analysis and Its Applications

, Volume 45, Issue 2, pp 99–116 | Cite as

Krichever formal groups

  • V. M. Buchstaber
  • E. Yu. Bunkova


On the basis of the general Weierstrass model of the cubic curve with parameters µ = (µ1, µ2, µ3, µ4, µ6), the explicit form of the formal group that corresponds to the Tate uniformization of this curve is described. This formal group is called the general elliptic formal group. The differential equation for its exponential is introduced and studied. As a consequence, results on the elliptic Hirzebruch genus with values in ℤ[µ] are obtained.

The notion of the universal Krichever formal group over the ring A Kr is introduced; its exponential is determined by the Baker-Akhiezer function Φ(t) = Φ(t; τ, g 2, g 3), where τ is a point on the elliptic curve with Weierstrass parameters (g 2, g 3). As a consequence, results on the Krichever genus which takes values in the ring A Kr ⊗ ℚ of polynomials in four variables are obtained. Conditions necessary and sufficient for an elliptic formal group to be a Krichever formal group are found.

A quasiperiodic function Ψ(t) = Ψ(t; v,w, µ) is introduced; its logarithmic derivative defines the exponential of the general elliptic formal group law, where v and w are points on the elliptic curve with parameters µ. For w ≠ ±v, this function has the branching points t = v and t = −v, and for w = ±v, it coincides with Φ(t; v, g 2, g 3) and becomes meromorphic. An addition theorem for the function Ψ(t) is obtained. According to this theorem, the function Ψ(t) is the common eigenfunction of differential operators of orders 2 and 3 with doubly periodic coefficients.

Key words

elliptic Hirzebruch genera addition theorems Baker-Akhiezer function deformed Lamé equation 


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  1. [1]
    V. M. Buchstaber, “The Chern-Dold Character in Cobordisms, I.,” Mat. Sb., 83:4 (1970), 575–595; English transl.: Math. USSR Sb., 12:4 (1970), 573–594.MathSciNetGoogle Scholar
  2. [2]
    V. M. Buchstaber, A. S. Mishchenko, and S. P. Novikov, “Formal groups and their role in the apparatus of algebraic topology,” Uspekhi Mat. Nauk, 26:2 (1971), 131–154; English transl.: Russian Math. Surveys, 26:2 (1971), 63–90.Google Scholar
  3. [3]
    V. M. Buchstaber, “Functional equations associated with addition theorems for elliptic functions, and two-valued algebraic groups,” Uspekhi Mat. Nauk, 45:3 (1990), 185–186; English transl.: Russian Math. Surveys, 45:3 (1990), 213–215.MathSciNetGoogle Scholar
  4. [4]
    V. M. Buchstaber and N. Ray, “Toric manifolds and complex cobordisms,” Uspekhi Mat. Nauk, 53:2 (1998), 139–140; English transl.: Russian Math. Surveys, 53:2 (1998), 371–373.MathSciNetGoogle Scholar
  5. [5]
    V. M. Buchstaber, T. E. Panov, and N. Ray, “Spaces of polytopes and cobordism of quasitoric manifolds,” Moscow Math. J., 7:2 (2007), 219–242; Scholar
  6. [6]
    V. M. Buchstaber, “The general Krichever genus,” Uspekhi Mat. Nauk, 65:5(395) (2010), 187–188; English transl.: Russian Math. Surveys, 65:5(395) (2010), 979–981.Google Scholar
  7. [7]
    V. M. Buchstaber and E. Yu. Bunkova, Elliptic formal group laws, integral Hirzebruch genera and Krichever genera,
  8. [8]
    V. M. Buchstaber and E. Yu. Bunkova, “Addition theorems, formal group laws and integrable systems,” in: XXIX Workshop on Geometrical Methods in Physics, AIP Conference Proceedings, vol. 1307, 2010, 33–44.Google Scholar
  9. [9]
    V. M. Buchstaber and D. V. Leykin, “Addition laws on Jacobian varieties of plane algebraic curves,” Trudy Mat. Inst. Steklova, 251 (2005), 54–126; English transl.: Proc. Math. Inst. Steklov, 251:4 (2005), 49–120.Google Scholar
  10. [10]
    E. Yu. Bunkova, “Addition theorem for a deformed Baker-Akhiezer function,” Uspekhi Mat. Nauk, 65:6(396) (2010), 183–184; English transl.: Russian Math. Surveys, 65:6 (2010), 1175–1177.MathSciNetGoogle Scholar
  11. [11]
    M. Hazewinkel, Formal Groups and Applications, Academic Press, New York-San Francisco-London, 1978.MATHGoogle Scholar
  12. [12]
    F. Hirzebruch, Komplexe Mannigfaltigkeiten, Proc. Int. Cong. Math. 1958, Cambridge Univ. Press, Cambridge, 1960.Google Scholar
  13. [13]
    F. Hirzebruch, Topological Methods in Algebraic Geometry, 3rd ed., Springer-Verlag, Berlin-New York-Heidelberg, 1966.MATHGoogle Scholar
  14. [14]
    F. Hirzebruch, Elliptic genera of level N for complex manifolds, Preprint MPI 88-24.Google Scholar
  15. [15]
    A. W. Knapp, Elliptic Curves, Math. Notes, vol. 40, Princeton Univ. Press, Princeton, NJ, 1992.MATHGoogle Scholar
  16. [16]
    I. M. Krichever, “Formal groups and the Atiyah-Hirzebruch formula,” Izv. Akad. Nauk SSSR, Ser. Math., 38:6 (1974), 1289–1304; English transl.: Math. USSR Izv., 8:6 (1974), 1271–1285.Google Scholar
  17. [17]
    I. M. Krichever, “Elliptic solutions of the Kadomtsev-Petviashvili equation and integrable systems of particles,” Funkts. Anal. Prilozhen., 14:4 (1980), 45–54; English transl.: Functional Anal. Appl., 14:4 (1980), 282–290.MathSciNetMATHGoogle Scholar
  18. [18]
    I. M. Krichever, “Generalized elliptic genera and Baker-Akhiezer functions,” Mat. Zametki, 47:2 (1990), 34–45; English transl.: Math. Notes, 47:2 (1990), 132–142.MathSciNetMATHGoogle Scholar
  19. [19]
    M. Lazard, “Sur les groupes de Lie formels `a un param`etre,” Bull. Soc. Math. France, 83 (1955), 251–274.MathSciNetMATHGoogle Scholar
  20. [20]
    J. Milnor, “On the cobordism ring Ω* and complex analogue, Part I,” Amer. J. Math., 82:3 (1960), 505–521.MathSciNetMATHCrossRefGoogle Scholar
  21. [21]
    S. P. Novikov, “Some problems in the topology of manifolds connected with the theory of Thom spaces,” Dokl. Akad. Nauk SSSR, 132:5 (1960), 1031–1034; English transl.: Soviet Math. Dokl., 1 (1960), 717–719.Google Scholar
  22. [22]
    S. P. Novikov, “Homotopy properties of Thom complexes,” Mat. Sb., 57(99):4 (1962), 407–442.MathSciNetGoogle Scholar
  23. [23]
    S. P. Novikov, “The methods of algebraic topology from the viewpoint of cobordism theory,” Izv. Akad. Nauk SSSR, Ser. Mat., 31:4 (1967), 855–951; English transl.: Math. USSR Izv., 1:4 (1967), 827–913.MathSciNetMATHGoogle Scholar
  24. [24]
    S. P. Novikov, “Adams operators and fixed points,” Izv. Akad. Nauk SSSR, Ser. Mat., 32:6 (1968), 1245–1263; English transl.: Izv. Akad. Nauk SSSR, 2:6 (1968), 1193–1211.MathSciNetMATHGoogle Scholar
  25. [25]
    S. Ochanine, “Sur les genres multiplicatifs définis par des intégrales elliptiques,” Topology, 26:2 (1987), 143–151.MathSciNetMATHCrossRefGoogle Scholar
  26. [26]
    D. Quillen, “On the formal group laws of unoriented and complex cobordism theory,” Bull. Amer. Math. Soc., 75:6 (1969), 1293–1298.MathSciNetMATHCrossRefGoogle Scholar
  27. [27]
    N. P. Smart, “The Hessian form of an elliptic curve,” in: CHES 2001, Lecture Notes in Comput. Sci., vol. 2162, Springer-Verlag, Berlin, 2001, 118–125.Google Scholar
  28. [28]
    R. E. Stong, Notes on Cobordism Theory, Princeton Univ. Press, Princeton, NJ, 1968.MATHGoogle Scholar
  29. [29]
    J. T. Tate, “The arithmetic of elliptic curves,” Invent. Math., 23:3–4 (1974), 179–206.MathSciNetMATHCrossRefGoogle Scholar
  30. [30]
    E. T. Whittaker and G. N. Watson, A Course of Modern Analysis, Reprint of 4th (1927) ed., Cambridge Univ. Press, Cambridge, 1996.MATHGoogle Scholar

Copyright information

© Springer-Verlag 2011

Authors and Affiliations

  1. 1.Steklov Mathematical Institute RASSt. PetersburgRussia

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