Theoretical and Mathematical Physics

, Volume 94, Issue 2, pp 142–149 | Cite as

Vector addition theorems and Baker-Akhiezer functions

  • V. M. Bukhshtaber
  • I. M. Krichever


Functional equations that arise naturally in various problems of modern mathematical physics are discussed. We introduce the concepts of anN-dimensional addition theorem for functions of a scalar argument and Cauchy equations of rankN for a function of ag-dimensional argument that generalize the classical functional Cauchy equation. It is shown that forN=2 the general analytic solution of these equations is determined by the Baker—Akhiezer function of an algebraic curve of genus 2. It is also shown that θ functions give solutions of a Cauchy equation of rankN for functions of ag-dimensional argument withN≤2 g in the case of a generalg-dimensional Abelian variety andNg in the case of a Jacobian variety of an algebra curve of genusg. It is conjectured that a functional Cauchy equation of rankg for a function of ag-dimensional argument is characteristic for θ functions of a Jacobian variety of an algebraic curve of genusg, i.e., solves the Riemann—Schottky problem.


Mathematical Physic Functional Equation Abelian Variety Algebraic Curve Vector Addition 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    A. L. Cauchy,Cours d'Analyse de l'Ecole Polyt., 1.Analyse Algebraique (1821), p. 103;Oeuvres Complets (2), Vol. 3, p. 98.Google Scholar
  2. 2.
    N. H. Abel, “Méthode générale pour trouver des fonctions d'une seule quantite variable, lorsqu'une propriété de ces fonctions est exprimee par une equation entre deux variables,”Magazin for Naturvidenskaberne, Aargang I, Bind 1, Cristiana (1823);Oever Completes, Vol. 1, Cristiana (1881), p. 1.Google Scholar
  3. 3.
    G. Frobenius and Stikelberger, “Ueber die Addition und Multiplication der elliptischen Functionen,”J. Reine Angew. Math.,88, 146 (1880).Google Scholar
  4. 4.
    H. F. Baker, “Note on the foregoing paper ‘Commutative ordinary differential operators’,”Proc. R. Soc. London,118, 584 (1928).Google Scholar
  5. 5.
    J. L. Burchnal and T. W. Chaundy, “Commutative ordinary differential operators, I, II,”Proc. London Math. Soc.,21, 420 (1922);Proc. R. Soc. London,118, 557 (1928).Google Scholar
  6. 6.
    I. M. Krichever, “Algebro—geometric construction of Zakharov—Shabat equations and their periodic solutions,”Dokl. Akad. Nauk SSSR,2, 291 (1976).Google Scholar
  7. 7.
    I. M. Krichever, “Integration of nonlinear equations by the methods of algebraic geometry,”Funktsional. Analiz i Ego Prilozhen.,11, Ser. 1, 15 (1977).Google Scholar
  8. 8.
    B. A. Dubrovin, V. B. Matveev, and S. P. Novikov, “Nonlinear equations of Korteweg—de Vries type, finite-gap operators, and Abelian varieties,”Usp. Mat. Nauk,31, 55 (1976).Google Scholar
  9. 9.
    V. E. Zakharov, S. V. Manakov, S. P. Novikov, and L. P. Pitaevskii,The Theory of Solitons. The Inverse Scattering Method [in Russian], Nauka, Moscow (1980).Google Scholar
  10. 10.
    I. M. Krichever and S. P. Novikov, “Holomorphic bundles over algebraic curves. Nonlinear equations,”Usp. Mat. Nauk,35, No. 6 (1980).Google Scholar
  11. 11.
    B. A. Dubrovin, “Theta function and nonlinear equations,”Usp. Mat. Nauk,36, 11 (1981).Google Scholar
  12. 12.
    B. A. Dubrovin, I. M. Krichever, and S. P. Novikov, “Integrable systems,” in:Reviews of Science and Technology, Fundamental Equations, Vol. 4 [in Russian], VINITI, Moscow (1985), p. 179.Google Scholar
  13. 13.
    I. M. Krichever, “Spectral theory of two-dimensional periodic operators and its applications,”Usp. Mat. Nauk,44, 121 (1989).Google Scholar
  14. 14.
    I. M. Krichever, “Elliptic solutions of the Kadomtsev—Petviashvili equation and integrable systems of particles,”Funktsional. Analiz i Ego Prilozhen.,14, 45 (1980).Google Scholar
  15. 15.
    F. Calogero, “One-dimensional many-body problems with pair interactions whose exact ground-state wave function is of product type,”Lett. Nuovo Cimento,13, 507 (1975).Google Scholar
  16. 16.
    M. Bruschi and F. Calogero, “The Lax representation for an integrable class of relativistic dynamical systems,”Commun. Math. Phys.,109, 481 (1987).Google Scholar
  17. 17.
    M. Bruschi and F. Calogero, “General analytic solution of certain functional equations of addition type,”Sian J. Math. Anal.,21, 1019 (1990).Google Scholar
  18. 18.
    A. M. Perelomov,Integrable Systems of Classical Mechanics and Lie Algebras [in Russian], Nauka, Moscow (1990).Google Scholar
  19. 19.
    V. M. Buchstaber,Report on Scientific Activity During Visit at the MPI from 04.06.92 to 07.04.92, MPI, Bonn (1992).Google Scholar
  20. 20.
    F. Hirzebruch,Topological Methods in Algebraic Geometry, Springer-Verlag, New York (1966).Google Scholar
  21. 21.
    I. M. Krichever, “Generalized elliptic genera and Baker—Akhiezer functions,”Mat. Zametki,47, 132 (1990).Google Scholar
  22. 22.
    S. Ochanine, “Sur les genres multiplicatifs difinis par des integrales elliptiques,” in:Topology,26, 143 (1987).Google Scholar
  23. 23.
    E. Witten, “Elliptic genera and quantum field theory,”Commun. Math. Phys.,109, 525 (1987).Google Scholar
  24. 24.
    C. Taubes, “S 1-actions and elliptic genera,”Commun. Math. Phys. 122, 455 (1989).Google Scholar
  25. 25.
    R. Bott and C. Taubes, “On the rigidity theorem of Witten,”J. Am. Math. Soc.,2, 137 (1989).Google Scholar
  26. 26.
    V. M. Bukhshtaber and A. N. Kholodov, “Formal groups, functional equations, and generalized cohomology theories,”Mat. Sb.,69, 77 (1991).Google Scholar
  27. 27.
    V. M. Bukhshtaber, “Functional equations associated with addition theorems for elliptic functions and two-valued algebraic groups,”Usp. Mat. Nauk,45, 213 (1990).Google Scholar
  28. 28.
    T. Shiota, “Characterization of Jacobian varieties in terms of soliton equations,”Inv. Math.,83, 333 (1986).Google Scholar

Copyright information

© Plenum Publishing Corporation 1993

Authors and Affiliations

  • V. M. Bukhshtaber
  • I. M. Krichever

There are no affiliations available

Personalised recommendations