Abstract
We consider a wide class of models of plane algebraic curves, so-called (n, s)-curves. The case (2, 3) is the classical Weierstrass model of an elliptic curve. On the basis of the theory of multivariate sigma functions, for every pair of coprime n and s we obtain an effective description of the Lie algebra of derivations of the field of fiberwise Abelian functions defined on the total space of the bundle whose base is the parameter space of the family of nondegenerate (n, s)-curves and whose fibers are the Jacobi varieties of these curves. The essence of the method is demonstrated by the example of Weierstrass elliptic functions. Details are given for the case of a family of genus 2 curves.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
V. I. Arnold, Singularities of Caustics and Wave Fronts, Kluwer Academic Publishers Group, Dordrecht, 1990.
V. I. Arnold, S. M. Gusein-Zade, and A. N. Varchenko, Singularities of Differentiable Maps I, II, Birkhauser, Boston-Basel-Stuttgart, 1985, 1988.
H. F. Baker, Abelian Functions, Cambridge University Press, Cambridge, 1995.
V. M. Buchstaber, V. Z. Enolskii, and D. V. Leykin, “Hyperelliptic Kleinian functions and applications,” in: Solitons, Geometry and Topology: On the Crossroad, Adv. Math. Sci., Amer. Math. Soc. Transl. Ser. 2, vol. 179, Amer. Math. Soc., Providence, RI, 1997, 1–34.
V. M. Buchstaber, V. Z. Enolskii, and D. V. Leykin, “Kleinian functions, hyperelliptic Jacobians and applications,” Rev. Math. Math. Phys., 10:2 (1997), 3–120.
V. M. Buchstaber, D. V. Leykin, and V. Z. Enolskii, “Rational analogs of Abelian functions,” Funkts. Anal. Prilozhen., 33:2 (1999), 1–15; English transl.: Funct. Anal. Appl., 33:2 (1999), 83–94.
V. M. Buchstaber and D. V. Leykin, “Polynomial Lie algebras,” Funkts. Anal. Prilozhen., 36:4 (2002), 18–34; English transl.: Funct. Anal. Appl., 36:4 (2002), 267–280.
V. M. Buchstaber and D. V. Leykin, “Heat equations in a nonholonomic frame,” Funkts. Anal. Prilozhen., 38:2 (2004), 12–27; English transl.: Funct. Anal. Appl., 38:2 (2004), 88–101.
V. M. Buchstaber and D. V. Leykin, “Differentiation of Abelian functions with respect to parameters,” Uspekhi Mat. Nauk, 62:4 (2007), 153–154; English transl.: Russian Math. Surveys, 62: 4 (2007), 787–789.
B. A. Dubrovin, “Geometry of 2D topological field theories,” in: Lecture Notes in Math., vol. 1620, 1994, 120–348.
B. A. Dubrovin and S. P. Novikov, “A periodic problem for the Korteweg-de Vries and Sturm-Liouville equations. Their connection with algebraic geometry,” Dokl. Akad. Nauk SSSR, 219:3 (1974), 531–534; English transl.: Soviet Math. Dokl., 15 (1974), 1597–1601.
F. G. Frobenius and L. Stickelberger, “Über die Differentiation der elliptischen Funktionen nach den Perioden und Invarianten,” J. Reine Angew. Math., 92 (1882), 311–337.
A. B. Givental, “Displacement of invariants of groups that are generated by reflections and are connected with simple singularities of functions,” Funkts. Anal. Prilozhen., 14:2 (1980), 4–14; English transl.: Funct. Anal. Appl., 14:2 (1980), 81–89.
K. Weierstrass, “Zur Theorie der elliptischen Funktionen,” in: Mathematische Werke, Bd. 2, Teubner, Berlin, 1894, 245–255.
K. Weierstrass, Abelsche Funktionen, Gesammelte Werke, Bd. 4, 1904.
V. M. Zakalyukin, “Reconstructions of wave fronts depending on one parameter,” Funkts. Anal. Prilozhen., 10:2 (1976), 69–70; English transl.: Funct. Anal. Appl., 10:2 (1976), 139–140.
Author information
Authors and Affiliations
Corresponding author
Additional information
To Israel Moiseevich Gelfand with admiration and deep gratitude
__________
Translated from Funktsional’nyi Analiz i Ego Prilozheniya, Vol. 42, No. 4, pp. 24–36, 2008
Original Russian Text Copyright © by V. M. Buchstaber and D. V. Leykin
Rights and permissions
About this article
Cite this article
Buchstaber, V.M., Leykin, D.V. Solution of the problem of differentiation of Abelian functions over parameters for families of (n, s)-curves. Funct Anal Its Appl 42, 268–278 (2008). https://doi.org/10.1007/s10688-008-0040-4
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10688-008-0040-4