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European Journal of Mathematics

, Volume 4, Issue 1, pp 93–112 | Cite as

Differentiation of genus 3 hyperelliptic functions

  • Elena Yu. Bunkova
Research Article
  • 73 Downloads

Abstract

We give an explicit solution to the problem of differentiation of hyperelliptic functions in genus 3 case. It is a genus 3 analogue of the result of Frobenius and Stickelberger (J Reine Angew Math 92:311–337, 1882). Our method is based on the series of works by Buchstaber, Enolskii and Leikin. First we introduce a polynomial map \(p:\mathbb {C}^{3g} \rightarrow \mathbb {C}^{2g}\) . Next for \(g = 1,2,3\) we provide 3g polynomial vector fields in \(\mathbb {C}^{3g}\) projectable for p and describe their polynomial Lie algebras. Finally we obtain the corresponding derivations of the field of hyperelliptic functions.

Keywords

Abelian functions Elliptic functions Jacobians Hyperelliptic curves Hyperelliptic functions Lie algebra of derivations Polynomial vector fields 

Mathematics Subject Classification

14H52 32N99 33E05 58J26 

Notes

Acknowledgements

The author thanks Victor M. Buchstaber for fruitful discussions of the results.

References

  1. 1.
    Arnold, V.I.: Singularities of Caustics and Wave Fronts. Mathematics and Its Applications (Soviet Series), vol. 62. Kluwer, Dordrecht (1990)CrossRefGoogle Scholar
  2. 2.
    Baker, H.F.: On the hyperelliptic sigma functions. Amer. J. Math. 20(4), 301–384 (1898)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Buchstaber, V.M.: Polynomial dynamical systems and Korteweg–de Vries equation. Proc. Steklov Inst. Math. 294, 176–200 (2016)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Buchstaber, V.M., Enolskiĭ, V.Z., Leĭkin, D.V.: Hyperelliptic Kleinian functions and applications. In: Buchstaber, V.M., Novikov, S.P. (eds.) Solitons, Geometry and Topology: On the Crossroad. American Mathematical Society Translations Series 2, vol. 179, pp. 1–33. American Mathematical Society, Providence (1997)Google Scholar
  5. 5.
    Buchstaber, V.M., Enolskii, V.Z., Leikin, D.V.: Kleinian functions, hyperelliptic Jacobians and applications. In: Novikov, S.P., Krichever, I.M. (eds.) Reviews in Mathematics and Mathematical Physics, vol. 10, 2nd edn, pp. 3–120. Gordon and Breach, London (1997)Google Scholar
  6. 6.
    Buchstaber, V.M., Enolskii, V.Z., Leykin, D.V.: Multi-dimensional sigma-functions (2012). arXiv:1208.0990
  7. 7.
    Buchstaber, V.M., Leykin, D.V.: Polynomial Lie algebras. Funct. Anal. Appl. 36(4), 267–280 (2002)MathSciNetCrossRefGoogle Scholar
  8. 8.
    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. Appl. 42(4), 268–278 (2008)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Buchstaber, V.M., Mikhailov, A.V.: Infinite-dimensional Lie algebras determined by the space of symmetric squares of hyperelliptic curves. Funct. Anal. Appl. 51(1), 2–21 (2017)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Dubrovin, B.A., Novikov, S.P.: A periodic problem for the Korteweg-de Vries and Sturm-Liouville equations. Their connection with algebraic geometry. Dokl. Akad. Nauk SSSR 219(3), 531–534 (1974) (in Russian)Google Scholar
  11. 11.
    Frobenius, F.G., Stickelberger, L.: Ueber die Differentiation der elliptischen Functionen nach den Perioden und Invarianten. J. Reine Angew. Math. 92, 311–337 (1882)MathSciNetMATHGoogle Scholar
  12. 12.
    Whittaker, E.T., Watson, G.N.: A Course of Modern Analysis. Cambridge Mathematical Library. Reprint of the Fourth Edition. Cambridge University Press, Cambridge (1996)Google Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Steklov Mathematical Institute of Russian Academy of SciencesMoscowRussia

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