Manuscripta Mathematica

, Volume 134, Issue 3–4, pp 273–308 | Cite as

Division polynomials and canonical local heights on hyperelliptic Jacobians

  • Yukihiro UchidaEmail author


We generalize the division polynomials of elliptic curves to hyperelliptic Jacobians over the complex numbers. We construct them by using the hyperelliptic sigma function. Using the division polynomial, we describe a condition that a point on the Jacobian is a torsion point. We prove several properties of the division polynomials such as determinantal expressions and recurrence formulas. We also study relations among the sigma function, the division polynomials, and the canonical local height functions.

Mathematics Subject Classification (2000)

Primary 14H40 Secondary 11G10 11G50 


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  1. 1.
    Adams W.W., Loustaunau P.: An Introduction to Gröbner Bases. Graduate Studies in Mathematics, vol. 3. American Mathematical Society, Providence (1994)Google Scholar
  2. 2.
    Baker H.F.: Abelian Functions: Abel’s Theorem and the Allied Theory of Theta Functions. Cambridge University Press, Cambridge (1897)Google Scholar
  3. 3.
    Baker H.F.: On a system of differential equations leading to periodic functions. Acta Math. 27, 135–156 (1903)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Baker H.F.: An Introduction to the Theory of Multiply Periodic Functions. Cambridge University Press, Cambridge (1907)zbMATHGoogle Scholar
  5. 5.
    Buchstaber V.M., Enolskii V.Z., Leykin D.V.: Kleinian functions, hyperelliptic Jacobians and applications. Rev. Math. Math. Phys. 10, 1–125 (1997)Google Scholar
  6. 6.
    Buchstaber V.M., Enolskii V.Z., Leykin D.V.: A recursive family of differential polynomials generated by the Sylvester identity and addition theorems for hyperelliptic Kleinian functions. Funct. Anal. Appl. 31, 240–251 (1997)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Bukhshtaber V.M., Leikin D.V., Enol’skii V.Z.: σ-Functions of (n, s)-curves. Russ. Math. Surv. 54, 628–629 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Call G.S., Silverman J.H.: Canonical heights on varieties with morphisms. Compost. Math. 89, 163–205 (1993)zbMATHMathSciNetGoogle Scholar
  9. 9.
    Cantor D.G.: On the analogue of the division polynomials for hyperelliptic curves. J. Reine Angew. Math. 447, 91–145 (1994)zbMATHMathSciNetGoogle Scholar
  10. 10.
    Caspary F.: Zur Theorie der Thetafunctionen mehrerer Argumente. J. Reine Angew. Math. 96, 324–326 (1884)Google Scholar
  11. 11.
    Frobenius G.: Ueber Thetafunctionen mehrerer Variabeln. J. Reine Angew. Math. 96, 100–122 (1884)Google Scholar
  12. 12.
    Grant D.: Formal groups in genus two. J. Reine Angew. Math. 411, 96–121 (1990)zbMATHMathSciNetGoogle Scholar
  13. 13.
    Grayson, D.R., Stillman, M.E.: Macaulay 2, a software system for research in algebraic geometry.
  14. 14.
    Hindry M., Silverman J.H.: Diophantine Geometry: An Introduction. Graduate Texts in Mathematics, vol. 201. Springer-Verlag, New York (2000)Google Scholar
  15. 15.
    Kanayama N.: Division polynomials and multiplication formulae of Jacobian varieties of dimension 2. Math. Proc. Camb. Philos. Soc. 139, 399–409 (2005)zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Kanayama N.: Corrections to “Division polynomials and multiplication formulae in dimension 2”. Math. Proc. Camb. Philos. Soc. 149, 189–192 (2010)zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Lang S.: Introduction to Algebraic and Abelian Functions, 2nd edn. Graduate Texts in Mathematics, vol. 89. Springer-Verlag, New York (1982)Google Scholar
  18. 18.
    Lang S.: Fundamentals of Diophantine Geometry. Springer-Verlag, New York (1983)zbMATHGoogle Scholar
  19. 19. Maxima, a Computer Algebra System. Version 5.18.1, (2009)
  20. 20.
    Mumford D.: Tata Lectures on Theta I. Progress in Mathematics, vol. 28. Birkhäuser, Boston (1983)Google Scholar
  21. 21.
    Mumford D.: Tata Lectures on Theta II. Progress in Mathematics, vol. 43. Birkhäuser, Boston (1984)Google Scholar
  22. 22.
    Nakayashiki, A.: On algebraic expressions of sigma functions for (n, s) curves. arXiv:0803.2083v1 [math.AG]Google Scholar
  23. 23.
    Noro, M., et al.: A computer algebra system Risa/Asir.
  24. 24.
    Ônishi Y.: Determinant expressions for Abelian functions in genus two. Glasg. Math. J. 44, 353–364 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Ônishi Y.: Determinantal expressions for hyperelliptic functions in genus three. Tokyo J. Math. 27, 299–312 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    Ônishi Y.: Determinant expressions for hyperelliptic functions (with an appendix by Shigeki Matsutani). Proc. Edinb. Math. Soc. 48, 705–742 (2005)zbMATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    Schmitt S., Zimmer H.G.: Elliptic Curves: A Computational Approach. de Gruyter Studies in Mathematics 31. Walter de Gruyter, Berlin (2003)Google Scholar
  28. 28.
    Silverman J.H.: Advanced Topics in the Arithmetic of Elliptic Curves. Graduate Texts in Mathematics, vol. 151. Springer-Verlag, New York (1994)Google Scholar
  29. 29.
    van der Waerden B.L.: Algebra, vol. 2. Ungar, New York (1970)Google Scholar
  30. 30.
    Weierstrass, K.: Zur Theorie der Jacobi’schen Functionen von mehreren Veränderlichen. Sitzung. Königlich Preuss. Akad. Wiss. Berl. 505–508 (1882); Mathematische Werke III, Johnson Reprint Corporation, New York, pp. 155–159 (1967)Google Scholar
  31. 31.
    Whittaker E.T., Watson G.N.: A Course of Modern Analysis, 4th edn. Cambridge University Press, Cambridge (1927)zbMATHGoogle Scholar
  32. 32.
    Yoshitomi K.: On height functions on Jacobian surfaces. Manuscripta Math. 96, 37–66 (1998)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of ScienceKyoto UniversityKyotoJapan

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