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A generalized Kiepert formula for C ab curves

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Abstract

Kiepert (1873) and Brioschi (1864) published algebraic equations for the n-division points of an elliptic curve, in terms of the Weierstrass ℘-function and its derivatives with respect to a uniformizing parameter, or another elliptic function, respectively. We generalize both types of formulas for a compact Riemann surface which, outside from one point, has a smooth polynomial equation in the plane, in the sense that we characterize the points whose n-th multiple in the Jacobian belongs to the theta divisor.

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References

  1. E. Arbarello, M. Cornalba, P. A. Griffiths and J. Harris, Geometry of Algebraic Curves. Vol. I, Springer-Verlag, New York, 1985.

    MATH  Google Scholar 

  2. S. Arita, An addition algorithm in Jacobian of C ab curves, Discrete Applied Mathematics 130 (2003), 13–31.

    Article  MATH  MathSciNet  Google Scholar 

  3. Ch. Birkenhake and P. Vanhaecke, The vanishing of the theta function in the KP direction: a geometric approach, Compositio Mathematica 135 (2003), 323–330.

    Article  MATH  MathSciNet  Google Scholar 

  4. H. W. Braden, V. Z. Enolskii and A. N. W. Hone, Bilinear recurrences and addition formulae for hyperelliptic sigma functions, Journal of Nonlinear Mathematical Physics 12, suppl. 2 (2005), 46–62.

    Article  MATH  MathSciNet  Google Scholar 

  5. F. Brioschi, Sur quelques formules pour la multiplication des fonctions elliptiques, Comptes Rendus Mathématique. Académiedes Sciences. Paris 59 (1864), 769–775.

    Google Scholar 

  6. V. M. Bukhshtaber, D. V. Leikin, and V. Z. Enol’skiii, Rational analogues of abelian functions, Functional Analysis and its Applications 33 (1999), 83–94.

    Article  MATH  Google Scholar 

  7. J. L. Burchnall and T. W. Chaundy, Commutative ordinary differential operators, Proceedings of the Royal Society of Edinburgh. Section A 118 (1928), 557–583.

    Google Scholar 

  8. D. G. Cantor, On the analogue of the division polynomials for hyperelliptic curves, Journal fur die Reine und Angewandte Mathematik 447 (1994), 91–145.

    Article  MATH  MathSciNet  Google Scholar 

  9. J. Denef and F. Vercauteren, Counting points on C ab curves using Monsky-Washnitzer cohomology, Finite Fields and their Applications 12 (2006), 78–102.

    Article  MATH  MathSciNet  Google Scholar 

  10. H. M. Farkas and I. Kra, Riemann Surfaces, Second edition. Graduate Texts in Mathematics 71, Springer-Verlag, New York, 1992.

    Google Scholar 

  11. R. Garnier, Sur une classe de systems différentiels Abéliens déduits de la thé orie des équations linéaires, Rendiconti del Circolo Matematico di Palermo 43 (1918), 155–191.

    Article  Google Scholar 

  12. L. Kiepert, Wirkliche Ausführung der ganzzahligen Multiplication der elliptischen Functionen, Journal für die Reine und Angewandte Mathematik 76 (1873), 21–33.

    Google Scholar 

  13. S. Matsutani, Elliptic and hyperelliptic solutions of discrete Painlevé I and its Extensions to third difference equation, Physics Letters. A 300 (2002), 233–242.

    Article  MATH  MathSciNet  Google Scholar 

  14. S. Matsutani, Recursion relation of hyperelliptic psi-functions of genus two, Integral Transforms and Special Functions 14 (2003), 517–527.

    Article  MATH  MathSciNet  Google Scholar 

  15. S. Miura, Algebraic geometric codes on certain plane curves, Trans. IEICE J75-A (1992), 1735–1745.

    MathSciNet  Google Scholar 

  16. A. Nakayashiki, On the cohomology of theta divisor of hyperelliptic Jacobians, in Integrable Systems, Topology, and Physics. A Conference on Integrable Systems in Differential Geometry, University of Tokyo, Japan, July 17–21, 2000 (M. Guest et al. eds.), Contemp. Math. 309, AMS, Providence, RI., 2002, pp. 177–183.

    Google Scholar 

  17. Y. Ônishi, Determinant expressions for hyperelliptic functions, Proceeding of the Edinburgh Mathematical Society 48 (2005), 1–38.

    Article  Google Scholar 

  18. Y. Ônishi, Determinant expressions in Abelian functions for purely trigonal curves of degree four, preprint. math.NT/0503696 (2005).

  19. E. Previato, Generalized Weierstrass ℘-functions and KP flows in affine space, Commentarii Mathematici Helvetici 62 (1987), 292–310.

    Article  MATH  MathSciNet  Google Scholar 

  20. H. Weber, Lehrbuch der Algebra III, Friedr. Vieweg & Sohn, 1908.

  21. E. T. Whittaker and G. N. Watson, A Course of Modern Analysis, fourth edition, Cambridge University Press, Cambridge, 1927.

    MATH  Google Scholar 

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Authors and Affiliations

  1. 8-21-1 Higashi-Linkan, Sagamihara, 228-0811, Japan

    Shigeki Matsutani

  2. Institut Mittag-Leffer, The Royal Swedish Academy of Sciences, Djursholm, Sweden

    Emma Previato

  3. Department of Mathematics and Statistics, Boston University, Boston, MA, 02215-2411, USA

    Emma Previato

Authors
  1. Shigeki Matsutani
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  2. Emma Previato
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Correspondence to Shigeki Matsutani.

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Matsutani, S., Previato, E. A generalized Kiepert formula for C ab curves. Isr. J. Math. 171, 305–323 (2009). https://doi.org/10.1007/s11856-009-0051-8

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  • DOI: https://doi.org/10.1007/s11856-009-0051-8

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