Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 9 (2013), 046, 14 pages      arXiv:1303.2878

On Addition Formulae for Sigma Functions of Telescopic Curves

Takanori Ayano a and Atsushi Nakayashiki b
a) Department of Mathematics, Osaka University, Toyonaka, Osaka 560-0043, Japan
b) Department of Mathematics, Tsuda College, Kodaira, Tokyo 187-8577, Japan

Received March 13, 2013, in final form June 14, 2013; Published online June 19, 2013

A telescopic curve is a certain algebraic curve defined by m−1 equations in the affine space of dimension m, which can be a hyperelliptic curve and an (n,s) curve as a special case. We extend the addition formulae for sigma functions of (n,s) curves to those of telescopic curves. The expression of the prime form in terms of the derivative of the sigma function is also given.

Key words: sigma function; tau function; Schur function; Riemann surface; telescopic curve; gap sequence.

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  1. Ayano T., Sigma functions for telescopic curves, Osaka J. Math., to appear, arXiv:1201.0644.
  2. Baker H.F., Abelian functions. Abel's theorem and the allied theory of theta functions, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1995.
  3. Braden H.W., Enolski V.Z., Fedorov Yu.N., Dynamics on strata of trigonal Jacobians and some integrable problems of rigid body motion, arXiv:1210.3596.
  4. Brauer A., On a problem of partitions, Amer. J. Math. 64 (1942), 299-312.
  5. Brauer A., Seelbinder B.M., On a problem of partitions. II, Amer. J. Math. 76 (1954), 343-346.
  6. Buchstaber V.M., Enolski V.Z., Leykin D.V., Multi-dimensional sigma functions, arXiv:1208.0990.
  7. Bukhshtaber V.M., Enolski V.Z., Leykin D.V., Rational analogues of abelian functions, Funct. Anal. Appl. 33 (1999), 83-94.
  8. Eilbeck J.C., Enolski V.Z., Gibbons J., Sigma, tau and Abelian functions of algebraic curves, J. Phys. A: Math. Theor. 43 (2010), 455216, 20 pages, arXiv:1006.5219.
  9. Fay J.D., Theta functions on Riemann surfaces, Lecture Notes in Mathematics, Vol. 352, Springer-Verlag, Berlin, 1973.
  10. Harnad J., Enolski V.Z., Schur function expansions of KP τ-functions associated to algebraic curves, Russ. Math. Surv. 66 (2011), 767-807, arXiv:1012.3152.
  11. Kirfel C., Pellikaan R., The minimum distance of codes in an array coming from telescopic semigroups, IEEE Trans. Inform. Theory 41 (1995), 1720-1732.
  12. Klein F., Ueber hyperelliptische Sigmafunctionen, Math. Ann. 27 (1886), 431-464.
  13. Klein F., Ueber hyperelliptische Sigmafunctionen (Zweite Abhandlung), Math. Ann. 32 (1888), 351-380.
  14. Komeda J., Matsutani S., Previato E., The sigma function for Weierstrass semigroups <3,7,8> and <6,13,14,15,16>, arXiv:1303.0451.
  15. Korotkin D., Shramchenko V., On higher genus Weierstrass sigma-function, Phys. D 241 (2012), 2086-2094, arXiv:1201.3961.
  16. Matsutani S., Sigma functions for a space curve (3,4,5) type with an appendix by J. Komeda, arXiv:1112.4137.
  17. Micale V., Olteanu A., On the Betti numbers of some semigroup rings, Matematiche (Catania) 67 (2012), 145-159, arXiv:1111.1433.
  18. Miura S., Linear codes on affine algebraic curves, Trans. IEICE J81-A (1998), 1398-1421.
  19. Nakayashiki A., On algebraic expressions of sigma functions for (n,s) curves, Asian J. Math. 14 (2010), 175-211, arXiv:0803.2083.
  20. Nakayashiki A., Sigma function as a tau function, Int. Math. Res. Not. 2010 (2010), no. 3, 373-394, arXiv:0904.0846.
  21. Nakayashiki A., Yori K., Derivatives of Schur, tau and sigma functions on Abel-Jacobi images, in Symmetries, Integrable Systems and Representations, Springer Proceedings in Mathematics & Statistics, Vol. 40, Editors K. Iohara, S. Morier-Genoud, B. Remy, Springer-Verlag, London, 2013, 429-462, arXiv:1205.6897.
  22. Nishijima D., Order counting algorithm for Fermat curves and Klein curves using p-adic cohomology, Master's Thesis, Osaka University, 2008.
  23. Ônishi Y., Determinant expressions for hyperelliptic functions (with an Appendix by Shigeki Matsutani), Proc. Edinb. Math. Soc. (2) 48 (2005), 705-742, math.NT/0105189.
  24. Sato M., Sato Y., Soliton equations as dynamical systems on infinite-dimensional Grassmann manifold, in Nonlinear Partial Differential Equations in Applied Science (Tokyo, 1982), North-Holland Math. Stud., Vol. 81, Editors P.D. Lax, H. Fujita, G. Strang, North-Holland, Amsterdam, 1983, 259-271.

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