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

SIGMA 12 (2016), 086, 21 pages      arXiv:1603.09569      https://doi.org/10.3842/SIGMA.2016.086

### On Jacobi Inversion Formulae for Telescopic Curves

Takanori Ayano
Osaka City University, Advanced Mathematical Institute, 3-3-138 Sugimoto, Sumiyoshi-ku, Osaka, 558-8585, Japan

Received May 06, 2016, in final form August 23, 2016; Published online August 27, 2016

Abstract
For a hyperelliptic curve of genus $g$, it is well known that the symmetric products of $g$ points on the curve are expressed in terms of their Abel-Jacobi image by the hyperelliptic sigma function (Jacobi inversion formulae). Matsutani and Previato gave a natural generalization of the formulae to the more general algebraic curves defined by $y^r=f(x)$, which are special cases of $(n,s)$ curves, and derived new vanishing properties of the sigma function of the curves $y^r=f(x)$. In this paper we extend the formulae to the telescopic curves proposed by Miura and derive new vanishing properties of the sigma function of telescopic curves. The telescopic curves contain the $(n,s)$ curves as special cases.

Key words: sigma function; inversion of algebraic integrals; vanishing of sigma function; Riemann surface; telescopic curve.

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References

1. Arbarello E., Cornalba M., Griffiths P.A., Harris J., Geometry of algebraic curves. Vol. I, Grundlehren der Mathematischen Wissenschaften, Vol. 267, Springer-Verlag, New York, 1985.
2. Ayano T., Sigma functions for telescopic curves, Osaka J. Math. 51 (2014), 459-480, arXiv:1201.0644.
3. Ayano T., Nakayashiki A., On addition formulae for sigma functions of telescopic curves, SIGMA 9 (2013), 046, 14 pages, arXiv:1303.2878.
4. Baker H.F., Abel's theorem and the allied theory including the theory of the theta functions, Cambridge University Press, Cambridge, 1897.
5. Buchstaber V.M., Enolski V.Z., Leykin D.V., Kleinian functions, hyperelliptic Jacobians and applications, Rev. Math and Math. Phys. 10 (1997), no. 2, 1-125, solv-int/9603005.
6. Bukhshtaber V.M., Enolskii V.Z., Leykin D.V., Rational analogues of abelian functions, Funct. Anal. Appl. 33 (1999), 83-94.
7. Bukhshtaber V.M., Enolskii V.Z., Leykin D.V., Multi-dimensional sigma functions, arXiv:1208.0990.
8. Eilbeck J.C., Enolskii V.Z., Leykin D.V., On the Kleinian construction of abelian functions of canonical algebraic curves, in SIDE III - Symmetries and Integrability of Difference Equations (Sabaudia, 1998), CRM Proc. Lecture Notes, Vol. 25, Amer. Math. Soc., Providence, RI, 2000, 121-138.
9. Eilers K., Modular form representation for periods of hyperelliptic integrals, SIGMA 12 (2016), 060, 13 pages, arXiv:1512.06765.
10. Enolski V., Hartmann B., Kagramanova V., Kunz J., Lämmerzahl C., Sirimachan P., Inversion of a general hyperelliptic integral and particle motion in Hořava-Lifshitz black hole space-times, J. Math. Phys. 53 (2012), 012504, 35 pages, arXiv:1106.2408.
11. Enolski V.Z., Hackmann E., Kagramanova V., Kunz J., Lämmerzahl C., Inversion of hyperelliptic integrals of arbitrary genus with application to particle motion in general relativity, J. Geom. Phys. 61 (2011), 899-921, arXiv:1011.6459.
12. Jorgenson J., On directional derivatives of the theta function along its divisor, Israel J. Math. 77 (1992), 273-284.
13. Klein F., Ueber hyperelliptische Sigmafunctionen, Math. Ann. 27 (1886), 431-464.
14. Klein F., Ueber hyperelliptische Sigmafunctionen, Math. Ann. 32 (1888), 351-380.
15. Komeda J., Matsutani S., Previato E., The sigma function for Weierstrass semigoups $\langle3, 7, 8\rangle$ and $\langle6, 13, 14, 15, 16\rangle$, Internat. J. Math. 24 (2013), 1350085, 58 pages, arXiv:1303.0451.
16. Matsutani S., Komeda J., Sigma functions for a space curve of type $(3,4,5)$, J. Geom. Symmetry Phys. 30 (2013), 75-91, arXiv:1112.4137.
17. Matsutani S., Previato E., Jacobi inversion on strata of the Jacobian of the $C_{rs}$ curve $y^r=f(x)$, J. Math. Soc. Japan 60 (2008), 1009-1044.
18. Matsutani S., Previato E., Jacobi inversion on strata of the Jacobian of the $C_{rs}$ curve $y^r=f(x)$. II, J. Math. Soc. Japan 66 (2014), 647-692, arXiv:1006.1090.
19. Miura S., Linear codes on affine algebraic curves, Trans. IEICE J81-A (1998), 1398-1421.
20. Mumford D., Tata lectures on theta. I, Progress in Mathematics, Vol. 28, Birkhäuser Boston, Inc., Boston, MA, 1983.
21. Nakayashiki A., On algebraic expressions of sigma functions for $(n,s)$ curves, Asian J. Math. 14 (2010), 175-211, arXiv:0803.2083.
22. Nakayashiki A., Yori K., Derivatives of Schur, tau and sigma functions on Abel-Jacobi images, in Symmetries, Integrable Systems and Representations, Springer Proc. Math. Stat., Vol. 40, Springer, Heidelberg, 2013, 429-462, arXiv:1205.6897.
23. Ônishi Y., Complex multiplication formulae for hyperelliptic curves of genus three, Tokyo J. Math. 21 (1998), 381-431.
24. Ônishi Y., Determinant expressions for hyperelliptic functions, Proc. Edinb. Math. Soc. 48 (2005), 705-742, math.NT/0105189.
25. Suzuki J., Klein's fundamental second kind 2-form for the $C_{ab}$ curves, Talk at 2014 Mathematical Society of Japan Autumn Meeting.