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

SIGMA 12 (2016), 086, 21 pages      arXiv:1603.09569

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

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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