On Jacobi Inversion Formulae for Telescopic Curves

TL;DR

This paper extends Jacobi inversion formulae and sigma function properties from hyperelliptic and $(n,s)$ curves to the broader class of telescopic curves, enriching the algebraic geometry of these curves.

Contribution

It generalizes Jacobi inversion formulae and sigma function vanishing properties to telescopic curves, which include $(n,s)$ curves as special cases.

Findings

Derived new Jacobi inversion formulae for telescopic curves.

Established novel vanishing properties of sigma functions for these curves.

Extended algebraic tools for a broader class of algebraic curves.

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.