On Jacobi Inversion Formulae for Telescopic 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 yr=f(x), which are special cases of (n,s) curves, and derived new vanishing properties of the sigma function of the curves yr=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.