On Addition Formulae for Sigma Functions of Telescopic Curves

TL;DR

This paper extends addition formulae for sigma functions from specific algebraic curves to more general telescopic curves, providing new explicit expressions and relations involving the prime form.

Contribution

It introduces generalized addition formulae for sigma functions applicable to telescopic curves, broadening the scope beyond previously studied $(n,s)$ curves.

Findings

Derived explicit addition formulae for telescopic curves

Expressed the prime form in terms of sigma function derivatives

Extended known results to a wider class of algebraic curves

Abstract

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.