Derivatives of Schur, Tau and Sigma Functions on Abel-Jacobi Images

TL;DR

This paper investigates derivatives of Schur, tau, and sigma functions via the Abel-Jacobi map, deriving new properties, expressions, and addition formulas for these functions on algebraic curves.

Contribution

It introduces novel derivative properties of sigma functions on (n,s) curves and generalizes addition formulas, linking prime forms to sigma function derivatives.

Findings

Derived new properties of sigma function derivatives.

Expressed prime form in terms of sigma derivatives.

Generalized addition formulas for hyperelliptic sigma functions.

Abstract

We study derivatives of Schur and tau functions from the view point of the Abel-Jacobi map. We apply the results to establish several properties of derivatives of the sigma function of an (n,s) curve. As byproducts we have an expression of the prime form in terms of derivatives of the sigma function and addition formulae which generalize those of Onishi for hyperelliptic sigma functions.