Multi-variable sigma-functions: old and new results

TL;DR

This paper explores multi-variable sigma functions of hyperelliptic curves, providing new theta-constant representations of periods, generalizations of Rosenhain formulae, and methods to construct differential operators for hyperelliptic functions.

Contribution

It introduces novel theta-constant formulas for periods and a method to construct differential operators for hyperelliptic sigma-functions, extending existing theories.

Findings

Derived theta-constant expressions for periods of hyperelliptic curves.

Generalized Rosenhain type formulae for first kind periods.

Constructed differentiation operators for genus 1 and 2 hyperelliptic functions.

Abstract

We consider multi-variable sigma function of a genus $g$ hyperelliptic curve as a function of two group of variables -jacobian variables and parameters of the curve. In the theta-functional representation of sigma-function, the second group arises as periods of first and second kind differentials of the curve. We develop representation of periods in terms of theta-constants. For the first kind periods, generalizations of Rosenhain type formulae are obtained, whilst for the second kind periods theta-constant expressions arepresented which are explicitly related to the fixed co-homology basis.We describe a method of constructing differentiation operators for hyperelliptic analogues of $\zeta$- and $\wp$-functions on the parameters of the hyperelliptic curve. To demonstrate this method, we gave the detailed construction of these operators in the cases of genus 1 and 2.