Rosenhain-Thomae formulae for higher genera hyperelliptic curves

TL;DR

This paper generalizes Rosenhain's formula, which relates periods of hyperelliptic curves to theta constants, to higher genera using the second Thomae formula, expanding the understanding of period relations in algebraic geometry.

Contribution

It extends Rosenhain's classical formula to higher genus hyperelliptic curves via the second Thomae formula, providing a broader framework for period computations.

Findings

Generalization of Rosenhain formula to higher genera

Application of second Thomae formula to derive new relations

Enhanced understanding of theta constants in hyperelliptic curves

Abstract

Rosenhain's famous formula expresses the periods of first kind integrals of genus two hyperelliptic curves in terms of $\theta$-constants. In this paper we generalize the Rosenhain formula to higher genera hyperelliptic curves by means of the second Thomae formula for derivated non-singular $\theta$-constants.