Periods of second kind differentials of (n,s)-curves

TL;DR

This paper extends classical results on elliptic integrals of the second kind to higher genus algebraic curves, deriving new formulas for periods of (n,s)-curves using projective connection comparisons.

Contribution

It introduces a method to generalize period expressions from elliptic to higher genus (n,s)-curves via projective connection comparisons, producing new Thomae and Rosenhain-type formulas.

Findings

Derived new formulas for genus two hyperelliptic curves.

Showed the method's potential extension to higher genera and non-hyperelliptic curves.

Connected classical elliptic integral results to more complex algebraic curves.

Abstract

For elliptic curves, expressions for the periods of elliptic integrals of the second kind in terms of theta-constants, have been known since the middle of the 19th century. In this paper we consider the problem of generalizing these results to curves of higher genera, in particular to a special class of algebraic curves, the so-called $(n,s)$-curves. It is shown that the representations required can be obtained by the comparison of two equivalent expressions for the projective connection, one due to Fay-Wirtinger and the other from Klein-Weierstrass. As a principle example, we consider the case of the genus two hyperelliptic curve, and a number of new Thomae and Rosenhain-type formulae are obtained. We anticipate that our analysis for the genus two curve can be extended to higher genera hyperelliptic curves, as well as to other classes of $(n,s)$ non-hyperelliptic curves.