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

Abstract

The problem of generalisation of classical expressions for periods of second kind elliptic integrals in terms of theta-constants to higher genera is studied. In this context special class of algebraic curves – (n, s)-curves is considered. It is shown that required representations can be obtained by comparison of equivalent expressions for projective connection by Fay-Wirtinger and Klein-Weierstrass. The case of genus two hyperelliptic curve is considered as a principle example and a number of new Thomae and Rosenhain-type formulae are obtained. We anticipate that the analysis undertaken for genus two curve can be extended to higher genera hyperelliptic curve as well to other classes of (n, s) non-hyperelliptic curves.