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Periods of second kind differentials of $$(n,s)$$-curves. (English) Zbl 1302.30053
Trans. Mosc. Math. Soc. 2013, 245-260 (2013) and Tr. Mosk. Mat. O.-va 74, No. 2, 297-315 (2013).
Summary: 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.

MSC:
 30F30 Differentials on Riemann surfaces 14H50 Plane and space curves
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