J. C. Eilbeck, K. Eilers, V. Z. Enolski, “Periods of second kind differentials of <nobr>$(n,s)$</nobr>-curves”, Tr. Mosk. Mat. Obs., 2013, Volume 74, Issue 2,Pages <nobr>297

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

J. C. Eilbeck^ab, K. Eilers^c, V. Z. Enolski^adb

^a Department of Mathematics, Heriot-Watt University, Edinburgh, UK
^b Maxwell Institute for Mathematical Sciences
^c Faculty of Mathematics, University of Oldenburg, Germany
^d Institute of Magnetism, National Academy of Sciences of Ukraine, Kiev, 03142, Ukraine

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. References: 33 entries.

Key words and phrases: moduli of algebraic curves, theta-constants, sigma-functions.

UDC: 515.178.2+517.958+514

MSC: 32G15, 14K25, 30F30

Received: 14.05.2013

Language: English