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 Tr. Mosk. Mat. Obs., 2013, Volume 74, Issue 2, Pages 297–315 (Mi mmo550)

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

J. C. Eilbeckab, K. Eilersc, V. Z. Enolskiadb

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.

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English version:
Transactions of the Moscow Mathematical Society, 2013, 74, 245–260

Bibliographic databases:

UDC: 515.178.2+517.958+514
MSC: 32G15, 14K25, 30F30
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Citation: J. C. Eilbeck, K. Eilers, V. Z. Enolski, “Periods of second kind differentials of $(n,s)$-curves”, Tr. Mosk. Mat. Obs., 74, no. 2, MCCME, M., 2013, 297–315; Trans. Moscow Math. Soc., 74 (2013), 245–260

Citation in format AMSBIB
\Bibitem{EilEilEno13} \by J.~C.~Eilbeck, K.~Eilers, V.~Z.~Enolski \paper Periods of second kind differentials of $(n,s)$-curves \serial Tr. Mosk. Mat. Obs. \yr 2013 \vol 74 \issue 2 \pages 297--315 \publ MCCME \publaddr M. \mathnet{http://mi.mathnet.ru/mmo550} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=3235799} \zmath{https://zbmath.org/?q=an:1302.30053} \elib{https://elibrary.ru/item.asp?id=21369373} \transl \jour Trans. Moscow Math. Soc. \yr 2013 \vol 74 \pages 245--260 \crossref{https://doi.org/10.1090/s0077-1554-2014-00218-1} \scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84960097646} 

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This publication is cited in the following articles:
1. Keno Eilers, “Modular Form Representation for Periods of Hyperelliptic Integrals”, SIGMA, 12 (2016), 060, 13 pp.
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