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


Authors: J. C. Eilbeck, K. Eilers and V. Z. Enolski
Original publication: Trudy Moskovskogo Matematicheskogo Obshchestva, tom 74 (2013), vypusk 2.
Journal: Trans. Moscow Math. Soc. 2013, 245-260
MSC (2010): Primary 32G15, 14K25, 30F30
DOI: https://doi.org/10.1090/S0077-1554-2014-00218-1
Published electronically: April 9, 2014
MathSciNet review: 3235799
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Abstract: 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.


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Additional Information

J. C. Eilbeck
Affiliation: Department of Mathematics and Maxwell Institute, Heriot-Watt University, Edinburgh, United Kingdom
Email: J.C.Eilbeck@hw.ac.uk

K. Eilers
Affiliation: Faculty of Mathematics, University of Oldenburg, Oldenburg, Germany
Email: keno.eilers@hotmail.de

V. Z. Enolski
Affiliation: School of Mathematics and Maxwell Institute, Edinburgh University, Edinburgh, UK, on leave from the Institute of Magnetism, National Academy of Sciences of Ukraine, Kiev, 03142, Ukraine
Email: Viktor.Enolskiy@ed.ac.uk

DOI: https://doi.org/10.1090/S0077-1554-2014-00218-1
Keywords: Moduli of algebraic curves, theta-constants, sigma-functions
Published electronically: April 9, 2014
Dedicated: Dedicated to the 70th birthday of Victor Buchstaber
Article copyright: © Copyright 2014 J.C.Eilbeck, K.Eilers, V.Z.Enolski

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