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

J. C. Eilbeck, K. Eilers, V. Z. Enolski

Research output: Contribution to journalArticlepeer-review

3 Citations (Scopus)
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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.
Original languageEnglish
Pages (from-to)245-260
Number of pages16
JournalTransactions of the Moscow Mathematical Society
Publication statusPublished - 2013


  • math.CV
  • math-ph
  • math.AG
  • math.MP
  • 14K25, 14H45, 14K20 (Primary) 37K20 (Secondary)


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