
SIGMA 12 (2016), 060, 13 pages arXiv:1512.06765
https://doi.org/10.3842/SIGMA.2016.060
Modular Form Representation for Periods of Hyperelliptic Integrals
Keno Eilers
Faculty of Mathematics, University of Oldenburg, CarlvonOssietzkyStr. 911, 26129 Oldenburg, Germany
Received December 22, 2015, in final form June 17, 2016; Published online June 24, 2016
Abstract
To every hyperelliptic curve one can assign the periods of the integrals over the holomorphic and the meromorphic differentials. By comparing two representations of the socalled projective connection it is possible to reexpress the latter periods by the first. This leads to expressions including only the curve's parameters $\lambda_j$ and modular forms. By a change of basis of the meromorphic differentials one can further simplify this expression. We discuss the advantages of these explicitly given bases, which we call Baker and Klein basis, respectively.
Key words:
periods of second kind differentials; thetaconstants; modular forms.
pdf (418 kb)
tex (33 kb)
References

Baker H.F., An introduction to the theory of multiply periodic functions, Cambridge University Press, Cambridge, 1897.

Baker H.F., An introduction to the theory of multiply periodic functions, Cambridge University Press, Cambridge, 1907.

Buchstaber V.M., Enolski V.Z., Leykin D.V., Kleinian functions, hyperelliptic Jacobians and applications, Rev. Math. Math. Phys. 10 (1997), 3120.

Eilbeck J.C., Eilers K., Enolski V.Z., Periods of second kind differentials of $(n,s)$curves, Trans. Moscow Math. Soc. (2013), 245260, arXiv:1305.3201.

Enolski V., Hartmann B., Kagramanova V., Kunz J., Lämmerzahl C., Sirimachan P., Inversion of a general hyperelliptic integral and particle motion in HořavaLifshitz black hole spacetimes, J. Math. Phys. 53 (2012), 012504, 35 pages, arXiv:1011.6459.

Farkas H.M., Kra I., Riemann surfaces, Graduate Texts in Mathematics, Vol. 71, SpringerVerlag, New York  Berlin, 1980.

Fay J.D., Theta functions on Riemann surfaces, Lecture Notes in Math., Vol. 352, SpringerVerlag, Berlin — New York, 1973.

Klein F., Ueber hyperelliptische Sigmafunctionen, Math. Ann. 27 (1886), 431464.

Klein F., Ueber hyperelliptische Sigmafunctionen, Math. Ann. 32 (1888), 351380.

Korotkin D., Shramchenko V., On higher genus Weierstrass sigmafunction, Phys. D 241 (2012), 20862094, arXiv:1201.3961.

