### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

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, Carl-von-Ossietzky-Str. 9-11, 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 so-called 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; theta-constants; modular forms.

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