Modular Form Representation for Periods of Hyperelliptic Integrals

TL;DR

This paper develops a method to express periods of hyperelliptic integrals solely in terms of curve parameters and modular forms, using special bases for differentials to simplify calculations.

Contribution

It introduces explicit Baker and Klein bases for differentials that streamline the computation of hyperelliptic periods in terms of curve parameters and modular forms.

Findings

Explicit formulas for periods in terms of curve parameters and modular forms.

Comparison of different bases simplifies the representation of periods.

Enhanced computational approaches for hyperelliptic integrals.

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.