Computing period matrices and the Abel-Jacobi map of superelliptic curves

TL;DR

This paper introduces a numerical algorithm for computing period matrices and the Abel-Jacobi map of superelliptic curves, enabling high-precision calculations for complex algebraic curves.

Contribution

The authors develop a new algorithm that uses rigorous numerical integration methods to compute period matrices and Abel-Jacobi maps for superelliptic curves with high accuracy.

Findings

Achieves thousands of digits of precision in computations.

Applicable to large genus curves.

Uses Gauss and Double-Exponential integration methods.

Abstract

We present an algorithm for the computation of period matrices and the Abel-Jacobi map of complex superelliptic curves given by an equation $y^m=f(x)$. It relies on rigorous numerical integration of differentials between Weierstrass points, which is done using Gauss method if the curve is hyperelliptic ($m=2$) or the Double-Exponential method. We take linear combination of these integrals to obtain the actual periods on a symplectic basis of loops. The algorithm is implemented and makes it possible to reach thousands of digits accuracy even on large genus curves.