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

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.