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

Pascal Molin (1, 2, 3) , Christian Neurohr (4)

1 IMJ-PRG (UMR_7586) - Institut de Mathématiques de Jussieu - Paris Rive Gauche
2 OURAGAN - OUtils de Résolution Algébriques pour la Géométrie et ses ApplicatioNs
3 ANR - Agence Nationale de la Recherche
4 Institut für Mathematik [Oldenburg]

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. The algorithm is implemented and makes it possible to reach thousands of digits accuracy even on large genus curves.

Domaines: Théorie des nombres [math.NT]; Géométrie algébrique [math.AG]

Pascal Molin, Christian Neurohr. Computing period matrices and the Abel-Jacobi map of superelliptic curves. Mathematics of Computation, 2019. ⟨hal-02416012⟩