Computational approach to compact Riemann surfaces

A purely numerical approach to compact Riemann surfaces starting from plane algebraic curves is presented. A set of generators of the fundamental group for the complement of the critical points in the complex plane is constructed from circles around these points and connecting lines obtained from a minimal spanning tree. The monodromies are computed by solving the defining equation of the algebraic curve on collocation points along these contours and by analytically continuing the roots. The collocation points are chosen to correspond to Chebychev collocation points for an ensuing Clenshaw–Curtis integration of the holomorphic differentials which gives the periods of the Riemann surface with spectral accuracy. At the singularities of the algebraic curve, Puiseux expansions computed by contour integration on the circles around the singularities are used to identify the holomorphic differentials. The Abel map is also computed with the Clenshaw–Curtis algorithm and contour integrals. Siegel's algorrithm is applied to approximately identify a symplectic transformation to the fundamental domain in order to allow for an efficient computation of multi-dimensional theta functions.