Christian Neurohr (Oldenburg, with Stefan Hellbusch)

16 July 2015 - W01 0-006 (Wechloy), 4 pm c.t.

Computation of the period matrix of compact Riemann surfaces

In this talk I will present an algorithm for computing the period matrix of a compact Riemann surface given by an affine plane curve over the rational numbers. For this we use a variety of numerical and symbolic methods to compute some necessary objects, such as the local monodromy and a symplectic homology basis. Central tasks are the analytical continuation of paths in the complex plane onto the Riemann surface and the numerical integration of differential forms along such paths. The aim is to implement efficient and provable algorithms with which period matrices can be calculated with very high numerical precision. Direct applications in number theory include the calculation of the Abel-Jacobi mapping and theta functions. The implementation is part of a magma package as part of our project "Algorithms on Riemann surfaces".