Crossing the Transcendental divide: from Schottky groups to algebraic curves
- Samantha Fairchild (MPI MiS, Leipzig)
Though the uniformization theorem guarantees an equivalence of Riemann surfaces and smooth algebraic curves, moving between analytic and algebraic representations is inherently transcendental. We construct a family of non-hyperelliptic surfaces of genus at least 3 where we know the Riemann surface as well as properties of the canonical embedding, including a nontrivial symmetry group and a real structure with the maximal number of connected components (an M-curve). I will also share some numerical approximations where we try to detect the underlying algebraic curve through sampling. This is based on joint work with Ángel David Ríos Ortiz.