Crossing the transcendental divide: from Schottky groups to algebraic curves
Abstract of paper. Though the uniformization theorem guarantees an equivalence of Riemann surfaces and smooth algebraic curves, moving between analytic and algebraic representations is inherently transcendental. Our analytic curves identify pairs of circles in the complex plane via free groups of Möbius transformations called Schottky groups. We construct a family of non-hyperelliptic surfaces of genus \(g\geq 3\) 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). We then numerically approximate the algebraic curve and Riemann matrices underlying our family of Riemann surfaces.
Jupyter Notebook
All code used in the numerical experiments is contained in the following notebook, either by clicking the link below or downloading directly: Schottky2Curves.ipynb.