Crossing the transcendental divide: from Schottky groups to algebraic curves

• Main Authors: Fairchild, Samantha, Ortiz, Ángel David Ríos
• Format: Journal Article
• Language: English
• Published: 19-01-2024
• Subjects: Mathematics - Algebraic Geometry; Mathematics - Geometric Topology
• DOI: 10.48550/arxiv.2401.10801

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 M\"obius
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.