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arXiv · 2401.10801

Crossing the transcendental divide: from Schottky groups to algebraic curves

Samantha Fairchild·Ángel David Ríos Ortiz
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Abstract

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.

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Samantha Fairchild, Ángel David Ríos Ortiz. 2024-01-19. Crossing the transcendental divide: from Schottky groups to algebraic curves. https://arxiv.org/abs/2401.10801

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