Crossing the transcendental divide: from translation surfaces to algebraic curves

Türkü Özlüm Çelik, Samantha Fairchild, and Yelena Mandelshtam

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Submission date: 29. Nov. 2022
paper submitted to: Experimental mathematics
MSC-Numbers: 14H42, 30F10, 32G15
Link to arXiv: See the arXiv entry of this preprint.

Abstract:
We study constructing an algebraic curve from a Riemann surface given via a translation surface, which is a collection of finitely many polygons in the plane with sides identified by translation. We use the theory of discrete Riemann surfaces to give an algorithm for approximating the Jacobian variety of a translation surface whose polygon can be decomposed into squares. We first implement the algorithm in the case of L-shaped polygons where the algebraic curve is already known. The algorithm is then implemented for a family of translation surfaces called Jenkins--Strebel representatives that, until now, lived squarely on the analytic side of the transcendental divide between Riemann surfaces and algebraic curves. Using Riemann theta functions, we give numerical experiments and resulting conjectures up to genus 5.