Crossing the transcendental divide: from translation surfaces to algebraic curves

TL;DR

This paper develops an algorithm to construct algebraic curves from translation surfaces, bridging the gap between the analytic and algebraic perspectives of Riemann surfaces, with numerical experiments up to genus 5.

Contribution

It introduces a novel algorithm for approximating Jacobian varieties from translation surfaces, including implementation for specific polygon types and genus, advancing the computational connection between Riemann surfaces and algebraic curves.

Findings

Algorithm successfully approximates Jacobian for square-decomposed polygons

Numerical experiments conducted up to genus 5

Conjectures proposed based on Riemann theta functions

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 also implemented in any genus for specific examples of Jenkins-Strebel representatives, a dense family of translation surfaces 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.