Crossing the transcendental divide: from translation surfaces to algebraic curves

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.

arXiv commentfinal version; 33 pages, 7 figures, comments welcome!

References in corpus (9)

• Connected components of the moduli spaces of Abelian differentials with prescribed singularities
• Discrete Riemann Surfaces and the Ising model
• Computing Theta Functions with Julia
• Discrete complex analysis on planar quad-graphs
• Discrete Riemann surfaces based on quadrilateral cellular decompositions
• Effective reconstruction of generic genus 4 curves from their theta hyperplanes
• A short introduction to translation surfaces, Veech surfaces, and Teichmueller dynamics
• Effective reconstruction of generic genus 5 curves from their theta hyperplanes
• Crossing the transcendental divide: from translation surfaces to algebraic curves

Cited by in corpus (1)

• Crossing the transcendental divide: from translation surfaces to algebraic curves