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.

KEYWORD:

• Translation surfaces
• Riemann surfaces
• algebraic curves
• discrete Riemann surfaces
• Riemann theta functions

Acknowledgments

We wish to thank Felix Günther for sharing his knowledge of discrete Riemann surfaces, Bernd Sturmfels for connecting us on this project, and the anonymous referee for detailed and constructive feedback. We also want to thank Nils Bruin, Christophe Ritzenthaler, and André Uschmajew for useful conversations.

Declaration of Interest

The authors report there are no competing interests to declare.

Additional information

Funding

T.Ö.Ç was supported by the Turkish Scientific and Technological Research Council (TÜBİTAK) - Project number 121C039. S.F. was supported by the Deutsche Forschungsgemeinschaft (DFG)—Projektnummer 445466444. Y.M. was partially supported by NSF grant DGE 2146752.