Composite Genus One Belyi Maps

TL;DR

This paper presents an efficient method for explicitly computing genus one Belyi maps by composing elliptic curve coverings with simpler maps and isogenies, resulting in new dessins with applications in physics.

Contribution

It introduces a novel approach combining elliptic curve coverings, genus zero Belyi maps, and isogenies to compute genus one Belyi maps explicitly.

Findings

Generated many new explicit dessins on the doubly periodic plane.

Realized several dessins in physics as brane-tilings.

Provided an efficient computational method for genus one Belyi maps.

Abstract

Motivated by a demand for explicit genus 1 Belyi maps from theoretical physics, we give an efficient method of explicitly computing genus one Belyi maps by (1) composing covering maps from elliptic curves to the Riemann sphere with simpler (univariate) genus zero Belyi maps, as well as by (2) composing further with isogenies between elliptic curves. This gives many new explicit dessins on the doubly periodic plane, including several which have been realized in the physics literature as so-called brane-tilings in the context of quiver gauge theories.