Genus One Belyi Maps by Quadratic Correspondences

Abstract

We present a method of obtaining a Belyi map on an elliptic curve from that on the Riemann sphere. This is done by writing the former as a radical of the latter, which we call a quadratic correspondence, with the radical determining the elliptic curve. With a host of examples of various degrees we demonstrate that the correspondence is an efficient way of obtaining genus one Belyi maps. As applications, we find the Belyi maps for the dessins d'enfant which have arisen as brane-tilings in the physics community, including ones, such as the so-called suspended pinched point, which have been a standing challenge for a number of years.

27 pages, 29 figures; Some sections expanded and pictures adjusted