Belyi maps on elliptic curves and dessin d'enfant on the torus

10/13/2015 - 12:15pm
10/13/2015 - 1:10pm
Edray Goins (Purdue University)

A Belyi map $\beta: P^1(C) \to P^1(C)$ is a rational function with at most three critical values; we may assume these values are ${0, 1, \infty }$. A Dessin d'Enfant is a planar bipartite graph obtained by considering the preimage of a path between two of these critical values, usually taken to be the line segment from 0 to 1. Such graphs can be drawn on the sphere by composing with stereographic projection: $\beta^{-1} ([0,1]) \subseteq P^1(C) \simeq S^2(R)$. Replacing $P^1$ with an elliptic curve $E$, there is a similar definition of a Belyi map $\beta: E(C) \to P^1(C)$. The corresponding Dessin d'Enfant can be drawn on the torus by composing with an elliptic logarithm: $\beta^{-1} ([0,1]) \subseteq E(C) \simeq T^2(R)$. In this talk, we discuss the problems of (1) constructing examples of Belyi maps for elliptic curves and (2) drawing Dessins d'Enfants on the torus. This work is part of PRiME (Purdue Research in Mathematics Experience) with Leonardo Azopardo, Sofia Lyrintzis, Bronz McDaniels, Maxim Millan, Yesid Sanchez Arias, Danny Sweeney, and Sarah Thomaz with assistance by Hongshan Li and Avi Steiner.

Millikan 2099, Pomona College