Critical Points of Toroidal Bely\uı Maps

Abstract

A Belyi map $\beta :{\mathbb{P}}^1(\mathbb{C})\to {\mathbb{P}}^1(\mathbb{C})$ is a rational function with at most three critical values; we may assume these values are $\{0,1,\infty \}$. Replacing ${\mathbb{P}}^1$ with an elliptic curve $E:y^2=x^3+Ax+B$, there is a similar definition of a Belyi map $\beta :E(\mathbb{C})\to {\mathbb{P}}^1(\mathbb{C})$. Since $E(\mathbb{C})\simeq {\mathbb{T}}^2(\mathbb{R})$ is a torus, we call $(E,\beta )$ a Toroidal Belyi pair. There are many examples of Belyi maps $\beta :E(\mathbb{C})\to {\mathbb{P}}^1(\mathbb{C})$ associated to elliptic curves; several can be found online at LMFDB. Given such a Toroidal Belyi map of degree $N$, the inverse image $G={\beta }^{-1}(\{0,1,\infty \})$ is a set of $N$ elements which contains the critical points of the Belyi map. In this project, we investigate when $G$ is contained in $E(\mathbb{C}{)}_{\text{tors}}$. This is work done as part of the Pomona Research in Mathematics Experience (NSA H98230-21-1-0015).