Critical Points of Toroidal Bely\u{\i} Maps

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 $E: \ y^2 = x^3 + A \, x + B$, 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 $G = \beta^{-1} \bigl( \{ 0, \, 1, \, \infty \} \bigr)$ 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).