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

TL;DR

This paper explores the critical points of Belyi maps on elliptic curves, called Toroidal Belyi pairs, focusing on when these points are torsion points on the elliptic curve.

Contribution

It investigates conditions under which the critical points of Toroidal Belyi maps are contained within the torsion subgroup of the elliptic curve.

Findings

Identifies criteria for critical points to be torsion points

Provides examples from the LMFDB database

Contributes to understanding of Belyi maps on elliptic curves

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 $E: \ y^2 = x^3 + A \, x + B$, 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).