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 joint work with Tesfa Asmara (Pomona College), Erik Imathiu-Jones (California Institute of Technology), Maria Maalouf (California State University at Long Beach), Isaac Robinson (Harvard University), and Sharon Sneha Spaulding (University of Connecticut).  This was work done as part of the Pomona Research in Mathematics Experience (NSA H98230-21-1-0015).