CSUN Algebra, Number Theory, and Discrete Mathematics Seminar

A Belyĭ map $\beta :{\mathbb{P}}^1(ℂ)\to {\mathbb{P}}^1(ℂ)$ is a rational function with at most three critical values; we may assume these values are $\{0,\mathrm{\hspace{0.17em}1},\mathrm{\infty }\}$. Replacing ${\mathbb{P}}^1$ with an elliptic curve $E:y^2=x^3+Ax+B$, there is a similar definition of a Belyĭ map $\beta :E(ℂ)\to {\mathbb{P}}^1(ℂ)$. Since $E(ℂ)\simeq {𝕋}^2(ℝ)$ is a torus, we call $(E,\beta )$ a Toroidal Belyĭ pair.

There are many examples of Belyĭ maps $\beta :E(ℂ)\to {\mathbb{P}}^1(ℂ)$ associated to elliptic curves; several can be found online at LMFDB. Given such a Toroidal Belyĭ map of degree $N$, the inverse image $G={\beta }^{-1}\left(\{0,\mathrm{\hspace{0.17em}1},\mathrm{\infty }\}\right)$ is a set of $N$ elements which contains the critical points of the Belyĭ map. In this project, we investigate when $G$ is contained in $E{(ℂ)}_{\text{tors}}$.

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).