Quasi-Critical Points of Toroidal Belyǐ Maps

A Belyǐ map β : P¹(C) → P¹(C) is a rational function with at most three critical values; we may assume these values are {0,1,∞}.  Replacing P¹ with an elliptic curve E: y² = x³ + Ax + B, there is a similar definition of a Belyǐ map β : E(C) → P¹(C).  Since E(C) ≃ T²(R) is a torus, we call (E, β) a Toroidal Belyǐ pair.

There are many examples of Belyǐ maps β : E(C) → P1(C) associated to elliptic curves; several can be found online at LMFDB. Given such a Toroidal Belyǐ map of degree N, the inverse image G = β^-1({0,1, ∞}) is a set of N elements which contains the critical points of the Belyǐ map. In this project, we investigate when G is contained in E(C)_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).