Critical Points of Toroidal Bely\u{\i} Maps
Tesfa Asmara, Edray Herber Goins, Erik Imathiu-Jones, Maria Maalouf,, Isaac Robinson, and Sharon Sneha Spaulding

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 is a rational function with at most three critical values; we may assume these values are . Replacing with an elliptic curve , there is a similar definition of a Belyi map . Since is a torus, we call a Toroidal Belyi pair. There are many examples of Belyi maps associated to elliptic curves; several can be found online at LMFDB. Given such a Toroidal Belyi map of degree , the inverse image is a set of elements which contains the critical points of the Belyi map. In this project, we investigate when is contained in…
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Taxonomy
TopicsAlgebraic Geometry and Number Theory · Mathematical Dynamics and Fractals · Advanced Differential Equations and Dynamical Systems
