2022/12/21 by Tesfa Asmara, Asmara, Tesfa, Edray Herber Goins +9
Mathematics · #11G32 #14H52 #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Mathematical Dynamics and Fractals
paper · pdf · doi:10.48550/arxiv.2212.11373
openalex publication_date 2022/12/21 · openalex created_date 2023/01/04 · openalex updated_date 2026/07/28
A Belyi 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 + A x + B, there is a similar definition of a Belyi map β: E(ℂ) → ℙ1(ℂ). Since E(ℂ) ≃ \mathbb T2(\mathbb R) is a torus, we call (E, β) a Toroidal Belyi pair. There are many examples of Belyi maps β: E(ℂ) → \mathbb P1(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 = β-1 ( \ 0, 1, ∞ \ ) 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(ℂ)tors. This is work done as part of the Pomona Research in Mathematics Experience (NSA H98230-21-1-0015).