2024/02/26 by You-Cheng Chou, Chou, You-Cheng, Chin-Lung Wang +3
Mathematics · #Advanced Differential Equations and Dynamical Systems #Advanced Topics in Algebra #Algebraic structures and combinatorial models #Differential Geometry (math.DG) #FOS: Mathematics
paper · pdf · doi:10.48550/arxiv.2402.16286
openalex publication_date 2024/02/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We give a complete characterization of the classical Lamé equations y'' = (n(n + 1)\wp(z) + B)y, n ∈ \Bbb R, B ∈ \Bbb C on flat tori Eτ= \Bbb C/(\Bbb Z + \Bbb Z τ) with finite monodromy groups M. Beuker--Waall had shown that such n must lie in a finite number of arithmetic progressions ni + \Bbb N ⊂ \Bbb Q and they determined all corresponding M. By combining the theory of dessin d'enfants with the geometry of spherical tori, we prove the existence of (B, τ) for each such n and provide a description of all such (n, B, τ, M). In particular, for a given (n, M) with n \not∈ \tfrac12 + \Bbb Z, we prove the finiteness of (B, τ) and derive an explicit counting formula of them. (The case n ∈ \tfrac12 + \Bbb Z is a classical result due to Brioschi--Halphen--Crawford.) The main ingredients in this work are (1) the definition and classification of basic spherical triangles with finite monodromy and (2) the process of attaching cells corresponding to n ↦ n + 1 which reduces the problem to the basic case.