Characterization and enumeration on Lamé equations with finite monodromy
Journal
Transactions of the American Mathematical Society
Journal Volume
378
Journal Issue
8
Start Page
5279
End Page
5304
ISSN
1088-6850
0002-9947
Date Issued
2025-05-29
Author(s)
Abstract
We give a complete characterization of the classical Lamé equations y’’ = (n(n + 1)℘(z) + B)y, n ∈ R, B ∈ C on flat tori Eτ = C/(Z + Z τ) with finite monodromy groups M. Beukers–Waall had shown that such n must lie in a finite number of arithmetic progressions ni+ N ⊂ 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 ∉ 1/2 + Z, we prove the finiteness of (B, τ) and derive an explicit counting formula of them. (The case n ∈ 1/2 + 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.
SDGs
Publisher
American Mathematical Society (AMS)
Type
journal article