Characterization and enumeration on Lam\'e equations with finite monodromy

We give a complete characterization of the classical Lam\'e equations y'' = (n(n + 1)(z) + B)y, n ∈ R, B ∈ C on flat tori E_τ = C/( Z + Z\,τ) with finite monodromy groups M. Beuker--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.