Characterization and enumeration on Lamé equations with finite monodromy

Abstract

We give a complete characterization of the classical Lamé equations $y'' = (n(n + 1)\wp(z) + B)y$, $n \in \Bbb R$, $B \in \Bbb C$ on flat tori $E_τ= \Bbb C/(\Bbb Z + \Bbb Z\,τ)$ with finite monodromy groups $M$. Beuker--Waall had shown that such $n$ must lie in a finite number of arithmetic progressions $n_i + \Bbb N \subset \Bbb Q$ 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 $n \not\in \tfrac{1}{2} + \Bbb Z$, we prove the finiteness of $(B, τ)$ and derive an explicit counting formula of them. (The case $n \in \tfrac{1}{2} + \Bbb Z$ 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 $n \mapsto n + 1$ which reduces the problem to the basic case.

Figures (12)

• Figure 1: A demonstration of the decomposition of a balanced spherical triangle into hemispheres and a basic triangle. All the unshaded regions in the figure are hemispheres, and the shaded region is the basic spherical triangle.
• Figure 2: Demonstration of all possible positions of the circumcenter (black) and its corresponding basic triangle, given three vertices (red) not lying on a great circle.
• Figure 4: A fundamental domain of $(T,x)$ with $n=1$. The image of the red path under the map $f$ is actually a smooth arc of length $\ell_2+\ell_3$ on the unit circle.
• Figure 5: The number of Lamé equations with unitary monodromy and with given parameter $2s, 2t \pmod 1$ for $n\in\mathbb Z$.
• Figure 6: The number of Lamé equations with unitary monodromy with given parameters $s, t \pmod 1$, and with odd $n = 2l - 1$ (left) or even $n = 2l$ (right).

Theorems & Definitions (41)

• Theorem 1.1: Klein_1877Baldassarri_Dwork_1979Baldassarri_1980
• Theorem 1.2: Brioschi–Halphen–Crawford Crawford_1895
• Theorem 1.3: Dahmen_2007_Dessin
• Theorem 1.4: Chen_Kuo_Lin_2021Wu_2022
• Theorem 1.5: Beukers--Waall Beukers_Waall_2004
• Remark 1.6
• Definition 2.1
• Definition 2.2
• Proposition 2.3
• proof