Generalized Lam\'e equation with finite monodromy

TL;DR

This paper classifies all finite monodromy groups of generalized Lamé equations on tori, providing explicit parameters and using dessins d'enfants to systematically construct solutions with finite monodromy.

Contribution

It offers a complete classification of finite projective monodromy groups for generalized Lamé equations, including explicit parameters and a dessin-based construction method.

Findings

List of all finite monodromy group types and parameters.

Complete classification for equations with 1 or 2 singular points.

A systematic dessin construction and gluing procedure.

Abstract

In this paper, we study the algebraic form of the symmetric generalized Lam\'e equations which have finite projective monodromy groups. In particular, we consider equations with $3$ regular singular points on a flat torus $T$ which takes the form \begin{equation*} \begin{split} \frac{d^2 y}{dz^2}-\left[n_1(n_1+1)(\wp(z+a) +\wp(z-a))\right.\left. +A_1(\zeta(z+a) - \zeta(z-a))+n_0(n_0+1) \wp(z)+B\right]y=0, \end{split} \end{equation*} where $n_1, n_0 \in \Bbb R$, $A_1,B \in \Bbb C$, and $\wp$ is the Weierstrass elliptic function. We give a complete list of all the group types that occur as the finite projective monodromy groups on the algebraic form and give the corresponding parameters $n_0$ and $n_1$. For equations with only $1$ or $2$ regular singular points, we further determine their monodromy group types. The main tool used is the Grothendieck correspondence which gives a bijection between Belyi pairs and dessin d'enfants. By Klein's theorem, we may regard generalized Lam\'e equations with finite monodromy as pullbacks of the hypergeometric equations. In this paper, we will restrict our cases with the assumption that the pullback maps are Belyi functions. Under this setup, our main results consist of a systematic construction on the required dessin. In particular a gluing procedure of dessin will be developed to enable inductive constructions of the dessin.