Generalized Lamé equation with finite monodromy

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.