Counting Integral Lamé Equations by Means of Dessins d'Enfants

Abstract

We obtain an explicit formula for the number of Lamé equations (modulo scalar equivalence) with index $n$ and projective monodromy group of order $2N$, for given $n \in \Z$ and $N \in \N$. This is done by performing the combinatorics of the `dessins d'enfants' associated to the Belyi covers which transform hypergeometric equations into Lamé equations by pull-back.