Counting integral Lamé equations by means of dessins d’enfants

Dahmen, Sander

doi:10.1090/S0002-9947-06-03924-9

Counting integral Lamé equations by means of dessins d’enfants
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by Sander R. Dahmen PDF

Trans. Amer. Math. Soc. 359 (2007), 909-922 Request permission

Abstract:

We obtain an explicit formula for the number of Lamé equations (modulo linear changes of variable) with index $n$ and projective monodromy group of order $2N$, for given $n \in \mathbb {Z}$ and $N \in \mathbb {N}$. 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.

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