Counting Integral Lamé Equations by Means of Dessins d'Enfants

Details Counting Integral Lamé Equations by Means of Dessins d'Enfants Counting Integral Lamé Equations by Means of Dessins d'Enfants

2003-11-27

arxiv.org/abs/math/0311510v1

Mathematics

Classical Analysis and ODEs

with 9 figures

Scientific paper

We obtain an explicit formula for the number of Lam\'e 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 Lam\'e equations by pull-back.

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