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Counting Integral Lamé Equations by Means of Dessins d'Enfants

2003/11/27 by Sander R. Dahmen, Sander Dahmen, Dahmen, Sander
Mathematics · #12H20 #34L40 #34M15 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Functional Equations Stability Results #History and Theory of Mathematics #math.CA #msc:12H20 #msc:34L40 #msc:34M15

paper · pdf · doi:10.48550/arxiv.math/0311510

with 9 figures

arxiv created 2003/11/27 · openalex publication_date 2003/11/27 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 ∈ \Z and N ∈ \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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