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Counting integral Lamé equations by means of dessins d'enfants

Author(s): Sander R. Dahmen
Journal: Trans. Amer. Math. Soc. 359 (2007), 909-922.
MSC (2000): Primary 34L40, 34M15; Secondary 11F11, 14H30
Posted: September 12, 2006
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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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Additional Information:

Sander R. Dahmen
Affiliation: Department of Mathematics, Utrecht University, Budapestlaan 6, 3584 CD Utrecht, The Netherlands
Email: dahmen@math.uu.nl

DOI: 10.1090/S0002-9947-06-03924-9
PII: S 0002-9947(06)03924-9
Keywords: Lam\'{e} equation, algebraic solution, monodromy, dessin d'enfants
Received by editor(s): June 25, 2004
Received by editor(s) in revised form: January 21, 2005
Posted: September 12, 2006
Copyright of article: Copyright 2006, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

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