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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Counting integral Lamé equations by means of dessins d'enfants


Author: Sander R. Dahmen
Journal: Trans. Amer. Math. Soc. 359 (2007), 909-922
MSC (2000): Primary 34L40, 34M15; Secondary 11F11, 14H30
Published electronically: September 12, 2006
MathSciNet review: 2255201
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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: http://dx.doi.org/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
Published electronically: September 12, 2006
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.