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Transactions of the American Mathematical Society
ISSN 1088-6850(e) ISSN 0002-9947(p)

     

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
MathSciNet review: 2255201
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Abstract | References | Similar articles | Additional information

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.

References:

[Bal]
F. Baldassarri, On Algebraic Solutions of Lamé's Differential Equation, J. Differential Equations, 41:44-58, 1981. MR 0626620 (84a:34006)

[BD]
F. Baldassarri and B. Dwork, On Second Order Differential Equations with Algebraic Solutions, Amer. J. Math., 101:42-76, 1979.MR 0527825 (81d:34002)

[BW]
Frits Beukers and Alexa van der Waall, Lamé Equations with Algebraic Solutions, J. Differential Equations, 197:1-25, 2004.MR 2030146 (2004j:34202)

[Chi]
F. Chiarellotto, On Lamé Operators which are Pull-Backs of Hypergeometric Ones, Trans. Amer. Math. Soc., 347(8):2753-2780, 1995.MR 1308004 (96e:34008)

[Cop]
F. Coppi, Tesi di Laurea, Universitá di Padova, 1992.

[Hec]
E. Hecke, Zur Theorie der Elliptischen Modulfunktionen, Math. Ann., 97:210-242, 1926. MR 1512360

[Lit1]
R. Litcanu, Counting Lamé Differential Operators, Rend. Sem. Mat. Univ. Padova, 107:191-208, 2002. MR 1926211 (2003h:34186)

[Lit2]
-, Lamé Operators with Finite Monodromy, to appear.

[LZ]
S. G. Lando and A. K. Zvonkin, Graphs on Surfaces and their Applications, Encyclopaedia of Mathematical Sciences, Vol. 141, Springer, Berlin, 2004. MR 2036721 (2005b:14068)

[Poo]
E. G. C. Poole, Introduction to the Theory of Linear Differential Equations, Oxford Univ. Press, London, 1936. MR 0111886 (22:2746)

[vdW]
Alexa van der Waall, Lamé Equations with Finite Monodromy, Universiteit Utrecht, Utrecht, 2002, Thesis. On-line reference: http://www.library.uu.nl/digiarchief/dip/diss/ 2002-0530-113355/inhoud.htm.

[WW]
E. T. Whittaker and G. N. Watson, A Course of Modern Analysis, Cambridge Univ. Press, Cambridge, 1950, Reprint of the 1927 fourth edition.MR 1424469 (97k:01072)

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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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