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Computation of highly ramified coverings

Authors: Raimundas Vidunas and Alexander V. Kitaev
Journal: Math. Comp. 78 (2009), 2371-2395
MSC (2000): Primary 57M12, 34M55; Secondary 33E17
Published electronically: February 11, 2009
MathSciNet review: 2521293
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Abstract: An almost Belyi covering is an algebraic covering of the projective line, such that all ramified points except one simple ramified point lie above a set of 3 points of the projective line. In general, there are 1-dimensional families of these coverings with a fixed ramification pattern. (That is, Hurwitz spaces for these coverings are curves.) In this paper, three almost Belyi coverings of degrees 11, 12, and 20 are explicitly constructed. We demonstrate how these coverings can be used for computation of several algebraic solutions of the sixth Painlevé equation.

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Additional Information

Raimundas Vidunas
Affiliation: Department of Mathematics, Kyushu University, Fukuoka 812-8581, Japan
Address at time of publication: Department of Mathematics, Kobe University, Rokko-dai 1-1, Nada-ku, Kobe 657-8501, Japan

Alexander V. Kitaev
Affiliation: School of Mathematics and Statistics, University of Sydney, NSW 2006, Australia
Address at time of publication: Steklov Mathematical Institute, Fontanka 27, St. Petersburg 191023, Russia

Keywords: Belyi map, dessin d'enfant, the Painlev\'e VI equation.
Received by editor(s): June 21, 2007
Received by editor(s) in revised form: October 16, 2008
Published electronically: February 11, 2009
Additional Notes: The first author was supported by the 21st Century COE Programme “Development of Dynamic Mathematics with High Functionality” of the Ministry of Education, Culture, Sports, Science and Technology of Japan.
The second author was supported by JSPS grant-in-aide No. $14204012$.
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.