On computing Belyi maps
Publications mathématiques de Besançon. Algèbre et théorie des nombres, no. 1 (2014), pp. 73-131.

We survey methods to compute three-point branched covers of the projective line, also known as Belyĭ maps. These methods include a direct approach, involving the solution of a system of polynomial equations, as well as complex analytic methods, modular forms methods, and p-adic methods. Along the way, we pose several questions and provide numerous examples.

Nous donnons un aperçu des méthodes actuelles pour le calcul des revêtements de la droite projective ramifiés en au plus trois points, connus sous le nom de morphismes de Belyĭ. Ces méthodes comprennent une approche directe, se ramenant à la solution d’un système d’équations polynomiales ainsi que des méthodes analytiques complexes, de formes modulaires et p-adiques. Ce faisant, nous posons quelques questions et donnons de nombreux exemples.

Received:
Published online:
DOI: 10.5802/pmb.5
Classification: 11G32,  11Y40
Keywords: Belyi maps, dessins d’enfants, covers, uniformization, computational algebra
Sijsling, J. 1; Voight, J. 2

1 Mathematics Institute, Zeeman Building, University of Warwick, Coventry CV4 7AL, UK
2 Department of Mathematics and Statistics, University of Vermont, 16 Colchester Ave, Burlington, VT 05401, USA; Department of Mathematics, Dartmouth College, 6188 Kemeny Hall, Hanover, NH 03755, USA
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Sijsling, J.; Voight, J. On computing Belyi maps. Publications mathématiques de Besançon. Algèbre et théorie des nombres, no. 1 (2014), pp. 73-131. doi : 10.5802/pmb.5. http://www.numdam.org/articles/10.5802/pmb.5/

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