On computing Belyi maps. (English) Zbl 1364.11127

Sijsling, J.; Voight, J.

Numéro consacré au trimestre “Méthodes arithmétiques et applications”, automne 2013. Besançon: Presses Universitaires de Franche-Comté. Publications Mathématiques de Besançon. Algèbre et Théorie des Nombres 2014, 1, 73-131 (2014).

Summary: We survey methods to compute three-point branched covers of the projective line, also known as Belyi 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.
For the entire collection see [Zbl 1301.11002].

MSC:

11G32	Arithmetic aspects of dessins d’enfants, Belyĭ theory
11Y40	Algebraic number theory computations
14H57	Dessins d’enfants theory

Keywords:

Belyi maps; dessins d’enfants; covers; uniformization; computational algebra