Computation and rationality of Belyi functions in genus zero. (Calcul et rationalité de fonctions de Belyi en genre 0.) (French) Zbl 0791.11059

Summary: Belyi functions are algebraic functions with only three ramification values. In the genus zero case, we present some practical methods for computing such functions. We also prove that, unless the automorphism group of such a function is cyclic of even order, its moduli field is also a definition field, and we give explicit constructions for such a model. We give some example of a covering ramified over 3 points which has no model over its field of moduli.
All that leads us to looking for rational points in genus zero extensions.


11R32 Galois theory
11Y40 Algebraic number theory computations
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