Couveignes, Jean-Marc Computation and rationality of Belyi functions in genus zero. (Calcul et rationalité de fonctions de Belyi en genre 0.) (French) Zbl 0791.11059 Ann. Inst. Fourier 44, No. 1, 1-38 (1994). 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. Cited in 1 ReviewCited in 23 Documents MSC: 11R32 Galois theory 11Y40 Algebraic number theory computations Keywords:coverings; Belyi functions; algebraic functions; rational points in genus zero extensions × Cite Format Result Cite Review PDF Full Text: DOI Numdam EuDML References: [1] [Arms] , Groups and Symmetry, U.T.M. Springer Verlag (1988). · Zbl 0663.20001 [2] [Bely] , On Galois extensions of the maximal cyclotomic field, Izvestiya Ak. Nauk. SSSR, ser. mat., 43-2 (1979), 269-276. · Zbl 0409.12012 [3] [Birc] , Arithmetic of noncongruence subgroups, non publié. · Zbl 0930.11024 [4] [Grot] , Esquisse d’un programme, non publié. · Zbl 0901.14001 [5] [Joux] , Thèse sur les applications de LLL en cryptographie et théorie de la complexité. Sous la direction de Jacques Stern, à paraître. [6] [LeLeLo] , . et , Factoring Polynomials with rational coefficients, Math. Ann., 261 (1982), 515-534. · Zbl 0488.12001 [7] [8] , Gauge field theory and complex geometry, Grundlehren der Math. Wissenschaften, 289, Springer (1988 · Zbl 0823.14017 [8] [9] , Automorphisms of simple Lie superalgebras, Math. USSR Izvestiya, 24 (1985 · Zbl 0137.02601 [9] [ShVo] et , Drawing curves over number fields. Papers in honour of A. Grothendieck. The Grothendieck Festschrift, Birkhauser, 1990, 199-229. · Zbl 0790.14026 [10] A. Weil, The field of definition of a variety, Amer. J. Math., 78 (1956), 509-524.0072.1600118,601a · Zbl 0072.16001 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.