Properties of the degree of Belyi functions. (Proprieétés du degré des morphismes de Belyi.)
By a famous theorem of Belyi, a complex projective algebraic curve \(C\) can be defined over a number field if and only if and only if there exists a finite morphism \(\beta : C \to {\mathbb P}^1\) non-ramified outside \(\{0,1,\infty\}\). Assume that \(C\) is defined over a number field and call \(\beta: C \to {\mathbb P}^1\) a Belyi morphism for \(C\) if it satisfies the preceding assumptions. One defines the Belyi degree \(B(C)\) of the curve \(C\) as the minimum of the degrees of the Belyi morphisms for \(C\). Similarly, if \(a\in C\) is a point, one defines the Belyi degree \(B(a)\) of \(a\) as the minimum of the degrees of the Belyi morphisms \(\beta\) for \(C\) such that \(a\in \beta^{-1}(\{0,1,\infty\})\). Following an idea of Bogomolov and Szpiro, the paper uses the Belyi degrees to start the construction of a theory of heights on the moduli spaces of curves. The first theorem states that, for any real number \(M>1\), the number of isomorphism classes of curves \(C\) with \(B(C)\leq M\) is finite. The bound for this number is explicit and is given in terms of the combinatorial properties of the ``dessins d'enfant'' associated to the Belyi morphisms via the Grothendieck correspondence. A second result states that the set of points \(a\in C\) such \(B(a)\leq M\) is finite and can be effectively bounded. For the case \(C={\mathbb P}^1\), an explicit bound is given. Finally, a bound for the Belyi degree of an algebraic point of \({\mathbb P}^1\) in terms of its degree and its normalized height is given.
scientific article; zbMATH DE number 2176491