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We explicitly bound the Faltings height of a curve over Q polynomially in its Belyi degree. Similar bounds are proven for three other Arakelov invariants: the discriminant, Faltings' delta invariant and the self-intersection of the dualizing sheaf. Our results allow us to explicitly bound Arakelov invariants of modular curves, Hurwitz curves and Fermat curves in terms of their genus. Moreover, as an application, we show that the Couveignes-Edixhoven-Bruin algorithm to compute coefficients of modular forms for congruence subgroups of SL2(Z) runs in polynomial time under the Riemann hypothesis for zeta-functions of number fields. This was known before only for certain congruence subgroups. Finally, we use our results to prove a conjecture of Edixhoven, de Jong and Schepers on the Faltings height of a cover of the projective line with fixed branch locus. Our proof uses Merkl's method for bounding Arakelov-Green's functions to deal with archimedean contributions. In the appendix, Peter Bruin proves an explicit version of Merkl's results., 44 pages. Appendix by Peter Bruin. To be published in Algebra and Number Theory

We explicitly bound the Faltings height of a curve over the field of algebraic numbers in terms of the Belyi degree. Similar bounds are proven for three other Arakelov invariants: the discriminant, Faltings' delta invariant and the self-intersection of the dualizing sheaf. Our results allow us to explicitly bound these Arakelov invariants for modular curves, Hurwitz curves and Fermat curves. Moreover, as an application, we show that the Couveignes-Edixhoven-Bruin algorithmtime under the Riemann hypothesis for zeta-functions of number fields. This was known before only for certain congruence subgroups. Finally, we utilize our results to prove a conjecture of Edixhoven, de Jong and Schepers on the Faltings height of a branched cover of the projective line over the ring of integers.; On borne explicitement la hauteur de Faltings d'une courbe sur le corps de nombres algèbriques en son degré de Belyi. Des résultats similaires sont démontré pour trois autres invariants arakeloviennes : le discriminant, l'invariant delta et l'auto-intersection de omega. Nos résultats nous permettent de borner explicitement les invariantes arakeloviennes des courbes modulaires, des courbes de Fermat et des courbes de Hurwitz. En plus, comme application, on montre que l'algorithme de Couveignes-Edixhoven-Bruin est polynomial sous l’hypothèse de Riemann pour les fonctions zeta des corps de nombres. Ceci était connu uniquement pour certains sous-groupes de congruence. Finalement, on utilise nos résultats pour démontrer une conjecture de Edixhoven, de Jong et Schepers sur la hauteur de Faltings d'un revêtement ramifié de la droite projective sur l'anneau des entiers.

We bound Arakelov invariants of curves in terms of their Belyi degree. We give three applications which are algorithmic, geometric and Diophantine of nature, respectively