Polynomial bounds for Arakelov invariants of Belyi curves

Abstract

We explicitly bound the Faltings height of a curve over $\overline{\mathbb{Q}}$ 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 dualising sheaf. Our results allow us to explicitly bound these Arakelov invariants for 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 ${SL}_2\left(\mathbb{Z}\right)$ 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 ${\mathbb{P}}_{\mathbb{Z}}^1$ with fixed branch locus.