Polynomial bounds for Arakelov invariants of Belyi curves

Abstract

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.

arXiv comment44 pages. Appendix by Peter Bruin. To be published in Algebra and Number Theory

References in corpus (5)

• Stable reduction of finite covers of curves
• On the arithmetic self-intersection numbers of the dualizing sheaf for Fermat curves of prime exponent
• Algebraic curves uniformized by congruence subgroups of triangle groups
• On the arithmetic self-intersection number of the dualizing sheaf on arithmetic surfaces
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Cited by in corpus (12)

• New explicit formulas for Faltings' delta-invariant
• Numerical calculation of three-point branched covers of the projective line
• Modularity and integral points on moduli schemes
• On computing Belyi maps
• Heights, ranks and regulators of abelian varieties
• Integral points on Hilbert moduli schemes
• Semi-stable models of modular Curves $X_0(p^2)$ and some arithmetic applications
• An effective Arakelov-theoretic version of the hyperbolic isogeny theorem
• Bounds for Green's functions on noncompact hyperbolic Riemann orbisurfaces of finite volume
• The minimal regular model of a Fermat curve of odd squarefree exponent and its dualizing sheaf
• The Belyi degree is computable
• An effective version of Belyi's theorem in positive characteristic