Polynomial bounds for Arakelov invariants of Belyi curves

Javanpeykar, Ariyan

doi:10.2140/ant.2014.8.89

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Polynomial bounds for Arakelov invariants of Belyi curves

Ariyan Javanpeykar

Appendix: Peter Bruin

Vol. 8 (2014), No. 1, 89–140

DOI: 10.2140/ant.2014.8.89

Abstract

We explicitly bound the Faltings height of a curve over $\bar{ℚ}$ 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 ${SL}_{2} (ℤ)$ runs in polynomial time under the Riemann hypothesis for $ζ$ -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 $ℙ_{ℤ}^{1}$ with fixed branch locus.

Keywords

Arakelov theory, Arakelov–Green functions, Wronskian differential, Belyi degree, arithmetic surfaces, Riemann surfaces, curves, Arakelov invariants, Faltings height, discriminant, Faltings' delta invariant, self-intersection of the dualising sheaf, branched covers

Mathematical Subject Classification 2010

Primary: 14G40

Secondary: 11G30, 11G32, 11G50, 14H55, 37P30

Milestones

Received: 22 June 2012

Revised: 27 February 2013

Accepted: 17 April 2013

Published: 20 April 2014

Authors

Ariyan Javanpeykar
Mathematisch Instituut Universiteit Leiden 2717 GA Leiden Netherlands

Peter Bruin
Institut für Mathematik Universität Zürich Winterthurerstrasse 190 CH-8057 Zürich Switzerland

Vol. 8, No. 1, 2014