Vol. 8, No. 1, 2014

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ISSN: 1944-7833 (e-only)
ISSN: 1937-0652 (print)
Polynomial bounds for Arakelov invariants of Belyi curves

Ariyan Javanpeykar

Appendix: Peter Bruin

Vol. 8 (2014), No. 1, 89–140
Abstract

We explicitly bound the Faltings height of a curve over ¯ 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  SL2() 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 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