Vol. 5, No. 8, 2011

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ISSN: 1944-7833 (e-only)
ISSN: 1937-0652 (print)
Conjecture de Shafarevitch effective pour les revêtements cycliques

Robin de Jong and Gaël Rémond

Vol. 5 (2011), No. 8, 1133–1143
Abstract

On donne une borne supérieure explicite en fonction de K, S, g pour la hauteur de Faltings de la jacobienne d’une courbe C de genre g, définie sur un corps de nombres K et ayant bonne réduction en dehors d’un ensemble fini S de places de K, pourvu que C puisse s’écrire comme un revêtement cyclique de degré premier de la droite projective. La preuve repose sur le fait que les birapports des points de branchement du revêtement sont des S-unités, donc de hauteur bornée, et donnent un modèle plan de C.

We give an explicit upper bound in terms of K, S, g for the Faltings height of the jacobian of a curve C of genus g, defined over a number field K and with good reduction outside a finite set S of places of K under the condition that C can be written as a cyclic cover of prime order of the projective line. The proof rests on the fact that the cross ratios of the branch points of the cover are S-units, thus of bounded height, and give a plane model of C.

Keywords
conjecture de Shafarevitch, courbe, revêtement, hauteur, réduction, birapport, Shafarevich conjecture, curve, cover, height, reduction, cross ratio
Mathematical Subject Classification 2010
Primary: 11G30
Milestones
Received: 6 January 2011
Revised: 7 March 2011
Accepted: 7 March 2011
Published: 5 June 2012
Authors
Robin de Jong
Mathematisch Instituut
Universiteit Leiden
PO Box 9512
2300 RA Leiden
Netherlands
Gaël Rémond
Institut Fourier, UMR 5582
Université Grenoble I
BP 74
38402 Saint-Martin-d’Hères Cedex
France