Comptes Rendus
no. 7-8
Number Theory/Dynamical Systems
A remark on the effective Mordell Conjecture and rational pre-images under quadratic dynamical systems
[Une remarque sur la Conjecture de Mordell effective et les pré-images rationnelles par les systèmes dynamiques quadratiques]

Présenté par : Christophe Soulé

X.W.C. Faber1
1 Department of Mathematics and Statistics, McGill University, Montréal, Qc H3A 2K6, Canada
Comptes Rendus. Mathématique, Volume 348 (2010) no. 7-8, pp. 355-358.

Soit b un point cible rationnel, et soit c un nombre rationnel. Pour le système dynamique quadratique fc(x)=x2+c, il a été montré que le nombre de points rationnels dans l'orbite inverse de b est borné indépendamment du choix du paramètre rationnel c. Dans cette courte Note, nous étudions la dépendance en le point cible b, en supposant une forme forte de la Conjecture de Mordell.

Fix a rational basepoint b and a rational number c. For the quadratic dynamical system fc(x)=x2+c, it has been shown that the number of rational points in the backward orbit of b is bounded independent of the choice of rational parameter c. In this short Note we investigate the dependence of the bound on the basepoint b, assuming a strong form of the Mordell Conjecture.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2010.02.010
X.W.C. Faber 1

1 Department of Mathematics and Statistics, McGill University, Montréal, Qc H3A 2K6, Canada 
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X.W.C. Faber. A remark on the effective Mordell Conjecture and rational pre-images under quadratic dynamical systems. Comptes Rendus. Mathématique, Volume 348 (2010) no. 7-8, pp. 355-358. doi : 10.1016/j.crma.2010.02.010. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2010.02.010/

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[4] Xander Faber; Benjamin Hutz On the number of rational iterated pre-images of the origin under quadratic dynamical systems, 2008 (preprint) | arXiv

[5] Xander Faber; Benjamin Hutz; Patrick Ingram; Rafe Jones; Michelle Manes; Thomas J. Tucker; Michael E. Zieve Uniform bounds on pre-images under quadratic dynamical systems, Math. Res. Lett., Volume 16 (2009) no. 1, pp. 87-101

[6] Gerd Faltings Endlichkeitssätze für abelsche Varietäten über Zahlkörpern, Invent. Math., Volume 73 (1983) no. 3, pp. 349-366

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