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

Faber, X.W.C.1
1 Department of Mathematics and Statistics, McGill University, Montréal, Qc H3A 2K6, Canada
Comptes Rendus. Mathématique, Tome 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
Faber, X.W.C. 1

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

[1] Bombieri, Enrico The Mordell conjecture revisited, Ann. Sc. Norm. Sup. Pisa Cl. Sci. (4), Volume 17 (1990) no. 4, pp. 615-640

[2] Call, Gregory S.; Silverman, Joseph H. Canonical heights on varieties with morphisms, Compos. Math., Volume 89 (1993) no. 2, pp. 163-205

[3] de Diego, Teresa Théorème de Faltings (conjecture de Mordell) pour les familles algébriques de courbes, C. R. Acad. Sci. Paris Sér. I Math., Volume 323 (1996) no. 2, pp. 175-178

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

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

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

[7] Hindry, Marc; Silverman, Joseph H. Diophantine Geometry, Graduate Texts in Mathematics, vol. 201, Springer-Verlag, New York, 2000 (An introduction)

[8] Vojta, Paul Siegel's theorem in the compact case, Ann. of Math. (2), Volume 133 (1991) no. 3, pp. 509-548

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