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A remark on the effective Mordell conjecture and rational pre-images under quadratic dynamical systems. (English. French summary) Zbl 1264.37053
Summary: Fix a rational base point \(b\) and a rational number \(c\). For the quadratic dynamical system \(f_c(x)=x^{2}+c\), 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 base point \(b\), assuming a strong form of the Mordell conjecture.

MSC:
37P05 Arithmetic and non-Archimedean dynamical systems involving polynomial and rational maps
37P30 Height functions; Green functions; invariant measures in arithmetic and non-Archimedean dynamical systems
11G50 Heights
14G05 Rational points
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[1] Bombieri, Enrico, The Mordell conjecture revisited, Ann. sc. norm. sup. Pisa cl. sci. (4), 17, 4, 615-640, (1990) · Zbl 0722.14010
[2] Call, Gregory S.; Silverman, Joseph H., Canonical heights on varieties with morphisms, Compos. math., 89, 2, 163-205, (1993) · Zbl 0826.14015
[3] de Diego, Teresa, Théorème de faltings (conjecture de Mordell) pour LES familles algébriques de courbes, C. R. acad. sci. Paris Sér. I math., 323, 2, 175-178, (1996) · Zbl 0858.11033
[4] Faber, Xander; Hutz, Benjamin, On the number of rational iterated pre-images of the origin under quadratic dynamical systems, (2008), preprint · Zbl 1242.14019
[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., 16, 1, 87-101, (2009) · Zbl 1222.11086
[6] Faltings, Gerd, Endlichkeitssätze für abelsche varietäten über zahlkörpern, Invent. math., 73, 3, 349-366, (1983) · Zbl 0588.14026
[7] Hindry, Marc; Silverman, Joseph H., Diophantine geometry, Graduate texts in mathematics, vol. 201, (2000), Springer-Verlag New York, (An introduction) · Zbl 0948.11023
[8] Vojta, Paul, Siegel’s theorem in the compact case, Ann. of math. (2), 133, 3, 509-548, (1991) · Zbl 0774.14019
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