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On Dependence of Rational Points on Elliptic Curves

C. R. Math. Rep. Acad. Sci. Canada Vol. 38 (2) 2016, pp. 75-84
Vol.38 (2) 2016
Mohammad Sadek Details
(Received: 2015-04-15 , Revised: 2015-08-18 )
(Received: 2015-04-15 , Revised: 2015-08-18 )

Mohammad Sadek,Department of Mathematics and Actuarial Science, American University in Cairo, Cairo, Egypt; e-mail: mmsadek@aucegypt.edu

Abstract/Résumé:

Let \(E\) be an elliptic curve defined over \(Q\). Let \(\Gamma\) be a subgroup of \(E(Q)\) and \(P\in E(Q)\). In \cite{Arithmetic}, it was proved that if \(E\) has no nontrivial rational torsion points, then \(P\in\Gamma\) if and only if \(P\in \Gamma\) mod \(p\) for finitely many primes \(p\). In this note, assuming the General Riemann Hypothesis, we provide an explicit upper bound on these primes when \(E\) does not have complex multiplication and either \(E\) is a semistable curve or \(E\) has no exceptional prime.

Soit \(E\) une courbe elliptique définie sur \(Q\). Soit \( \Gamma\) un sous-groupe de \( E(Q) \) et \( P \in E (Q) \). Dans \cite{Arithmetic}, il on a prouvé que si \( E \) n’a pas de points de torsion rationels non trivials, alors \( P \in \Gamma \) si et seulement si \( P \in \Gamma \) mod \( p \) pour un nombre fini de nombres premiers \( p \). Dans cette note, supposant l’hypothèse général de Riemann, nous fournissons une borne-supérieure explicite sur ces nombres premiers quand \( E \) n’a pas de multiplication complexe et soit \( E \) est une courbe semi-stable soit \( E \) n’a aucun nombre premier exceptionnel.

Keywords: elliptic curves, linear dependence, rational points

AMS Subject Classification: Elliptic curves over global fields, Rational points 11G05, 14G05

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