On dependence of rational points on elliptic curves

Abstract

Let $E$ be an elliptic curve defined over $\mathbb{Q}$. Let $\Gamma$ be a subgroup of $E(\mathbb{Q})$ and $P\in E(\mathbb{Q})$. In [1], 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.