Effective results on linear dependence for elliptic curves

Abstract

Given a subgroup $\Gamma$ of rational points on an elliptic curve $E$ defined over $\mathbf{Q}$ of rank $r\ge 1$ and any sufficiently large $x\ge 2$, assuming that the rank of $\Gamma$ is less than $r$, we give upper and lower bounds on the canonical height of a rational point $Q$ which is not in the group $\Gamma$ but belongs to the reduction of $\Gamma$ modulo every prime $p\le x$ of good reduction for $E$.