#### Vol. 295, No. 1, 2018

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Effective results on linear dependence for elliptic curves

### Min Sha and Igor E. Shparlinski

Vol. 295 (2018), No. 1, 123–144
##### Abstract

Given a subgroup $\Gamma$ of rational points on an elliptic curve $E$ defined over $ℚ$ 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 $E$.

##### Keywords
elliptic curve, linear dependence, pseudolinearly dependent point, pseudomultiple, canonical height
##### Mathematical Subject Classification 2010
Primary: 11G05, 11G50