Effective results on linear dependence for elliptic curves

Min Sha and Igor E. Shparlinski

Vol. 295 (2018), No. 1, 123–144

DOI: 10.2140/pjm.2018.295.123

Abstract

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

Milestones

Received: 10 June 2016

Revised: 18 October 2017

Accepted: 29 January 2018

Published: 13 March 2018