Linear $x$-coordinate relations of triples on elliptic curves

Abstract

For an elliptic curve $E$ defined over the field $\mathbb{C}$ of complex numbers, we classify all translates of elliptic curves in $E^3$ such that the $x$-coordinates satisfy a linear equation. This classification enables us to establish a relation between the rank of finite rank subgroups of $E$ and triples in $E$ whose $x$-coordinates are linearly related. The method of proof integrates complex analytic techniques on elliptic curves with results of Gao, Ge and K\"uhne on Uniform Mordell-Lang Conjecture for subvarieties in abelian varieties.