Linear x-coordinate relations of triples on elliptic curves

Abstract

For an elliptic curve E defined over the field $C$ 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 Kühne on Uniform Mordell-Lang Conjecture for subvarieties in abelian varieties.