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arXiv · 2310.17592

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

Jerson Caro·Natalia Garcia-Fritz
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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.

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Jerson Caro, Natalia Garcia-Fritz. 2023-10-26. Linear $x$-coordinate relations of triples on elliptic curves. https://arxiv.org/abs/2310.17592

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