Linear x-coordinate relations of triples on elliptic curves

2023/10/26 by Jerson Caro, Caro, Jerson, Natalia Garcia‐Fritz +1 · 1 citation
Mathematics · #Mathematics and Applications #Functional Equations Stability Results #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.2310.17592

Abstract

For an elliptic curve E defined over the field ℂ of complex numbers, we classify all translates of elliptic curves in E³ 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.

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