Elliptic Curves with Long Arithmetic Progressions Have Large Rank

Abstract

Abstract For any family of elliptic curves over the rational numbers with fixed $j$-invariant, we prove that the existence of a long sequence of rational points whose $x$-coordinates form a nontrivial arithmetic progression implies that the Mordell–Weil rank is large, and similarly for $y$-coordinates. We give applications related to uniform boundedness of ranks, conjectures by Bremner and Mohanty, and arithmetic statistics on elliptic curves. Our approach involves Nevanlinna theory as well as Rémond’s quantitative extension of results of Faltings.