Elliptic curves with long arithmetic progressions have large rank

TL;DR

This paper demonstrates that elliptic curves with long arithmetic progressions in their rational points tend to have high rank, linking geometric properties to algebraic complexity and providing new insights into rank distribution.

Contribution

It establishes a connection between long arithmetic progressions on elliptic curves and large Mordell-Weil rank, using advanced Nevanlinna theory and Rémond's methods.

Findings

Long arithmetic progressions imply high rank in elliptic curves.

Results support conjectures on uniform boundedness of ranks.

Applications include insights into elliptic curve statistics and conjectures.

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 non-trivial 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 R\'emond's quantitative extension of results of Faltings.