Patterns on elliptic curves beyond Bremner's conjecture

TL;DR

This paper explores generalized patterns in the images of finite rank subgroups of elliptic curves, extending previous results on arithmetic progressions and providing rank-dependent bounds for various configurations.

Contribution

It isolates a pattern principle from prior work to establish uniform bounds for diverse patterns in elliptic curve images, beyond just arithmetic progressions.

Findings

Derived rank-dependent bounds for various patterns in elliptic curve images.

Extended the pattern principle to include geometric progressions, shifts, and M"obius orbits.

Connected the results to previous conjectures and theorems in elliptic curve theory.

Abstract

In the late 1990's, Bremner conjectured that long arithmetic progressions among the $x$-coordinates of rational points of an elliptic curve $E$ over $\mathbb{Q}$ should force the rank of $E$ to be large. This conjecture (and a broad generalization of it) was proved by the authors two decades later, by combining Nevanlinna theory and the Uniform Mordell--Lang theorem of Gao--Ge--K\"uhne. The proof inspired subsequent work by the authors where a generalization of the Bogomolov--Fu--Tschinkel conjecture was proved by similar means. In this note we isolate a flexible pattern principle implicit in the latter work, obtaining rank-dependent (but otherwise uniform) bounds for more general patterns in the image of finite rank subgroups of elliptic curves under maps to the projective line. These patterns include, for instance, arithmetic progressions, geometric progressions, additive shifts, multiplicative shifts, and M\"obius orbits.