Elliptic curves and rational points in arithmetic progression

TL;DR

This work conditional on Lang's conjecture proves Bremner's conjecture in full generality: for any elliptic curve E/\mathbb{Q} of Mordell–Weil rank r and any finite sequence of rational points P_1,\dots,P_N with x-coordinates in arithmetic progression, one has N \ll A^r for an effectively computable constant A. The authors develop gap principles for rational points on elliptic curves and leverage spherical-code bounds derived from canonical heights to bound the size of such sequences, splitting analysis into small and large x-regions and establishing a key counting lemma for points of small canonical height. The approach avoids transcendental Nevanlinna theory, relying instead on Diophantine geometry tools, and yields an effectively computable constant under an effective Lang constant c_L, with potential unconditional paths via uniform-height results. The results reinforce a deep connection between arithmetic progression constraints on x-coordinates and the Mordell–Weil rank, offering a conceptually accessible framework for understanding the Bremner conjecture conditional on Lang’s conjecture.

Abstract

Let $E/\mathbb{Q}$ be an elliptic curve. We consider a finite sequence of rational points $\{P_1,\ldots,P_N\}$ whose $x$-coordinates form an arithmetic progression in $\mathbb{Q}$. Under the assumption of Lang's conjecture on lower bounds for canonical height functions, we prove that the length $N$ of such sequences satisfies the upper bound $\ll A^r$, where $A$ is an absolute constant and $r$ is the Mordell-Weil rank of $E/\mathbb{Q}$.

Theorems & Definitions (29)

• Conjecture 1.1
• Theorem 1.2
• Conjecture 1.3
• Theorem 1.4
• Remark 1.5
• Theorem 1.6
• Theorem 1.7
• Theorem 2.1
• proof
• Theorem 2.2