A note on Bremner's conjecture and uniformity

Abstract: In 1998, Bremner conjectured that elliptic curves over the rationals having long sequences of different rational points whose $x$-coordinates are in arithmetic progression, have large rank. This conjecture was proved some years ago in a strong form as a consequence of previous work by the authors, by a combination of Nevanlinna theory and the uniform Mordell--Lang conjecture of Gao--Ge--Kühne. In particular, if the ranks of elliptic curves over the rationals are uniformly bounded, then so are the lengths of the aforementioned arithmetic progressions. In this note we give a more direct proof of this last statement, which only uses the uniform Mordell--Lang conjecture for curves (due to Dimitrov--Gao--Habegger) and avoids the technicalities of our original argument with Nevanlinna theory.

• The paper establishes that if elliptic curves over ℚ have uniformly bounded Mordell–Weil ranks, then the lengths of arithmetic progressions of rational points are uniformly bounded.
• It replaces complex, height-dependent tools like Nevanlinna theory with a streamlined approach based on the height-uniform Mordell theorem to obtain bounds independent of the j-invariant.
• The result bridges conjectural uniformity in abelian varieties with explicit structural constraints on elliptic curves, highlighting key implications for arithmetic geometry.

Uniform Bounds for Arithmetic Progressions on Elliptic Curves and Bremner's Conjecture

Introduction

This essay provides a technical review of the paper "A note on Bremner's conjecture and uniformity" (2604.04850), which investigates the interrelation between the existence and length of arithmetic progressions of rational points on elliptic curves and the Mordell--Weil rank, culminating in new proofs of uniform boundedness conditional on uniform boundedness of ranks. The exposition delineates the methods by which the authors streamline previous, more intricate arguments and positions the work in the context of major conjectures and recent advances in arithmetic geometry.

Historical Context and Motivation

Bremner's conjecture posits a correlation between the existence of long sequences of rational points with $x$-coordinates in arithmetic progression (AP) on elliptic curves $E/\mathbb{Q}$ and the (typically large) Mordell–Weil rank of $E(\mathbb{Q})$. Specifically, Bremner observed empirically that such progression points tend to be linearly independent, indirectly suggesting that curves endowed with long APs exhibit large ranks.

The question of whether arbitrary long APs can exist on any elliptic curve (Bremner's uniformity question) is fundamentally connected to conjectures about the uniform boundedness of Mordell--Weil ranks, and relates directly to the framework of the Mordell, Mordell–Lang, and Bombieri–Lang conjectures.

Previous Results and Technical Advances

A strong form of Bremner's conjecture is established, demonstrating that for any elliptic curve $E/\mathbb{Q}$ of rank $r$, there is an absolute constant $C>1$ such that any AP of rational points on $E$ has length bounded by $C^{r+1}$. Prior proofs leveraged Nevanlinna theory and height-dependent quantitative Mordell–Lang bounds. Subsequent developments, notably the work of Gao–Ge–Kühne, replaced these with height-independent bounds, thus yielding statements with constants independent of the $j$-invariant.

A further contribution of (2604.04850) is a simplified and more direct argument eschewing the technicalities of Nevanlinna theory, relying instead on the height-uniform Mordell theorem of Dimitrov, Gao, and Habegger (DGH). The current work thus achieves unconditional, uniform bounds on AP length—conditional only on uniform boundedness of ranks.

Main Theorem and Proof Outline

The central statement is: if the ranks of all elliptic curves over $\mathbb{Q}$ are uniformly bounded by an integer $E/\mathbb{Q}$0, then the maximal length of a rational $E/\mathbb{Q}$1-coordinate AP on any $E/\mathbb{Q}$2 is uniformly bounded.

The proof proceeds as follows:

1. Given an elliptic curve $E/\mathbb{Q}$3, any sufficiently long AP of rational points yields a rational map from a genus 2 (hyperelliptic) curve $E/\mathbb{Q}$4 to $E/\mathbb{Q}$5.
2. The construction ensures that $E/\mathbb{Q}$6 is an isogeny factor of $E/\mathbb{Q}$7.
3. By the DGH height-uniform Mordell bound, the number of rational points on $E/\mathbb{Q}$8 is bounded in terms of the rank of $E/\mathbb{Q}$9, which is in turn bounded in terms of the rank of $E(\mathbb{Q})$0.
4. If one assumes a uniform bound $E(\mathbb{Q})$1 on the ranks of elliptic curves, this provides a uniform bound on the AP length.

This argument avoids technical Diophantine approximation machinery and yields a succinct conditional link: Uniform boundedness of elliptic curve ranks $E(\mathbb{Q})$2 uniform boundedness of AP lengths.

Theoretical and Practical Implications

This result provides a concrete instance in which uniformity phenomena for rational points on higher-genus curves—here, constructed from APs on elliptic curves—reduce to uniformity questions for abelian varieties. The key ingredient is the decomposition of Jacobians of explicit genus 2 curves induced by arithmetic or structural properties of the AP on $E(\mathbb{Q})$3.

The method exemplifies the transfer of uniformity from higher-genus curves (where unconditional results like DGH are available) to the subtler setting of abelian varieties (where uniform boundedness of rank remains open and central). It shows that the growth of AP lengths is tightly controlled by the Mordell–Weil rank, and that no infinite APs can exist without unbounded rank growth.

Practically, the result places an obstruction to searches for "unusually long" arithmetic progressions on elliptic curves: it is not merely a matter of computational or example enumeration, but ultimately of profound Diophantine uniformity principles.

Future Directions

A uniform bound for ranks over $E(\mathbb{Q})$4 (the uniformity conjecture) remains elusive, so establishing the absolute bound on AP lengths unconditionally is not yet feasible. However, progress on uniformity for ranks—analytic, arithmetic, or heuristic—would immediately reflect on such concrete questions as AP lengths. Extensions to other number fields or other functions (beyond $E(\mathbb{Q})$5-coordinates) are implied and, as noted, follow by similar lines using the established techniques.

Further computational and theoretical investigation may exploit the explicitness of DGH-type bounds (see, e.g., (Yu et al., 2 Feb 2026)), potentially yielding effective constants for AP lengths in families of curves.

Conclusion

The note establishes a clean and rigorous equivalence: as soon as Mordell--Weil ranks of elliptic curves are known to be uniformly bounded, so too are the lengths of arithmetic progressions of rational points on these curves. By circumventing Nevanlinna theoretic subtleties and focusing on the modern, height-uniform approach to Mordell–Lang type theorems, the result provides a transparent path from high-level conjectural uniformity to explicit structural constraints on rational points in arithmetic progression. This situates Bremner's question directly within the ongoing effort to understand and quantify uniformity phenomena in arithmetic geometry.