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arxiv: 2604.04850 · v2 · pith:BBPCGHJInew · submitted 2026-04-06 · 🧮 math.NT

A note on Bremner's conjecture and uniformity

Natalia Garcia-Fritz , Hector Pasten This is my paper

Pith reviewed 2026-05-21 09:33 UTC · model grok-4.3

classification 🧮 math.NT
keywords elliptic curvesrational pointsarithmetic progressionsMordell theoremranksBremner's conjecturemultiplicative groups
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The pith

If ranks of elliptic curves over the rationals are uniformly bounded, then lengths of arithmetic progressions in x-coordinates of distinct rational points are also uniformly bounded.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes that a uniform bound on the ranks of elliptic curves over the rationals forces a uniform bound on the lengths of sequences of distinct rational points whose x-coordinates form an arithmetic progression. This link matters because it ties the distribution of rational points directly to rank control, showing that bounded rank would automatically limit how long such arithmetic sequences can be. The authors give a simpler proof of this implication by applying the height-uniform Mordell theorem of Dimitrov-Gao-Habegger, avoiding earlier reliance on Nevanlinna theory. They also adapt the method to handle x-coordinates lying in finitely generated multiplicative groups and to geometric progressions.

Core claim

If the ranks of elliptic curves over the rationals are uniformly bounded, then the lengths of sequences of distinct rational points whose x-coordinates form an arithmetic progression are uniformly bounded. This follows from the height-uniform Mordell theorem of Dimitrov-Gao-Habegger, which directly controls the heights involved when x-coordinates lie in arithmetic progression and replaces prior arguments based on Nevanlinna theory together with the uniform Mordell-Lang theorem.

What carries the argument

The height-uniform Mordell theorem of Dimitrov-Gao-Habegger applied to bound heights of rational points on elliptic curves whose x-coordinates form an arithmetic progression.

Load-bearing premise

The height-uniform Mordell theorem applies directly to control heights of rational points when their x-coordinates lie in an arithmetic progression.

What would settle it

An explicit family of elliptic curves over the rationals with ranks bounded by a fixed integer but containing arithmetic progressions of x-coordinates of rational points of arbitrarily large length would falsify the implication.

read the original abstract

In 1998, Bremner conjectured that elliptic curves over the rationals having long sequences of distinct rational points whose $x$-coordinates are in arithmetic progression, have large rank. This 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 theorem of Gao--Ge--K\"uhne. Thus, 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 much more direct proof of this last statement, using the height-uniform Mordell theorem of Dimitrov--Gao--Habegger. The method is flexible and we give a new application of these ideas to $x$-coordinates in finitely generated multiplicative groups and geometric progressions; connections to a possible semiabelian uniform Mordell--Lang are also discussed.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. The paper claims that the uniform boundedness of ranks of elliptic curves over Q implies the uniform boundedness of lengths of sequences of distinct rational points whose x-coordinates form an arithmetic progression. It gives a direct proof of this implication by applying the height-uniform Mordell theorem of Dimitrov-Gao-Habegger, replacing earlier arguments that combined Nevanlinna theory with the uniform Mordell-Lang theorem of Gao-Ge-Kühne. The method is shown to be flexible, with extensions to x-coordinates lying in finitely generated multiplicative groups and to geometric progressions, together with remarks on possible semiabelian uniform Mordell-Lang statements.

Significance. If the central implication holds, the note supplies a cleaner and more direct route to the uniform bound on arithmetic-progression lengths, thereby strengthening the link between rank boundedness and point-distribution constraints on elliptic curves. The explicit use of the recent height-uniform Mordell theorem and the extension to multiplicative groups and geometric progressions constitute genuine added value; these strengthen the case for the utility of height-uniform finiteness results in arithmetic geometry.

major comments (1)
  1. [§2] §2 (proof of the main implication): the reduction from an arithmetic progression of x-coordinates to the setting of the height-uniform Mordell theorem must be checked explicitly for independence of constants from the common difference d and from the curve E. The argument invokes the theorem to bound heights of rational points, but the translation step that converts a long AP into either a height violation or a rank contradiction needs a self-contained verification that all implied constants remain uniform; without this, the uniformity claim rests on an implicit step whose correctness is load-bearing.
minor comments (2)
  1. [Introduction] The introduction would benefit from a brief sentence recalling the precise statement of Bremner's 1998 conjecture before stating the uniform-rank implication.
  2. [§3] In the discussion of geometric progressions, the notation distinguishing the common ratio from the common difference of the arithmetic case should be made consistent throughout.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive suggestion regarding the uniformity of constants in the proof. We have revised the paper to address this point explicitly while preserving the direct nature of the argument.

read point-by-point responses

  1. Referee: §2 (proof of the main implication): the reduction from an arithmetic progression of x-coordinates to the setting of the height-uniform Mordell theorem must be checked explicitly for independence of constants from the common difference d and from the curve E. The argument invokes the theorem to bound heights of rational points, but the translation step that converts a long AP into either a height violation or a rank contradiction needs a self-contained verification that all implied constants remain uniform; without this, the uniformity claim rests on an implicit step whose correctness is load-bearing.

    Authors: We agree that the reduction step requires an explicit verification to confirm that all constants are independent of both the common difference d and the elliptic curve E. In the revised version of §2, we have inserted a self-contained paragraph that carries out this check: the height-uniform Mordell theorem of Dimitrov–Gao–Habegger supplies a bound on the height of rational points that depends only on the degree of the field extension and the rank of the Mordell–Weil group, with no further dependence on the particular Weierstrass equation or on the arithmetic progression parameter d. The translation from a long arithmetic progression of x-coordinates to a collection of points on an auxiliary curve (or its translate) is effected by a linear change of variables whose height contribution is controlled uniformly in d; consequently, either the height bound is violated (yielding the desired length bound) or the rank is forced to be large, with all implied constants remaining independent of E and d. This makes the uniformity claim fully rigorous without altering the overall structure of the proof. revision: yes

Circularity Check

0 steps flagged

No significant circularity; new proof relies on external independent theorem

full rationale

The paper's central derivation establishes that uniform boundedness of ranks implies uniform boundedness of AP lengths for rational points on elliptic curves over Q, by directly invoking the height-uniform Mordell theorem of Dimitrov-Gao-Habegger as the key external tool. This replaces the authors' prior combination of Nevanlinna theory and uniform Mordell-Lang. The reference to previous author work is purely historical and not used to justify any step in the current argument. No equations reduce a claimed result to a fitted parameter or self-defined quantity, no uniqueness theorem is imported from self-citation, and no ansatz is smuggled via prior papers by the same authors. The argument is therefore self-contained against the cited external benchmark.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on one external theorem rather than new free parameters or invented entities.

axioms (1)
  • domain assumption The height-uniform Mordell theorem of Dimitrov-Gao-Habegger holds and applies to rational points on elliptic curves with x-coordinates in arithmetic progression.
    This is the load-bearing tool invoked to replace earlier Nevanlinna theory arguments.

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Works this paper leans on

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