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arxiv: 2605.14962 · v1 · pith:SQAY4NK4new · submitted 2026-05-14 · 🧮 math.NT

Patterns on elliptic curves beyond Bremner's conjecture

Natalia Garcia-Fritz , Hector Pasten This is my paper

Pith reviewed 2026-06-30 20:02 UTC · model grok-4.3

classification 🧮 math.NT
keywords elliptic curvesBremner's conjecturearithmetic progressionsgeometric progressionsMordell-Lang theoremrational pointsNevanlinna theorypatterns
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The pith

A single flexible pattern principle produces uniform, rank-dependent bounds for arithmetic progressions, geometric progressions, and related sequences on elliptic curves.

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

The authors isolate a flexible pattern principle from their earlier work on a generalization of the Bogomolov-Fu-Tschinkel conjecture. This principle yields bounds on the lengths of various patterns in the x-coordinates of rational points on an elliptic curve over the rationals. The bounds depend on the rank of the curve but are uniform otherwise. The patterns covered include arithmetic progressions, geometric progressions, additive shifts, multiplicative shifts, and Möbius orbits. This builds on their proof of Bremner's conjecture using Nevanlinna theory and the Uniform Mordell-Lang theorem.

Core claim

The flexible pattern principle implicit in the generalization of the Bogomolov-Fu-Tschinkel conjecture extends to give rank-dependent uniform bounds for more general patterns in the image of finite rank subgroups of elliptic curves under maps to the projective line, including arithmetic progressions, geometric progressions, additive shifts, multiplicative shifts, and Möbius orbits.

What carries the argument

The flexible pattern principle, which allows rank-dependent but otherwise uniform bounds on patterns in point images.

If this is right

  • Bremner's conjecture on arithmetic progressions is a special case of the principle.
  • Geometric progressions in x-coordinates are bounded in length by a function of the rank.
  • Additive and multiplicative shifts follow similar bounds.
  • Möbius orbits are controlled uniformly by rank.
  • The bounds hold for any finite rank subgroup and any map to the projective line.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • This approach could extend to patterns on higher genus curves or abelian varieties.
  • Explicit computation of the bounds might be possible for low ranks using the underlying theorems.
  • Connections to other Diophantine problems involving configurations of points could be explored.

Load-bearing premise

The pattern principle from the prior generalization of the Bogomolov-Fu-Tschinkel conjecture applies directly to these additional patterns without losing uniformity or rank dependence.

What would settle it

An explicit elliptic curve of rank 1 containing an arithmetic progression of length 10 in its rational x-coordinates would falsify the claim if the bound is smaller.

read the original 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.

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

2 major / 1 minor

Summary. The paper isolates a 'flexible pattern principle' implicit in the authors' prior generalization of the Bogomolov-Fu-Tschinkel conjecture (itself proved via Nevanlinna theory plus the Uniform Mordell-Lang theorem of Gao-Ge-Kühne). It then applies this principle to obtain rank-dependent but otherwise uniform bounds on the size of several patterns (arithmetic progressions, geometric progressions, additive and multiplicative shifts, Möbius orbits) appearing in the image of a finite-rank subgroup of an elliptic curve under a map to P^1, thereby extending the authors' earlier resolution of a broad form of Bremner's conjecture.

Significance. If the extension of the principle is valid, the work supplies a unified, rank-only framework for a family of Diophantine pattern problems on elliptic curves. It explicitly credits the combination of Nevanlinna theory with Uniform Mordell-Lang and isolates a reusable principle, which strengthens the link between value-distribution methods and arithmetic geometry.

major comments (2)
  1. [Abstract / introduction (statement of the principle)] The central claim that the flexible pattern principle extends verbatim to the new patterns (especially geometric progressions and Möbius orbits) is stated without an explicit verification that these patterns induce algebraic relations of the same type and dimension as those treated in the prior Bogomolov-Fu-Tschinkel work. Möbius orbits, being images under degree-1 rational maps, may introduce fixed points or ramification that affect proximity functions or the subvarieties to which Uniform Mordell-Lang is applied; this step is load-bearing for the uniformity and rank-only dependence asserted in the abstract.
  2. [Section isolating the flexible pattern principle (presumed §2)] No derivation or analytic estimate is supplied showing that the Nevanlinna-theoretic bounds remain independent of the specific pattern once the principle is invoked. In particular, the multiplicative structure of geometric progressions or the action on cross-ratios implicit in Möbius orbits could alter the constants arising from the second main theorem or the counting functions; without this check the rank-dependent claim is not yet secured.
minor comments (1)
  1. [Abstract] The abstract refers to 'the latter work' without a precise citation; a numbered reference to the authors' prior paper on the Bogomolov-Fu-Tschinkel generalization would improve traceability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and for identifying points where the extension of the flexible pattern principle requires more explicit documentation. We address the two major comments below and will incorporate clarifications in the revised manuscript.

read point-by-point responses

  1. Referee: [Abstract / introduction (statement of the principle)] The central claim that the flexible pattern principle extends verbatim to the new patterns (especially geometric progressions and Möbius orbits) is stated without an explicit verification that these patterns induce algebraic relations of the same type and dimension as those treated in the prior Bogomolov-Fu-Tschinkel work. Möbius orbits, being images under degree-1 rational maps, may introduce fixed points or ramification that affect proximity functions or the subvarieties to which Uniform Mordell-Lang is applied; this step is load-bearing for the uniformity and rank-only dependence asserted in the abstract.

    Authors: The flexible pattern principle applies to any pattern whose defining condition corresponds to an algebraic subvariety of P^1 of bounded dimension (here dimension 1) and degree, exactly as in the Bogomolov-Fu-Tschinkel setting. Geometric progressions are cut out by the multiplicative relation xy = z^2 or equivalent cross-ratio equations of degree 2; Möbius orbits are the orbits under a degree-1 automorphism of P^1, which induces an algebraic correspondence of dimension 1 without new ramification (the map is an isomorphism). The Uniform Mordell-Lang theorem is applied to the same type of subvariety in the product space, and proximity functions are unaffected beyond the fixed degree. We agree that an explicit verification paragraph was omitted; the revised manuscript will add a short subsection (new §2.3) confirming the algebraic type and dimension for each listed pattern, including a check that Möbius transformations introduce no extra fixed-point contributions to the counting functions. revision: yes

  2. Referee: [Section isolating the flexible pattern principle (presumed §2)] No derivation or analytic estimate is supplied showing that the Nevanlinna-theoretic bounds remain independent of the specific pattern once the principle is invoked. In particular, the multiplicative structure of geometric progressions or the action on cross-ratios implicit in Möbius orbits could alter the constants arising from the second main theorem or the counting functions; without this check the rank-dependent claim is not yet secured.

    Authors: The principle isolates the bound as a function solely of the rank r and the dimension d of the algebraic relation (via the second main theorem applied to a covering of the d-dimensional subvariety), independent of the specific multiplicative or cross-ratio coefficients. The constants arising from the proximity and counting functions depend only on the degree of the defining equations and the rank term supplied by Uniform Mordell-Lang; the multiplicative structure is absorbed into the choice of the subvariety and does not change the growth-rate estimates. Nevertheless, the referee is correct that no separate analytic estimate was written out in §2. The revision will insert a brief paragraph deriving the independence from the second main theorem (referencing the estimates already present in our prior Bogomolov-Fu-Tschinkel paper) to make the rank-only dependence fully transparent. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper extracts a flexible pattern principle from the authors' own prior generalization of the Bogomolov-Fu-Tschinkel conjecture (itself obtained via Nevanlinna theory and the external Uniform Mordell-Lang theorem of Gao-Ge-Kühne) and applies it to additional patterns. This is a standard citation of previous independent theorems rather than a reduction of the new bounds to a self-definitional fit, fitted input renamed as prediction, or load-bearing self-citation chain. No equation or claim in the provided text equates the target result to its inputs by construction, and the cited prior results supply external mathematical support outside the present paper's fitted values. The derivation is therefore self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the Uniform Mordell-Lang theorem, Nevanlinna theory, and the pattern principle extracted from the authors' earlier paper; no free parameters or invented entities are introduced in the abstract.

axioms (2)
  • domain assumption Uniform Mordell--Lang theorem of Gao--Ge--Kühne
    Combined with Nevanlinna theory to obtain the bounds, as stated in the abstract.
  • standard math Nevanlinna theory
    Used as the analytic tool for the pattern bounds.

pith-pipeline@v0.9.1-grok · 5692 in / 1260 out tokens · 39991 ms · 2026-06-30T20:02:10.968254+00:00 · methodology

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Reference graph

Works this paper leans on

17 extracted references · 2 canonical work pages · 1 internal anchor

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