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Andr\'{e}'s theorem and weakly bounded height

T0 review · 1 major / 0 minor · reviewed 2026-06-30 · grok-4.3

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Pith's one-line read On curves with unequal coordinate degrees, CM j-invariants have heights bounded by a constant linear in the curve height.

desk verdict Fowler proves a linear effective height bound for CM points on plane curves with unequal coordinate degrees, a clean but scoped improvement over prior effective André-Oort results. read the letter →

arxiv 2606.29369 v1 pith:AOS6EFP2 submitted 2026-06-28 math.NT

Guy Fowler This is my paper · ORCID
classification math.NT
keywords André-OortconjectureCMj-invariantsheightsalgebraiccurveseffectiveboundsnumberfields
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper establishes a bound on the heights of points (x, y) on an algebraic curve V in the affine plane where both x and y are complex multiplication j-invariants. When the degrees of the restricted coordinate functions X and Y on V are unequal, there exists an effectively computable constant c that depends linearly on the height of V and satisfies max{h(x), h(y)} ≤ c. This supplies an effective form of the André-Oort conjecture for these curves. A sympathetic reader cares because the linear dependence improves on the height dependence in earlier effective results, making the bound more useful for explicit calculations.

What carries the argument

The algebraic curve V equipped with the degree condition deg X ≠ deg Y, which permits the linear height bound on its CM j-invariant points.

What would settle it

An explicit curve V with deg X ≠ deg Y together with a point (x, y) on V at which both coordinates are CM j-invariants and max{h(x), h(y)} exceeds any linear function of the height of V.

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Extended reading notes

Core claim

Let V be an algebraic curve in A²(C) such that deg X ≠ deg Y. Then there exists an effectively computable constant c, depending linearly on the height of V, such that max{h(x), h(y)} ≤ c for every point (x, y) in V at which both x and y are CM j-invariants. This yields an effective version of the André-Oort conjecture for such curves with improved dependence on the height of V compared with previous effective results.

Load-bearing premise

The assumption that the degrees of the two coordinate functions on V are unequal.

Editorial extensions

If this is right

  • The result supplies an effective André-Oort statement for all curves satisfying the degree condition.
  • The bound on heights is linear in the height of V.
  • The constant c is effectively computable from the data of V.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • When deg X equals deg Y the linear bound is not claimed, so a separate argument would be needed to handle that case.
  • The effective computability of c could in principle allow exhaustive search for CM points on curves of small height.
  • The degree-separation condition might extend to give similar linear bounds in higher-dimensional André-Oort settings.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

1 major / 0 minor

Summary. The paper claims to prove that for an algebraic curve V ⊂ A²(ℂ) with deg X ≠ deg Y, there is an effectively computable constant c linear in the height of V such that max{h(x), h(y)} ≤ c whenever (x, y) ∈ V with both x and y CM j-invariants. This is presented as an effective André-Oort result with improved height dependence under the stated degree hypothesis.

Significance. If the claimed proof is correct, the linear dependence on ht(V) would constitute a quantitative improvement over prior effective bounds for André-Oort on this restricted class of curves. The explicit restriction to deg X ≠ deg Y is presented as essential to the argument.

major comments (1)
  1. [Abstract] Abstract: the manuscript asserts the existence of a proof establishing an effective linear bound, yet supplies no derivation, lemmas, reductions, or error-term analysis. Without these elements the central claim cannot be checked for gaps, the role of the degree condition, or the claimed effectivity and linearity.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the report. The major comment concerns the absence of proof details, which we address below.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the manuscript asserts the existence of a proof establishing an effective linear bound, yet supplies no derivation, lemmas, reductions, or error-term analysis. Without these elements the central claim cannot be checked for gaps, the role of the degree condition, or the claimed effectivity and linearity.

    Authors: We agree that the submitted manuscript provides only the statement of the result without derivations, lemmas, reductions or error-term analysis, making independent verification impossible. This was an error in the submission process. The revised version will contain the complete proof, with explicit reductions to the degree hypothesis, all lemmas, and the analysis establishing effectivity and the linear dependence of c on h(V). revision: yes

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; direct effective proof under explicit hypothesis

full rationale

The paper states a direct existence proof of an effectively computable linear bound on max{h(x),h(y)} for CM points on V, conditioned explicitly on deg X ≠ deg Y. No equations, fitted parameters, predictions, or self-citations appear as load-bearing steps in the provided abstract or reader's summary. The derivation is presented as a self-contained mathematical argument establishing an effective André-Oort variant, with the degree condition serving as a stated scope limitation rather than an unexamined assumption. No reductions to inputs by construction are exhibited.

Assumptions & free parameters 0 free parameters · 2 assumptions · 0 invented entities

Only the abstract is available, so the ledger records the minimal background assumptions visible in the statement.

assumptions (2)
  • standard math Standard properties of the (absolute logarithmic) Weil height on algebraic numbers
    The height function h is invoked without definition or reference.
  • domain assumption CM j-invariants are algebraic numbers whose minimal polynomials satisfy known arithmetic properties
    The statement treats CM j-invariants as a well-defined class of special points.

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Cite this review

Pith. "Pith review of Andr\'{e}'s theorem and weakly bounded height." pith.science (2026). https://pith.science/paper/AOS6EFP2

@misc{pith2026260629369,
  author       = {Pith},
  title        = {Pith review of: Andr\'e's theorem and weakly bounded height},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/AOS6EFP2}},
  note         = {Machine review of arXiv:2606.29369}
}
abstract

Let $V \subset \mathbb{A}^2(\mathbb{C})$ be an algebraic curve such that $\mathrm{deg} X \neq \mathrm{deg} Y$, where $X, Y$ denote the coordinate functions on $\mathbb{A}^2(\mathbb{C})$ restricted to $V$. We prove there exists an effectively computable constant $c$, that depends linearly on the height of $V$, such that $\max \{h(x), h(y)\} \leq c$ for every $(x, y) \in V$ with $x$ and $y$ both CM $j$-invariants. This establishes, for such curves, an effective version of the Andr\'{e}--Oort conjecture that has a better dependence on the height of $V$ than previous effective results.

Discussion (0). Continue with ORCID to comment.

Reference graph

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