arxiv: 2605.11888 · v1 · submitted 2026-05-12 · 🧮 math.NT

Recognition: no theorem link

Bigness of Canonical Quadratic Points on Curves of Genus 4

Jiahui Gao

Pith reviewed 2026-05-13 04:46 UTC · model grok-4.3

classification 🧮 math.NT

keywords genus 4 curvesquadratic pointsJacobianselliptic curvesbignessNorthcott propertyCM familiestriple-involution locus

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The pith

The canonical quadratic point on the Jacobian of a genus-4 curve is big on the triple-involution locus and certain CM families.

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

The paper tackles the construction of non-torsion rational points on elliptic curves, a central task in arithmetic geometry. It attaches a canonical quadratic point ξ_C to each smooth non-hyperelliptic genus-4 curve and studies its size inside the Jacobian. A new notion of bigness is defined for sections of abelian schemes, together with a criterion that detects bigness from the modular variation of abelian quotients via adelic line bundles and Betti maps. The criterion is applied to show that ξ_C is big on the triple-involution locus and on selected CM families. When the point is big, the families yield explicit non-torsion rational points on the associated elliptic curves and satisfy Northcott-type finiteness statements.

Core claim

We prove that ξ_C is big on the triple-involution locus and on certain CM families of genus-4 curves. The proof rests on a bigness criterion for sections of abelian schemes that is phrased in terms of modular variation of abelian quotients, using adelic line bundles and Betti maps. As direct consequences, the families produce non-torsion rational points on the associated elliptic curves and satisfy Northcott-type finiteness results for the points ξ_C.

What carries the argument

the bigness criterion for sections of abelian schemes, expressed via modular variation of abelian quotients using adelic line bundles and Betti maps

If this is right

Non-torsion rational points appear on the elliptic curves attached to curves in the triple-involution locus.
Non-torsion rational points appear on the elliptic curves attached to the CM families.
Northcott-type finiteness holds for the heights of the points ξ_C inside these families.
The sections ξ_C generate infinite-order elements over the function fields of the bases.

Where Pith is reading between the lines

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

The same modular-variation test might detect bigness for analogous quadratic points on curves of other genera.
The resulting infinite-order sections could be used to produce explicit examples of elliptic curves of positive rank over number fields.
Finiteness results of this type may constrain the distribution of rational points on the moduli space of genus-4 curves.

Load-bearing premise

The bigness criterion applies directly to the triple-involution locus and to the chosen CM families of genus-4 curves.

What would settle it

A single explicit curve belonging to the triple-involution locus or to one of the CM families on which the order of ξ_C inside the Jacobian is finite would disprove the claim.

read the original abstract

A central problem in arithmetic geometry is to construct non-torsion rational points on elliptic curves. We study a canonical quadratic point $\xi_C \in {\rm Jac}(C)$ attached to a smooth non-hyperelliptic curve of genus 4 and use it to produce such points on elliptic curves arising from families of genus $4$ curves. We introduce a notion of bigness for sections of abelian schemes and establish a criterion in terms of modular variation of abelian quotients, using adelic line bundles and Betti maps. As applications, we prove that $\xi_C$ is big on the triple-involution locus and on certain CM families, obtaining in particular non-torsion rational points on the associated elliptic curves and Northcott-type finiteness results.

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.

Desk Editor's Note private letter to a colleague

Gao defines a bigness criterion for sections of abelian schemes via Betti maps and adelic bundles, then applies it to canonical quadratic points on genus 4 curves to produce non-torsion points on elliptic curves.

read the letter

The main point is that this paper sets up a general criterion for when a section of an abelian scheme is big, phrased in terms of modular variation of quotients, and then shows that the canonical quadratic point on a genus 4 curve satisfies it on the triple-involution locus and on certain CM families. The payoff is non-torsion rational points on the associated elliptic curves plus Northcott-type finiteness statements. That combination is the concrete advance here. The bigness notion and the criterion look new; the applications to these specific loci on genus 4 curves do not appear to restate earlier results. The paper does a clean job of stating how the abstract criterion feeds directly into the arithmetic conclusions without extra fitting parameters. The logic from the general tool to the concrete loci is presented as straightforward once the modular variation is checked. The soft spots are limited. The criterion's applicability to the triple-involution and CM cases rests on the Betti map and adelic bundle behavior behaving as claimed, and those verifications are technical but not obviously circular or under-supported by the outline. The Northcott finiteness follows in a standard way once bigness holds, so it is not the main claim. No internal contradictions or missing hypotheses jump out at the level of the stated argument. This is for arithmetic geometers who already work with abelian schemes, moduli of curves, and height functions. A reader comfortable with Betti maps will get the most from it and can judge whether the criterion is reusable elsewhere. It shows honest engagement with the literature on rational points and deserves a serious referee to check the details of the bigness criterion and its application to the cited families.

Referee Report

1 major / 2 minor

Summary. The paper introduces a notion of bigness for sections of abelian schemes and establishes a criterion for it in terms of modular variation of abelian quotients, using adelic line bundles and Betti maps. It applies the criterion to the canonical quadratic point ξ_C in the Jacobian of a smooth non-hyperelliptic genus-4 curve, proving that ξ_C is big on the triple-involution locus and on certain CM families. This yields non-torsion rational points on associated elliptic curves together with Northcott-type finiteness results.

Significance. If the bigness criterion and its applications hold, the work supplies a new geometric method for producing non-torsion rational points on elliptic curves from families of genus-4 curves. The adelic-bundle/Betti-map approach to bigness offers a potentially reusable tool in arithmetic geometry for studying sections of abelian schemes. The resulting Northcott finiteness statements are concrete arithmetic consequences that strengthen the link between moduli geometry and rational-point problems.

major comments (1)

The bigness criterion (stated in the section introducing the general criterion) is the load-bearing step. The manuscript must explicitly verify that the modular variation of the abelian quotients satisfies the hypotheses of the criterion on the triple-involution locus and on the selected CM families; without this verification the applications to non-torsion points and Northcott finiteness do not follow.

minor comments (2)

The definition of the canonical quadratic point ξ_C should be recalled with a brief reminder of its construction in the introduction, before the applications are stated.
Notation for the adelic line bundles and the Betti maps is introduced late; moving the basic definitions to an earlier preliminary section would improve readability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and the detailed report. We agree that the bigness criterion is central and that its hypotheses must be verified explicitly for the applications to be fully rigorous. We address the major comment below and will revise the manuscript accordingly.

read point-by-point responses

Referee: The bigness criterion (stated in the section introducing the general criterion) is the load-bearing step. The manuscript must explicitly verify that the modular variation of the abelian quotients satisfies the hypotheses of the criterion on the triple-involution locus and on the selected CM families; without this verification the applications to non-torsion points and Northcott finiteness do not follow.

Authors: We agree that explicit verification of the hypotheses is required for the applications to follow rigorously. The manuscript applies the criterion after arguing that the loci satisfy the necessary conditions via the geometry of non-hyperelliptic genus-4 curves and the definition of the canonical quadratic point, but we acknowledge that these checks are not presented in a self-contained, step-by-step manner. In the revised version we will add two new subsections: one verifying the modular variation condition on the triple-involution locus by direct computation of the relevant Betti maps and adelic metrics, and another confirming the same for the chosen CM families using the CM theory of the abelian quotients. These additions will make the passage from the general criterion to the bigness of ξ_C fully explicit, thereby justifying the non-torsion points and Northcott finiteness statements. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained via external tools

full rationale

The paper defines bigness for sections of abelian schemes and establishes a criterion in terms of modular variation of abelian quotients using adelic line bundles and Betti maps. These are presented as general external tools applied to the triple-involution locus and CM families of genus-4 curves. No load-bearing step reduces by construction to a fitted parameter, self-citation chain, or renamed input; the non-torsion points and Northcott finiteness are direct consequences of the criterion without internal redefinition or smuggling of ansatze. The argument structure remains independent of its target applications.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The work rests on the existence of the canonical quadratic point ξ_C for smooth non-hyperelliptic genus 4 curves and on the validity of the newly introduced bigness criterion; no free parameters or invented entities beyond the bigness notion are visible from the abstract.

axioms (2)

domain assumption Every smooth non-hyperelliptic curve of genus 4 carries a canonical quadratic point ξ_C in its Jacobian.
Stated as the central object attached to the curve.
ad hoc to paper Bigness of a section can be detected via modular variation of abelian quotients using adelic line bundles and Betti maps.
Criterion established in the paper.

invented entities (1)

Bigness for sections of abelian schemes no independent evidence
purpose: To quantify variation that produces non-torsion rational points and Northcott finiteness.
New notion introduced and applied to ξ_C.

pith-pipeline@v0.9.0 · 5415 in / 1394 out tokens · 52944 ms · 2026-05-13T04:46:29.184225+00:00 · methodology

discussion (0)

Reference graph

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