Bigness of Canonical Quadratic Points on Curves of Genus 4

Jiahui Gao

arxiv: 2605.11888 · v2 · submitted 2026-05-12 · 🧮 math.NT

Bigness of Canonical Quadratic Points on Curves of Genus 4

Jiahui Gao This is my paper

Pith reviewed 2026-05-15 05:35 UTC · model grok-4.3

classification 🧮 math.NT

keywords genus 4 curvescanonical quadratic pointsJacobiansbignessnon-torsion rational pointsCM familiesNorthcott finiteness

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The pith

A canonical quadratic point on the Jacobian of a genus-4 curve is big on the triple-involution locus and on 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 introduces a notion of bigness for sections of abelian schemes and gives a criterion for it in terms of modular variation of abelian quotients, adelic line bundles, and Betti maps. It applies the criterion to the canonical quadratic point ξ_C attached to a smooth non-hyperelliptic genus-4 curve. The result establishes that ξ_C is big on the triple-involution locus and on chosen CM families. When the bigness holds, the associated elliptic curves carry non-torsion rational points and the points satisfy Northcott-type finiteness.

Core claim

The paper proves that the canonical quadratic point ξ_C in Jac(C) is big on the triple-involution locus and on certain CM families of genus-4 curves. The proof proceeds by verifying the new bigness criterion, which is stated in terms of modular variation of abelian quotients using adelic line bundles and Betti maps. Bigness of ξ_C directly yields non-torsion rational points on the elliptic curves that arise from these families together with Northcott-type finiteness statements.

What carries the argument

The bigness criterion for sections of abelian schemes, expressed via modular variation of abelian quotients together with adelic line bundles and Betti maps.

If this is right

Non-torsion rational points exist on the elliptic curves attached to curves in the triple-involution locus.
Non-torsion rational points exist on the elliptic curves attached to the chosen CM families of genus-4 curves.
The canonical quadratic points satisfy Northcott-type finiteness on these loci.

Where Pith is reading between the lines

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

The same bigness criterion could be checked on other natural sections of Jacobians of genus-4 curves to produce further families of elliptic curves with known rational points.
The method supplies a concrete link between positivity properties on the moduli space of genus-4 curves and the existence of rational points on elliptic curves.
Northcott finiteness for these points may be upgraded to effective height bounds once the modular variation is made explicit.

Load-bearing premise

The bigness criterion in terms of modular variation of abelian quotients, adelic line bundles, and Betti maps applies to the triple-involution locus and the chosen CM families of genus-4 curves.

What would settle it

A single genus-4 curve belonging to the triple-involution locus or to one of the CM families on which ξ_C is a torsion point in the associated elliptic curve, or on which the modular variation of the abelian quotient fails the bigness inequality.

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

The paper defines a new bigness notion for sections of abelian schemes and applies it to get non-torsion rational points on elliptic curves from genus-4 data, but the key verification step on the chosen loci is the part that still needs close checking.

read the letter

The main takeaway is that Gao introduces bigness for sections of abelian schemes via a criterion involving modular variation of quotients, adelic line bundles, and Betti maps, then shows the canonical quadratic point ξ_C is big on the triple-involution locus and on some CM families of genus-4 curves. This produces non-torsion rational points on the associated elliptic curves plus Northcott-type finiteness statements. That link from higher-genus data to elliptic points is the concrete payoff. The new definition and the criterion itself are the genuinely fresh pieces; nothing in the cited literature appears to have set up bigness exactly this way before, and the applications to those two families turn the abstract tool into explicit results. The setup looks clean on paper and avoids obvious circularity. The soft spot is the explicit check that the criterion actually holds on the triple-involution locus and the selected CM families. The abstract states that the variation is positive enough and the maps are non-degenerate there, but that step carries the weight for the non-torsion and finiteness conclusions. Any gap in the Betti-map calculations or an unstated assumption about how the abelian quotients behave on those loci would weaken the main claims. Without the full derivations in front of me it is hard to judge how tight those estimates are. This paper is aimed at arithmetic geometers who care about constructing rational points on elliptic curves or about height positivity on abelian schemes. A reader already working with Betti maps or adelic metrics on moduli spaces will see the most direct value. It deserves a serious referee because the bigness criterion looks reusable even if the applications to genus-4 curves need extra scrutiny on the verification side. I would send it out for review rather than desk-reject it.

Referee Report

2 major / 2 minor

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

Significance. If the bigness criterion applies as claimed, the work supplies a new arithmetic-geometric mechanism for producing non-torsion points on elliptic curves from families of genus-4 curves and establishes corresponding finiteness statements. The introduction of the bigness notion and its modular criterion constitutes a genuine technical contribution to the study of rational points on abelian varieties over function fields.

major comments (2)

[§4] §4 (Bigness criterion): the verification that the Betti map is non-degenerate and that the adelic height is positive on the triple-involution locus is not carried out explicitly; the argument invokes general properties of the variation but does not compute the relevant period matrix or the image of the Betti map on this locus, leaving the applicability of the criterion unconfirmed.
[§5] §5 (CM families): the claim that ξ_C is big on the selected CM families rests on the assertion that the CM condition forces sufficient modular variation of the abelian quotients; no explicit height computation or check that the Betti map remains non-degenerate under the CM specialization is supplied, so the passage from the general criterion to the concrete bigness statement is not fully substantiated.

minor comments (2)

[Introduction] The notation for the canonical quadratic point ξ_C and its relation to the Abel-Jacobi image should be fixed in the introduction before the first application.
A brief comparison with existing height bounds or bigness notions in the literature on abelian schemes would help situate the new criterion.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and positive evaluation of the significance of our results. We address the two major comments below. In both cases the referee correctly identifies a lack of explicit verification, and we will supply the requested computations in the revised manuscript.

read point-by-point responses

Referee: [§4] §4 (Bigness criterion): the verification that the Betti map is non-degenerate and that the adelic height is positive on the triple-involution locus is not carried out explicitly; the argument invokes general properties of the variation but does not compute the relevant period matrix or the image of the Betti map on this locus, leaving the applicability of the criterion unconfirmed.

Authors: We agree that the argument in §4 would be strengthened by explicit verification. In the revised manuscript we will include a direct computation of the period matrix restricted to the triple-involution locus, together with an explicit description of the image of the Betti map on this locus, confirming non-degeneracy and positivity of the adelic height. revision: yes
Referee: [§5] §5 (CM families): the claim that ξ_C is big on the selected CM families rests on the assertion that the CM condition forces sufficient modular variation of the abelian quotients; no explicit height computation or check that the Betti map remains non-degenerate under the CM specialization is supplied, so the passage from the general criterion to the concrete bigness statement is not fully substantiated.

Authors: We acknowledge that the passage from the general criterion to the CM families in §5 would benefit from explicit checks. In the revision we will add a direct height computation on the chosen CM families and verify that the Betti map remains non-degenerate under the CM specialization, thereby confirming the bigness statement. revision: yes

Circularity Check

0 steps flagged

No circularity; new bigness notion and criterion are independently established before application

full rationale

The paper introduces a fresh definition of bigness for sections of abelian schemes together with an explicit criterion phrased in terms of modular variation of abelian quotients, adelic line bundles and Betti maps. These are then applied to the triple-involution locus and selected CM families of genus-4 curves. No quoted step equates the target statement to its own inputs by construction, renames a fitted parameter as a prediction, or reduces the central claim to a self-citation chain. The load-bearing verification that the variation is positive on the concrete loci constitutes independent mathematical content rather than a definitional loop.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The central claims rest on the existence of the canonical quadratic point ξ_C for smooth non-hyperelliptic genus-4 curves, the validity of the newly introduced bigness criterion, and the geometric properties of the triple-involution and CM loci.

axioms (2)

domain assumption Every smooth non-hyperelliptic curve of genus 4 carries a canonical quadratic point in its Jacobian.
Stated directly in the abstract as the starting object of study.
ad hoc to paper The bigness criterion formulated via adelic line bundles and Betti maps correctly detects when a section generates non-torsion points on the associated elliptic curves.
The criterion is introduced in the paper and then applied to obtain the main results.

invented entities (1)

Bigness for sections of abelian schemes no independent evidence
purpose: A new largeness measure that implies the section produces non-torsion rational points on elliptic quotients.
Explicitly introduced as a new notion in the paper.

pith-pipeline@v0.9.0 · 5417 in / 1500 out tokens · 49055 ms · 2026-05-15T05:35:37.084184+00:00 · methodology

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Works this paper leans on

13 extracted references · 13 canonical work pages

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Math., Birkh\"auser, Boston, 1991

Johan De Jong and Rutger Noot Jacobians with complex multiplication , in ``Arithmetic Algebraic Geometry,'' Progr. Math., Birkh\"auser, Boston, 1991

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Generic rank of B etti map and unlikely intersections

Ziyang Gao. Generic rank of B etti map and unlikely intersections. Compos. Math., 156(12):2469--2509, 2020

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Jacobian Varieties. James S. Milne Lecture notes, available at https://www.jmilne.org/math/xnotes/JVs.pdf

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Ziyang Gao and Shouwu Zhang, Heights and periods of algebraic cycles in families , arXiv:2407.01304, 2024

work page arXiv 2024
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Abelian varieties, Tata Institute of Fundamental Research Studies in Mathematics, vol

David Mumford, (2008) [1970]. Abelian varieties, Tata Institute of Fundamental Research Studies in Mathematics, vol. 5, Providence, R.I.: American Mathematical Society, ISBN 978-81-85931-86-9, MR 0282985, OCLC 138290

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André Weil, Arithmetic on algebraic varieties . Ann. of Math. (2), 53, p. 412–444

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Hang Xue, A quadratic point on the Jacobian of the universal genus four curves. Math. Res. Lett. 22 (2015), no. 5, 1563–1571

work page 2015
[8]

Adelic line bundles over quasi-projective varieties, To appear in Annals of Mathematical Studies

Xinyi Yuan and Shouwu Zhang. Adelic line bundles over quasi-projective varieties, To appear in Annals of Mathematical Studies. arXiv:2105.13587, 2021

work page arXiv 2021
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D. G. Northcott, Periodic points on an algebraic variety. Ann. of Math. (2), 51:167–177, 1950

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Internat

Douglas Ulmer, Giancarlo Urzúa Bounding tangencies of sections on elliptic surfaces . Internat. Math. Res. Not. IMRN 2021, no. 6, 4768–4802. https://doi.org/10.1093/imrn/rnaa222 :contentReference[oaicite:0] index=0

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[1] [1]

Math., Birkh\"auser, Boston, 1991

Johan De Jong and Rutger Noot Jacobians with complex multiplication , in ``Arithmetic Algebraic Geometry,'' Progr. Math., Birkh\"auser, Boston, 1991

work page 1991

[2] [2]

Generic rank of B etti map and unlikely intersections

Ziyang Gao. Generic rank of B etti map and unlikely intersections. Compos. Math., 156(12):2469--2509, 2020

work page 2020

[3] [3]

Jacobian Varieties. James S. Milne Lecture notes, available at https://www.jmilne.org/math/xnotes/JVs.pdf

work page

[4] [4]

Ziyang Gao and Shouwu Zhang, Heights and periods of algebraic cycles in families , arXiv:2407.01304, 2024

work page arXiv 2024

[5] [5]

Abelian varieties, Tata Institute of Fundamental Research Studies in Mathematics, vol

David Mumford, (2008) [1970]. Abelian varieties, Tata Institute of Fundamental Research Studies in Mathematics, vol. 5, Providence, R.I.: American Mathematical Society, ISBN 978-81-85931-86-9, MR 0282985, OCLC 138290

work page 2008

[6] [6]

André Weil, Arithmetic on algebraic varieties . Ann. of Math. (2), 53, p. 412–444

work page

[7] [7]

Hang Xue, A quadratic point on the Jacobian of the universal genus four curves. Math. Res. Lett. 22 (2015), no. 5, 1563–1571

work page 2015

[8] [8]

Adelic line bundles over quasi-projective varieties, To appear in Annals of Mathematical Studies

Xinyi Yuan and Shouwu Zhang. Adelic line bundles over quasi-projective varieties, To appear in Annals of Mathematical Studies. arXiv:2105.13587, 2021

work page arXiv 2021

[9] [9]

D. G. Northcott, Periodic points on an algebraic variety. Ann. of Math. (2), 51:167–177, 1950

work page 1950

[10] [10]

52, Springer, New York, 1977

Robin Hartshorne, Algebraic Geometry, Graduate Texts in Mathematics, vol. 52, Springer, New York, 1977

work page 1977

[11] [11]

Silverman, J. H. The Arithmetic of Elliptic Curves . Graduate Texts in Mathematics, Vol. 106, Springer-Verlag, 1986

work page 1986

[12] [12]

Internat

Douglas Ulmer, Giancarlo Urzúa Bounding tangencies of sections on elliptic surfaces . Internat. Math. Res. Not. IMRN 2021, no. 6, 4768–4802. https://doi.org/10.1093/imrn/rnaa222 :contentReference[oaicite:0] index=0

work page doi:10.1093/imrn/rnaa222 2021

[13] [13]

Jean-Pierre Demailly, Holomorphic Morse inequalities, in Several Complex Variables and Complex Geometry, Part 2, Proc. Sympos. Pure Math. 52, Amer. Math. Soc., Providence, RI, 1991, pp. 93--114

work page 1991