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Examples of effectivity for integral points on certain curves of genus 2

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Pith's one-line read For many families of genus-2 curves with one point removed, a complex-analytically dense set of fibres have effectively computable integral points over every number field.

desk verdict Genuinely new method and an important main theorem; the two-dimensional theorem needs a repaired proof. read the letter →

arxiv 2411.17930 v2 pith:USEZ3HD3 submitted 2024-11-26 math.NT math.AG

Pietro Corvaja , Davide Lombardo , Umberto Zannier This is my paper · ORCID
classification math.NTmath.AG MSC 14G0511G3014H2511D25
keywords IntegralpointsdiophantineequationsBilu'smethodseffectivityalgebraiccurvesabelianschemestorsionvaluesBettimap
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 that for many families of smooth projective genus-2 curves with a single point removed, the integral points on the open curve are effectively computable for a complex-analytically dense set of algebraic parameter values, over every number field. The engine is a torsion-value criterion: each curve is pulled back to a degree-3 étale cover whose Jacobian contains a square-elliptic factor, and the differences of the three points at infinity define a section of that square-elliptic abelian scheme. Exactly when this section is torsion does Bilu's effectivity criterion apply. The authors show the section is generically not torsion and, using the theory of the Betti map, that its torsion values form a dense analytic set. They obtain a general three-dimensional statement and a complete result for the explicit two-parameter family $y^4 + ay^2 + xy + x^3 + bx^2 = 0$.

What carries the argument

The central object is the section $\sigma = (\phi(p_1)-\phi(p_2), \phi(p_2)-\phi(p_3))$ of the square-elliptic abelian scheme $E^2 \to T$, where $E$ is the elliptic curve $w^3 - 3Q(x)w - 2P(x) = 0$ obtained as the quotient of the degree-3 étale cover $\tilde{Y}$ by the lifted hyperelliptic involution, and the points $p_i$ are the three preimages of the removed point $q$. The construction depends on the parametrisation of 3-torsion points of the Jacobian by decompositions $f = P^2 - Q^3$, which gives the cover and the elliptic factor. The argument then has two pillars: Proposition 4.1 shows $\sigma$ is generically non-torsion, and the Betti-map density theorem (invoked through [ACZ20], [Gao20] and [CMZ18]) shows that for a non-torsion section of a square-elliptic scheme the torsion locus is complex-analytically dense. At any parameter where $\sigma$ is torsion, Bilu's criterion applies to the cover, yielding effectivity for the integral points of the original affine curve.

What would settle it

For the quartic family $y^4 + ay^2 + xy + x^3 + bx^2 = 0$, compute the section $\sigma$ along a curve in the parameter plane where the auxiliary elliptic curve $E$ has a fixed j-invariant: if for some $j_0$ the section took only one value on an irreducible component, the conclusion of Theorem 1.9 would fail. The paper rules this out for $j_0 = 1$ by a finite-field count, so repeating that count for another $j_0$ (or checking the published code) would either confirm or overturn the claim.

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Extended reading notes

Core claim

Write each genus-2 curve in hyperelliptic form $y^2 = f(x)$ with $f(x) = P(x)^2 - Q(x)^3$; this decomposition is equivalent to choosing a 3-torsion point on the Jacobian, and it produces a cyclic étale triple cover $\tilde{Y} \to \tilde{X}$. Quotienting $\tilde{Y}$ by the lifted hyperelliptic involution gives an elliptic curve $E$ with affine model $w^3 - 3Q(x)w - 2P(x) = 0$. If $\tilde{X}$ is punctured at a non-special point $q$ whose preimages $p_1,p_2,p_3$ lie over one value of $x$, the differences $\phi(p_1)-\phi(p_2)$ and $\phi(p_2)-\phi(p_3)$ in $E$ form a section $\sigma$ of the square-elliptic scheme $E^2$ over the family base. The paper proves that this section is not identically torsion and that, for every point where it is torsion, Bilu's criterion yields an effective computation of the integral points on $\tilde{X}\setminus\{q\}$. The density of torsion points follows from Betti-map theory: for a non-torsion section of a square-elliptic abelian scheme over a base of dimension at least two, the torsion locus is complex-analytically dense. The theorems then express the effectivity conclusion for a dense set of fibres of any family whose moduli map is dominant (Theorem 1.6) and for the quartic family $y^4 + ay^2 + xy + x^3 + bx^2 = 0$ (Theorem 1.9).

Load-bearing premise

The proof depends on the theorem that a non-torsion section of a square-elliptic abelian scheme over a base of dimension at least two acquires torsion at a complex-analytically dense set of algebraic points; if that density statement were invalid, the set $\Sigma$ and both main theorems would not follow.

Editorial extensions

If this is right

  • For every parameter in the dense set $\Sigma$, the S-integral points on the corresponding affine genus-2 curve are finite and can be listed explicitly, over any number field and any finite set of places S.
  • The quartic family $y^4 + ay^2 + xy + x^3 + bx^2 = 0$ contains infinitely many members $(a,b)$ for which the integral points are effectively computable, giving a positive answer to a family of equations that had no known algorithm even for special parameter values.
  • The dense set $\Sigma$ is itself effective: one can compute a point of $\Sigma$ inside any prescribed disk in the parameter space, and one can decide whether any given algebraic point belongs to $\Sigma$.
  • The method produces examples where the torsion order—and hence the degree of the auxiliary equation solved—tends to infinity, so the effectivity cannot be obtained from a single universal equation covering the whole family.

Reading between the lines

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

  • The construction suggests a template: any family of curves admitting a finite étale cover whose Jacobian contains a non-isotrivial square-elliptic factor and a generically non-torsion section should yield, via the same Betti-map density argument, a dense set of parameters with effective integral points; testing this template on higher-genus or multi-punctured curves is a natural next step.
  • The explicit examples where the two components of the torsion section have different orders (2 and 3 in Section 7.7) indicate that the method tolerates asymmetric sections; searching for parameters where the two orders are coprime would force the image curve in $\mathbb{G}_m^2$ to have degree at least the product of the orders, producing instances of genuinely high complexity that cannot be captur
  • Because the dense set $\Sigma$ is described by countably many explicit algebraic equations and membership in it is decidable, the paper's method could in principle be implemented for small torsion orders; a practical implementation that computes the integral points for one of the listed examples would be a concrete test of the algorithm's reach.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 3 minor

Summary. The paper develops a method to prove effectivity of integral points on affine curves X = X̃ \ {q}, where X̃ is a smooth genus-2 curve and q is a generically non-special point. The method attaches a degree-3 étale cover Y → X̃ to a 3-torsion section of the Jacobian and applies Bilu's criterion: if the three points of Y above q differ by torsion in Jac(Y), then the integral points of X are effectively computable over every number field. The authors relate this torsion condition to a section σ of a square-elliptic scheme E² and prove that in several families there is a complex-analytically dense set of parameters where the condition holds. The main abstract result is Theorem 1.6 for 3-dimensional families dominant over the moduli space M2; the paper further claims the same density for the explicit two-parameter family y⁴ + ay² ± xy ± x³ + bx² = 0 (Theorem 1.9 / Theorem 5.12), and gives several explicit examples supported by Magma code.

Significance. If Theorem 1.6 is correct, it is a substantial advance: it produces a dense set of genus-2 curves with a single removed point for which integral points can be effectively determined via Bilu's criterion, a phenomenon not expected from the negative heuristics of Landesman–Poonen. The paper is also careful about models over number fields and about the distinction between complex-analytic density and p-adic non-density. The computational parts are accompanied by a public Magma repository, which is a clear strength. However, the proof of the two-dimensional-family theorem has gaps that currently leave Theorem 1.9/5.12 not fully established; these gaps do not affect Theorem 1.6, which is the main conceptual contribution.

major comments (3)
  1. [Section 5.1, proof of Theorem 5.9, case (2)] The sentence 'Since the elliptic scheme in question is non-constant, it is well-known that the set of points where σ is torsion is dense in the base' is false without an additional hypothesis. For example, let T = C × D, let B = π_C^*E_0 for a non-isotrivial elliptic curve E_0 over C, and take σ = π_C^*s for a non-torsion section s. Then B is non-constant and σ is non-torsion, but the torsion locus is a union of fibers C_{t_0} × D over the (typically non-dense) set of t_0 where s is torsion. Such a section is constant on the fibers of the j-map, which is exactly alternative (2). The proof must therefore split the relative-dimension-1 case according as the Betti rank of σ in B is 2 or 1; in the rank-2 case density follows from [ACZ20], while in the rank-1 case one must prove that σ falls into alternative (2). As written, Theorem 5.9 and hence Theorem 5.12 are not fully established; Theorem 1.6 is unaffected.
  2. [Section 5.1, beginning of proof of Theorem 5.9] The assertion 'Proposition 5.3 implies that there exists i ∈ {1,2,3} such that E_{i,t} → T is non-isotrivial' is not a formal consequence of Proposition 5.3 as stated. Proposition 5.3 only proves that the map J: A → A³ is dominant and generically étale; a dominant morphism between smooth 3-folds can have 2-dimensional fibers over a special point. Thus a 2-dimensional subvariety T of A could, a priori, lie inside a single fiber of J, making all three j-invariants constant on T, while T still maps to a 2-dimensional image in M₂. The proof needs to rule out this possibility (for instance by proving that the relevant fibers of J have dimension at most 1, or by treating the isotrivial case separately). Without such an argument, the case analysis in Theorem 5.9 is incomplete.
  3. [Section 5.1, proof of Theorem 5.12] The finite-field specialisation step used to rule out alternative (2) of Theorem 5.9 needs a more precise justification. The text argues that if σ were constant on every irreducible component of T''_0, then after reduction modulo p it would take at most as many values as the number of geometric components, and hence finding more than 630 distinct values over a finite field would contradict the degree bound 630. This reasoning requires that the reduction process preserve the relevant non-constancy and the component count, which is not automatic when the schemes are not smooth or the prime divides denominators. Since this computational check is the only step that excludes case (2) in the application, the manuscript should either give a rigorous reduction argument or provide the exact finite-field data (prime, model, and certification of the 630 bound and of the distinct-value count) in a way that can be checked directly from the supplied Magma code.
minor comments (3)
  1. [Theorems 1.9 and 5.12] The quartic family is stated with +xy + x³ in Theorem 1.9 and with −xy − x³ in Theorem 5.12; the two forms are isomorphic under x ↦ −x, but the statements should be reconciled so that the reader is not left with an apparent contradiction.
  2. [Page 20, proof of Lemma 2.19] The text says 'Denote by [Z : W : W] projective coordinates', which is presumably a typo; the projective coordinates should have three distinct entries.
  3. [Remark 3.8(B)] The notation '(a, b, c) ∈ Z' with a + b + c = 0 is used before the equivalence classes of the points are formally introduced; a sentence clarifying that the relation is a divisor relation on Jac(Y) would improve readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the dense set Σ is defined by torsion of a geometric section, and the Betti-density inputs are independent published theorems rather than fitted outputs.

full rationale

The paper's derivation is not circular. In Theorem 1.6, the dense set Σ is defined as the locus where the section σ=(φ(p1)−φ(p2), φ(p2)−φ(p3)) of the square-elliptic scheme is torsion, and torsion is a geometric property independent of the effectivity conclusion. The proof of §3.2 does not assume effectivity: it uses Bilu's criterion (Theorem 3.5) as a one-way sufficient condition, and the torsion condition is obtained from external Betti-map results ([Zan12], [ACZ20, Prop. 2.1.1], [Gao20, (1.4)], and [CMZ18, Thm 1.1]). Although several of these are authored by Corvaja and/or Zannier, they are published theorems with independent proofs and parameter-free assumptions (e.g., a non-torsion section of an abelian scheme over a base of dimension at least 2), not restatements of the present paper's conclusion. Proposition 4.1 independently rules out the degenerate case where the section is identically torsion. The explicit examples in §7 are verified by computation and are not back-fitted predictions. One genuine non-circular concern should be flagged: in §5.1, case (2) of the proof of Theorem 5.9, the sentence "Since the elliptic scheme in question is non-constant, it is well-known that the set of points where σ is torsion is dense in the base" is not generally valid (for example, a section pulled back from a non-isotrivial elliptic surface over a curve can have torsion values only along finitely many fibers). This is a gap in the proof of Theorem 5.9 rather than a circular reduction, so it does not raise the circularity score.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The paper's central claim rests on standard algebraic geometry and on published Betti-map theorems. The main new mathematical content is the construction of the covers and the verification of non-torsion; the density of torsion values is imported from the literature. No parameters are fitted to data. The computational MAGMA checks are shipped in the GitHub repository, but were not re-run here, so their correctness is assumed.

assumptions (5)
  • domain assumption Bilu's criterion for effectivity of integral points on curves admitting two independent morphisms to G_m
    Used as the ultimate engine of effectivity; cited from [Bil95] and stated as Theorems 3.1, 3.2, 3.5. The paper does not re-prove the underlying S-unit effective theorem.
  • domain assumption Betti map density theorems: non-torsion algebraic sections of abelian schemes become torsion on a complex-analytically dense set under non-isotriviality conditions
    Invoked in §3.2 and §5.1 via [ACZ20, Proposition 2.1.1], [Gao20, (1.4)] and [CMZ18, Theorem 1.1]. This is the key external input that produces the dense set Σ.
  • domain assumption CMZ18 Theorem 1.1 on finiteness of torsion hypersurfaces
    Used in Lemma 4.2 to prove Proposition 4.1 with finitely many exceptions. It confirms the generic non-torsion of the section, a step that is otherwise proved by a direct geometric argument in Proposition 4.1.
  • standard math Siegel's theorem, Chevalley-Weil, and Baker's linear forms in logarithms
    Background reduction steps and the basis of Proposition 3.4; the paper relies on these standard Diophantine tools without re-proving them.
  • domain assumption Correctness of the shipped MAGMA computations
    The proof of Theorem 5.12 depends on a finite-field specialization check and the determinant computation in Proposition 5.3; the code is in [CLZ24] but was not re-run for this review.

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Pith. "Pith review of Examples of effectivity for integral points on certain curves of genus 2." pith.science (2026). https://pith.science/paper/USEZ3HD3

@misc{pith2026241117930,
  author       = {Pith},
  title        = {Pith review of: Examples of effectivity for integral points on certain curves of genus 2},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/USEZ3HD3}},
  note         = {Machine review of arXiv:2411.17930}
}
read the original abstract

We consider families of smooth projective curves of genus 2 with a single point removed and study their integral points. We show that in many such families there is a dense set of fibres for which the integral points can be effectively determined. Our method is based on the construction of degree-3 \'etale covers of such curves of genus 2 and the study of the torsion values of sections of certain doubly elliptic abelian schemes.

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Reviewed August 12, 2026 · model on record in the stance chip above.