{"id":"11054341-df4a-41d8-96b1-c3cd5fc3ddf5","arxiv_id":"2411.17930","paper_version":2,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"For many one-punctured genus-2 curve families, a complex-analytically dense set of fibers has effectively computable integral points, proved via torsion density of sections of doubly elliptic schemes.","lead":"This paper finds a dense set of genus-2 curves with one point removed whose integral points can be effectively computed over any number field. It develops a new method based on degree-3 unramified covers and torsion values in elliptic schemes, giving the first such results outside the standard hyperelliptic cases.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 5.9's proof contains an unjustified density claim in the relative-dimension-1 case: a non-constant elliptic subscheme with a pulled-back section need not have dense torsion values.","rationale":"The reader identified the external Betti-map density theorem as the weakest assumption, and that is indeed a load-bearing external input. However, my review found a more concrete internal gap in the proof of Theorem 5.9: the relative-dimension-1 case asserts density from non-constancy of the elliptic scheme, which is insufficient. A pulled-back non-isotrivial elliptic scheme with a pulled-back non-torsion section forms a natural counterexample to the assertion as stated; such an example would land in case (2) of the theorem, so the theorem may still be true, but the proof must explicitly separate the constant-on-fibers case. Since Theorem 5.12 is proved by ruling out case (2) computationally, its conclusion may be correct, but the current proof of Theorem 5.9 is incomplete and the paper should be accepted only if this gap is patched or explicitly bypassed for the families used. I therefore recommend a conditional acceptance rather than a firm rejection or an unqualified accept.","tokens_in":60,"tokens_out":29055,"duration_ms":648803,"concrete_test":"For the explicit family in Theorem 5.12, compute numerically the real rank of the derivative of the Betti map β∘σ at a generic complex parameter (using period integrals). If the rank is 4, the density conclusion is true even if the proof of Theorem 5.9 needs repair. Additionally, test whether a non-isotrivial elliptic scheme pulled back from a curve, with the section also pulled back, can occur in the geometric setup of Theorem 5.9; if it can, the 'well-known' assertion is false and the case split is mandatory.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The weak point is in Section 5.1, in the proof of Theorem 5.9. After applying Gao's formula, the authors consider the case where the generic special closure has relative dimension 1 and write: '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.' This assertion is not generally valid. Let T = C × D, let B = π_C^*E_0 be a non-isotrivial elliptic surface pulled back from a curve, and let σ = π_C^*s be the pullback of a non-torsion section s. Then B is non-constant and σ is non-torsion, but σ(t) is torsion only when s(c) is torsion, which is typically a non-dense union of fibers. The theorem's own case (2) is exactly the constant-on-iso-j-fibers situation, so the proof must split: either σ varies on the fibers of the j-map and the Betti rank is 2, or σ is constant on those fibers and case (2) holds. As written, the proof skips this split, so Theorem 5.9, and hence the application Theorem 5.12, is not fully established. The main Theorem 1.6 is not affected, but a central advertised family of results is.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":22,"tokens_out":23353,"duration_ms":346806,"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":[{"comment":"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.","section":"Section 5.1, proof of Theorem 5.9, case (2)"},{"comment":"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.","section":"Section 5.1, beginning of proof of Theorem 5.9"},{"comment":"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.","section":"Section 5.1, proof of Theorem 5.12"}],"minor_comments":[{"comment":"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.","section":"Theorems 1.9 and 5.12"},{"comment":"The text says 'Denote by [Z : W : W] projective coordinates', which is presumably a typo; the projective coordinates should have three distinct entries.","section":"Page 20, proof of Lemma 2.19"},{"comment":"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.","section":"Remark 3.8(B)"}],"recommendation":"major_revision","confidential_remarks":"The paper fits the journal's scope and Theorem 1.6 is an interesting and well-motivated result. My main concern is that the proof of the two-dimensional-family theorem is not yet rigorous: the density assertion in the relative-dimension-1 case is false as stated, and the inference from Proposition 5.3 to non-isotriviality of one factor needs justification. These are likely repairable, but they currently leave a central advertised family of results unproved. I did not find evidence of circularity: Proposition 4.1 independently rules out identically torsion sections, and the Betti-map density is an external geometric input rather than the conclusion being assumed. The sign inconsistency between Theorems 1.9 and 5.12 should be fixed in revision."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper delivers the first effective integral point computations for a dense set of genus-2 curves with one point at infinity. That is the punchline. The construction is genuinely new: degree-3 etale covers from 3-torsion sections, reduction of Bilu's criterion to torsion of a section of a square-elliptic abelian scheme, and Betti-map density to find the dense set. It also corrects a gap in [DD19] and broadens [GGW23] to number fields. The geometric core (Section 2, Propositions 2.13 and 4.1) is clean and rigorous. The modular-parametrisation lemma (Prop 5.3) is checked by explicit calculation, and the authors ship MAGMA code for the computations. The main Theorem 1.6 looks solid: the Betti-density input is standard in the 3-dimensional case and the section is proved generically non-torsion.\n\nThe real soft spot is the proof of Theorem 5.9. In the relative-dimension-1 case, the paper asserts that a non-constant elliptic scheme has a dense set of torsion values for any non-torsion section. That is not true in general: a section pulled back from a curve on the base can have torsion points only along a non-dense union of fibres. The theorem's own case (2) is precisely the constant-on-j-fibers situation, but the proof does not split into 'section varies on j-fibers' (where the Betti rank is maximal and density follows) and 'section is constant on j-fibers' (which gives case (2)). As written, the proof of Theorem 5.9 is incomplete, and because Theorem 5.12 depends on it, that advertised application is not yet fully established. This should be fixable with a short extra argument, but it is a genuine gap. The computational check in 5.12 (degree 630, more than 630 distinct specializations) is reproducible, though I haven't run it.\n\nWho is this for: anyone working on Diophantine effectivity, Bilu's method, or unlikely intersections. It deserves a serious referee. I would recommend acceptance after the authors fix the gap in 5.9 and either repair the theorem or clearly state it as conditional on that split.","headline":"Genuinely new method and an important main theorem; the two-dimensional theorem needs a repaired proof.","tokens_in":44271,"tokens_out":10127,"would_cite":true,"duration_ms":85588,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14G05","11G30","14H25","11D25"],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["Integral points","diophantine equations","Bilu's methods","effectivity","algebraic curves","abelian schemes","torsion values","Betti map"],"falsifier":"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.","tokens_in":81,"feed_emoji":"🧮","tokens_out":11693,"duration_ms":700638,"temperature":0.7,"pith_summary":"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 $y^4 + ay^2 + xy + x^3 + bx^2 = 0$.","feed_headline":"Dense set of genus-2 curves gets effective integral points","feed_subtitle":"A triple cover turns the integral-point search into a torsion problem, effective over every number field","key_machinery":"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.","core_discovery":"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).","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Provides Bilu's criterion, which converts a rank-2 group of regular functions to $\\mathbb{G}_m$ on a cover into an effective determination of S-integral points.","marker":"[Bil95]"},{"why":"Supplies Proposition 2.1.1, the Betti-map density criterion used to show the torsion locus of the section is complex-analytically dense.","marker":"[ACZ20]"},{"why":"Supplies formula (1.4) for the generic rank of the Betti map, used to handle two-dimensional bases in Theorem 5.9.","marker":"[Gao20]"},{"why":"Gives the torsion-hypersurface theorem for abelian schemes, used in the proof of Theorem 1.6 to get a dense set where the section is torsion.","marker":"[CMZ18]"},{"why":"Provides the explicit universal family of genus-2 curves with two 3-torsion points (Theorem 5.1), used to prove the j-invariants map is dominant (Proposition 5.3).","marker":"[BFT14]"},{"why":"Introduces the Betti map formalism on which the density arguments rest.","marker":"[Zan12]"}],"fun_headline_variants":["Triple cover turns genus-2 integral points into torsion check","Effective integral points on dense fibers via torsion sections","Triple cover yields effective integral points on genus-2 curves","Dense set of genus-2 fibers gets effective points via triple cover","Torsion sections make integral points effective on many genus-2 curves"],"cache_read_input_tokens":46336,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Triple cover turns genus-2 integral points into torsion check","Effective integral points on dense fibers via torsion sections","Triple cover yields effective integral points on genus-2 curves","Dense set of genus-2 fibers gets effective points via triple cover","Torsion sections make integral points effective on many genus-2 curves"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00096,"raw_usage":{"total_tokens":4098,"prompt_tokens":962,"completion_tokens":3136,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":578,"completion_tokens_details":{"reasoning_tokens":3049}},"tokens_in":578,"tokens_out":3136,"duration_ms":22324,"temperature":1.0,"reasoning_tokens":3049,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T11:41:23.377819+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[],"review_version":1}