{"id":"df0cafb2-ab73-4ea0-af59-2511bc48c94f","arxiv_id":"2601.17142","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"At least X^{6.5+o(1)} of the X^7 genus-2 models of height ≤ X have Jacobian rank at least 1, and at least X^{5+o(1)} have rank at least 2.","lead":"This paper proves unconditional lower bounds on how many genus-2 curves over Q have Jacobians with Mordell-Weil rank at least 1 or 2, when curves are ordered by coefficient height: rank≥1 occurs for a set of logarithmic density at least 13/14, and rank≥2 for at least 5/7. A reader might care because these are among the first quantitative density results for high-rank Jacobians in genus 2, and they give a theoretical basis for sampling curves used in post-quantum hyperelliptic","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Unverified LMFDB rank-2 witness in Prop 2.8 is load-bearing for the 5/7 rank≥2 density; recomputing it would settle the central gap.","rationale":"The reader's weakest_assumption flagged the same LMFDB dependence; I agree that the Prop 2.8 witness is the critical one for Theorem 1(3). I would only partial-agree with bundling Remark 2.2 as equally load-bearing for the density claims: the 13/14 rank≥1 lower bound survives even if some h-components have torsion α, because the h=0 component alone contributes ≍X^{13/2} models and the log-density calculation ignores constant factors. However, the abstract's 'almost all ... have rank ≥1' and Prop 2.5 as stated do require the 15 witnesses. The elimination step deg(F_n)≪n^2 in Prop 2.5 is also under-justified, but a weaker polynomial bound in n would suffice, so it is less likely to be fatal. Thus the verdict remains CONDITIONAL pending independent verification of the database witnesses.","tokens_in":22108,"tokens_out":14255,"duration_ms":158809,"concrete_test":"Run an independent Magma/Sage verification: (i) match the Igusa invariants of C0: y^2=x^6+8x^5+10x^4+10x^3+5x^2+2x+1 against LMFDB label 15625.a.15625.1; (ii) compute MordellWeilGroup(Jac(C0)) and, on the returned generators, the height-pairing determinant for the classes α = G1−2G2 and β = G2. If the rank is <2 or the determinant is 0, Prop 2.8 collapses; if rank=2 and the determinant is nonzero, the 5/7 bound is supported. The same script should also regenerate the 15 omitted specializations from Remark 2.2 and verify that each has trivial rational torsion and c=4a_6+h_3^2 ∈ (Q*)^2.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The 5/7 lower bound (Corollary 2.11) rests entirely on Proposition 2.8, which asserts Z-independence of α_univ and β_univ on the 5-parameter family U∞_{1,1}. The only witness for this generic statement is a single specialization C0: y^2 = x^6+8x^5+10x^4+10x^3+5x^2+2x+1, with the assertion from LMFDB:15625.a.15625.1 that Jac(C0)(Q) ≅ Z^2, generated by G1 = 2(0:-1:1)-(1:-1:0)-(1:1:0) and G2 = (0:-1:1)-(1:1:0). From this the authors derive α_a0 = G1−2G2 and β_a0 = G2. The paper supplies no certificate, no 2-descent output, and no check that the LMFDB model corresponds exactly to this Weierstrass equation. If the actual rank is 1, or if the generator labels are for a different model, then the displayed relation does not contradict a generic relation mα_univ+nβ_univ=0, and the Néron-specialization step in Prop 2.9 no longer produces rank ≥2. Since the whole X^5-counting family and the density 5/7 depend on this one database entry, it is the most load-bearing external premise. (The 15 unlisted specializations in Remark 2.2 are a related reproducibility gap; they affect the overstated 'almost all' formulation and the statement of Prop 2.5, but the numerical 13/14 density can already be obtained from the h=0 component alone.)","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the distribution of Mordell--Weil ranks of Jacobians of genus-2 curves over Q ordered by the naive height of coefficients of integral Weierstrass models. It introduces a universal family of curves y^2+h(x)y=f(x) with h in {0,1}^4 and studies the divisor class alpha=[infty_+ - infty_-]; it proves that the torsion locus is Zariski closed and, modulo a computational witness, that the universal section is non-torsion. Combining explicit torsion bounds (Gaudron--Remond, Kieffer/Pazuki) with a polynomial-counting argument, it obtains that torsion specializations are O(X^6 (log X)^6) out of about X^{13/2} models with two rational points at infinity, giving logarithmic density at least 13/14 for rank >= 1. On the subfamily U^infty_{1,1}, two independent sections are produced via Neron specialization, yielding rank >= 2 for almost all fibers and hence logarithmic density at least 5/7. A split-Jacobian construction gives density at least 2/21, and twist families of split Jacobians give positive proportions of rank-2 twists. The paper also discusses applications to Regev's quantum algorithm for hyperelliptic curve cryptography.","tokens_in":22491,"tokens_out":16708,"duration_ms":174958,"significance":"If correct, the results give quantitative unconditional lower bounds showing that high-rank genus-2 Jacobians are abundant in the coefficient-box ordering, contrasting with zero-proportion heuristics for natural families. The main geometric framework is standard and mostly sound: the torsion-locus argument, the use of Neron's specialization theorem, and the split-Jacobian constructions are all reasonable. The paper is also explicit about its h=0 witness and points to a GitHub repository. However, the two central computational inputs are not self-contained: the 5/7 rank>=2 bound depends on one LMFDB rank-2/generator assertion, and the 13/14 'almost all' statement depends on fifteen unlisted analytic-rank witnesses. The analytic-rank witnesses do not by themselves unconditionally prove non-torsion. These issues are fixable, but they are load-bearing for the statements as written.","major_comments":[{"comment":"The rank>=2 logarithmic density 5/7 is entirely supported by Proposition 2.8, whose proof reduces to the single specialization C0 and the LMFDB assertion Jac(C0)(Q) = Z^2 with generators G1,G2. No certificate, 2-descent output, or verification that LMFDB:15625.a.15625.1 matches the displayed equation is included. If the rank were 1 or the generators were for a different model, the displayed relation would not contradict a generic relation m*alpha+n*beta=0, and Proposition 2.9 would not produce rank>=2. Please supply a reproducible computation or an independent proof of this rank-2 assertion; as it stands this is a load-bearing external premise.","section":"Prop. 2.8 and Cor. 2.11"},{"comment":"The proof that alpha_univ is non-torsion on all 16 h-components is incomplete. For h=0 a citation is given, but for the remaining 15 components the sole evidence is an unlisted LMFDB search for a specialization of analytic rank 1 and trivial torsion. Analytic rank 1 does not by itself imply the algebraic non-torsion of alpha (BSD is not assumed), and the examples are explicitly not listed. Since Proposition 2.5 and Corollary 2.6 use the properness of V_n on every component, the 'almost all' conclusion for S^box_1(X) is not established for all h. Either list/prove the 15 witnesses or restrict these statements to h=0; the 13/14 numerical density can be obtained from the h=0 component alone.","section":"Remark 2.2 and Prop. 2.5"}],"minor_comments":[{"comment":"The displayed definition of proportion has a typo: 'R_{\\ge0} S\\{\\infty\\}' should be 'R_{\\ge0}\\cup\\{\\infty\\}'. The same issue appears in Definition 2.","section":"Section 1, Definition 1"},{"comment":"The phrase 'The former has cardinality X^{13/2}' should read 'has cardinality \\asymp X^{13/2}', with an implied constant. Also clarify that for Theorem 1(2) one uses the h=0 component inside C_1(X), since S_1(X) counts pairs (f,h).","section":"Proof of Cor. 2.7"},{"comment":"The sentence 'Since V_n is proper...' should say 'proper closed subset'. The elimination step producing a nonzero polynomial F_n in the a-coordinates deserves a brief justification: the reader needs to see why the image of V_n under the finite map to A^7 remains a proper closed subset.","section":"Prop. 2.5"},{"comment":"If the additional 15 examples are not listed, at least provide LMFDB labels or a script output in the GitHub repository so that the analytic-rank-1 and trivial-torsion assertions are checkable by the reader.","section":"Remark 2.2 / GitHub"},{"comment":"After setting a6=1, the cover u^2=4 splits; please specify which u-component (u=2 or u=-2) is meant when defining the restrictions of alpha and beta. This affects the labelling of infty_+ and infty_-.","section":"Notation for U^infty_{1,1}"}],"recommendation":"major_revision","confidential_remarks":"The central risk is the reliance on LMFDB computations without certificates. I would be willing to accept after the rank-2 assertion for C0 is made reproducible and the status of the 15 analytic-rank witnesses is clarified (either listed or the statements restricted to h=0). The cryptographic discussion is somewhat independent from the main density results and could be shortened if the paper is too long."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"First, the headline: this is a strong paper with real new results — the 13/14 lower bound for rank ≥1 and 5/7 for rank ≥2 in the naive-height ordering over genus-2 curves are new, and the universal-section method for counting non-torsion classes is a genuinely useful idea. The paper deserves a serious referee.\n\nWhat is new: the construction of a 5-parameter family U∞_{1,1} where two universal sections α_univ and β_univ are shown Z-independent by a single specialization, then Néron specialization plus Serre's thin-set bound gives rank ≥2 for almost all members. The torsion-locus count (Prop 2.1–2.5) is standard material but assembled well: algebraic torsion loci, Faltings-height bounds from Kieffer/Pazuki, and Gaudron–Rémond give the X^6 log^6 X bound that powers the 13/14 density. The split-Jacobian twist results (Props 2.20–2.22) are also new and correct as far as I can see. They even point to a GitHub repo for scripts, which is good practice.\n\nSoft spots, in proportion:\n\nThe 5/7 rank≥2 bound rests entirely on one LMFDB assertion: that the Jacobian of C0: y^2 = x^6+8x^5+10x^4+10x^3+5x^2+2x+1 is Z^2 and the displayed generators G1, G2 are as stated. If that entry is wrong — wrong model, wrong generators, or actual rank 1 — then Prop 2.8's independence conclusion collapses, and with it the 5/7 bound. The paper gives no 2-descent certificate. This is easy to verify externally, but as written it is a load-bearing unsupported premise. The stress-test note has this exactly right.\n\nThe 15 unlisted LMFDB specializations in Remark 2.2 are a milder version of the same issue. They back the 'almost all models with two rational points at infinity have rank ≥1' proportion claim in Cor 2.7. The 13/14 logarithmic density itself is safe — you can get it from the h=0 component alone — so this only affects the stronger almost-all formulation, not the main numerical bound.\n\nTwo smaller quibbles: the elimination step in Prop 2.5 (deg F_n ≪ n^2) is asserted without proof; it's probably fine, and even a weaker bound like n^4 would still make the final count O(X^6 log^10 X), which still works. And the abstract says 'positive proportion of rank-2 twists' where Prop 2.21 actually gives positive proportion of twists with rank ≡ 2 mod 4, which is not exactly rank 2. Fix the wording.\n\nOverall: the core contributions are likely correct, the gaps are fixable, and the paper will be worth a careful referee. Send it to a number theorist with access to Magma/PARI to check the witness, and it should be publishable.","headline":"New and likely-correct density bounds for rank ≥1 and ≥2 genus-2 Jacobians; the 5/7 bound leans on a single LMFDB entry that needs verification.","tokens_in":23065,"tokens_out":4636,"would_cite":true,"duration_ms":46346,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11G10","11G30","11G50","14H40","94A60"],"pacs":[],"model":"deepseek-v4-flash","headline":"When genus-2 curves are ordered by coefficient height, rank-1 Jacobians have logarithmic density at least 13/14 and rank-2 Jacobians at least 5/7.","keywords":["genus-2 curves","Jacobian rank","logarithmic density","universal divisor class","specialization theorem","split Jacobian","quadratic twists","hyperelliptic curve cryptography"],"falsifier":"Independently compute the rank and a generator basis of the Jacobian of y^2 = x^6 + 8x^5 + 10x^4 + 10x^3 + 5x^2 + 2x + 1: finding rank 1, or finding that the two listed divisor classes are linearly dependent, would invalidate the 5/7 rank-2 density bound. Likewise, for each of the fifteen coefficient vectors h in {0,1}^4, verify the asserted existence of a specialization of analytic rank 1 and trivial torsion; a single failure would break the 'almost all' torsion bound for that component and degrade the 13/14 statement.","tokens_in":21940,"feed_emoji":"🧮","tokens_out":11537,"duration_ms":98034,"temperature":0.7,"pith_summary":"The paper asks how common it is for the group of rational points on the Jacobian of a genus-2 curve over Q to have positive rank, ordering curves by the naive height of the coefficients of their integral Weierstrass models. Using a universal Jacobian over the parameter space of models with two rational points at infinity, it shows the divisor class [∞+ − ∞−] is non-torsion for almost all such models, giving an unconditional logarithmic density of at least 13/14 for Jacobians of rank at least 1. For rank at least 2 it constructs an explicit five-parameter subfamily with two independent rational sections, and the classical specialization theorem shows independence fails only on a thin set, yielding an unconditional logarithmic density of at least 5/7. A separate split-Jacobian construction gives a 2/21 lower bound, and twist families of split Jacobians are shown to produce rank-2 twists in positive proportion. A reader should care because these are unconditional existence results for high-rank Jacobians in a natural ordering, and they directly bound the search cost of finding curves suitable for a recent quantum discrete-logarithm algorithm in hyperelliptic curve cryptography.","feed_headline":"Curve count finds rank-1 genus-2 Jacobians at 13/14 log-density","feed_subtitle":"A universal divisor class plus the specialization theorem proves both bounds and feeds quantum discrete-log attacks.","key_machinery":"The central mechanism is the universal divisor class α_univ = [∞+ − ∞−] on the Jacobian of the universal genus-2 curve over the parameter space U∞ of integral models y^2+h(x)y=f(x) with h-coefficients in {0,1} and with the leading coefficient of 4f+h^2 a square. The proof combines three ingredients: the loci V_n = {n α_univ = 0} are Zariski closed while α_univ itself is non-torsion; torsion points on any specialization have order bounded by a power of log H, via theta-height and stable-height comparison; and a box count shows the union of V_n for n up to (log X)^2 contributes only X^6(log X)^6 points against X^{13/2} models. For rank 2, the subfamily U∞_{1,1} with a6=a0=1 carries two section","core_discovery":"The central claim is that positive-rank Jacobians of genus-2 curves over Q are not sparse in the natural coefficient-height box C1(X) = {y^2=f(x): H(f)≤X}. On the parameter space of integral models whose two points at infinity are rational, the universal divisor class α_univ = [∞+ − ∞−] is shown to be non-torsion, and its torsion specializations are so rare that almost all of the X^{13/2} such models give Jacobians of rank at least 1. For rank at least 2, the paper exhibits a five-parameter subfamily y^2 = x^6 + a_5x^5 + ... + a_1x + 1 carrying two independent rational sections, so by specialization theory all but a thin exceptional set of these curves have Jacobian rank at least 2; since th","pith_inferences":["If rank parity equidistributes in the subfamily with two rational points at infinity, as the authors note is plausible, the rank-2 logarithmic density should match the 13/14 rank-1 value; checking this in numerical tables for increasing X would be a direct test of that heuristic.","The rank-1 proof depends only on one non-torsion specialization; replacing the unlisted data used for the other 15 choices of h by an explicit finite list would make the theorem independent of database lookups.","The X^{13/2} count of models with square leading coefficient suggests that other geometrically forced rational sections—not just the two points at infinity—could produce analogous positive logarithmic densities for higher-rank sets in genus 2 and higher genus.","Because high-rank rational lifts are exactly what the lattice-based quantum algorithm consumes, the practical bottleneck shifts from existence to small canonical height: the tables of generators and heights in the paper point toward optimizing that second step."],"forward_implications":["Sampling models uniformly from C1(X), a rank-1 Jacobian is found within O(X^{1/2+o(1)}) trials and a rank-2 Jacobian within O(X^{2+o(1)}) trials, replacing naive X^7 search costs.","The 13/14 and 5/7 lower bounds are the first unconditional positive logarithmic densities for these rank sets in this ordering, giving a quantitative explanation for the abundance of high-rank curves in small-coefficient databases.","The counted curves can be reduced modulo primes to supply the multi-scalar multiplication inputs for the recent lattice-based quantum discrete-logarithm algorithm, so the search cost for suitable lifts ceases to be the bottleneck.","In split-Jacobian families satisfying the hypotheses, double quadratic twists equidistribute over ranks 0, 1, 2 with proportions 1/4, 1/2, 1/4, and quadratic twists of E×E have rank at least 2 for at least half of squarefree d, so high-rank twists are found in constant time.","The same universal-section argument extends to genus g≥1, giving rank at least 1 on a subfamily of logarithmic density at least (4g+5)/(4g+6) in the analogous height box."],"fun_headline_variants":["Rank-1 genus-2 Jacobians: logarithmic density 13/14","Most genus-2 Jacobians have rank ≥1 (density 13/14)","Rank-2 genus-2 curves: density at least 5/7","Quantum crypto shake-up: genus-2 Jacobians often rank ≥1","13/14 log-density: genus-2 Jacobians rank ≥1"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The proof's load-bearing external premise is that the recorded specializations it cites are accurate: the displayed rank-2 curve really has Jacobian rank exactly 2 with the two listed divisor classes independent, and for each of the fifteen remaining coefficient choices h there really is a specialization with analytic rank 1 and trivial torsion; if any one of these assertions fails, the corresponding density conclusion is not established.","fun_headline_variants_meta":{"raw":{"variants":["Rank-1 genus-2 Jacobians: logarithmic density 13/14","Most genus-2 Jacobians have rank ≥1 (density 13/14)","Rank-2 genus-2 curves: density at least 5/7","Quantum crypto shake-up: genus-2 Jacobians often rank ≥1","13/14 log-density: genus-2 Jacobians rank ≥1"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000709,"raw_usage":{"total_tokens":3084,"prompt_tokens":850,"completion_tokens":2234,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":594,"completion_tokens_details":{"reasoning_tokens":2143}},"tokens_in":594,"tokens_out":2234,"duration_ms":14374,"temperature":1.0,"reasoning_tokens":2143,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-03T08:28:26.926179+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Independently compute the rank and a generator basis of the Jacobian of y^2 = x^6 + 8x^5 + 10x^4 + 10x^3 + 5x^2 + 2x + 1: finding rank 1, or finding that the two listed divisor classes are linearly dependent, would invalidate the 5/7 rank-2 density bound. Likewise, for each of the fifteen coefficient vectors h in {0,1}^4, verify the asserted existence of a specialization of analytic rank 1 and trivial torsion; a single failure would break the 'almost all' torsion bound for that component and degrade the 13/14 statement.","supporting_citations":[],"review_version":1}