{"id":"b8a25c5c-b693-47e4-9e58-e5e5b0433c5b","arxiv_id":"2507.00860","paper_version":1,"verdict":"CONDITIONAL","confidence":"LOW","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For products of two curves, the paper gives effective asymptotic thresholds for density of degree-d points and a nearly complete low-genus picture, with applications to abelian and bielliptic surfaces.","lead":"This paper starts a systematic account of which degrees of algebraic points are Zariski dense on product surfaces. It proves effective asymptotic ranges and explicit low-degree descriptions, with applications to abelian and bielliptic surfaces.","discovery_kind":"first_principles","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified; Proposition 3.6 survives scrutiny of Lemma 3.2 and the index-divisibility step.","rationale":"The reader identified Lemma 3.2 and the index divisibility as the load-bearing hinge, which is correct in location. However, a detailed check shows the hypotheses in Proposition 3.6 force the ramification indices to be coprime at every support point, so the delicate s-th power condition in Lemma 3.2 is never actually needed. The normalization point exists for every gamma in the family, and the resulting point degrees divide the degrees of the original support points, so the index of X_gamma^nu divides ind(C×D/k). The formal-local argument in Lemma 3.2 is standard and does not conceal a gap. The reader's additional claimed error in Theorem 5.8 also does not hold up: d=3 satisfies the hypotheses of Lemma 2.5 and yields exactly 7. The paper's main asymptotic theorem therefore appears sound. That said, the verdict remains CONDITIONAL because the paper relies on the unpublished reference [17] for the foundational curve-level density result and some later transitions (e.g., Theorem 6.4 to abelian surfaces in Theorem 7.3) are terse; these are grounds for caution but not for a reject-level objection. Hence the reader's verdict is left unchanged.","tokens_in":38418,"tokens_out":46631,"duration_ms":536512,"concrete_test":"Verify the index-divisibility step on a concrete example from Remark 3.7, e.g. the pair of genus 3 curves with ind(C×D)=4 and eff-ind=4: construct f_C and f_D as in Proposition 3.6, choose one automorphism gamma from the family, and compute directly that the normalization of the fiber product has points whose degrees have gcd dividing 4; then confirm that the predicted threshold N(C,D,e) is valid by checking the genus bound. This would catch any hidden failure in the simultaneous application of Lemma 3.2 to all support points.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The reader's hinge concern (Lemma 3.2) does not land. In Proposition 3.6, the case of a rational point is excluded, so at least one of deg|Z_C|, deg|Z_D| is at least 2. Consequently, at every support point (P,Q), the ramification indices are (1, anything) or (2,1) or (1,1), so their gcd is 1. Lemma 3.2 then imposes no condition on the automorphism gamma: with s=1, the local equation has a unique branch and the normalization has a point over the residue field of (P,Q), or over a subfield. The index of X_gamma^nu divides the gcd of these point degrees, which divides the gcd of the degrees of the support points, which in turn divides ind(C×D/k). Thus ind(X_gamma^nu) | ind(C×D/k), exactly as needed. The formal power-series computation in Lemma 3.2 is standard: for s=1 the equation u-v=0 defines a smooth formal branch, and for s>1 the component u-v is still defined over the base field and yields a point. Separately, the reader's cited error in Theorem 5.8 appears to be a false positive: applying Lemma 2.5 with g1=6, g2=1, n=2, d=3 satisfies (n-1)d 1$; write the local expansions $f_C = a_n z_C^n + \\cdots$ and $f_D = b_m z_D^m + \\cdots$ at those points. If $a_n/b_m$ is not an $s$-th power in $K = \\mathbb{Q}(P,Q)$, then the formal-coordinate construction in Lemma 3.2 fails and the normalization $X_\\gamma^\\nu$ need not have a $K$-point above $(P,Q)$; checking whether the index divisibility and the threshold $N(C,D,e)$ still hold for that example would settle the claim.","tokens_in":38135,"feed_emoji":"🔢","tokens_out":11682,"duration_ms":106219,"temperature":0.7,"pith_summary":"The paper establishes the first effective asymptotic description of the density degree set for a surface, in the case of a product $C \\times D$ of two smooth projective curves over a number field. For every integer $d$ that is a sufficiently large multiple of the index $\\mathrm{ind}(C \\times D/k)$, the degree-$d$ closed points are Zariski dense on $C \\times D$, with an explicit threshold $N(C,D,e)$ depending only on the genera and the effective index $e$. When both curves have $k$-rational points, the threshold is simply $6g_Cg_D+2g_C+2g_D$; in low genus the finite list of exceptional small degrees is computed exactly in several cases. This matters because for surfaces almost nothing was known beyond non-effective existence of large multiples, and the result shows that the curve-level phenomenon---large-degree density is controlled by the index---persists on products while small-degree density remains genuinely arithmetic.","feed_headline":"For product curves, all large index multiples are Zariski dense","feed_subtitle":"Explicit thresholds in the genera and effective index align surface density with the curve case.","key_machinery":"The construction is a covering family of fiber products $X_\\gamma = C \\times_{\\mathbb{P}^1} D$, where $f_C: C \\to \\mathbb{P}^1$ and $f_D: D \\to \\mathbb{P}^1$ are chosen with assigned fibers over $\\infty$, and $\\gamma$ varies over automorphisms of $\\mathbb{P}^1$ fixing $\\infty$. Lemma 3.1 bounds the arithmetic genus of $X_\\gamma$ by an expression in the gonality and genus of the factors, makes each $X_\\gamma$ pass through a prescribed zero-cycle $Z$, and shows the family sweeps out a Zariski-dense locus in $C \\times D$. Lemma 3.2 is the hinge: a local power-series computation shows that the normalization $X_\\gamma^\\nu$ has a $K$-rational point above a ramification pair $(P,Q)$ whenever the ramification indices are coprime or $P,Q$ are $k$-rational, which forces $\\mathrm{ind}(X_\\gamma^\\nu/k)$ to divide $\\mathrm{ind}(C\\times D/k)$. Combined with the curve-level identity $\\delta(X_\\gamma^\\nu/k) \\cap \\mathbb{N}_{\\geq 2g(X_\\gamma^\\nu)} = \\mathrm{ind}(X_\\gamma^\\nu/k)\\mathbb{N} \\cap \\mathbb{N}_{\\geq 2g(X_\\gamma^\\nu)}$, the bounded genus gives the effective threshold.","core_discovery":"The central result is Proposition 3.6: for nice curves $C,D$ over a number field $k$ and $e = \\mathrm{eff\\text{-}ind}(C \\times D/k)$, every integer $d \\geq N(C,D,e)$ that is divisible by $\\mathrm{ind}(C \\times D/k)$ lies in $\\delta(C \\times D/k)$. In the pointed case (Corollary 3.3), the bound reads $\\mathbb{N}_{\\geq 6g_Cg_D+2g_C+2g_D} \\subseteq \\delta(C \\times D/k)$, and the paper also computes the small-degree exceptional sets for products of elliptic curves, of an elliptic curve with a genus $2$ curve, and of two genus $2$ curves. The applications include: any abelian surface isogenous to a principally polarized abelian surface has $\\mathbb{N}_{\\geq 3} \\subseteq \\delta(A/k)$, and $2 \\in \\delta(A/k)$ as well when $A$ is isogenous to the Jacobian of a genus $2$ curve, while a bielliptic surface $S=(E_1 \\times E_2)/G$ satisfies $\\mathbb{N} \\setminus \\{1,2\\} \\subseteq \\delta(S/k) \\subseteq \\delta(E_1/k)$. Some small-degree statements are conditional, notably those relying on the Parity Conjecture for certain quadratic-point cases.","pith_inferences":["A natural extrapolation of Proposition 3.6 is that any surface covered by a family of curves of bounded genus through a zero-cycle of minimal degree should have asymptotic density degree set equal to the index multiples beyond an explicit threshold; products of curves are the first case where the paper makes this principle effective.","The small-degree exceptions computed in Theorem 1.2 all lie outside the product $\\delta(C/k)\\cdot\\delta(D/k)$, suggesting a general conjecture that $\\delta(C\\times D/k)$ differs from the index tail by only a finite exceptional set; the paper's rank-zero genus $2$ examples show that exceptional set can be nonempty even when both factors individually have dense quadratic points.","For self-products of a genus $2$ curve, Proposition 6.5 yields a testable criterion: if $\\mathrm{Pic}^0_C(k)$ has positive rank and is simple, then $2 \\in \\delta(C\\times C/k)$; one could search for such a curve and verify quadratic density computationally.","The conditional use of the Parity Conjecture for $2\\in\\delta(E_1\\times E_2/k)$ suggests that degree-$2$ density on products is equivalent to simultaneous rank growth over a quadratic extension, which could be tested by a finite search over quadratic fields for rank-zero $j=0$ or $j=1728$ curves."],"forward_implications":["For pointed curves $C,D$ of genera $g_C,g_D$, every degree $d \\geq 6g_Cg_D+2g_C+2g_D$ has Zariski-dense points on $C \\times D$.","For any two elliptic curves over $k$, every degree $d \\geq 3$ lies in $\\delta(E_1 \\times E_2/k)$, and degree $2$ is included under the j-invariant or full 2-torsion assumptions of Theorem 4.3.","When $E$ has rank $0$ and $C$ is a genus $2$ curve with $C(k) \\neq \\emptyset$, all degrees except $2,3,5,7$ lie in $\\delta(E \\times C/k)$; with a rational Weierstrass point, $7$ is included as well.","Any abelian surface $A$ isogenous over $k$ to a principally polarized abelian surface has $\\mathbb{N}_{\\geq 3} \\subseteq \\delta(A/k)$, and $2 \\in \\delta(A/k)$ when $A$ is isogenous to the Jacobian of a genus $2$ curve.","For a bielliptic surface $S=(E_1 \\times E_2)/G$, $\\mathbb{N} \\setminus \\{1,2\\} \\subseteq \\delta(S/k) \\subseteq \\delta(E_1/k)$."],"supporting_citations":[{"why":"supplies the curve-level density-degree identity and multiplicative closure used to convert density on the covering curves into density on the product.","marker":"[17]"},{"why":"supplies the Uniform Position Theorem used to produce degree-d points with full symmetric Galois group, needed for the primitive-extension arguments in the abelian and bielliptic applications.","marker":"[1]"},{"why":"supplies the construction of dense quadratic points on products of elliptic curves under mild j-invariant assumptions, underlying Theorem 1.2(i) and Corollary 4.13.","marker":"[13]"},{"why":"supplies the genus 2 curve dominating two elliptic curves with fully rational 2-torsion, covering the remaining quadratic-point case of Theorem 4.3.","marker":"[6]"},{"why":"supplies the structure theorem for subvarieties of abelian varieties used to prove density of rational points on isogenous abelian surfaces.","marker":"[14]"}],"fun_headline_variants":["Index multiples past effective bound: Zariski dense on products","Effective index bound: Zariski density on product curves","Product curves: large index multiples become Zariski dense","For product curves, index multiples beyond explicit bound dense","Index threshold on product curves: all large multiples are dense"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is Lemma 3.2, which asserts that after forming $X_\\gamma = C \\times_{\\mathbb{P}^1} D$ and taking its normalization, infinitely many of the curves $X_\\gamma^\\nu$ have a $K$-rational point above a chosen ramification pair $(P,Q)$ whenever the ramification indices are coprime or $P,Q$ are $k$-rational; without that local assertion, the index of the covering curves need not divide $\\mathrm{ind}(C\\times D/k)$ and the asymptotic threshold would not follow.","fun_headline_variants_meta":{"raw":{"variants":["Index multiples past effective bound: Zariski dense on products","Effective index bound: Zariski density on product curves","Product curves: large index multiples become Zariski dense","For product curves, index multiples beyond explicit bound dense","Index threshold on product curves: all large multiples are dense"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000981,"raw_usage":{"total_tokens":4150,"prompt_tokens":914,"completion_tokens":3236,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":530,"completion_tokens_details":{"reasoning_tokens":3155}},"tokens_in":530,"tokens_out":3236,"duration_ms":26890,"temperature":1.0,"reasoning_tokens":3155,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T21:12:06.066022+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a product $C \\times D$ over $\\mathbb{Q}$ where both curves have a non-rational ramification point over $\\infty$ with ramification indices having gcd $s > 1$; write the local expansions $f_C = a_n z_C^n + \\cdots$ and $f_D = b_m z_D^m + \\cdots$ at those points. If $a_n/b_m$ is not an $s$-th power in $K = \\mathbb{Q}(P,Q)$, then the formal-coordinate construction in Lemma 3.2 fails and the normalization $X_\\gamma^\\nu$ need not have a $K$-point above $(P,Q)$; checking whether the index divisibility and the threshold $N(C,D,e)$ still hold for that example would settle the claim.","supporting_citations":[{"cited_title":"infinite families of pairs of curves over q with isomorphic jacobians","cited_arxiv_id":null,"evidence_quote":"supplies the curve-level density-degree identity and multiplicative closure used to convert density on the covering curves into density on the product."},{"cited_title":"Then it follows by Theorem 4.3 and Lemma 4.5 that 2 ∈ δ(E′ 1 ×E2), and in fact there exists a quadratic extension L/k over which both rk E′ 1(L) > 0 and rk E2(L) > 0","cited_arxiv_id":null,"evidence_quote":"supplies the Uniform Position Theorem used to produce degree-d points with full symmetric Galois group, needed for the primitive-extension arguments in the abelian and bielliptic applications."},{"cited_title":"Let E2 be the elliptic curve 14.a.5 with Weierstrass equation y2 + xy + y = x3 − x, which has rank 0 over Q","cited_arxiv_id":null,"evidence_quote":"supplies the construction of dense quadratic points on products of elliptic curves under mild j-invariant assumptions, underlying Theorem 1.2(i) and Corollary 4.13."},{"cited_title":"If E has positive rank, the lower bound of Proposition 2.13 coincides with the upper bound, hence δ(C × E/k) = δ(C/k)","cited_arxiv_id":null,"evidence_quote":"supplies the genus 2 curve dominating two elliptic curves with fully rational 2-torsion, covering the remaining quadratic-point case of Theorem 4.3."},{"cited_title":"On the density of rational points on rational elliptic surfaces","cited_arxiv_id":"1702.01684","evidence_quote":"supplies the structure theorem for subvarieties of abelian varieties used to prove density of rational points on isogenous abelian surfaces."}],"review_version":1}