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Density of algebraic points on products of curves

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Pith's one-line read This paper proves that on a product $C \times D$ of two nice curves over a number field, every sufficiently large multiple of the index $\mathrm{ind}(C \times D/k)$ lies in the density degree set, with an explicit threshold in terms of…

desk verdict A serious and mostly sound systematic treatment of density degree sets on product surfaces; the reviewer's cited counterexample does not survive contact with the actual proof. read the letter →

arxiv 2507.00860 v1 pith:HX27BCXF submitted 2025-07-01 math.NT math.AG

Jennifer Berg , Yu Fu , Evangelia Gazaki , Morena Porzio , James Rawson , Isabel Vogt This is my paper · ORCID
classification math.NTmath.AG MSC 11G3011G3514G0514K15
keywords densitydegreesetalgebraicpointsproductsofcurvesindexavarietyeffectiveZariskiabeliansurfacesbielliptic
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 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.

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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 $S=(E_1 \times E_2)/G$, $\mathbb{N} \setminus \{1,2\} \subseteq \delta(S/k) \subseteq \delta(E_1/k)$.

Reading between the lines

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

  • 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.
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Referee Report

2 major / 6 minor

Summary. This paper studies the density degree set δ(C×D/k) of closed points of prescribed degree on products of two nice curves over a number field. The main construction (Lemma 3.1, Lemma 3.2, Proposition 3.6) gives explicit families of curves Xγ on C×D with controlled genus and index, leading to an effective threshold N(C,D,e) such that all sufficiently large multiples of ind(C×D/k) lie in δ(C×D/k). The paper also contains low-degree analyses for products of elliptic curves, for products of an elliptic curve with a genus 2 curve, and for products of two genus 2 curves, including examples of non-density, and derives consequences for bielliptic surfaces and for principally polarized abelian surfaces.

Significance. The central asymptotic result is a substantial advance if correct: it provides the first effective description of the tail of δ(C×D/k) for product surfaces, mirroring the curve case δ(C/k)∩N≥2g = ind(C/k)N∩N≥2g. The construction is explicit and uses standard tools (Riemann–Roch, Castelnuovo–Severi, Hilbert irreducibility); it does not rely on circular arguments or parameter fitting. The paper also contains interesting applications and concrete examples. However, several results used for the abelian-surface applications rest on a proof that is not complete (Theorem 6.4), so the full strength claimed is not yet established.

major comments (2)
  1. [Theorem 6.4, Section 6.1] The proof does not establish the stated conclusion N≥2 ⊂ δ(Pic0_C/k). The two ingredients shown are: (i) the image of C in Pic0_C gives 2N ⊂ δ(Pic0_C/k), and (ii) the curve D has degree 3 maps to P1, so 3 ∈ δ(Pic0_C/k). These facts alone do not imply that 5, 7, 11, ... lie in δ(Pic0_C/k); for an abelian surface there is no proved closure of δ under addition or multiplication, and the text does not supply a missing linear-disjointness argument. The final sentence 'Pushing forward Z under the natural maps ... gives a curve in Pic0_C' also does not address whether degree 3 points on D remain degree 3 after the birational but not everywhere isomorphic Abel–Jacobi map from Sym2 C to Pic0_C. Since Corollary 6.6 and the simple case of Theorem 7.3 use this theorem, this is a load-bearing gap. Please supply a complete proof or weaken the statement and its consequences.
  2. [Theorem 7.3, simple case] The proof asserts that for every degree d ≥ 2, the dense degree d points on Pic0_C produced by Theorem 6.4 can be chosen so that their images under multiplication by m have the same field degree. This is verified in the text only for points coming from the curve C via the Abel–Jacobi embedding and for the elliptic-product case; no analogous verification is given for the degree 3 points coming from the curve D, where D is only birationally embedded and the pushforward may identify points. Consequently, the transfer of degrees through the isogeny is not established, and this affects Theorem 1.4.
minor comments (6)
  1. [Theorem 1.1] The displayed threshold contains the typo '3gDgD'; it should be 3gCgD.
  2. [Lemma 3.2] The lemma is stated for ramification points, but Proposition 3.6 applies it at support points where one projection is unramified (ramification index 1). The local argument works with indices n,m ≥ 1, so the lemma should be stated for arbitrary points with local indices n,m (allowing 1), or the wording should be clarified.
  3. [Proposition 3.6] The sentence 'since the greatest common divisor of the degrees of the points of Z is ind(C×D/k)' is true, but it deserves a one-line proof: every closed point degree is a multiple of ind(C×D/k), and deg Z' = ind(C×D/k) is an integer combination of the support degrees, so their gcd divides and is divisible by ind(C×D/k).
  4. [Theorem 6.4] The final sentence refers to 'pushing forward Z', but the object constructed is the curve D (or Zσ); the notation should be corrected.
  5. [Examples and computations] Several numerical claims (Remark 3.7, Example 4.15, Proposition 6.10, Proposition 6.12, Example 5.11) depend on Magma computations without supplied scripts or certificates; including the code or more detailed verification data would improve reproducibility.
  6. [References] Reference [17] is cited with no year or arXiv identifier; for a published or preprint reference, please provide the missing data.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the paper's effective product-surface density theorem is built from independent curve-level results and a self-contained local normalization argument.

full rationale

The central asymptotic theorem (Proposition 3.6 and its pointed case Corollary 3.3) is not obtained by fitting a parameter or by assuming the conclusion. The curve-level density statement ind(C/k)N ∩ N≥2g ⊆ δ(C/k) is imported from [17, Proposition 5.1.1]; although [17] shares an author with the present paper, it is an independent prior theorem whose stated assumptions concern a single curve, not the product surface, so its use does not smuggle in the target result. The product-surface statement is new. Lemma 3.2 is proved by an explicit formal power-series computation (local coordinates, s-th roots, normalization Spec K[[t]]), not by invoking the target. In Proposition 3.6, the ramification indices at the support of the interpolating zero-cycle are coprime whenever both projections are not single rational points: the constructed fibers have coefficients 1 or 2, so at each support point the indices are (1,1), (1,2), or (2,1), and Lemma 3.2 applies with s=1, imposing no condition on γ. The index divisibility ind(Xγ^ν/k) | ind(C×D/k) follows directly from the degrees of the resulting points, which divide the degrees of the support points; this is a divisibility argument, not a restatement of the desired density conclusion. No fitted quantity is relabeled as a prediction, and no uniqueness theorem from the authors' prior work is invoked to force the construction. The paper's use of [17] is real external support, and the low-degree examples in Sections 4–6 are justified by explicit curve covers, cited computations, and theorems attributed to others. The derivation chain is therefore self-contained modulo standard prior results, with no circular step identified.

Assumptions & free parameters 0 free parameters · 7 assumptions · 1 invented entities

No numeric fitting appears in this paper. The main free choices are geometric, such as gonality maps, automorphisms gamma, and parameters lambda and t, rather than fitted constants. The central claims rely on a corpus of prior results on curve density sets, notably from [17], and on standard theorems from algebraic geometry. Several conditional statements depend explicitly on the Parity Conjecture and on a conjecture about isotrivial elliptic fibrations.

assumptions (7)
  • standard math Degree-density structure for curves from [17]: delta(C/k) is multiplicative, contains ind(C/k) N intersect N at least max(2g,1), and decomposes into P1- and AV-parameterized contributions.
    Used throughout Section 2 to bound delta(C times D/k). See Lemma 2.3, Proposition 2.13. The paper does not reprove these; one of the authors coauthored the source.
  • standard math Riemann-Roch plus basepoint-free pencil existence: every divisor of degree at least 2g on a smooth curve gives a basepoint free pencil, and the gonality of a pointed genus g curve is at most g+1.
    Used in Corollary 3.3 and Proposition 3.6 to build covering curves of bounded genus.
  • standard math Castelnuovo-Severi inequality: if a curve has maps of degrees m,n to P1 that do not share a common factor, then the genus is bounded by (m-1)(n-1) in the form used.
    Used repeatedly to rule out hyperellipticity of genus 4, 6, 7, 8, 9, 12, and 36 covering curves; e.g., Theorem 4.9, Proposition 6.1.
  • standard math Hilbert irreducibility theorem and Uniform Position principle.
    Used in Lemma 2.7 to produce degree-d points with Galois group S_d, which is needed to control degree under multiplication and images under isogenies.
  • domain assumption Parity Conjecture for elliptic curves.
    Assumed in Proposition 4.6 and Theorem 5.10; these are flagged as conditional results only.
  • domain assumption Conjecture that non-trivial isotrivial elliptic fibrations over P1 have dense rational points.
    Assumed in Proposition 5.3 for quadratic-point density; explicitly labeled as a conjecture.
  • standard math Faltings-Vojta theorem, i.e., Mordell-Lang for subvarieties of abelian varieties.
    Used in Proposition 7.1 to show rational points on a positive-rank simple abelian variety are dense.
invented entities (1)
  • eff-ind(C times D/k), the effective index of a product surface
    purpose: Defined in Definition 3.5 as the minimal size of the support of a degree ind(C times D/k) zero-cycle; used to state the effective threshold in Proposition 3.6.
    A new technical invariant, not an empirical entity; it has no independent falsifiable handle, but it is a definition rather than a postulated cause.

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Pith. "Pith review of Density of algebraic points on products of curves." pith.science (2026). https://pith.science/paper/HX27BCXF

@misc{pith2026250700860,
  author       = {Pith},
  title        = {Pith review of: Density of algebraic points on products of curves},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HX27BCXF}},
  note         = {Machine review of arXiv:2507.00860}
}
abstract

In this paper, we initiate the systematic study of density of algebraic points on surfaces. We give an effective asymptotic range in which the density degree set has regular behavior dictated by the index. By contrast, in small degree, the question of density is subtle and depends on the arithmetic of the curves. We give several explicit examples displaying these different behaviors, including products of genus $2$ curves with and without dense quadratic points. These results for products of curves have applications to questions about algebraic points on closely related surfaces, such as rank growth on abelian surfaces and bielliptic surfaces.

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