REVIEW 4 minor 16 references
Counting odd genus $2$ curves with a marked rational $3$-torsion point
T0 review · 0 major / 4 minor · reviewed 2026-07-13 · grok-4.5
Pith's one-line read The number of odd monic genus-2 curves over Q with a marked rational 3-torsion point on the Jacobian grows like a positive constant times X to the tenth.
desk verdict Solid first exact asymptotic for marked 3-torsion on odd genus-2 Jacobians; the ad-hoc P(1,2,3,4) compactification works and the proof is classical once the printed Magma non-vanishing is accepted. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
A weighted-homogeneous parametrisation of the pairs (f,T) by four polynomials Theta_0, Theta_1, Theta_2, Theta_3 of weights 4,6,8,10 on the space P(1,2,3,4). These polynomials define a morphism to the space of monic quintics and reduce the counting problem to lattice points of bounded height in a compact semi-algebraic region of R^4, subject to explicit congruence conditions for integrality and minimality.
What would settle it
Re-run the Gröbner-basis computation of the ideal generated by the four Theta polynomials (with A set to 1) over the integers or over the rationals; if the ideal does not contain a non-zero constant, or if an explicit common zero other than the origin is exhibited, the compactness of R(1) and therefore the volume argument fail.
Extended reading notes
Core claim
There exists a positive, effectively computable constant c such that the number of pairs (f,T), where f is a square-free minimal monic quintic of height at most X and T is a non-zero rational 3-torsion point on the Jacobian of y^2 = f(x), equals c X^10 + o(X^10). The constant is the product of the Euclidean volume of a compact region R(1) in R^4, a local density at 2 and 3, and the Euler product over primes p greater than or equal to 5 of (1 - p^{-10}).
Load-bearing premise
The four parameter polynomials have no common complex zero except the origin; this is verified only by a computer Gröbner-basis calculation, and if it fails then the counting region is no longer compact.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper counts monic square-free minimal degree-5 Weierstrass models of genus-2 curves over Q of naive height at most X that carry a marked non-zero rational 3-torsion point on the Jacobian. The main theorem asserts that the number of such pairs is c X^{10} + o(X^{10}), where the leading constant c = Vol(R(1)) eta_{2,3} ∏_{p≥5}(1-p^{-10}) is positive and effectively computable. The proof proceeds by constructing an explicit weighted-homogeneous parametrization of such pairs by points of the weighted projective space P(1,2,3,4), establishing compactness of the fundamental region R(1) via the absence of common zeros of the four coefficient forms, deriving local congruence conditions for p-integrality and p-minimality, and applying a geometry-of-numbers sieve.
Significance. This appears to be the first exact asymptotic for a moduli problem of genus-2 curves with prescribed level structure whose moduli space is three-dimensional rather than a curve. The construction of a base-point-free morphism from the compactification P(1,2,3,4) to the space of Weierstrass models is a concrete technical contribution that may serve as a template for other level structures. Strengths include the fully explicit polynomials Θ0–Θ3, the printed Magma script that verifies the key non-vanishing, the classical and checkable lattice-point and sieve arguments, and the effective computability of the constant. The result is therefore a solid, self-contained advance in arithmetic statistics of abelian surfaces.
minor comments (4)
- The constant eta_{2,3} is asserted to be effectively computable from the congruence conditions of Proposition 4.1, yet no numerical value or even a rough estimate is supplied. A short remark on the practical size of the modulus L_2 L_3, or a pointer to how one would enumerate the residue classes, would make the claim more concrete.
- In the proof of Proposition 4.2 the reduction for p=5 is handled by a separate system of four congruences; it would help the reader if those four polynomials were written out explicitly rather than left as “the same argument”.
- Remark 5.4 sketches the unmarked count and the thin-set argument for full 3-torsion of order 9. The reference to [4, Theorem 1.1] is appropriate, but a one-sentence reminder of what that theorem states for weighted projective stacks would improve readability.
- Typographical consistency: the title and abstract use “odd genus 2 curves” while the body speaks of “odd, monic, genus 2 curves”; a uniform phrase would be preferable. Also, the arXiv identifier in the header is 2607.09483 while the date line reads 10 Jul 2026—presumably a placeholder that should be corrected on publication.
Circularity Check
No circularity: asymptotic count is classical geometry-of-numbers on an explicit weighted-homogeneous parametrization, with volume and local densities defined independently of the count itself.
full rationale
The derivation chain is self-contained and non-circular. Proposition 2.1–2.2 construct an explicit weighted-homogeneous map (A,B,J,E)mapsto F_P from P(1,2,3,4) to monic degree-5 Weierstrass models carrying a marked rational 3-torsion point; the four forms Theta_i are written out in (2.6)–(2.9). Compactness of the region R(1) (Lemma 3.1) follows from the algebraic fact that the only common zero of the Theta_i over C is the origin, verified by a printed Magma Gröbner-basis computation (Appendix A) that produces the constant 9 in the ideal when A=1, together with an elementary hand check when A=0. Lattice-point asymptotics (Lemma 3.2) are Davenport’s Lipschitz principle applied to the weighted dilation of a compact semi-algebraic set. Local densities at p=2,3 are finite congruence conditions (Proposition 4.1) and at p>=5 reduce to the index-p^{10} sublattice of weighted multiples (Proposition 4.2), again using the same Gröbner non-vanishing. The sieve (Section 5) removes the non-minimal and singular loci by standard tail estimates; the leading constant is defined as Vol(R(1)) times the product of those densities and is not fitted to any data. No step reduces a claimed prediction to a fitted parameter, a self-definition, or a load-bearing self-citation; background citations (Davenport, Serre thin sets, earlier elliptic counts) are external and non-essential to the algebraic construction. The Magma script is an independent machine-checkable verification of a polynomial ideal membership, not a circular premise.
Assumptions & free parameters
assumptions (4)
- standard math Davenport’s Lipschitz principle for the number of lattice points in a compact semi-algebraic region of R^{4} (Lemma 3.2).
- domain assumption The four weighted forms heta_{0}, heta_{1}, heta_{2}, heta_{3} have no common zero over C except the origin (Lemma 3.1).
- domain assumption For p≥5, non-minimality of F_P is equivalent to p|A, p^{2}|B, p^{3}|J, p^{4}|E (Prop. 4.2).
- standard math Thin sets in weighted projective stacks contribute o(X^{10}) points (Remark 5.4, citing Chan–Loughran–Rome).
invented entities (2)
-
Weighted-homogeneous polynomials heta_{0}, heta_{1}, heta_{2}, heta_{3} of weights 10,8,6,4 on P(1,2,3,4)
independent evidence
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The compact region R(1) ⊂ R^{4} defined by H_ heta(P)≤1
independent evidence
Cite this review
Pith. "Pith review of Counting odd genus $2$ curves with a marked rational $3$-torsion point." pith.science (2026). https://pith.science/paper/V75PCCSX
@misc{pith2026260709483,
author = {Pith},
title = {Pith review of: Counting odd genus $2$ curves with a marked rational $3$-torsion point},
year = {2026},
howpublished = {\url{https://pith.science/paper/V75PCCSX}},
note = {Machine review of arXiv:2607.09483}
}
abstract
In this paper we count, ordered by naive height, the genus $2$ curves over the rationals which admit a monic Weierstrass model of odd degree and whose Jacobian has a marked rational $3$-torsion point.
Reference graph
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Reviewed July 13, 2026 · model on record in the stance chip above.
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