Kodaira-Neron statistics for rational elliptic curves with j-invariant 0 and 1728

Referee: [§3] §3 (height and enumeration): the paper orders curves by the naive height H(E) = max(|c₄|, |Δ|^{1/12}) on minimal Weierstrass models. For the j=0 family the scaling B ↦ u⁶B changes the model while preserving j; the minimality condition at p=3 (v₃(Δ) < 12 after admissible change of variables) restricts the admissible u. It is not shown that the induced distribution on v₃(B) for minimal models still admits the claimed natural density when ordered by H(E); the leading constant in the main asymptotic may therefore be incorrect.

Authors: We agree that the scaling action and minimality constraint at p=3 must be handled explicitly to confirm the natural density. In the manuscript the curves are enumerated by taking B ∈ ℤ such that y² = x³ + B is already minimal at 3, which restricts v₃(B) to a finite set of residue classes modulo 6. Because the height H(E) is a fixed power of |B|, the count reduces to a one-dimensional lattice-point problem with a periodic local condition at 3. This condition is independent of the global height bound and therefore preserves a positive natural density; the leading constant is simply multiplied by the measure of the admissible classes. We will add a short lemma in §3 that computes this density explicitly and verifies that the leading constant in the main theorems is unaffected. revision: yes

Referee: [Theorem 4.2] Theorem 4.2 (asymptotics for Kodaira symbols at p=3): the claimed proportions (e.g., 1/2 for I₀* and 1/6 for II) are derived from the p-adic valuations of B without an explicit error term or Tauberian argument that accounts for the global height bound. Because the family is thin, the usual sieve or lattice-point counting must be justified; the manuscript provides only a sketch and no verification that the error is o of the main term.

Authors: The proof of Theorem 4.2 proceeds by partitioning according to v₃(B) = k and applying the local Kodaira–Néron classification for each k; the global count of admissible B with |B| ≪ X is then summed. While the manuscript gives only a sketch, the underlying count is elementary: the number of integers B with |B| ≤ X and v₃(B) = k is (2X)/3^k + O(1), with the error uniform in k up to the height range. Summing the geometric series over k therefore yields an error O(1) for the main term, which is o(X). We will expand the proof of Theorem 4.2 to include this explicit error bound and the short summation argument. revision: partial

Referee: [§5.1] §5.1 (conditional statistics): when the torsion subgroup is fixed, the same height ordering is used, yet the torsion condition imposes congruence conditions on B (or A) that may correlate with v₃(B) (or v₂(A)). The paper does not prove that these correlations are negligible in the asymptotic; a separate major-term calculation or explicit example for the 3-torsion case would be needed.

Authors: When the torsion subgroup is fixed by a congruence condition modulo m with 3 ∤ m, the Chinese Remainder Theorem makes the condition on B mod m independent of v₃(B), so the joint density factors and the Kodaira proportions remain unchanged. For the 3-torsion case the condition is modulo 9 and does interact with v₃(B). We will add a short explicit computation in §5.1 that enumerates the admissible residue classes modulo 9, recomputes the weighted densities for each Kodaira symbol, and shows that the leading constants are modified by a computable factor (which we record). This supplies the requested major-term calculation and the concrete 3-torsion example. revision: yes