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The second moment of the size of the $2$-class group of monogenized cubic fields

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Pith's one-line read Over monogenized cubic fields ordered by height, the second moment of the 2-class group is at most 3 for totally real fields, at most 6 for complex fields, and at most 9 for the narrow class group; conditional on a tail estimate, these…

desk verdict A genuine new result: the first second-moment bounds for class groups in degree >2, with the exact values 3, 6, 9 matching the Siad–Venkatesh heuristics; the main soft spot is the unshown verification inside Proposition 3.4. read the letter →

arxiv 2506.05539 v1 pith:QK6RN4ZW submitted 2025-06-05 math.NT

Manjul Bhargava , Arul Shankar , Ashvin Swaminathan This is my paper
classification math.NT MSC 11R1611R2911G0511R37
keywords monogenizedcubicfields2-classgroupnarrowclasssecondmomentSelmerstructuresellipticcurvesbinaryformsheuristics
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

This paper tries to establish the first proven upper bounds for the second moment of a class group in degree greater than two. For monogenized cubic fields—cubic fields whose rings of integers are generated by a single specified element—ordered by height, it proves that the average of the squared size of the $2$-torsion subgroup of the class group is at most $3$ over totally real fields and at most $6$ over complex fields, and that the same moment for the narrow $2$-class group in the totally real case is at most $9$. Conditional on a tail estimate controlling very large class groups, these upper bounds are the exact values. If correct, the numbers confirm heuristics built on spin structures and show that imposing monogenicity changes not just the average but the spread of class group distributions.

What carries the argument

The central object is the unramified Selmer structure on the family of elliptic curves associated to monogenized cubic fields. A Selmer structure assigns to every place of $\mathbb{Q}$ a subgroup of local $2$-torsion cohomology, and the unramified one takes the subgroup of classes that become trivial on inertia. The argument depends on the theorem that this structure is sub-soluble: at every place, every everywhere-unramified class lies in the image of the local Kummer map. With that containment, the paper can apply the general second-moment inequality for Selmer structures, $\operatorname{Avg}^{(2)}(2;E;S) \le 1+6M(2;E;S)+8M(2;E;S)^2$, and the local masses computed from the five possible local étale algebra splittings carry the arithmetic.

What would settle it

Enumerate the five possible local splittings of a monogenized cubic étale algebra over $\mathbb{Q}_p$—the unramified cases $\mathbb{Q}_p^3$, $\mathbb{Q}_p \oplus \mathbb{Q}_{p^2}$, $\mathbb{Q}_{p^3}$ and the ramified cases $\mathbb{Q}_p \oplus F_{p,2}$, $F_{p,3}$—and for each compute the unramified subgroup of $H^1(\mathbb{Q}_p, E[2])$ alongside the image of the Kummer map. Proposition 3.4 asserts these coincide for every prime; one mismatch would sever the bridge from class groups to Selmer groups and invalidate the proof of the moment bounds.

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Extended reading notes

Core claim

The central claim is that the second moment of $\#\mathrm{Cl}_2(K)$ over monogenized cubic fields with $S_3$ normal closure, ordered by height, is at most $3$ in the totally real case, at most $6$ in the complex case, and the second moment of the narrow $2$-class group $\#\mathrm{Cl}_2^+(K)$ is at most $9$ in the totally real case; with a tail estimate, these are the exact values. The proof identifies $\mathrm{Cl}_2(K)$ with the Selmer group attached to the everywhere-unramified Selmer structure on the elliptic curve $E_f\colon y^2=f(x)$, where $f$ is the defining binary cubic form of the monogenized field. The crucial local theorem is that this unramified Selmer structure is sub-soluble, so the class group sits inside the usual $2$-Selmer group. Applying a general second-moment bound for Selmer groups, with local masses equal to $1/4$ (totally real) or $1/2$ (complex) at infinity and $1$ at every finite prime, yields the bounds $3$ and $6$. The narrow class group is handled by expressing $\#\mathrm{Cl}_2^+(K)$ through two sub-soluble Selmer structures and inclusion–exclusion, producing $9$.

Load-bearing premise

The proof depends on the local check that at every place, every unramified $2$-torsion cohomology class of the associated elliptic curve is already visible as a local point; if this failed at a single prime, the class group would not sit inside the $2$-Selmer group and the moment bound would not apply.

Editorial extensions

If this is right

  • For totally real monogenized cubic fields, the second moment of the $2$-class group is exactly $3$ if the tail estimate holds, so the distribution of pairs of $2$-torsion classes is fully constrained by the first moment plus this one number.
  • For complex monogenized cubic fields, the exact second moment would be $6$, twice the totally real value, matching the prediction of the spin-structure heuristics.
  • The narrow $2$-class group second moment in the totally real case is $9$; together with the known average size $5/2$, this pins down the variance of the narrow distribution.
  • The same upper bounds persist when the average is taken over monogenized cubic fields satisfying arbitrary prescribed local splitting conditions at finitely many primes, or certain families of such conditions at infinitely many primes.
  • These are the first proven second-moment bounds for class groups of number fields of degree greater than two, giving a concrete benchmark against which future distributional heuristics can be tested.

Reading between the lines

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

  • The Selmer-structure formulation suggests the same machinery could yield higher moments of $\mathrm{Cl}_2$ over monogenized cubic fields; Theorem 2.1 currently covers only first and second moments, but the orbit-parametrization strategy indicates a structured moment sequence rather than an ad hoc bound.
  • If the exactness conditional on the tail estimate is confirmed, the contrast with ordinary cubic fields (where heuristics predict second moments $15/8$ and $3$) quantifies how strongly monogenicity distorts $2$-class group statistics: it roughly doubles the second moment in both signatures.
  • The local sub-soluble check is finite in nature—five possible splittings of the local étale algebra—so a computer-assisted verification over all primes could strengthen confidence in Proposition 3.4 independently of the analytic arguments.
  • Analogous unramified Selmer structures may exist for monogenized fields of odd degree $n\ge 3$, where average-size results are known but second moments are not; this paper's local-mass framework is a natural template for those cases.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. This paper proves upper bounds for the second moment of the size of the 2-class group and narrow 2-class group of monogenized cubic fields ordered by height: at most 3 in the totally real case, at most 6 in the complex case, and at most 9 for the narrow 2-class group in the totally real case; conditional on a tail estimate, these bounds are asserted to be exact. The proof associates to a monogenized cubic field K a binary cubic form f and an elliptic curve E_f, realizes the dual 2-class group as an everywhere-unramified Selmer group, and applies a general theorem (Theorem 2.1) bounding the second moment of Selmer groups attached to sub-soluble 2-Selmer structures. The local masses are computed as 1/4 at the real place in the totally real case, 1/2 in the complex case, and 1 at all finite places; substitution into 1+6M+8M^2 yields 3, 6, and, after an inclusion-exclusion argument, 9.

Significance. If fully justified, the result would be the first proven second-moment bound for class groups in degree greater than two and would provide numerical confirmation of the Siad-Venkatesh spin-structure heuristics. The paper's main virtues are explicit: the constants 3, 6, and 9 are not fitted but emerge from local masses, the reduction from class groups to Selmer groups is conceptually clean, and the upper-bound statements are sharp enough to be falsified by computation. The full force of the paper, however, depends on the companion paper [9] for the Selmer-moment inequality and on a local sub-solubility statement (Proposition 3.4) whose proof is currently sketched rather than verified.

major comments (3)
  1. [§3.1, Proposition 3.4] Proposition 3.4 is the load-bearing bridge from class groups to Selmer moments, but its proof is incomplete. The sentence 'Since Condition 4 in [21, Lemma 5.1] is easily verified' does not constitute a verification, and the hypotheses needed to apply [21, Theorem 3.4] and obtain the isomorphism (6) are not checked. The subsequent five-case analysis also contains unjustified assertions, for example the claim in the ramified cases that E(M)[2] = E(Q_p)[2] with trivial τ-action. If H^1_ur(Q_p,E[2]) fails to be contained in the Kummer image for even one of the five local algebras in Lemma 3.3, then Sel_S(E_f) need not be contained in Sel_2(E_f), and Theorem 2.1 cannot be applied. Please supply a complete case-by-case proof or a precise reference covering exactly this family.
  2. [§2.3, Theorem 2.1] The third displayed formula in Theorem 2.1 is the quantitative engine of the paper, but its proof is a sketch that imports the crucial orbit count from the companion paper [9]. The asserted contribution 8M(2;E;S)^2 from pairs of nontrivial elements is not derived here, and the compatibility of the sub-soluble local conditions with the parametrization of special irreducible orbits is only asserted. Since the numerical bounds 3, 6, and 9 are obtained by substituting M=1/4 or 1/2 into 1+6M+8M^2, both the coefficient 8 and the square M^2 are load-bearing. The paper should either include the full proof of this formula or state the exact theorem from [9] on which it relies. In addition, the 'tail estimate' under which equality is claimed is never formulated; it should be stated explicitly so that the conditional part of Theorem 1.2 is checkable.
  3. [§3.2, Proposition 3.6] Equations (9) and (10) of Proposition 3.6 are not rigorously justified. The groups Sel_{\tilde S}(E_f)(E_f) and Sel_{\tilde S}(E_{f^*})(E_{f^*}) are subgroups of cohomology groups attached to different elliptic curves, so their union, intersection, and the displayed equality 'Sel_{\tilde S}(E_f)(E_f) = Sel_S(E_f)(E_f) or Sel_{\tilde S}(E_{f^*})(E_{f^*}) = Sel_S(E_f)(E_f)' are not meaningful without additional identifications. The intended inclusion-exclusion must be formulated via the bijections in (7) and (8), and the claimed intersection with Sel_S(E_f) must be proved. This gap directly affects Theorem 1.2(c), which depends on the size and square formulas (9)-(10).
minor comments (5)
  1. [§2.2, item (b)] The soluble local condition should read S_v = Im(δ_v), not Im(δ).
  2. [§1.4, Case 2] The text 'the average number of choices for π_2' should read α_2.
  3. [Equation (3)] The normalization for v=∞ reads 'Vol(Z^2\R^2)=1', which is unclear; presumably the intended statement is about Haar measure on R^2 normalized so that the quotient R^2/Z^2 has volume 1.
  4. [§3.1, Lemma 3.3 and Proposition 3.4] The phrases 'easily seen' and 'easily verified' should be replaced with a table of the five local possibilities for K_p and the corresponding sizes of H^1_ur and E(Q_p)[2], so that the local mass computation is independently checkable.
  5. [§3.2] The statement that for negative-definite g the curve z^2=g(x,y) is soluble over R 'up to quadratic twist' is imprecise; a negative-definite quartic has no real points, and the intended twist should be described explicitly.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; the second-moment bounds 3, 6, and 9 follow from the Selmer-moment formula and local masses, with no fitted constants.

full rationale

The derivation is self-contained in the relevant sense. Theorem 1.2 is obtained by applying Theorem 2.1, the general second-moment bound Avg^(2)(2;E;S) ≤ 1 + 6M + 8M^2 for acceptable and sub-soluble Selmer structures, to the unramified Selmer structure S defined in Section 3.1. Proposition 3.2 identifies Cl_2(K) with H^1_ur(Q,E_f[2]), Proposition 3.4 verifies sub-solubility, and the local masses are computed directly: M_∞ = 1/4 for E^+ and 1/2 for E^- from (11), and M_p = 1 from (12). Substituting gives 1 + 6(1/4) + 8(1/16) = 3 and 1 + 6(1/2) + 8(1/4) = 6, with 9 obtained similarly via inclusion-exclusion for the narrow class group. No parameter is fitted to the target moments; the output values emerge from the fixed formula and local group sizes. The reliance on the authors' earlier paper [9] for the Selmer second-moment inequality is real but non-circular: [9] treats ordinary 2-Selmer groups of elliptic curves, a different object, with stated assumptions that do not include the class-group conclusion, and the present paper extends and proves the needed Selmer-structure variant in Theorem 2.1. This is independent support, not a restatement of Theorem 1.2. The only fragile point is Proposition 3.4's appeal to Schaefer [21] with the sentence 'Condition 4 in [21, Lemma 5.1] is easily verified' without showing the verification; that is a correctness or completeness risk about an external input, not a circular reduction. The 'conditional on a tail estimate' clause is explicitly stated and not disguised as an unconditional exact value. Thus there are no circular steps of the kinds enumerated.

Assumptions & free parameters 0 free parameters · 8 assumptions · 0 invented entities

No free parameters are fitted; the constants 3, 6, 9 arise from substituting local masses 1/4 or 1/2 into a fixed inequality. The heavy lifting is outsourced to cited papers, especially the authors' own [9] and [7,8,5,6], which are treated as proven black boxes; the new content is the Selmer-structure interpretation, the sub-solubility lemma, and the local mass evaluations.

assumptions (8)
  • standard math Class field theory: bijection between unramified quadratic extensions of a number field and characters of its class group.
    Used in Propositions 3.2 and 3.5 to identify Cl*(K)[2] with H^1_ur(Q, E_f[2]).
  • standard math Kummer theory: H^1(F, Res_{K/F} mu_2) is isomorphic to K^times / K^{times 2}.
    Lemma 3.1, foundational for the cohomological parametrization.
  • standard math Geometry-of-numbers asymptotics for binary quartic and binary cubic forms with infinitely many local conditions (Bhargava-Shankar and related papers).
    Used in Theorem 2.1 to count irreducible locally soluble orbits; proved in [7,8,5,6].
  • standard math Second-moment inequality for 2-Selmer groups of elliptic curves, as proven in the companion paper [9].
    The black-box '8M^2' term in Theorem 2.1; not proved in this manuscript.
  • standard math Wood's theorem: a quartic ring with monogenic cubic resolvent is the global-section ring of a binary quartic form.
    Used in §3.2 to parametrize narrow-class-group characters by quartic forms.
  • standard math Schaefer's lemma: computation of the intersection of unramified cohomology with the Kummer image for elliptic curves over Q_p.
    Used in the proof of Proposition 3.4 to show sub-solubility.
  • standard math Bhargava-Gross: a class in H^1_{f-ur} is soluble at R if and only if the associated genus-1 curve has an R-point.
    Used in §3.2 to control the archimedean contribution to the narrow class group.
  • standard math Armitage-Frohlich: for totally real cubic K, the kernel of Cl^+(K)[2] -> Cl(K)[2] has order 1 or 2.
    Used in §3.2 to bound the archimedean image.

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Pith. "Pith review of The second moment of the size of the $2$-class group of monogenized cubic fields." pith.science (2026). https://pith.science/paper/QK6RN4ZW

@misc{pith2026250605539,
  author       = {Pith},
  title        = {Pith review of: The second moment of the size of the $2$-class group of monogenized cubic fields},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QK6RN4ZW}},
  note         = {Machine review of arXiv:2506.05539}
}
abstract

We prove that when totally real (resp., complex) monogenized cubic number fields are ordered by height, the second moment of the size of the $2$-class group is at most $3$ (resp., at most $6$). In the totally real case, we further prove that the second moment of the size of the narrow $2$-class group is at most $9$. This result gives further evidence in support of the general observation, first made in work of Bhargava--Hanke--Shankar and recently formalized into a set of heuristics in work of Siad--Venkatesh, that monogenicity has an altering effect on class group distributions. All of the upper bounds we obtain are tight, conditional on tail estimates.

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Reference graph

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