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Asymptotic independence of class-group 4-ranks in correlated pairs of imaginary quadratic fields

T0 review · 2 major / 3 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read The paper proves that the class-group $4$-ranks of $\mathbb{Q}(\sqrt{-d})$ and $\mathbb{Q}(\sqrt{-d_0 d})$ are asymptotically independent as $d$ varies, even though the two fields share every variable ramified prime.

desk verdict The main 4-rank independence theorem is solid and new; the biquadratic application rests on an unverified sign correction to an unpublished preprint and should be conditional. read the letter →

arxiv 2608.00387 v1 pith:A4QABK5V submitted 2026-08-01 math.NT

Yue Xu , Xiuwu Zhu This is my paper · ORCID
classification math.NT MSC 11R2911R1115B5260B20
keywords classgroup4-rankCohen-Lenstra-MartinetheuristicsbiquadraticfieldRedeimatrixasymptoticindependenceGaussian-binomialmomentstruncatedmomentinversion
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

Fix a squarefree multiplier $d_0>1$ and let $d$ range over squarefree integers coprime to $2d_0$. Although the two imaginary quadratic fields $\mathbb{Q}(\sqrt{-d})$ and $\mathbb{Q}(\sqrt{-d_0 d})$ ramify at exactly the same variable primes, the paper proves that their class-group $4$-ranks are asymptotically independent: over $d\le X$ the joint distribution converges in total variation to the product of two copies of the Cohen–Lenstra–Gerth distribution, with error $O((\log\log X)^{-c})$. The $4$-rank is the first level at which such a statement can hold, since genus theory forces a deterministic relation between the two raw $2$-ranks. The mechanism is a reduction to finite-field random matrix theory: after deleting one common prime, the two Rédei matrices share a common core, the fixed twist by $d_0$ acts as a diagonal perturbation, and the corank of a random symmetric matrix is shown to be asymptotically insensitive to a high-rank diagonal shift. If the main theorem is right, correlated ramification does not force correlated class-group statistics, and the paper uses this to deduce a limiting distribution for the $8$-rank of the biquadratic field $K(d)=\mathbb{Q}(\sqrt{d_0},\sqrt{-d})$.

What carries the argument

The central object is the pair of coupled, fixed-width bordered Rédei matrices attached to the two fields. A Rédei matrix for a quadratic field records the additive quadratic-residue symbols $[\Delta_i/p_j]$ among its prime-discriminant factors, and the Rédei–Reichardt criterion identifies the class-group $4$-rank with the corank of this matrix after deleting one variable row and column. Because the two fields share every variable ramified prime, deleting one common odd prime leaves a common core matrix; the fixed twist by $d_0$ appears as a diagonal perturbation, and the primes dividing $2d_0$ contribute a bounded number of border rows and columns. The argument has two stages. First, the box-method equidistribution theorem of [31] shows that the permutation-averaged distribution of the relevant residue symbols over admissible boxes is close to uniform, so the arithmetic problem becomes the joint corank distribution of the random matrix pair. Second, a quantitative truncated inversion of the Gaussian-binomial moment transform — the mixed moments $\mathbb{E}[\binom{Z_1}{u}_2\binom{Z_2}{v}_2]$ counting average kernel subspaces — transfers mixed-moment estimates through a growing order $D$ into an exponential total-variation bound $\ll\eta^r$. The key finite-field fact is that the corank pair of a uniform symmetric matrix and its high-rank diagonal perturbation is asymptotically distributed as $\pi_{\mathrm{CL}}\otimes\pi_{\mathrm{CL}}$; the borders and stratum data are uniformized over and absorbed into the constants.

What would settle it

For Theorem 1.1: with $d_0=2$ and $X$ of size around $10^9$–$10^{10}$, tabulate the empirical joint distribution of $(r_4(-d),r_4(-2d))$ over squarefree odd $d\le X$ and measure its total-variation distance to $\pi_{\mathrm{CL}}\otimes\pi_{\mathrm{CL}}$; the theorem predicts decay like $(\log\log X)^{-c}$, so a distance that does not shrink as $X$ grows would refute it. For Lemma 5.5: verify the Selmer dimension identity independently in the negative-twist range for a concrete real quadratic field $F$ (for instance $F=\mathbb{Q}(\sqrt{5})$ and small squarefree $|n|$) by computing the three Selmer groups in (82); any instance where the terminal $-1$ of [16, Theorem 3.4] is needed, rather than omitted, falsifies the correction and with it Theorem 1.3.

Watch

Extended reading notes

Core claim

The paper's principal theorem, Theorem 1.1, states that for every squarefree $d_0>1$ there is $c=c(d_0)>0$ with $d_{\mathrm{TV}}(\mu_X^{(d_0)},\pi_{\mathrm{CL}}\otimes\pi_{\mathrm{CL}})\ll_{d_0}(\log\log X)^{-c}$, where $\mu_X^{(d_0)}$ is the joint distribution of $(r_4(-d),r_4(-d_0 d))$ over squarefree $d\le X$ coprime to $2d_0$ and $\pi_{\mathrm{CL}}$ is the Cohen–Lenstra–Gerth distribution on $\mathbb{Z}_{\ge 0}$, with masses $\pi_{\mathrm{CL}}(m)=2^{-m^2}\eta_\infty(2)/\eta_m(2)^2$. The two $4$-ranks are therefore asymptotically independent, each with the Cohen–Lenstra–Gerth law. The authors conjecture the group-valued refinement that the corrected $2$-primary groups $2\mathrm{Cl}_{\mathbb{Q}(\sqrt{-d})}[2^\infty]$ and $2\mathrm{Cl}_{\mathbb{Q}(\sqrt{-d_0 d})}[2^\infty]$ are asymptotically independent with the Cohen–Lenstra measure on $2$-groups; Theorem 1.1 is the $4$-rank projection of this conjecture. Under the hypothesis that $\mathbb{Q}(\sqrt{d_0})$ has odd class number, Theorem 1.3 shows that for a density-one subset of $d$ extension of ideals induces $4\mathrm{Cl}_{K(d)}[2^\infty]\cong 2\mathrm{Cl}_{\mathbb{Q}(\sqrt{-d})}[2^\infty]\oplus 2\mathrm{Cl}_{\mathbb{Q}(\sqrt{-d_0 d})}[2^\infty]$, and Corollary 1.5 derives that the $8$-rank of $\mathrm{Cl}_{K(d)}$ has limiting distribution equal to the convolution $\pi_{\mathrm{CL}}*\pi_{\mathrm{CL}}$.

Load-bearing premise

The load-bearing premise for the biquadratic results (Theorem 1.3 and Corollary 1.5) is the paper's un-reproduced correction of the Selmer-dimension identity of [16, Theorem 3.4] for a real quadratic base field with negative twist — Lemma 5.5 asserts that the terminal $-1$ must be omitted in that sign range — and if this correction is wrong those results fail, although Theorem 1.1 itself does not depend on it.

Editorial extensions

If this is right

  • The joint distribution of $(r_4(-d),r_4(-d_0 d))$ over squarefree $d\le X$ coprime to $2d_0$ converges in total variation to $\pi_{\mathrm{CL}}\otimes\pi_{\mathrm{CL}}$ at rate $(\log\log X)^{-c}$: the two $4$-ranks are asymptotically independent despite sharing every variable ramified prime.
  • Projecting onto either coordinate recovers the known one-field Cohen–Lenstra–Gerth $4$-rank distribution, so the theorem is consistent with and refines the established marginals.
  • With $d_0=2$, the pairing $d\mapsto 2d$ runs over all imaginary quadratic fields (odd and even squarefree radicands, respectively), and their $4$-ranks are asymptotically independent under this natural pairing.
  • Assuming odd class number of $\mathbb{Q}(\sqrt{d_0})$, the $8$-rank of $\mathrm{Cl}_{K(d)}$ has limiting distribution $\pi_{\mathrm{CL}}*\pi_{\mathrm{CL}}$ with total-variation convergence, and a density-one subfamily satisfies $\mathrm{rk}_8\,\mathrm{Cl}_{K(d)}=r_4(-d)+r_4(-d_0 d)$.
  • If Conjecture 1.2 holds, the corrected $2$-primary groups are jointly two independent Cohen–Lenstra $2$-groups, and Theorem 1.3 turns this into a corrected Cohen–Lenstra–Martinet prediction for $4\mathrm{Cl}_{K(d)}[2^\infty]$ in the biquadratic family.

Reading between the lines

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

  • The mechanism isolates a random-matrix principle that outlives this family: the corank of a uniform symmetric matrix over $\mathbb{F}_2$ is asymptotically independent of a diagonal perturbation whose support has positive density (Theorem 1.6). I take this as evidence that fixed-multiplier pairings of quadratic fields exhibit asymptotic independence at all $2^j$-rank levels, not just $j=2$, exactly
  • The truncated Gaussian-binomial inversion is run at depth $D\asymp\sqrt{n}$ in the bordered case because the target moments decay like $2^{-u^2}$ (case II of Theorem 4.12). Since these moments at growing order determine the full finite kernel structure, the same input looks sufficient to prove the group-valued conjecture rather than only its $4$-rank projection, although the paper does not carry t
  • The $u_{-1}=0$ subfamily — $d$ with all primes $1\bmod 4$ — is left open at the bordered level (Remark 4.22): with zero symmetry defect the limiting mixed moments of bordered pairs are not computed, and the limit need not be $\pi_{\mathrm{sym}}\otimes\pi_{\mathrm{sym}}$ once corners are present. Computing those moments would yield a conditional independence statement for the all-$1\bmod 4$ family,
  • Remark 3.9 indicates the same apparatus can run under fixed Frobenius conditions; because the target distribution then changes (it is governed by conditional residue assignments), any such conditional independence theorem would be a genuinely new prediction rather than a repackaging of Theorem 1.1.
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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

2 major / 3 minor

Summary. The paper fixes a squarefree integer d0 > 1 and studies the joint distribution of the class-group 4-ranks of the two imaginary quadratic fields Q(√−d) and Q(√−d0d), where d ranges over squarefree integers coprime to 2d0. The principal theorem (Theorem 1.1) states that this joint distribution converges in total variation to πCL ⊗ πCL, the product of two Cohen–Lenstra–Gerth distributions, with error O((log log X)^−c). The proof combines Rédei–Reichardt matrix reduction, Smith's box-method equidistribution for residue symbols, and a new random-matrix theorem for the corank pair of two coupled bordered matrices sharing a common core. The paper also proves, under the hypothesis that Q(√d0) has odd class number, that for a density-one subset of d the map to 4Cl_{K(d)}[2∞] decomposes as a direct sum of the corrected 2-primary groups of the three quadratic subfields (Theorem 1.3), and derives the limiting 8-rank distribution for the biquadratic field (Corollary 1.5). The biquadratic part relies on Lemma 5.5, a sign-range correction to an identity of Koymans–Morgan–Smit [16] that is only sketched in the paper.

Significance. If correct, Theorem 1.1 is a substantial advance: it establishes asymptotic independence of 4-ranks in a naturally correlated family of imaginary quadratic fields, with an explicit total-variation rate, and it gives a corrected Cohen–Lenstra–Martinet prediction for biquadratic fields that is verified unconditionally at the 8-rank level. The central derivation is detailed and internally consistent: the target distributions are externally defined rather than fitted, the random-matrix limiting moments are computed independently, and the error terms are explicit throughout. The proof of Theorem 1.1 rests on the box-method transfer of Section 3 and the bordered random-matrix theorem of Section 4; I found no concrete flaw in that part. The advertised biquadratic applications, however, depend on Lemma 5.5, which is load-bearing and is not fully proved in the manuscript.

major comments (2)
  1. [§5.2.3, Lemma 5.5, Eq. (82)] This lemma is load-bearing for Theorem 1.3 and Corollary 1.5 through Proposition 5.6, but its proof is only a sketch. The paper asserts that the terminal '-1' in [16, Theorem 3.4] must be omitted when n < 0 and that 'the same generic set' used by Koymans–Morgan–Smit works, yet it does not reproduce the Selmer dimension computation or the verification that the generic-set conditions are unchanged in the negative sign range. Since [16] is an unpublished arXiv preprint, this correction cannot be checked by citation alone. I recommend either giving a complete proof of (82) in this paper or citing a published version of [16] that explicitly covers the real-base negative-twist case. This issue does not affect Theorem 1.1: Sections 3–4 and 5.1 never invoke Lemma 5.5.
  2. [§5.2.1, Lemma 5.1] Lemma 5.1, quoted from [16, Corollary 2.3], is used in the density-one proofs of the Hasse unit index and the 2-rank formula. The statement is standard and the proof is short, but because it is currently cited only from an unpublished source, it should be proved in the paper or replaced by a published reference. This is a secondary point relative to Lemma 5.5, but it is part of the same reliance on [16].
minor comments (3)
  1. [Throughout] The word 'sufficiently' is consistently misspelled as 'sufficiently' (see Definition 3.4, Proposition 3.5, and elsewhere); this should be corrected.
  2. [Title, page 1] The title appears as 'CORRELA TED P AIRS' with spurious spaces; the typesetting should be fixed.
  3. [Remark 4.22] The remark that bordered sign vectors with |u_-1|/r → 0, including u_-1 = 0, 'require separate analysis' could be misread as contradicting Theorem 1.1. It would help to state explicitly that in the arithmetic application these sign vectors have mass O(e^{-cr}) in the sign mixture (33), so their omission from the matrix theorem does not affect Theorem 1.1.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: Theorem 1.1 is derived from independent box-method and random-matrix estimates; the flagged Selmer correction is an external dependency, not a circular step.

full rationale

The paper's central claim, Theorem 1.1, asserts convergence of the joint 4-rank distribution to the externally defined Cohen–Lenstra–Gerth product πCL ⊗ πCL. This target is not fitted: πCL is the classical distribution (3), and the product is the conjectured independent limit. The proof proceeds by Smith's box method (Propositions 3.5 and 3.6) to replace arithmetic residue assignments by uniform random assignments, then by a self-contained random-matrix theorem (Theorem 4.13) for the joint corank of the bordered matrix pair. The random-matrix argument computes Gaussian-binomial mixed moments directly (Propositions 4.15 and Lemma 4.18) and applies a quantitative truncated inversion (Theorem 4.12). No parameter is tuned to force the limit, and the limiting moments are not imported from the conclusion. The only self-citation, to the authors' earlier work [36], is explicitly described as insufficient for the coupled problem and is used only for comparison, not as an input: 'These one-parameter methods do not directly close for the coupled bordered matrix pair above.' Section 5.2's Lemma 5.5 depends on Koymans–Morgan–Smit [16] with a claimed sign correction, and this dependency is load-bearing for Theorem 1.3 and Corollary 1.5 but not for Theorem 1.1. That is an external, partly sketched correctness concern, not circularity, because [16] is not defined in terms of the paper's own conclusions and the correction is asserted rather than fitted. Overall, the derivation chain is self-contained for the main independence theorem, and no step reduces by construction to its own input.

Assumptions & free parameters 2 free parameters · 6 assumptions · 0 invented entities

The central theorem rests on external machinery, mainly Smith's box method and standard analytic number theory. No new particles, forces, or arithmetic objects are postulated. The auxiliary constants are chosen to satisfy inequalities in Smith's framework, not tuned to the final distribution. The paper-specific modification of the Koymans-Morgan-Smit formula is the most delicate input and is not fully proved in the text.

free parameters (2)
  • Smith box-method auxiliary constants c1..c12, ccut, creg = Fixed numerical values, e.g., ccut = 1/200, c1 = 1000
    Chosen in Section 3 to satisfy Smith's abstract parameter inequalities. They do not enter the final theorem statement or the error exponent, and they are not fitted to data.
  • Rank threshold alpha = 1/4 in the proof of Theorem 1.1
    Any fixed alpha in (0,1/2) works throughout; the choice only controls the Hoeffding tail in Theorem 3.8 and is not fitted to the target distribution.
assumptions (6)
  • domain assumption Smith's box method for equidistribution of residue symbols and positive covering
    Used in Propositions 3.1, 3.3, 3.5 and 3.6 to transfer residue-symbol equidistribution to the arithmetic family. This is external heavy machinery from [31].
  • domain assumption Koymans-Morgan-Smit 4-rank formula and its Selmer framework for K(sqrt(n))
    Used in Lemma 5.5 and Proposition 5.6 for the biquadratic application. The paper modifies the formula for negative twists, and the cited source is an unpublished arXiv preprint.
  • domain assumption One-field Cohen-Lenstra-Gerth theorems of Gerth and Smith
    Used as marginal benchmark for the 4-rank distribution of a single imaginary quadratic field, and as target for the random matrix corank distribution.
  • standard math Sathe-Selberg and Erdős-Kac estimates for the number of prime factors
    Used in Proposition 3.1 to localize the family to an interval around log log X and to estimate exceptional prime-factor geometry.
  • standard math Landau separation theorem and Siegel exceptional zero bounds
    Used in Proposition 3.3 and Definition 3.4 to exclude boxes containing exceptional real characters.
  • standard math Kuroda's class number formula and the ambiguous class number formula
    Used in Section 5.2 to relate class numbers of K(d) to its quadratic subfields and to compute 2-ranks.

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Pith. "Pith review of Asymptotic independence of class-group 4-ranks in correlated pairs of imaginary quadratic fields." pith.science (2026). https://pith.science/paper/A4QABK5V

@misc{pith2026260800387,
  author       = {Pith},
  title        = {Pith review of: Asymptotic independence of class-group 4-ranks in correlated pairs of imaginary quadratic fields},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/A4QABK5V}},
  note         = {Machine review of arXiv:2608.00387}
}
abstract

Fix a squarefree integer $d_0>1$, and let $d$ range over the positive squarefree integers coprime to $2d_0$. Although $\mathbb{Q}(\sqrt{-d})$ and $\mathbb{Q}(\sqrt{-d_0d})$ share all variable ramified primes, we prove that their class-group $4$-ranks are asymptotically independent. Over the subfamily $d\le X$, their joint distribution converges in total variation to the product of two copies of the Cohen--Lenstra--Gerth distribution, with error bounded by a negative power of $\log\log X$. We further conjecture that the corrected $2$-primary groups $2\operatorname{Cl}_{\mathbb{Q}(\sqrt{-d})}[2^\infty]$ and $2\operatorname{Cl}_{\mathbb{Q}(\sqrt{-d_0d})}[2^\infty]$ are asymptotically independent, each with the Cohen--Lenstra distribution. Suppose in addition that the class number of $\mathbb{Q}(\sqrt{d_0})$ is odd. For a density-one subset of this family, we prove that extension of ideals to $K(d)=\mathbb{Q}(\sqrt{d_0},\sqrt{-d})$ induces $4\operatorname{Cl}_{K(d)}[2^\infty]\cong 2\operatorname{Cl}_{\mathbb{Q}(\sqrt{-d})}[2^\infty]\oplus 2\operatorname{Cl}_{\mathbb{Q}(\sqrt{-d_0d})}[2^\infty]$. Together with this decomposition, the group-valued conjecture predicts that $4\operatorname{Cl}_{K(d)}[2^\infty]$ is distributed as the direct sum of two independent Cohen--Lenstra $2$-groups, giving a corrected Cohen--Lenstra--Martinet distribution for the biquadratic family. Unconditionally, the $8$-rank of $\operatorname{Cl}_{K(d)}$ has limiting distribution given by the convolution of two copies of the Cohen--Lenstra--Gerth distribution. The proof combines Smith's box method with quantitative truncated Gaussian-binomial moment inversion for diagonally coupled, fixed-width bordered R\'edei matrices.

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