Hecke reciprocity and class groups

Ari Shnidman; Artane Siad

REVIEW 3 major objections 5 minor 14 references

Hecke reciprocity and class groups

T0 review · 3 major / 5 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read A reciprocity law for Hecke primes fixes the average 2-torsion class group sizes of pure cubic fields at 2 and 3/2.

desk verdict Strong paper with a real new idea and first exact averages, but the abstract and Theorem 1.2 swap the tame/wild averages, and the counting step in §7 needs an unverified weighting hypothesis fixed before I'd trust the numbers. read the letter →

arxiv 2506.13749 v2 pith:ZX47ZSJQ submitted 2025-06-16 math.NT

Ari Shnidman , Artane Siad This is my paper · ORCID

classification math.NT MSC 11R1611R2911R45

keywords classgroup2-torsionpurecubicfieldsHeckereciprocityKummerextensionsCohen-Martinetheuristicsarithmeticinvarianttheoryquadraticrefinementorbitcounting

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 claims that the average size of the 2-torsion subgroup of the class group of pure cubic fields $F_n=\mathbb{Q}(\sqrt[3]{n})$ is $2$ in the wild family ($n\not\equiv\pm 1 \pmod{9}$) and $3/2$ in the tame family ($n\equiv\pm 1 \pmod{9}$), when the fields are ordered by $n$. It derives both averages from a single parity law: an everywhere unramified quadratic extension of an odd-degree field must split in an even number of Hecke primes. That law explains why the tame family, which has exactly one Hecke prime above $3$, loses one local square-class and therefore has the smaller average. The same machinery gives average $3/2$ over $K=\mathbb{Q}(\sqrt{-3})$, matching the Cohen–Martinet prediction, and supplies new heuristics for families of $\Gamma$-extensions. The abstract prints the tame/wild labels in the opposite order from §7; the body's theorems are the ones summarized here.

What carries the argument

The load-bearing object is the Hecke ideal $H_{F/K}=\mathrm{Disc}_{F/K}\mathrm{Diff}_{F/K}^{-1}$, together with the associated Hecke primes, and the quadratic refinement $q_n\colon H^1(K,M_n)\to \mathrm{Br}(K)[2]$ attached to the Kummer extension $F_n=K(\sqrt[3]{n})$. The refinement sends a square-class $t$ to the class of the quadratic form $\frac{1}{3}\mathrm{Tr}_{F/K}(tx^2)$ in $H^1(K,\mathrm{SO}(3))\cong \mathrm{Br}(K)[2]$; its kernel selects the classes that correspond, via arithmetic invariant theory, to $G(K)$-orbits of pairs of binary cubic forms with $A_1=0$ and $A_3=n$, where $G=\mathrm{SL}_2^2/\mu_2$. The counting theorem of [ABS22] then turns local orbit counts into Euler products: the average is $1+2\prod_p \nu_p$, with Tamagawa number $2$, $\nu_p=1$ at $p\neq 3,\infty$, $\nu_\infty=1/2$, and $\nu_3=1$ in the wild family but $\nu_3=1/2$ in the tame family, because Hecke reciprocity keeps the nontrivial class inside the global Selmer group while placing it outside the local kernel of $q_{n,3}$.

What would settle it

For cubefree integers $n\leq X$, split the family by $n\equiv\pm 1 \pmod{9}$ versus $n\not\equiv\pm 1 \pmod{9}$, compute the average of $|\mathrm{Cl}_{\mathbb{Q}(\sqrt[3]{n})}[2]|$, and check whether the two averages tend to $3/2$ and $2$ respectively as $X\to\infty$; the paper's central claim fails if either limit differs. Equivalently, verifying the acceptability condition for $\phi(v)=1/m(v)$ would settle the completeness of the orbit-counting step.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is a parity constraint on 2-torsion classes. For an odd-degree extension $F/K$, define the Hecke ideal $H_{F/K}=\mathrm{Disc}_{F/K}\mathrm{Diff}_{F/K}^{-1}$; a Hecke prime is a prime whose exponent in this ideal is odd and that is locally inert in an unramified quadratic square-norm extension. The Hecke reciprocity theorem says that any unramified quadratic extension $F(\sqrt{t})/F$ with square norm has an even number of inert Hecke primes. In the family $\mathbb{Q}(\sqrt[3]{n})$, a tame prime above $3$ has splitting type $(121)$ and supplies exactly one Hecke prime, so reciprocity forbids the nontrivial unramified local square-class from occurring globally; this cuts the local factor at $3$ from $1$ to $1/2$ and turns the wild-family average $1+2\cdot(1/2)\cdot 1=2$ into the tame-family average $1+2\cdot(1/2)\cdot(1/2)=3/2$. Over $K=\mathbb{Q}(\sqrt{-3})$, odd degree Galois extensions are Hecke unramified, no obstruction appears, and the average is $3/2$, confirming the Cohen–Martinet prediction. The abstract states the correspondence with the wild/tame labels reversed; throughout §7 the wild family is $n\not\equiv\pm 1 \pmod{9}$ and the tame family is $n\equiv\pm 1 \pmod{9}$.

Load-bearing premise

The computation assumes that the orbit-weighting $\phi(v)=1/m(v)$, where $m(v)$ is the number of $G(\mathbb{Z})$-orbits inside a $G(\mathbb{Q})$-orbit, is an 'acceptable' weighting defined by congruence conditions, so that the counting theorem's Euler-product formula applies; the paper states this hypothesis but does not verify it, and Theorem 1.1 also rests on a sketched generalization of the counting theorem to imaginary quadratic fields.

Editorial extensions

If this is right

In the wild family $n\not\equiv\pm 1 \pmod{9}$, the average of $|\mathrm{Cl}_{\mathbb{Q}(\sqrt[3]{n})}[2]|$ over cubefree $n$ up to $X$ is $2$.
In the tame family $n\equiv\pm 1 \pmod{9}$, the average is $3/2$, so at least 50% of these fields have odd class number.
Over $K=\mathbb{Q}(\sqrt{-3})$, cubic Kummer extensions $K(\sqrt[3]{n})$ ordered by norm have average $|\mathrm{Cl}_{F/K}[2]| = 3/2$, confirming the predicted moment for $C_3$-extensions of $K$.
The Hecke-unramified versus Hecke-ramified split becomes a general heuristic: for $\Gamma$-extensions with even $|G|$, the predicted distribution on Hecke-unramified subfamilies shifts by one extra relation, so the pure $p$-th power families $\mathbb{Q}(\sqrt[p]{n})$ should repeat the dichotomy for every odd prime $p$.
Conjecture 1.7 makes explicit probabilities for class number one: about $0.5662$ for primes $n\equiv 8 \pmod{9}$ and about $0.3775$ for $n\equiv 2,5 \pmod{9}$, refining the old observed distinction in pure cubic fields.

Reading between the lines

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

If the abstract's reversed labels are a typo, the paper's headline claim should be quoted from §7; the appendix tables labeled type I (wild) and type II (tame) are consistent with the body's values $2$ and $3/2$.
The same mechanism should be visible in higher moments: if Hecke reciprocity acts as an extra relation on the alternating-matrix model of §9, then the full distribution of 2-ranks in the tame family should match a $u$-shifted Cohen–Lenstra–Martinet distribution, not just the first moment.
The Hecke-ideal parity constraint is not limited to Kummer families: any odd-degree family whose resolvent field is $\mathbb{Q}(i)$, $\mathbb{Q}(\sqrt{\pm 3})$, or another aberrant twist should display a 2-torsion boost, and the one-parameter family $x^3-3ax^2-3x+a$ with resolvent $\mathbb{Q}(\sqrt{3})$ provides a direct test.
Corollary 3.10 suggests that families with exactly one Hecke prime are the extremal case: one could test whether any transitive permutation group whose members generically have a single Hecke prime reproduces the tame-family average $3/2$ rather than the wild-family average $2$.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Short version: this is a genuine contribution, but the printed paper garbles its own main theorem and has a gap in the counting step that supports every numerical average. The body of §7 gives wild average 2 and tame average 3/2 for Q(∛n), and the tables match that. The abstract and Theorem 1.2 print the reverse. That's confusing but harmless once you decode it. The deeper problem is that Theorem 6.2 is applied to φ(v)=1/m(v) without checking that this function is acceptable and defined by congruence conditions in the sense of the theorem. The local factor ν3=1/2 in the tame case is exactly what produces the 3/2 average, so this is not cosmetic. Theorem 6.3, which carries the counting over Q(√-3), is only sketched. Both gaps are probably fixable, but as written they are load-bearing.

What's genuinely new: Hecke reciprocity as a constraint on Cl_F/K[2] is a nice idea, and the Γ-extension heuristics in §10 are a sensible way to organize the dichotomy. The first exact averages in these Kummer families are a real step beyond Ruth's upper bound. The paper is honest about which parts are conjectural.

I would send this to a serious referee. The referee should demand a proof that the weighting φ satisfies the hypotheses of Theorem 6.2, and a complete proof of Theorem 6.3 (or a statement that Theorem 1.1 depends on a conjecture). The label swap should be fixed in the abstract and introduction. After those changes, the central claims look solid to me.

Referee Report

3 major / 5 minor

Summary. The paper studies the average size of the 2-torsion of class groups in families of pure cubic fields F_n = Q(∛n), split according to whether the field is wildly or tamely ramified at 3, and in the analogous Kummer family over K = Q(√-3). The method is to identify Cl_{F/K}[2]^∨ with an unramified Selmer group for the Galois module M = ker(Res_{F/K} μ_2 → μ_2), to introduce a global 'Hecke reciprocity' constraint on the number of inert Hecke primes, and to parameterize Selmer elements by orbits of pairs of binary cubic forms with vanishing A_1-invariant. The counting of such orbits is imported from [ABS22]. The main numerical conclusions as printed in Section 7 are that the average of |Cl_F[2]| is 2 for wildly ramified n ≢ ±1 (mod 9) and 3/2 for tamely ramified n ≡ ±1 (mod 9); the abstract and Theorem 1.2 print the opposite labels. For K = Q(√-3), the paper claims average 3/2. Sections 9–11 propose heuristics for 2-torsion in families of Γ-extensions, including explicit predictions for pure cubic, quintic, and septic fields, with extensive Magma tables in Appendix A.

Significance. If the counting ingredients are fully supplied, the results would be a significant advance: Theorem 1.1 would be the first proven instance of the Sawin–Wood conjectural moment formula (1.1) in the case (G,K,V) = (C3, Q(√-3), F4), and Theorems 7.3/7.7 would give the first rigorous explanation of the Cohen–Martinet–Williams–Shanks dichotomy for pure cubic fields. The paper also contains a genuinely new structural idea, Hecke reciprocity, and proposes a concrete framework (gyroscopic families, aberrant Γ-groups) with falsifiable numerical predictions. The appendix supplies reproducible-looking data and explicit code availability statements. However, as printed the central average theorems depend on two unproven counting inputs—the admissibility of the weight φ = 1/m(v) in Theorem 6.2 and the sketched adaptation in Theorem 6.3—and the abstract/Theorem 1.2 are inconsistent with Section 7. The underlying algebraic framework is coherent, but the paper is not yet in a form where the advertised theorems are established.

major comments (3)

[§6, Theorem 6.2; §7, proofs of Theorems 7.3 and 7.7] The proof of Theorem 7.3 applies Theorem 6.2 to the weight φ(v) = 1/m(v), where m(v) is the number of G(Z)-orbits in the G(Q)-orbit of v. Theorem 6.2 requires φ to be an acceptable G(Z)-invariant function defined by congruence conditions, with an Euler product expansion (6.3). The paper never proves that 1/m(v) satisfies the required local conditions: in particular, it does not verify that m(v) is locally constant on Y(Z_p) in the relevant p-adic topology, nor that φ_p(y)=1 for all sufficiently large p when p^2 ∤ A_3(y). The Euler factors ν_p in Section 7 are obtained from this factorization, so the final averages in Theorems 7.3 and 7.7 are conditional on an unstated hypothesis about m(v). This is load-bearing because every numerical average in the paper is produced by these orbit counts.
[§6, Theorem 6.3; §8, Theorem 8.3] Theorem 1.1 depends on Theorem 6.3, whose proof is explicitly only a sketch: the text says the proof of [ABS22, Theorem 4.1] 'goes through essentially unchanged' and the preceding paragraph states that the general number-field adaptation has not been worked through. Theorem 8.3 uses Theorem 6.3 to obtain the main term X·(...) and the Euler product over primes of K = Q(√-3). As printed, the average 3/2 over the Kummer family over Q(√-3) is therefore not a proved theorem but a conditional statement pending a complete proof of Theorem 6.3 (or a precise reference establishing the analogue of the counting theorem over imaginary quadratic fields of class number one).
[Abstract and Theorem 1.2] The tame/wild terminology is reversed in the abstract and in Theorem 1.2 relative to Section 7. In Theorem 1.2, W(X) is defined as n ≡ ±1 (mod 9) and is assigned average 2, while T(X) is n ≢ ±1 (mod 9) with average 3/2. But in Section 7 the wild family is n ≢ ±1 (mod 9) (Theorem 7.3, average 2) and the tame family is n ≡ ±1 (mod 9) (Theorem 7.7, average 3/2). The abstract's 'wildly (resp. tamely) ramified fields ... average 3/2 (resp. 2)' prints the opposite assignment. This inconsistency affects the central claims exactly as stated in the abstract and theorem, and must be corrected.

minor comments (5)

[§7.2, after Proposition 7.4] The sentence 'because of the obstruction coming from Theorem 5.8' appears to refer to Proposition 7.4; Theorem 5.8 actually states that all classes in Sel^un_2(F_n) lie in ker(q_n), so the local obstruction at 3 is not coming from Theorem 5.8 as written.
[§8.1, Proposition 8.1 and proof of Theorem 8.3] The notation 'p = p' in 'the only possible Hecke prime is p = p' is confusing because p denotes both a prime of K and the rational prime 3; please use a distinguishing notation such as p_3.
[§5.1, Lemma 5.6] In the proof of Theorem 5.5 the step 'v(p) is odd, by Lemma 5.6' is terse: Lemma 5.6 gives b ≡ -p (mod K×2), and one needs to spell out that in the totally ramified case the valuation of b is odd. Adding one sentence would make the local computation easier to check.
[Appendix A] The tables report observed moments to four decimal places, but the text does not state the standard error or the exact number of fields used for each individual moment. Since the agreement for large |H| is visibly imperfect, reporting the error bars or field counts would strengthen the numerical evidence.
[§6, Theorem 6.2] The statement of Theorem 6.2 counts 'irreducible' G(Z)-orbits, but the application in Section 7 uses all orbits corresponding to unramified vectors; the paper should clarify that every orbit arising from Sel^un_2(F_n) is irreducible, or that the counting theorem applies to the appropriate subset.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular derivation found: the averages are computed from Selmer-group orbit counts and explicit local factors, not from the class-group values being predicted.

full rationale

The derivation chain is self-contained with respect to circularity: Theorem 2.1 identifies |Cl_F[2]| with |Selun_2(F)|, Corollaries 7.2 and 7.6 biject Selun_2(F_n) with G(Q)-orbits on the quadric A1=0, A3=n, and the counting formula of Theorem 6.2 (imported from [ABS22]) estimates those orbits with |A3|<X. The numerical averages 2 and 3/2 are then obtained from explicit local factors nu_p, nu_3=1/2, and the Tamagawa number tau(G)=2; no parameter is fitted to the predicted class-group averages. The self-citation [ABS22], coauthored by Shnidman, is load-bearing but independent: it is a separate orbit-counting result whose assumptions do not include the target class-group average, so it does not reduce the derivation to its own conclusion. The unverified admissibility of phi(v)=1/m(v) in the application of Theorem 6.2, the sketched proof of Theorem 6.3 for imaginary quadratic fields, and the reversal of the tame/wild labels between the abstract/Theorem 1.2 and Section 7 are correctness and completeness concerns, not circularity: none of them makes the proof equivalent to its inputs by construction. Hence the paper receives a circularity score of 0.

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

No free parameters are introduced; the numerical outcomes are derived rather than fitted. The paper's new mathematical notions, such as Hecke primes and Γ-extensions, are definitions from existing class field theory rather than speculative physical entities. The load-bearing unproved assumptions are the applicability of the cited counting theorem to the weighted orbit count and the sketched generalization to imaginary quadratic fields.

assumptions (6)

standard math Class field theory and Hecke's theorem on the different being a square in the class group
Used in §2-3 to identify Cl∨[2] with unramified Selmer groups and to prove Hecke reciprocity, especially Corollary 3.6 and Theorem 3.8.
standard math Arithmetic invariant theory parametrization, Proposition 6.1
Gives the bijection between G(K)-orbits on Y(K)_n and ker(q_n); cited from [BG14].
domain assumption Counting theorem [ABS22, Theorem 4.1]
Central input to Theorems 7.3 and 7.7; it is cited as a black box and not proved in this paper.
domain assumption Generalization of the counting theorem to class-number-one imaginary quadratic fields, Theorem 6.3
Sketch only, yet supports Theorem 1.1 over K = Q(√-3); the circle method, divisor bound, and geometric sieve over OK are not written.
ad hoc to paper The weighted orbit-counting function phi = 1/m(v) is acceptable and defined by congruence conditions
Needed to apply Theorem 6.2 in the proofs of Theorems 7.3 and 7.7; no verification is given in the text.
domain assumption Malle and Sawin-Wood heuristics are the starting point for the conjectural section
Used in §9-11 to formulate predictions; not needed for the proofs of Theorems 1.1 and 1.2.

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Pith. "Pith review of Hecke reciprocity and class groups." pith.science (2026). https://pith.science/paper/ZX47ZSJQ

@misc{pith2026250613749,
  author       = {Pith},
  title        = {Pith review of: Hecke reciprocity and class groups},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZX47ZSJQ}},
  note         = {Machine review of arXiv:2506.13749}
}

abstract

We compute the average size of $\mathrm{Cl}_F[2]$ in the family of cubic fields $F = \mathbb{Q}(\sqrt[3]{n})$. Specifically, as $F$ varies over the subfamily of wildly (resp. tamely) ramified fields $\mathbb{Q}(\sqrt[3]{n})$, the average size of $\mathrm{Cl}_F[2]$ is $3/2$ (resp. $2$). This tame/wild dichotomy is not accounted for by the class group heuristics in the literature. Analogously, when the extensions $F = K(\sqrt[3]{n})$ of $K = \mathbb{Q}(\sqrt{-3})$ are ordered by the norm of $n \in \mathcal{O}_K$, we show that the average size of $\mathrm{Cl}_F[2]$ is $3/2$, as is predicted by the Cohen--Martinet heuristics for $C_3$-extensions of $K$. Underlying our proofs is a reciprocity law for the relative class groups $\mathrm{Cl}_{F/K}[2]$ of odd degree extensions of number fields $F/K$. This leads us to propose class group heuristics for families of $K$-extensions with a fixed Galois $K$-group that explains the aberrant behavior in the family $\mathbb{Q}(\sqrt[3]{n})$ and predicts similar behavior in other special families. The other main ingredient is the work of Alp\"oge--Bhargava--Shnidman on the number of integral $G(\mathbb{Q})$-orbits in a $G$-invariant quadric with bounded invariants.

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Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.

[1] [8]

32 [LST20] Michael Lipnowski, Will Sawin, and Jacob Tsimerman,Cohen–lenstra heuristics and bilinear pairings in the presence of roots of unity, available athttps://arxiv.org/abs/2210.10730,

arXiv

[2] [9]

Lemke Oliver, Jiuya Wang, and Melanie Matchett Wood,The average size of3-torsion in class groups of2-extensions, available athttps://arxiv.org/abs/2110.07712,

[LWW21] Robert J. Lemke Oliver, Jiuya Wang, and Melanie Matchett Wood,The average size of3-torsion in class groups of2-extensions, available athttps://arxiv.org/abs/2110.07712,

arXiv

[3] [13]

[SSS25] Arul Shankar, Artane Siad, and Ashvin A

[Sia20b] , Monogenic fields with odd class number Part II: even degree, 2020, available at https:// arxiv.org/abs/2011.08842. [SSS25] Arul Shankar, Artane Siad, and Ashvin A. Swaminathan, Counting integral points on symmetric varieties with applications to arithmetic statistics, Proc. Lond. Math. Soc. (3) 130 (2025), no. 4, Paper No. e70039. MR 4891959 [S...

arXiv 2025

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Swaminathan, A new parametrization for ideal classes in rings defined by binary forms, and applications, J

[Swa23] Ashvin A. Swaminathan, A new parametrization for ideal classes in rings defined by binary forms, and applications, J. Reine Angew. Math.798 (2023), 143–191. MR 4579703 [The23] The PARI Group, Univ. Bordeaux, PARI/GP version 2.15.4, 2023, available from http://pari. math.u-bordeaux.fr/. [TW86] M. Tennenhouse and H. C. Williams, A note on class-numb...

work page 2023

[5] [1974]

MR 427267 [WS79] H. C. Williams and Daniel Shanks, A note on class-number one in pure cubic fields, Math. Comp.33 (1979), no. 148, 1317–1320. MR 537977 Institute for Adv anced Study, 1 Einstein Drive, Princeton NJ 08540 Email address: ari.shnidman@gmail.com Institute for Adv anced Study, 1 Einstein Drive, Princeton NJ 08540 Email address: ajsiad@ias.edu 33

work page 1979

[6] [1978]

Cohen and H

MR 522835 [CL84] H. Cohen and H. W. Lenstra, Jr., Heuristics on class groups of number fields, Number theory, Noord- wijkerhout 1983 (Noordwijkerhout, 1983), Lecture Notes in Math., vol. 1068, Springer, Berlin, 1984, pp. 33–62. MR 756082 [CM87] H. Cohen and J. Martinet, Class groups of number fields: numerical heuristics, Math. Comp. 48 (1987), no. 177, 1...

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[7] [1979]

[Ser84] , L’invariant de Witt de la formeTr(x2), Comment. Math. Helv.59 (1984), no. 4, 651–676. MR 780081 [Sia20a] Artane Siad, Effect of monogenicity on2-torsion in the class group of number fields of odd degree, available athttps://arxiv.org/abs/2011.08834,

arXiv 1984

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