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This paper predicts, conditional on a general conjectural framework, that the number of Galois extensions of Q with Galois group the order-64 Heisenberg group and discriminant at most X is asymptotic to (1/2)(C0 + Cβ) X^(1/32) (log X)^8, wi

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2026-08-02 08:14 UTC pith:33SH5LJC

load-bearing objection Explicit two-Euler-product constant for Heis_4 that would show failed independence of local events, but the numbers rely on the authors' own unproven LS conjectures and on LMFDB completeness plus uncommitted code; still worth refereeing. the 3 major comments →

arxiv 2607.06476 v2 pith:33SH5LJC submitted 2026-07-07 math.NT

A refined Malle conjecture for Heisenberg groups

Jack B. Miller , Tim Santens This is my paper
classification math.NT MSC 11N4511R3211R3414G1220D15
keywords Heisenberg groupMalle's conjectureleading constantBrauer-Manin obstructionEuler productsindependence of local events2-adic massGalois discriminant
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper aims to pin down the leading constant in the refined counting conjecture for Galois extensions of Q with group the order-64 Heisenberg group, ordered by discriminant. The prediction is that the constant is not one Euler product but a sum of two, C0 + Cβ, each spelled out explicitly, and that the same mechanism produces a nonzero correlation between ramification at two different odd primes — a failure of the usual assumption that local field-counting events are asymptotically independent. If the underlying conjectural framework is correct, this gives the first example of a non-concentrated regular permutation group for which a transcendental obstruction forces a genuinely two-part constant and a failure of almost-everywhere independence. A secondary technical contribution is an efficient algorithm for computing the 2-adic contribution to the constant for a broad family of two-step nilpotent 2-groups.

Core claim

On the paper's terms, the central discovery is that the partially unramified Brauer group of the classifying stack of Heis_4 — the order-64 Heisenberg group over the field of 4 elements acting regularly on itself — has four elements, of which the two nontrivial classes are transcendental rather than algebraic. Under the general leading-constant conjecture the paper relies on, those classes split the v-adic local masses into two different Euler products; summing over the Brauer group gives c = (C0 + Cβ)/2, with Cβ approximately 0.633 C0. The same computation predicts that the joint probability that two odd primes both ramify is not the product of the individual probabilities; the gap is 36 C0

What carries the argument

The central mechanism is the partially unramified Brauer group Br^e_C BHeis_4 — the subgroup of Brauer classes marked by the order-2 elements of Heis_4 — together with its v-adic Brauer transforms. The nontrivial Brauer classes act as a transcendental obstruction that splits the global constant into two Euler products. The 2-adic transforms are computed through a finite universal quotient Γ^L_{Q2} of order 256, with presentation ⟨σ_{-1}, σ_5, σ_2 | σ_{-1}^2 = [σ_5, σ_2]⟩ and an explicit lower-numbering ramification filtration with breaks at -1, 1, 5, 13, 29; homomorphisms from Γ_{Q2} into any relevant 2-group are then encoded by admissible triples of elements satisfying the same relation.

Load-bearing premise

Everything explicit in the conjecture depends on the prior leading-constant conjectures the paper cites (Conjecture 1.1 is derived as a special case, not proved unconditionally), together with the completeness of a database of degree-16 2-adic fields used to fix the ramification filtration of the universal quotient of order 256.

What would settle it

Compute the 2-adic mass τ_{disc,2}(BHeis_4) and the two nontrivial Brauer transforms by exhaustive enumeration of Galois 2-adic étale algebras of degree 64 — or by an independent computation of the finite quotient Γ^L_{Q2} with its ramification filtration that avoids the database used in the paper. If the values differ from the explicit numbers stated in Corollary 5.6, the predicted constants fail. Alternatively, an unconditional count of Heis_4-extensions whose leading term disagrees with (C0 + Cβ)/2 · X^(1/32) (log X)^8 would falsify the conjecture itself.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

  • If the framework's conjectures hold, the number of Galois Heis_4-extensions of Q with discriminant at most X is explicitly predicted as (1/2)(C0 + Cβ) X^(1/32) (log X)^8, with both Euler products required.
  • For distinct odd primes p1 and p2, ramification events have nonzero covariance of size 36 C0 Cβ/(C0 + Cβ)^2 · p1^(-3/2) p2^(-3/2) (1 + O(p1^(-1) + p2^(-1))), so a single Euler product cannot describe the global count.
  • Within the same framework, the regular permutation group Heis_{2^n} satisfies independence of local events if and only if n = 1; for every n > 1 the group fails almost-everywhere independence.
  • For every finite 2-group of nilpotency class at most 2 and exponent at most 4, the 2-adic mass and Brauer transforms are computable in terms of admissible triples and the explicitly described finite quotient Γ^L_{Q2}.
  • For Heis_4, and more generally for any regular group where the minimal-index elements are the p-torsion elements for the smallest prime p dividing the group order, no thin set of fields needs to be removed before the predicted asymptotic applies.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If the reduction is correct, the leading constant in the refined counting conjecture is not just a product of local factors over places; the failure shows that global Brauer-class averaging can be an essential component of the constant for an actual discriminant height, not only for specially constructed balanced heights.
  • The universal finite quotient Γ^L_{Q2} computed here can be reused to write down leading constants for other two-step nilpotent 2-groups of exponent 4, making the paper's algorithmic claim directly testable by applying the same admissible-triple enumeration to those groups.
  • Because the asymptotic grows only like X^(1/32) (log X)^8, direct numerical verification from counting fields is far out of reach; a more realistic falsification path is an independent computation of the 2-adic Brauer transform, or a proof of the underlying conjectures for nilpotent groups via character sums, which the paper suggests is plausible.
  • The 2-adic computation relies on the completeness of a database of degree-16 2-adic fields for its ramification filtration; an independent group-theoretic derivation of Γ^L_{Q2}'s filtration would either confirm or correct the stated 2-adic constants, which is the one computational component not justified from first principles.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. The paper formulates Conjecture 1.1, an explicit prediction for the number of Galois Heis_4-extensions of Q ordered by discriminant: the leading constant is predicted to be 1/2(C0 + Cβ) X^{1/32} (log X)^8 with explicit constants α0, αβ and Euler products, together with a prediction that Heis_4 ≤ S_64 fails almost-everywhere independence of local events. The derivation (Theorem 2.3) reduces this to [LS24, Conj. 9.3 and 9.10]. The paper computes the relevant partially unramified Brauer groups, their residues, the odd-prime local factors, and gives a finite presentation of the pro-(class≤2, exponent≤4) completion of Gal(Q2) (Theorem 1.5), which is then used to compute the 2-adic Brauer transforms. It also sketches an algorithm for other 2-groups of nilpotency class 2 and exponent dividing 4.

Significance. If correct, this gives a rare explicit example where a discriminant-height (balanced) ordering of a non-concentrated permutation group has a leading constant that is not a single Euler product, and where almost-everywhere independence of local events fails. The group-theoretic computations are substantial and appear correct: the classification via quadratic forms (Lemma 3.13), the residue computation (Lemma 4.5), and the odd-prime local factors (Section 4.3) are clean and well-motivated. The paper is transparent that the main prediction is conditional on the Loughran–Santens conjectures, and it is best read as a concrete specialization and computational test of that framework. The explicit numerical content, however, rests on two external computational inputs that are not fully verified in the manuscript: the GitHub code behind Corollary 5.6 and the LMFDB completeness assertion behind Proposition 6.11.

major comments (3)
  1. [§5, Corollary 5.6] The proof of the 2-adic Brauer transform is 'See the verification code in the accompanying GitHub repository.' These values are load-bearing: the constants α0 and αβ in Conjecture 1.1 are obtained directly from bτ_disc,2(BHeis4;β). A journal proof should be self-contained or include a verifiable computational appendix—for example, the finite sum from Corollary 5.5 together with the explicit ramification filtration from Proposition 6.11—so that the referee can check the result without relying on an external repository whose version and output are not fixed. As it stands, the numerical prediction is not independently verifiable from the paper.
  2. [§6.4, Proposition 6.11] The proof of the ramification filtration relies on the assertion that LMFDB contains a complete list of degree-16 2-adic fields, and Table 1 reads conductor exponents from LMFDB labels. This completeness is asserted, not proven. If LMFDB were incomplete, some conductor exponent in the flag W^• could differ, changing the relative discriminant exponents, the lower ramification breaks, and ultimately d(g−1,g5,g2) in Theorem 5.4, hence the constants in Corollary 5.6 and Conjecture 1.1. Please supply an independent algebraic derivation of these conductor exponents (for instance via explicit Kummer extensions and local class field theory) or include a machine-checkable exhaustive computation with a formal completeness argument. A citation to LMFDB is not sufficient for a proof.
  3. [General framing, Theorem 2.3] The paper's main prediction is conditional on [LS24, Conj. 9.3 and 9.10], which are unproved conjectures by Loughran and the second author. This is not a logical circularity, but it means the paper does not prove Malle's conjecture for Heis4; it derives a prediction from the LS framework. The authors state this in the introduction, but the abstract and the opening of Section 2 could make the conditional status even more prominent, so that the paper is not misread as a theorem. This is a clarity issue, not a technical flaw.
minor comments (5)
  1. [Conjecture 1.1 and Theorem 2.3] The constants α0 and αβ are typeset with ambiguous radical notation: '4√2' can be read as 4·√2. Please use \sqrt[4]{2} or an equivalent unambiguous notation throughout.
  2. [Conjecture 1.1 and Theorem 2.3] The denominators '271' appear to be intended as 2^7; likewise '1/249' in the proof of Theorem 2.3 seems to be 2^4. Please correct the typesetting.
  3. [§6.4, Table 1] The row for χ5,2 has empty entries in the Fχ, Eχ, Gal(E'χ/Q2) columns and a parenthetical '(4)' for the conductor exponent. The text explains the deduction, but the table would be much clearer if those entries were filled in explicitly.
  4. [§2.1, Lemma 2.4] The proof uses a footnote saying that [LS24, Lem. 6.31] 'technically provides an isomorphism ... but the argument given there also works.' Please expand this: the reader should not have to reconstruct the argument from a reference.
  5. [References] The bibliography entry [LMFDB] gives only a general URL and access date. Since specific database entries (e.g., 2.2.4.16b1.1) are used in the proof, please list the exact identifiers or provide stable links so the computation can be reproduced.

Circularity Check

0 steps flagged

No significant circularity: the central prediction is an explicitly conditional specialization of the authors' own framework, supported by nontrivial local computations and external database queries.

full rationale

The two apparent dependencies do not constitute circularity. First, Theorem 2.3 is explicitly conditional: "Assume [LS24, Conj. 9.3] for the base field Q and permutation group Heis_4 ≤ S_64. Then the explicit asymptotic counting statement in Conjecture 1.1 holds." This is a one-way implication from a stated general conjecture to a special case; the special case is not used to justify the conjecture, and no parameter is fitted from the target asymptotic. The self-cited nature of [LS24] (Loughran–Santens) is a provenance and conditionality concern, but the paper does not present that conjecture as an externally proven theorem or use a uniqueness theorem to forbid alternatives. Second, the explicit 2-adic constants are not built into Conjecture 1.1 by definition. They are computed from the ramification filtration of Proposition 6.11, whose proof invokes an external completeness assertion: "[LMFDB] contains a complete list of all 2-adic number fields of degree ≤ 16," and from the GitHub verification code referenced in Corollary 5.6 ("See the verification code in the accompanying GitHub repository"). If the LMFDB completeness claim or the code were wrong, the constants would change, but that is a computational/reproducibility risk, not a circular reduction. The covariance computation in Theorem 2.3(2) is a direct algebraic consequence of having two distinct Euler products with C_β ≠ 0; it is not an assumed conclusion. The paper's contribution—computing the 2-adic Brauer transforms, deriving the flag W^•, and specializing the LS24 framework—contains independent content beyond restating the input.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 0 invented entities

No numbers are fitted to data. The constants α0, αβ and the Euler products are computed from group theory, Brauer groups, and local sums; the 2-adic transform is evaluated by explicit enumeration (code), not by fitting. No new particles, forces, dimensions, or entities are introduced; the Brauer-Manin obstruction is an existing notion, and Heis_4 is a known group.

axioms (3)
  • domain assumption [LS24, Conj. 9.3 and 9.10] for Heis_4 over Q with Galois discriminant
    Unproven conjecture by Loughran and Santens; the paper's Conjecture 1.1 is a special case of it (Theorem 2.3). The framework is not independently proved.
  • domain assumption LMFDB completeness for degree-16 2-adic fields and discriminant data (Prop. 6.11)
    The ramification filtration on G_{2,2}(Q2) and hence the 2-adic mass is inferred from an LMFDB API query; no independent enumeration proof is included.
  • standard math Demushkin presentation and Hilbert-symbol table for Gal(Q2/Q2) (Lemma 6.10)
    Imported from [NSW08]; standard local class field theory used to justify the canonical presentation.

pith-pipeline@v1.3.0-alltime-deepseek · 25112 in / 16358 out tokens · 146168 ms · 2026-08-02T08:14:01.190298+00:00 · methodology

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read the original abstract

Based on a conjecture of Loughran and the second author, we give an explicit prediction for the leading constant in Malle's conjecture for Galois $\mathcal{H}$-extensions of $\mathbb{Q}$ ordered by discriminant, where $\mathcal{H}$ is the $3\times 3$ Heisenberg group over $\mathbb{F}_4$. The predicted leading constant is not a single Euler product, but rather a sum of two distinct Euler products. Our methods also give an efficient algorithm for computing the conjectural Loughran-Santens leading constant for many $2$-groups of nilpotency class $2$.

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