Pith. sign in

REVIEW 3 major objections 4 minor 2 cited by

Malle's Conjecture for Galois octic fields over $\mathbb Q$

T0 review · 3 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read This paper proves the strong form of Malle's conjecture for octic D4-fields over Q, giving exact asymptotic counts with explicit leading constants.

desk verdict Major result on Malle's conjecture for D4 octics, likely correct, but the write-up has load-bearing omissions and a suspect lemma statement. read the letter →

arxiv 2505.23690 v2 pith:Y2QGND34 submitted 2025-05-29 math.NT

Arul Shankar , Ila Varma This is my paper
classification math.NT MSC 11R4511R3211R1611M41
keywords Malle'sconjectureocticD4fieldsdiscriminantasymptoticsdoubleDirichletserieslocalmassformulaHeckeL-functionssubconvexityGaloisnumber
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 establishes the strong form of Malle's conjecture for octic number fields over $\mathbb Q$ whose Galois group is $D_4$, the symmetries of a square. Counting such fields by absolute discriminant, the authors prove that the totally real ones grow like $\frac{56+3\sqrt2}{256}\prod_{p\ge 3}(1+\frac3p+\frac1{p^{3/2}})(1-\frac1p)^3\,X^{1/4}\log^2 X$, and the complex ones grow with twice this constant. This is the first exact asymptotic for a non-concentrated family whose Galois group is neither abelian nor symmetric, and the constant obeys the Malle-Bhargava principle of being a product of local masses. Combined with known counts for the other order-8 Galois groups, it implies that 100 percent of Galois octic fields, ordered by discriminant, have Galois group $C_2\times C_2\times C_2$.

What carries the argument

The central object is the double Dirichlet series $\Phi_\Lambda(s,t;K)=\sum_L q_{\sigma^2}(L,K)^{-s}q_\tau(L,K)^{-t}$ over quadratic extensions $L$ of a quadratic field $K$, together with the Euler products $D_{\Lambda}(s,t;K,\alpha)$ obtained by modifying each local factor. Setting $s=t$ orders extensions by the radical of the relative discriminant norm, the invariant governing the discriminant of the octic Galois closure, while $s=2t$ orders by the norm itself. The analytic engine is the uniform subconvexity bound of Lemma 4.11 for the underlying Hecke $L$-functions, which turns the double poles into the $X^{1/4}\log^2 X$ main term with a power-saving error.

What would settle it

Compute, for a sequence of quadratic fields $K$ with $|\Delta(K)|$ tending to infinity, the central value $D_\Lambda(1/2+\epsilon,1/2+\epsilon;K,\alpha)$ for each admissible $\alpha$ and compare with the bound $(|\Delta(K)|P(\Lambda)\operatorname{Im}(s))^{1/4-\delta}$ asserted in Lemma 4.11. A single sequence violating this bound, or a smoothed count whose error term behaves like $X^{1/2+\epsilon}|\Delta(K)P(\Lambda)|^{1/4}$ rather than with the power saving $\delta$, would break the error estimates of Propositions 4.13 and 5.3 and hence the proof of the asymptotic.

Watch

Extended reading notes

Core claim

The central claim is Theorem 1: the number $N_8^{(r)}(D_4,X)$ of totally real octic $D_4$-fields with absolute discriminant at most $X$ is asymptotic to $\frac{56+3\sqrt2}{256}\prod_{p\ge 3}(1+\frac3p+\frac1{p^{3/2}})(1-\frac1p)^3\,X^{1/4}\log^2 X$, and the complex count has the same shape with constant $\frac{56+3\sqrt2}{128}$. The proof works by counting quartic $D_4$-fields $L$, whose normal closures are exactly the octic $D_4$-fields, through their quadratic subfield $K$; the discriminant of the octic closure is essentially $\Delta(K)^4$ times the fourth power of the radical of the relative discriminant norm. Because the radical forces a non-polynomial ordering, the counting function for each $K$ is encoded in a double Dirichlet series whose diagonal specialization counts by the radical and whose other specialization counts by the norm. Nontrivial Hecke characters contribute mainly to $V_4$-fields and are separated from the $D_4$ contribution, leaving the trivial character's double pole to supply the $X^{1/4}\log^2 X$ main term. Theorem 2 extends the asymptotic to any finite set of local splitting conditions, with the constant given by a product of local masses.

Load-bearing premise

The load-bearing premise is Lemma 4.11, quoted from [21] without proof: for every quadratic field and every character in the relevant family, a certain associated $L$-function admits a subconvexity bound with a fixed power saving and conductor dependence $(|\Delta(K)|P(\Lambda)\operatorname{Im}(s))^{1/4-\delta}$; if this uniformity fails for one sequence of fields, the power-saving error estimates and the final asymptotic do not follow.

Editorial extensions

If this is right

  • The strong form of Malle's conjecture now holds for every Galois octic group: $C_8$, $C_2\times C_4$, $C_2^3$, $Q_8$, and $D_4$, with explicit constants.
  • Among Galois octic fields ordered by discriminant, 100 percent have Galois group $C_2\times C_2\times C_2$.
  • For any finite set of local splitting conditions, the number of octic $D_4$-fields has the predicted product-of-local-masses constant, as stated in Theorem 2.
  • The nontrivial Hecke characters of quadratic fields, which individually could contribute $X^{1/4}\log X$, are shown to contribute mainly to $V_4$-fields and are negligible for the $D_4$ count, so the $D_4$ main term is governed entirely by the trivial character.

Reading between the lines

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

  • Editorial extension: because the counting is driven by the diagonal specialization $s=t$ of the double Dirichlet series, the same machinery should produce asymptotics for orderings intermediate between norm and radical, with constants varying continuously in the exponents.
  • Editorial extension: the sharp separation of $V_4$ contributions from $D_4$ contributions suggests a general mechanism: in families where an abelian subfamily contributes a positive proportion, the Malle-Bhargava constant will be correct only after that subfamily's mass is removed.
  • Editorial extension: the only unproved input, the uniform subconvexity bound over all quadratic fields, is testable by computing the smoothed sums of Lemma 4.12 for fields of large discriminant; a direct check for $|\Delta(K)|$ up to a moderate bound would test the load-bearing uniformity assumption.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The paper claims to prove the strong form of Malle's conjecture for octic D4-fields over Q, giving explicit asymptotic formulas with leading constants for both totally real and complex fields (Theorem 1), together with a more general local-specifications counting result (Theorem 2). The method counts D4-pairs (L,K) where L is a quadratic extension of a quadratic field K, expresses the relevant Dirichlet series as sums of Hecke L-functions, introduces double Dirichlet series to handle the radical ordering, and reduces the main term to an average of residues over quadratic fields. The final constants are computed from local mass tables and are shown to satisfy the Malle--Bhargava principle, despite the failure of that principle for quartic D4-fields ordered by discriminant.

Significance. If the arguments are correct, this is a landmark result: it provides the first asymptotic with an exact constant for a non-concentrated family of number fields whose Galois group is neither abelian nor symmetric, and it settles the question of whether a positive proportion of Galois octic extensions have non-abelian Galois group in the negative. The proof is a substantial technical tour de force, and the explicit leading constant emerging from local masses without fitted parameters is a strong positive check of the Loughran--Santens conjectures. The paper also gives credit to the prior published work [3] on D4-fields ordered by conductor, which is used as a black box for several counting inputs.

major comments (3)
  1. [§4.1, Lemma 4.11] Lemma 4.11 is stated with a uniform bound in the t-aspect that is not justified by the cited source [21]. For the trivial character α=1, the function D_Λ(s,s;K,1) is, up to absolutely convergent factors, ζ(s)^3 L(s,χ_K) (or ζ(s)^2 ζ_K(s)). The displayed bound |Δ(K)P(Λ)Im(s)|^{1/4−δ} on Re(s)=1/2+ε would imply a Lindelöf-type bound |ζ(1/2+it)|^3 ≪ |t|^{1/4−δ} for large |t|, which is not a consequence of the GL2 subconvexity results in [21] and is stronger than what follows from the Weyl bound. The subsequent estimates in Propositions 4.13, 5.3, 5.4, and Theorem 6.1 all use Lemma 4.11, so this is a load-bearing point. Since the later applications only need |t| ≪ X^ε (as in Lemma 4.8), the lemma should be restricted to this bounded t-range, with the t-aspect absorbed into X^ε; as stated, the lemma is not established and the proof of the power-saving errors does not currently go through.
  2. [§6, proof of Theorem 6.1] The error term displayed in the proof of Theorem 6.1 has a sign error that makes the argument invalid. The text bounds the total number of D4-pairs by O_{A,ε}( X^{1/4}(log X)^2 / T^{3/2−ε} + X^{1/4}/(log X)^{A/4−2} · T^{1+δ−ε} ). The second term grows with T, so summing over T > Z diverges and cannot yield the claimed O(X^{1/4}(log X)^2 / Z^{δ−ε}). Substituting the dyadic height relation Y X_{σ2} X_τ ≍ X^{1/4}/T^{3/2} into the second error term of Theorem 5.4 (which is T^{1/2−δ+ε} X/(log X)^{A/4}) gives instead T^{-1−δ+ε} X^{1/4}/(log X)^{A/4}, which would sum acceptably. As printed, Theorem 6.1 is unproved, and since Theorem 6.1 is used to pass from finite-total-ramification collections to general acceptable collections in Theorem 6.2 and hence to Theorems 1 and 2, the final asymptotic is not established by the displayed proof.
  3. [§5.3, Lemma 5.5 and proof of Theorem 5.1] The proof of Lemma 5.5, which computes the average of the residues of D^{(I)}_Λ(s;K) and is essential for determining the leading constant in Theorem 5.1, is only sketched with the sentence 'We omit the details of the proof since they are identical to those of the previous subsection.' The subsequent application of 'Theorems 6.9 and 7.9 of [3]' is also stated without the necessary verification that those theorems apply with the local conditions and error uniformity required here. A similar omission occurs at the end of the proof of Theorem 5.1, where the tiling argument is deferred to '[3, Theorem 6]'. Because the constant of proportionality in Theorem 1 is a central claim, these omitted details should be supplied or referenced with full precision, rather than left as a gap in the argument.
minor comments (4)
  1. [Throughout] The paper contains numerous typographical slips, including 'Datskowski–Wright' for Datskovsky–Wright, repeated words such as 'we we see' in the proof of Proposition 4.5, and inconsistent hyphenation of 'Malle–Bhargava'. These should be corrected in a final revision.
  2. [§2.2, Definition 2.5] The notation S_K(Λ) and S = S_K^{ram}(Λ) is introduced but used rather densely; a short table or diagram relating S(Λ), S_K(Λ), and S would improve readability.
  3. [§4.3, Lemma 4.12] In the statement of Lemma 4.12, the error term 'X^{1/2+ε}|Δ(K)P(Λ)|^{1/4−δ}' appears with an unbalanced parenthesis in the quadratic/biquadratic case; this should be corrected for clarity.
  4. [§6, proof of Theorem 6.2] The phrase 'By additivity, the general result clearly follows from the result on discriminant stable collections' is too terse; a one-sentence justification that every acceptable collection can be partitioned into finitely many discriminant-stable pieces would be helpful.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the leading constant is derived from Dirichlet-series residues and local mass computations, with self-cited prior work serving as independent published input on a different ordering.

full rationale

I walked the derivation chain from the counting functions in §3–§4 to the asymptotic in Theorem 1 and found no step in which a claimed prediction reduces by construction to a fitted input or to a self-citation. The discriminant identity Δ_S(M)=H_S(L,K)^4 (Lemma 2.4) is an algebraic relation between invariants, not an assumed asymptotic. The double Dirichlet series Φ_Λ(s,t;K) and D_{Λ}(s,t;K,α) are defined from local invariants; their poles and residues are analyzed internally in §4, and no parameter is fitted to the target X^{1/4}\log^2 X count. The final constant is obtained by evaluating local masses (Tables 4 and 5) and by averaging residues (Proposition 5.6), with the single-pole average supplied by Lemma 5.5. That lemma invokes Theorems 6.9 and 7.9 of [3], a prior published paper by two of the present authors; however, [3] concerns quartic D4-fields ordered by Artin conductor, a different invariant, and its theorems are stated and proved there rather than assumed from the present work. Under the stated rules, this is independent support, not load-bearing circularity. The uniform subconvexity bound Lemma 4.11, quoted from [21], is also an external input; whether its t-aspect or uniformity is fully justified is a correctness and assumption risk, not a circularity risk, because the asymptotic conclusion is not an input to that lemma. No equation in the paper is identical to the target result by definition, and no fitted value is renamed as a prediction. Therefore the appropriate circularity score is 0.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No fitted parameters; the constants come from local Euler factors. The paper depends on standard class field theory, Brauer-Siegel, uniform subconvexity bounds, and prior published results of [3] by three authors including two of the current authors. These are external inputs rather than assumptions whose truth the paper's claim depends on in a circular way.

assumptions (4)
  • standard math Global class field theory parametrization of quadratic extensions of a number field (Prop 3.4).
    Used to express the Dirichlet series counting quadratic extensions L/K as a sum over quadratic characters.
  • domain assumption Uniform subconvexity bound for Hecke L-functions over quadratic fields: |D(s,s;K,α)| ≪ (|Δ(K)|P(Λ)|Im(s)|)^{1/4-δ} for some absolute δ>0 (Lemma 4.11).
    Quoted from [21] and used to control error terms in Sections 4.3, 5.2, and 6. The uniformity over the family is not proven in this paper.
  • domain assumption Results of [3, Theorems 6.9 and 7.9] and [3, Theorem 6] on counting D4-fields ordered by Artin conductor.
    Used without proof in Lemma 5.5 and the proof of Theorem 5.1 to evaluate the average of residues determining the main constant.
  • standard math Brauer-Siegel theorem for the size of residues of Dedekind zeta functions of quadratic fields.
    Used in Proposition 5.2 and Theorem 5.4 to bound sums of residues.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Malle's Conjecture for Galois octic fields over $\mathbb Q$." pith.science (2026). https://pith.science/paper/Y2QGND34

@misc{pith2026250523690,
  author       = {Pith},
  title        = {Pith review of: Malle's Conjecture for Galois octic fields over $\mathbb Q$},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/Y2QGND34}},
  note         = {Machine review of arXiv:2505.23690}
}
abstract

We compute the asymptotic number of octic number fields whose Galois groups over $\mathbb Q$ are isomorphic to $D_4$, the symmetries of a square, when ordering such fields by their absolute discriminants. In particular, we verify the strong form of Malle's conjecture for such octic $D_4$-fields and obtain the constant of proportionality. Our result answers the question of whether a positive proportion of Galois octic extensions of $\mathbb Q$ have non-abelian Galois group in the negative. We further demonstrate that the constant of proportionality satisfies the Malle--Bhargava principle of being a product of local masses, despite the fact that this principle does {\em not} hold for discriminants of quartic $D_4$-fields. This is the first instance of asymptotics being recovered for a non-concentrated family of number fields of Galois group neither abelian nor symmetric. Previously, this was only known for abelian fields, degree-$n$ $S_n$-fields for $n=3,4,5$, and degree-$6$ $S_3$-fields.

Discussion (0). Sign in to comment.

Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. A refined Malle conjecture for Heisenberg groups

    math.NT 2026-07 conditional novelty 7.0 of 10

    The leading constant for Malle's conjecture for Heis_4-extensions of Q decomposes as a sum of two Euler products due to a transcendental Brauer–Manin obstruction, yielding the first discriminant-ordering failure of lo...

  2. Counting $D_4$-field extensions by multi-invariants

    math.NT 2025-07 conditional novelty 6.0 of 10

    For D4-Galois extensions of Q, the number with all four Gundlach multi-invariants bounded by X1, X2, X3, X4 is asymptotic to (27/8) times the Euler product over odd primes of (1-1/p)^4(1+4/p) times X1X2X3X4, provided ...

Reference graph

Works this paper leans on

25 extracted references · 23 canonical work pages · cited by 2 Pith papers

  1. [3]

    S. A. Altu˘ g, A. Shankar, I. Varma, and K. H. Wilson. The number ofD 4-fields ordered by conductor. J. Eur. Math. Soc. (JEMS), 23(8):2733–2785, 2021

    work page 2021

  2. [21]

    Michel and A

    P. Michel and A. Venkatesh. The subconvexity problem for GL 2.Publ. Math. Inst. Hautes ´Etudes Sci., (111):171–271, 2010

    work page 2010

  3. [1]

    B. Alberts. The weak form of Malle’s conjecture and solvable groups.Res. Number Theory, 6(1):Paper No. 10, 23, 2020

    work page 2020

  4. [2]

    Alberts, R

    B. Alberts, R. J. Lemke Oliver, J. Wang, and M. M. Wood. Inductive methods for counting number fields.arXiv preprint 2501.18574, 2025

    arXiv 2025

  5. [4]

    Bhargava

    M. Bhargava. The density of discriminants of quartic rings and fields.Ann. of Math. (2), 162(2):1031– 1063, 2005

    work page 2005

  6. [5]

    Bhargava

    M. Bhargava. Mass formulae for extensions of local fields, and conjectures on the density of number field discriminants.Int. Math. Res. Not. IMRN, (17):Art. ID rnm052, 20, 2007

    work page 2007

  7. [6]

    Bhargava

    M. Bhargava. The density of discriminants of quintic rings and fields.Ann. of Math. (2), 172(3):1559– 1591, 2010

    work page 2010

  8. [7]

    Bhargava and M

    M. Bhargava and M. M. Wood. The density of discriminants ofS 3-sextic number fields.Proc. Amer. Math. Soc., 136(5):1581–1587, 2008

    work page 2008

Show all 25 references
  1. [8]

    Cohen, F

    H. Cohen, F. Diaz y Diaz, and M. Olivier. Enumerating quartic dihedral extensions ofQ.Compositio Math., 133(1):65–93, 2002

    2002

  2. [9]

    Datskovsky and D

    B. Datskovsky and D. J. Wright. Density of discriminants of cubic extensions.J. Reine Angew. Math., 386:116–138, 1988

    1988

  3. [10]

    Davenport and H

    H. Davenport and H. Heilbronn. On the density of discriminants of cubic fields. II.Proc. Roy. Soc. London Ser. A, 322(1551):405–420, 1971

    1971

  4. [11]

    K. S. Kedlaya. Mass formulas for local Galois representations.Int. Math. Res. Not. IMRN, (17):Art. ID rnm021, 26, 2007. With an appendix by Daniel Gulotta

    2007

  5. [12]

    Kl¨ uners

    J. Kl¨ uners. ¨Uber die Asymptotik von Zahlk¨ orpern mit vorgegebener Galoisgruppe. Shaker, 2005

    2005

  6. [13]

    Kl¨ uners

    J. Kl¨ uners. The distribution of number fields with wreath products as Galois groups.Int. J. Number Theory, 8(3):845–858, 2012

    2012

  7. [14]

    Kl¨ uners and G

    J. Kl¨ uners and G. Malle. Counting nilpotent Galois extensions.J. Reine Angew. Math., 572:1–26, 2004

    2004

  8. [15]

    Koymans and C

    P. Koymans and C. Pagano. On Malle’s conjecture for nilpotent groups.Trans. Amer. Math. Soc. Ser. B, 10:310–354, 2023

    2023

  9. [16]

    Loughran and T

    D. Loughran and T. Santens. Malle’s conjecture and brauer groups of stacks, 2024

    2024

  10. [17]

    S. M¨ aki. On the density of abelian number fields.Ann. Acad. Sci. Fenn. Ser. A I Math. Dissertationes, (54):104, 1985

    1985

  11. [18]

    S. M¨ aki. The conductor density of abelian number fields.J. London Math. Soc. (2), 47(1):18–30, 1993

    1993

  12. [19]

    G. Malle. On the distribution of Galois groups.J. Number Theory, 92(2):315–329, 2002

    2002

  13. [20]

    G. Malle. On the distribution of Galois groups. II.Experiment. Math., 13(2):129–135, 2004

    2004

  14. [22]

    Shankar and F

    A. Shankar and F. Thorne. On the asymptotics of cubic fields ordered by general invariants. 2022. preprint. 29

    2022

  15. [23]

    C. L. Siegel. The average measure of quadratic forms with given determinant and signature.Ann. of Math. (2), 45:667–685, 1944

    1944

  16. [24]

    M. M. Wood. On the probabilities of local behaviors in abelian field extensions.Compos. Math., 146(1):102–128, 2010

    2010

  17. [25]

    D. J. Wright. Distribution of discriminants of abelian extensions.Proc. London Math. Soc. (3), 58(1):17– 50, 1989. ashankar@math.toronto.edu and ila@math.toronto.edu Department of Mathematics, University of Toronto, Toronto, ON, M5S 2E4, Canada 30

    1989

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.