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Counting abelian number fields with restricted ramification type

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Pith's one-line read Abelian number fields with restricted ramification type satisfy an explicit counting asymptotic.

desk verdict A serious and mostly well-built paper whose general-height result currently rests on a false local weight identity, so the unbalanced case is unproven as written. read the letter →

arxiv 2507.00448 v1 pith:IZLETSOV submitted 2025-07-01 math.NT

Julie Tavernier This is my paper · ORCID
classification math.NT MSC 11R4511R3211M4514G05
keywords Malle'sconjectureabeliannumberfieldsheightfunctionsramificationtypeBrauer-ManinobstructionTamagawameasuresBatyrev-Maninequidistribution
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 proves a general version of Malle's conjecture for finite abelian Galois groups: counting abelian number fields ordered by an arbitrary 'big' height function, with the generator of tame inertia restricted to lie in a given Galois-stable subset of the group, yields an asymptotic $cB^{a}(\log B)^{b-1}$ with explicit exponents and an explicit leading constant. The exponents are read off the weight function defining the height: $a$ is the reciprocal of the minimal weight in the allowed ramification set, and $b$ is the number of Galois orbits of minimal-weight elements. The leading constant is expressed both as sums of Euler products and, via the stack $BG$, as a Tamagawa measure of the adelic points cut out by a partially unramified Brauer group. A corollary proves strong equidistribution under infinitely many local conditions, giving a stacky version of a strong Grunwald problem with restricted ramification type.

What carries the argument

The machine has four parts. (1) The ramification type $\rho_{G,v}:\operatorname{Hom}(\Gamma_{k_v},G)\to G(-1)^{\Gamma_{k_v}}$, a homomorphism whose fibre at $0$ is the unramified characters, together with a $\Gamma_k$-invariant class function $w$ with $w(1)=0$, builds local heights $H_v=q_v^{w(\rho_{G,v})}$. (2) A height zeta function $F_R(s)=\sum f_R(\varphi)H(\varphi)^{-s}$ with the adelic indicator $f_R$, whose local Fourier transforms expand as $1+\lambda_x(v)q_v^{-s/a_R(H)}+\dots$, so by a Tauberian theorem the counts are governed by $a_R(H)$ and the mean $b_R(H)=|M_R(H)/\Gamma_k|$ of an $S$-frobenian function; Poisson summation (Proposition 2.9) transfers the asymptotic from Fourier coefficients back to the count with an explicit Euler-product constant. (3) Balanced heights, where $M_R(H)$ generates $G$, are handled directly; unbalanced heights are reduced to the balanced case by the quotient map $G\to G/\langle M_R(H)\rangle$ (Iitaka fibration) and a dominated-convergence argument along the fibres, using the Greenberg–Wiles formula to control the ramification of lifts. (4) The stack $BG$ repackages the constant as $|G|^{-1}a_R(H)^{b_R(H)-1}|\operatorname{Br}_{M_R(H)}BG/\operatorname{Br} k|\,\tau_H(W_{R,S}\cap BG(\mathbb A_k)^{\operatorname{Br}}_{M_R(H)})/|\widehat G(k)|(b_R(H)-1)!$, with $\tau_H$ a Tamagawa measure, which is what yields equidistribution.

What would settle it

For $G=\mathbb Z/4\mathbb Z$ with $w(1)=w(3)=1$, $w(2)=2$, take a place $v$ and characters $\tilde\psi$, $t$ with $\rho_{G,v}(\tilde\psi)=1$ and $\rho_{G,v}(t)=2$. Since $\rho_{G,v}$ is a homomorphism, $\rho_{G,v}(\tilde\psi+t)=3$; the factorization used in Lemma 3.5 would give $1=w(3)=w(1)w(2)=0$. Checking whether such local characters occur and whether the dominated-convergence bound can be derived without that factorization settles whether the general-height constant formula of Theorem 3.3 stands as proved.

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Extended reading notes

Core claim

The central claim is Theorem 1.1 (balanced case, Theorem 2.10; general case, Theorem 3.3): for a finite abelian group $G$, a number field $k$, a big height $H$ with weight function $w$, and a non-empty Galois-stable $R\subseteq G(-1)^*$, the number of homomorphisms $\varphi:\Gamma_k\to G$ with $H(\varphi)\le B$ and $\rho_{G,v}(\varphi_v)\in R\cup\{0\}$ for all $v\notin S$ is asymptotic to $c_{k,R,G,H}B^{a_R(H)}(\log B)^{b_R(H)-1}$, where $a_R(H)=(\min_{\gamma\in R}w(\gamma))^{-1}$, $b_R(H)=|M_R(H)/\Gamma_k|$ with $M_R(H)$ the minimal-weight elements of $R$, and $c_{k,R,G,H}$ is given explicitly by (2.9) in the balanced case and by the convergent sum (3.1) over fibres of the quotient $G\to G/\langle M_R(H)\rangle$ in general. The author reads this as a Batyrev–Manin-type statement for the stack $BG$, with the leading constant expressed as a Tamagawa measure of the partially unramified Brauer–Manin set, and derives strong equidistribution of these fields under infinitely many local conditions (Theorem 1.4) and a dense-image statement for a strong Grunwald problem with restricted ramification type (Corollary 1.5).

Load-bearing premise

The argument that handles unbalanced heights assumes that the weight of a lifted local character factors into the product of the weights of its two summands; the whole dominated-convergence step relies on that factorization.

Editorial extensions

If this is right

  • Corollary 1.2: setting $R=G(-1)^*$ gives the total count of $G$-extensions of bounded height, recovering the conductor and discriminant cases of Wood and Wright and extending them to arbitrary big heights.
  • Theorem 1.4 (strong equidistribution): for balanced heights, imposing infinitely many local conditions — any continuity set $W$ in the partial adelic space — gives a limit equal to the Tamagawa measure of $W\cap BG(\mathbb A_k)^{\operatorname{Br}}_{M_R(H)}$ divided by that of the Brauer–Manin set, so local behaviour is governed by the Brauer group.
  • Corollary 1.5: the image of $BG[k]$ in $BG(\mathbb A_k)^{\operatorname{Br}}_R$ is dense; in particular the strong Grunwald problem with restricted ramification type has an affirmative answer modulo Brauer–Manin obstruction, new except for $R=G(-1)$.
  • Example 1.8 shows the Brauer element $-4$ forbids $\mathbb Z/4\mathbb Z$-extensions with an odd number of ramified primes $p\equiv 3\bmod 4$ outside the allowed set, so the leading constant is a sum of two Euler products; restricting $R$ to $\{1,3\}$ removes the obstruction and collapses the constant to one Euler product.

Reading between the lines

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

  • One could test whether the same asymptotic shape holds for the non-surjective homomorphisms that are usually discarded: Theorem 2.10 says they are negligible under balanced heights, and the proof of Theorem 1.1 relies on that; an explicit secondary-term analysis for a small group like $\mathbb Z/4\mathbb Z$ would show how much of the constant they contribute.
  • Because the equidistribution result is stated for continuity sets in the strong-adic topology, it should imply variance bounds for families of local conditions under averaged heights, not just the first-order Malle–Bhargava quotient; comparing the Tamagawa-measure formula with Bhargava-style mass formulae could give a check of the leading constant in cases where both apply.
  • The stacky reading suggests that the number-field count is the rational-point count on $BG$ with a peaking at the minimal-weight strata; one could mirror the paper's argument for the balanced-in-fibres Iitaka fibration to other stacks with a toric or weighted-projective structure, where the Brauer group is no longer constant.
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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 / 4 minor

Summary. The paper proves asymptotic counting results for homomorphisms φ: Γ_k → G, where G is a finite abelian group, subject to a restriction on the local ramification type ρ_{G,v}(φ_v) lying in a given Galois-stable subset R of G(−1)^*, ordered by an arbitrary big height function H associated to a weight function w. The main theorem (Theorem 1.1) asserts an asymptotic of the form c B^{a_R(H)} (log B)^{b_R(H)-1} with an explicit constant; the balanced case (Theorem 2.10) is proven by harmonic analysis, an Euler product expansion, and Poisson summation, while the unbalanced case (Theorem 3.3) is treated by a fibre decomposition and a dominated-convergence argument. The paper also gives a stacky reformulation over BG, proves a strong equidistribution statement for balanced heights, and derives a Grunwald-type corollary. Worked examples with Z/4Z and Q illustrate the leading constants and Brauer-Manin obstructions.

Significance. The balanced case would be a substantial contribution: it establishes a general-height version of Malle's conjecture for abelian groups with restricted ramification type, with a leading constant expressed as a sum of Euler products that is not fitted to data. The stacky reformulation and equidistribution theorem for balanced heights provide a concrete bridge between analytic number theory and the Batyrev-Manin conjecture on stacks, confirming predictions of Loughran and Santens [18]. The paper is transparent about its reliance on [18] for the stacky framework, and there is no circularity: the constants are derived from the analytic arguments. However, the unbalanced case, which is essential for the unrestricted Theorem 1.1, rests on the flawed Lemma 3.5, so the full theorem as stated is not yet established.

major comments (2)
  1. [§3.2, Lemma 3.5, Eqs. (3.3)–(3.4)] The displayed identity w(ρ_{G,v}(ψ~+t)) = w(ρ_{G,v}(ψ~))·w(ρ_{G,v}(t)) is not a consequence of the definitions. Lemma 2.1(1) gives ρ_{G,v}(ψ~+t) = ρ_{G,v}(ψ~)+ρ_{G,v}(t), and w is an arbitrary class function with w(1)=0 (Definition 2.2); no multiplicativity is assumed. Taking v outside S_{H,0}, ψ~ unramified at v, and t of minimal ramification type γ at v yields a left-hand side w(γ)=a_R(H)^{-1} and a right-hand side w(1)w(γ)=0, so (3.4) is impossible. Because this identity is the step that transfers the height threshold and produces the uniform bound over ψ, the dominated-convergence interchange in Theorem 3.3 is not justified, and Theorem 1.1 for arbitrary big heights is not proven as written. The proof needs a correct comparison between w(ρ(ψ~+t)) and w(ρ(t)) (e.g., using additivity or convexity properties of w that are not present in the paper), or the unbalanced case must be handled differently.
  2. [§2.5, Proposition 2.9] The Poisson summation formula is used to pass from the asymptotic for the Fourier coefficients ∑_{n≤B} α_n(G,x) (Lemma 2.8) to the asymptotic for the height zeta function coefficients in Theorem 2.10, but its proof is omitted with only the remark that it is 'very similar' to [14, Prop. 3.9]. Since the height here involves an arbitrary weight function and the indicator f_R of restricted ramification type, the hypotheses of the cited result need to be checked explicitly; as it stands, this is a missing load-bearing proof. Please include a full proof or a precise statement of the variant with all conditions verified.
minor comments (4)
  1. [§2.3, Lemma 2.5] The notation Hom(O_v^×,G) is used for what appears to be the quotient of Hom(k_v^×,G) by the unramified characters; as written it literally denotes homomorphisms of O_v^×, for which the statement 'exactly one χ_v with ρ(χ_v)=1' is false. Please clarify the identification.
  2. [§1.1 and §2.1] The term 'big' is used for weights with w(γ)>0 for all γ≠1, but Theorem 1.1 also assumes R non-empty; since a_R(H) is not defined if R is empty, the text should state this explicitly.
  3. [Corollary 2.11] The notation C_{k,R,G,H_min} is confusing because the constant appears to depend only on the minimum height, whereas it depends on the full height function; please rename, for example C_{k,R,G,H}.
  4. [§4.5, Example 1.8(3)] The product over 'all places of Q(√2)' with factors (1−1/q_v)(1+1/q_v) would vanish at archimedean places if q_v is interpreted literally; please specify that the product is over non-archimedean places, or that archimedean factors are taken to be 1.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the asymptotic exponents and leading constants are derived from Euler-product calculations and standard Tauberian/Poisson-summation arguments, and the stacky reformulation is an independent interpretation rather than an input.

full rationale

The paper contains no fitted parameters that are later relabelled as predictions, and no load-bearing premise is justified only by a self-citation. The central analytic chain runs from the local Fourier transforms (Lemmas 2.4 and 2.5) to the Euler-product expansion of the height zeta function, the S-frobenian computation of b_R(H), Poisson summation (Proposition 2.9), and the Tauberian theorem (Theorem 2.7), with the leading constant expressed as a sum of explicit Euler products in (2.9). The unbalanced case (Theorem 3.3) is handled by applying the already-proved balanced Theorem 2.10 on the proper subgroup <M_R(H)> and by a dominated-convergence argument adapted from Koymans and Rome [17]; this is an application of a previous result to a smaller group, not an assumption of the target asymptotic. The extensive citations to [18] supply the stacky dictionary, the partially unramified Brauer group, and the Tamagawa-measure interpretation, but these are independent prior results by other authors, and the analytic theorem is not derived from them. The reported issue in Lemma 3.5, namely the displayed identity w(rho(psi~+t)) = w(rho(psi~)) w(rho(t)) with w(1)=0, is a mathematical-consistency concern about the uniform bound, not a circularity: it is a false or unsupported equation in a proof, not a reduction of the claimed output to its own input. Accordingly the circularity score is 0.

Assumptions & free parameters 1 free parameters · 6 assumptions · 0 invented entities

The central claim rests on standard reciprocity and Tauberian theorems and on a substantial body of results from Loughran-Santens [18], none of which are reproduced in the paper. There are no fitted numerical parameters and no new physical or geometric entities are postulated.

free parameters (1)
  • weight function w: G(-1) -> Q with w(1) = 0
    The theorem is stated for an arbitrary big class function w; the constant is an explicit function of w. It is a chosen input, not a number fitted to data.
assumptions (6)
  • standard math Local and global class field theory identify Hom(Gamma_k,G) with Hom(A^times/k^times,G) and Hom(k_v^times,G), and give the local Artin map and inertia isomorphisms.
    Used throughout Section 2, especially in the definition of ramification type and the Fourier transform setup.
  • standard math Delange's Tauberian theorem applies to the Dirichlet series with a pole of order b_R(H) at a_R(H).
    Theorem 2.7 extracts the B^{a_R(H)} (log B)^{b_R(H)-1} asymptotic from the height zeta function.
  • domain assumption Proposition 2.9, a Poisson summation formula for the global Fourier transform of f_R/H^s, whose proof is omitted as very similar to [14, Prop. 3.9].
    This is the bridge from local Euler products to the global height zeta function; it is not proved in the paper.
  • standard math Greenberg-Wiles formula and local Tate duality provide lifts of G/<M_R(H)>-extensions and the uniform bound in Lemma 3.5.
    Used in the unbalanced-height case; the proof is sketched and mainly cited to [22, Thm. 8.7.9].
  • domain assumption Frobenianity of lambda_x, local mass formulae, finiteness of the partially unramified Brauer group, and Tamagawa measure identities from [18] hold as stated.
    Theorem 2.10 uses [18, Rem. 8.13, Thm. 8.23]; Theorem 4.6 and Lemma 4.7 use [18, Cor. 6.30, Lem. 6.27, Thm. 6.29, Lem. 8.19, Lem. 10.22].
  • domain assumption Balancedness of H with respect to R makes X(k,R,H) finite, via equality with Br_{M_R(H)}BG / Br k.
    Proposition 4.3; the leading constant (2.9) is a finite sum only under this assumption.

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Pith. "Pith review of Counting abelian number fields with restricted ramification type." pith.science (2026). https://pith.science/paper/IZLETSOV

@misc{pith2026250700448,
  author       = {Pith},
  title        = {Pith review of: Counting abelian number fields with restricted ramification type},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/IZLETSOV}},
  note         = {Machine review of arXiv:2507.00448}
}
abstract

We count abelian number fields ordered by arbitrary height function whose generator of tame inertia is restricted to lie in a given subset of the Galois group, and find an explicit formula for the leading constant. We interpret our results as a version of the Batyrev-Manin conjecture on $BG$ and rephrase our result on number fields with restricted ramification type in terms of integral points on $BG$. We also prove that such number fields are equidistributed with respect to suitable collections of infinitely many local conditions.

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Forward citations

Cited by 1 Pith paper

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Works this paper leans on

30 extracted references · 23 canonical work pages · cited by 1 Pith paper

  1. [18]

    Loughran and T

    D. Loughran and T. Santens, Malle’s conjecture and Brauer groups of stacks , 2024, arXiv:2412.04196

    arXiv 2024

  2. [1]

    Alberts, A random group with local data realizing heuristics for number field counting, 2023, arXiv:2304.01323

    B. Alberts, A random group with local data realizing heuristics for number field counting, 2023, arXiv:2304.01323

    arXiv 2023

  3. [2]

    Alberts and E

    B. Alberts and E. O’Dorney,Harmonic analysis and statistics of the first Galois cohomology group, Res. Math. Sci.8 (2021), no. 3, Paper No. 50, 16

    work page 2021

  4. [3]

    Alberts, R.J

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

    arXiv 2025

  5. [4]

    Bhargava,The density of discriminants of quartic rings and fields, Ann

    M. Bhargava,The density of discriminants of quartic rings and fields, Ann. of Math.162 (2005), 1031–1062

    work page 2005

  6. [5]

    , Mass formulae for extensions of local fields, and conjectures on the density of number field discriminants, Int. Math. Res. Not. IMRN (2007), no. 17, Art. ID rnm052, 20

    work page 2007

  7. [6]

    of Math172 (2010), no

    , The density of discriminants of quintic rings and fields, Ann. of Math172 (2010), no. 3, 1559–1591 (English (US)). COUNTING ABELIAN NUMBER FIELDS WITH RESTRICTED RAMIFICATION TYPE 35

    work page 2010

  8. [7]

    Bhargava, A

    M. Bhargava, A. Shankar, and X. Wang,Geometry-of-numbers methods over global fields I: Pre- homogeneous vector spaces, 2015, arXiv:1512.03035

    arXiv 2015

Show all 30 references
  1. [8]

    Darda and T

    R. Darda and T. Yasuda,Torsors for finite group schemes of bounded height, J. Lond. Math. Soc. (2) 108 (2023), no. 3, 1275–1331

    2023

  2. [9]

    , The Batyrev-Manin conjecture for DM stacks, 2024, arXiv:2207.03645

    2024 arXiv

  3. [10]

    Délange,Généralisation du théorème de Ikehara, Ann

    H. Délange,Généralisation du théorème de Ikehara, Ann. Sci. Éc. Norm. Supér71 (1954), no. 3, 213–242 (fre)

    1954

  4. [11]

    J. S. Ellenberg, M. Satriano, and D. Zureick-Brown,Heights on stacks and a generalized Batyrev- Manin-Malle conjecture, Forum Math. Sigma11 (2023), Paper No. e14, 54

    2023

  5. [12]

    J. S. Ellenbergand A.Venkatesh,Counting extensions of function fields with bounded discriminant and specified Galois group, Geometric methods in algebra and number theory, Progr. Math., vol. 235, Birkhäuser Boston, Boston, MA, 2005, pp. 151–168

    2005

  6. [13]

    C. Frei, D. Loughran, and R. Newton,The Hasse norm principle for abelian extensions, Amer. J. Math. 140 (2018), no. 6, 1639–1685

    2018

  7. [14]

    , Number fields with prescribed norms, Comment. Math. Helv.97 (2022), no. 1, 133–181, With an appendix by Yonatan Harpaz and Olivier Wittenberg

    2022

  8. [15]

    Gundlach,Malle’s conjecture with multiple invariants, 2022, arXiv:2211.16698

    F. Gundlach,Malle’s conjecture with multiple invariants, 2022, arXiv:2211.16698

    2022 arXiv

  9. [16]

    Koymans and C

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

    2023

  10. [17]

    Koymans and N

    P. Koymans and N. Rome,A note on the Hasse norm principle, Bull. Lond. Math. Soc.56 (2024), no. 3, 1004–1013

    2024

  11. [19]

    Malle,On the distribution of Galois groups, J

    G. Malle,On the distribution of Galois groups, J. Number Theory92 (2002), no. 2, 315–329

    2002

  12. [20]

    II, Experiment

    , On the distribution of Galois groups. II, Experiment. Math.13 (2004), no. 2, 129–135

    2004

  13. [21]

    Narkiewicz,Number theory, World Scientific, 1983

    W. Narkiewicz,Number theory, World Scientific, 1983

    1983

  14. [22]

    Neukirch, A

    J. Neukirch, A. Schmidt, and K. Wingberg,Cohomology of number fields, Grundlehren der math- ematischen Wissenschaften, Springer Berlin Heidelberg, 2013

    2013

  15. [23]

    Peyre,Hauteurs et mesures de Tamagawa sur les variétés de Fano, Duke Math

    E. Peyre,Hauteurs et mesures de Tamagawa sur les variétés de Fano, Duke Math. J.79 (1995), no. 1, 101–218

    1995

  16. [24]

    Santens,Manin’s conjecture for integral points on toric varieties, 2023, arXiv:2312.13914

    T. Santens,Manin’s conjecture for integral points on toric varieties, 2023, arXiv:2312.13914

    2023 arXiv

  17. [25]

    Serre,Topics in Galois theory, Research Notes in Mathematics, vol

    J.P. Serre,Topics in Galois theory, Research Notes in Mathematics, vol. 1, Jones and Bartlett Publishers, 1992, Notes written by Henri Darmon

    1992

  18. [26]

    11, CRC Press, Boca Raton, FL, 2012

    , Lectures onNX (p), Chapman & Hall/CRC Research Notes in Mathematics, vol. 11, CRC Press, Boca Raton, FL, 2012

    2012

  19. [27]

    The Stacks project authors,Stacks project

  20. [28]

    Wiles, Modular elliptic curves and Fermat’s last theorem, Ann

    A. Wiles, Modular elliptic curves and Fermat’s last theorem, Ann. of Math. 141 (1995), no. 3, 443–551

    1995

  21. [29]

    Wood,On the probabilities of local behaviors in abelian field extensions, Compositio Math- ematica 146 (2010), no

    M.M. Wood,On the probabilities of local behaviors in abelian field extensions, Compositio Math- ematica 146 (2010), no. 1, 102–128

    2010

  22. [30]

    Wright, Distribution of discriminants of abelian extensions, Proc

    D.J. Wright, Distribution of discriminants of abelian extensions, Proc. London Math. Soc. 58 (1989), no. 1, 17–50. Julie Tavernier, Department of Mathematical Sciences, University of Bath, Claver- ton Down, Bath, BA2 7AY, UK Email address: jlt86@bath.ac.uk

    1989

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