Inductive methods for counting number fields

Brandon Alberts; Jiuya Wang; Melanie Matchett Wood; Robert J. Lemke Oliver

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Inductive methods for counting number fields

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Pith's one-line read A fiber-summation method converts weak subfield bounds into full asymptotic counts of number fields, proving many new cases of the Number Field Counting Conjecture and new counterexamples to its log-exponent prediction.

desk verdict A major new inductive method for Malle's conjecture that is likely correct, but with a black-box dependency on a corrected theorem that needs careful checking. read the letter →

arxiv 2501.18574 v1 pith:PSMBZHPM submitted 2025-01-30 math.NT

Brandon Alberts , Robert J. Lemke Oliver , Jiuya Wang , Melanie Matchett Wood This is my paper · ORCID

classification math.NT MSC 11R4511R3211R29

keywords numberfieldcountingdiscriminantasymptoticsGaloisextensionsclassgrouptorsionunramifiedcohomologywreathproductsnilpotentgroupsclosure

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 a general inductive method for counting number fields with fixed Galois closure group $G$, ordered by discriminant. It proves that if $G$ has a proper abelian normal subgroup $T$ and one has a mild averaged upper bound on the number of $G/T$-extensions weighted by certain unramified cohomology groups, then the number of $G$-extensions up to discriminant $X$ is asymptotic to $c X^{1/a(T)} (\log X)^{b-1}$; a parallel result handles wreath products $S_3 \wr B$. The input bounds are weak, in that they do not require most $T$-extensions over a $G/T$-extension to be $G$-extensions, so previously unreachable families fall out, including most nilpotent transitive groups in any permutation representation, iterated cyclic wreath products, and many $S_3$-wreath products. The method also produces infinitely many cases where the power of $\log X$ disagrees with the original prediction, recovering and extending the counterexample $C_3 \wr C_2$.

What carries the argument

The central mechanism is the pushforward fiber decomposition along the quotient map $q : G \to G/T$, viewed on surjective homomorphisms from the absolute Galois group: every $G$-extension lies in a fiber $q_*^{-1}(\pi)$ over a $G/T$-extension $\pi$, and the theorem adds these fibers together after checking a convergence criterion. The objects that make this work are the unramified cohomology groups $H^1_{\mathrm{ur}}(k,T(\pi))$, the Galois twist of $T$ by $\pi$, which generalize class group torsion and control the size of each fiber; the pushforward discriminant $q_* \mathrm{disc}$, which measures how a $G$-extension's discriminant bounds the underlying $G/T$-extension; and the index invariant $a(T)$ and orbit count $b(k,T(\pi))$ governing the asymptotic shape. Theorem 2.1 is the summation engine: precise fiber asymptotics plus a uniform upper bound with convergent sum force the total asymptotic, with a leading constant equal to the sum of the fiber constants.

What would settle it

Take a concrete abelian $T$, for example $T = C_3$ inside a sextic group with a fixed quadratic resolvent, and compute $\#\{\psi \in q_*^{-1}(\pi) : \mathrm{disc}\,\psi \leq X\}$ for a fixed $\pi$; if the ratio to $c X^{1/a(T)} (\log X)^{b(k,T(\pi))-1}$ fails to tend to 1 for some $\pi$, the fiber input used in Theorem 1.11 is false, and the main abelian case loses its foundation.

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Extended reading notes

Core claim

At the core is Theorem 1.11: for a finite transitive permutation group $G$ with abelian normal subgroup $T$, if the input bound (1.4) holds with $\theta < 1/a(T)$, then $\#F_{n,k}(G;X) \sim c(k,G) X^{1/a(T)} (\log X)^{\max_\pi b(k,T(\pi))-1}$, which is the Number Field Counting Conjecture for $G$ with explicit constants; if $\theta \geq 1/a(T)$, the same input yields the upper bound $X^{\theta+\varepsilon}$. Theorem 1.9 gives the analogous statement for $G = S_3 \wr B$, with the input an averaged 2-torsion class group bound and conclusion $\#F_{3m,k}(G;X) \sim c X$ when the exponent is below 2. Both theorems are consequences of a general fiber-summation theorem (Theorem 2.1): whenever each fiber of the pushforward $q_* : \mathrm{Sur}(G_k,G) \to \mathrm{Sur}(G_k,G/T)$ has an asymptotic with constant $c(\pi)$, a uniform upper bound $f(\pi)$, and $\sum f(\pi)$ converges, the total count is the sum of the $c(\pi)$. The paper verifies these fiber data for abelian $T$ using twisted abelian extension counts and for $T = S_3^m$ using cubic-extension counts with local conditions.

Load-bearing premise

The load-bearing premise is the exact asymptotic count, from the cited fiber-counting work, for abelian extensions with a prescribed twisted action and no local restrictions; the paper notes that work was corrected after an error, and argues this statement is unaffected, but if that argument fails the abelian branch of the method collapses.

Editorial extensions

If this is right

Corollary 1.2: for every finite nilpotent transitive group whose minimal-index elements generate an abelian subgroup, the Number Field Counting Conjecture holds over every number field; a machine computation shows this covers at least 2,686,926 of the 2,739,294 nilpotent transitive groups of degree at most 32.
Corollary 1.3: for any transitive $B$ with $\#F_{m,k}(B;X) \ll X^{1/2+1/(\ell-1)-\delta}$, the wreath product $C_n \wr B$ satisfies the conjecture, including regular, nilpotent, and large-rank finite simple Lie type base groups.
Corollary 1.4: iterated cyclic wreath products $C_{n_1} \wr \cdots \wr C_{n_r}$ satisfy the conjecture in four explicit families, including all Sylow $p$-subgroups and the counterexample $C_3 \wr C_2$.
Corollary 1.6: $S_3 \wr B$ satisfies the conjecture whenever the averaged 2-torsion bound in (1.1) has $\theta < 2$, giving the first results for nontrivial $S_3$-wreath products.
Corollary 1.8: the conjecture holds for at least 1665 transitive groups of degree up to 23 over $\mathbb{Q}$, 339 of them non-nilpotent.

Reading between the lines

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

The framework is deliberately modular: any future proof of the twisted fiber asymptotics for a nonabelian $T$, with control of how the constants depend on $\pi$, would automatically yield the counting conjecture for any group concentrated in $T$.
Because the input bound (1.4) is an average over $G/T$-extensions of $|H^1_{\mathrm{ur}}(k,T(\pi))|$, any new average bound on class group torsion, for instance on $|\mathrm{Cl}_F[\ell]|$, would lower $\theta$ and directly widen the families covered.
The log-exponent corrections proved for $C_\ell \wr C_d$ are driven by orbit counts of minimal-index elements under the $\pi$-twisted cyclotomic action; this suggests the original $b$-formula is systematically wrong for concentrated groups of this shape, not just in one example.
Theorem 9.1, which produces an admissible ordering for which every group with a nontrivial abelian normal subgroup has linear asymptotics, indicates the fiber-summation engine is an ordering-flexible counting principle, not a discriminant-specific trick.

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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

This is a substantial paper. The fiber-summation framework in Theorem 2.1 is a clean and powerful idea, and the authors use it to prove new cases of Malle's conjecture in impressive generality — nilpotent groups in arbitrary permutation representations, iterated wreath products of cyclic groups, and S3 ≀ B. The strength of the results (leading constants and log powers) is a step beyond prior inductive methods. The paper is also honest about what it does not do.

What is actually new: Theorem 2.1 itself, the summation method that adds fiber counts with uniform error bounds, is simple but seems to be the right abstraction. The execution for abelian T (Theorem 1.11) and for T = S3^m (Theorem 1.9) are genuinely new, not just reworkings. The applications to infinite families are remarkable in scope, and the computational benchmarks (e.g., 2,686,926 of 2,739,294 nilpotent transitive groups of degree ≤32) give a concrete sense of the reach. The code and data are available, which is appreciated.

My main concern is the black-box use of [AO21]. The proof of Theorem 1.11 is built on the Alberts–O'Dorney fiber-counting theorem, which has a published corrigendum [AO23]. Remark 6.2 argues that the no-local-restriction case is unaffected, and that argument is plausible — the error concerned nonviable local families, and the trivial class is always viable. But the authors do not quote the corrected theorem's hypotheses or demonstrate that the unrestricted setting meets them beyond the trivial-class remark. A referee should ask them to fill this in. If the corrected theorem requires an extra nonvanishing condition, the main asymptotic for abelian T would lose its lower bound. I do not think it does, but it deserves verification.

Secondary issues are minor. In the proof of Corollary 1.2, the comparison between disc(F(pi)/Q) and q_* disc(pi) is asserted without proof; this is needed only to verify a hypothesis for that corollary, and the gap looks repairable. In Lemma 6.3, the infinite-place contribution is bounded by |T[2]|^n, which is not quite right for real places where the fixed subgroup can be all of T; since constants may depend on k, this does not affect the result.

Who this is for: anyone working on counting number fields, Malle's conjecture, or arithmetic statistics more broadly. It will be a standard reference. My recommendation: engage with it, cite it, send it to a strong journal. The conditional on [AO21] is worth a careful read, but the framework is robust and the results are major.

Referee Report

2 major / 4 minor

Summary. The paper introduces an inductive method for counting number fields of fixed Galois closure group G by discriminant. The key framework (Theorem 2.1) decomposes the count of G-extensions into fibers over a quotient G/T, under hypotheses of precise fiber counts, uniform fiber bounds, and a convergence criterion. The main applications are Theorem 1.9 for wreath products S3 ≀ B and Theorem 1.11 for groups with an abelian normal subgroup T; both convert input bounds of the form (1.1) and (1.4) into asymptotics when the input exponent is below a threshold. These results are used to prove many new cases of Malle's Conjecture, including nilpotent groups, cyclic wreath products, iterated wreath products, and Klüners-style counterexamples, with supporting Magma computations reported in Corollary 1.8.

Significance. If the external input from [AO21]/[AO23] is fully valid in the no-local-restriction setting, this is a substantial advance: Theorem 2.1 gives a clean, flexible summation framework; Theorem 1.9 provides the first asymptotics for S3-wreath families; Theorem 1.11 supplies many new cases of Malle's Conjecture for concentrated groups, with explicit a- and b-values and a quantitative treatment of the class-group-torsion input. The paper is careful to state its input hypotheses as explicit bounds, and the computational census in Section 7 is a useful benchmark. The main risk is the black-box dependence on a corrected external theorem, which affects the central engine of Theorem 1.11.

major comments (2)

[§6, proof of Theorem 1.11, Remark 6.2] The 'precise counting of the fibers' hypothesis, Theorem 2.1(1), is supplied entirely by [AO21, Theorem 1.1 and Corollary 1.2] as a black box. The published version of [AO21] was corrected in [AO23], and Remark 6.2 asserts that the unrestricted local condition L_p = H^1(k_p, T(π)) is viable because the trivial class satisfies these local conditions. This is not a verification: the corrected theorem's precise hypotheses are not quoted, and no check is made of any additional nonvanishing, convergence, or meromorphy condition that [AO23] may impose on the generating Dirichlet series. Since Theorem 1.11(i) is the engine for the majority of the paper's new cases, the authors should either state the corrected theorem in full and prove that it applies with no local restrictions, or explicitly present Theorem 1.11 as conditional on that external input.
[§7.3, proof of Corollary 1.2] The step 'p | disc(F(π)/Q) iff p | q_* disc(π)' is asserted without proof, and the displayed comparison |disc(F(π)/Q)|^{ε'} ≪ |q_* disc(π)|^{ε} is not immediate from the definition of the pushforward discriminant in (5.2). This comparison is used to pass from the pointwise bound of Corollary 1.14(i) to the hypothesis (1.4), so Corollary 1.2 is not fully proved as written. Provide a proof of the ramification-support comparison, or replace it with a proved upper bound that suffices for the summation.

minor comments (4)

[Lemma 6.3] In the final displayed inequality, the product over infinite places is bounded by |T[2]|^n, but for a real place the fixed subgroup H^0(k_p, T(π)) can be all of T (for example when T is central), so the displayed bound is not valid as stated. A correct uniform bound is |T|^{[k:Q]}; since constants are allowed to depend on k and G, this does not affect the main results.
[Theorem 1.11] In part (i), the constant is written as c(K,G), while everywhere else and in the surrounding text it is c(k,G). This should be made uniform.
[§1.6, Notation] The notation list defines Sur(GK,G), using 'GK' instead of 'G_k'; this is inconsistent with the rest of the paper and should be corrected.
[§5.1] The line 'S /lessn⋊tequalS′ /lessn⋊tequalG' contains a formatting artifact and should be replaced by proper subgroup notation such as S ≤ S' ≤ G.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the central theorems are conditional summations of externally established fiber counts; self-cited inputs are independent published results.

full rationale

The derivation chain is conditional and not circular. Theorem 2.1 is a summation identity: if each fiber q_*^{-1}(pi) has asymptotic c(pi) X^{1/a}(log X)^{b-1}, uniform upper bounds f(pi) with convergent sum, then summing fibers gives the total asymptotic. This is a straightforward analytic lemma, not a disguised prediction. Theorem 1.11 verifies Theorem 2.1's hypotheses: the 'precise counting of fibers' condition is supplied by the externally published Alberts-O'Dorney theorem [AO21, Theorem 1.1 and Corollary 1.2] for abelian T, not by the conclusion of Theorem 1.11; the required uniformity is proved in Theorem 6.1 via Lemmas 6.3-6.6; and convergence follows from the explicit input bound (1.4). Similarly, Theorem 1.9 obtains fiber counts from S3-extensions via [DW88] and uniformity from [LOWW21]. The applications (Corollaries 1.2-1.7, 7.1-7.2, 9.1) all feed previously established upper bounds for G/T-extensions and class-group torsion into Theorems 1.9 and 1.11; none of these bounds is fitted to the target asymptotic or defined in terms of it. The only notable author-overlap input is [AO21]/[AO23] (Alberts is a coauthor of both papers). This is load-bearing for Theorem 1.11(i), and Remark 6.2 argues that the corrigendum's restriction to local conditions does not affect the no-local-restriction setting; whether that argument is fully correct is a correctness/verification question, not a circularity. No equation in the paper is equivalent by construction to its own output, and no fitted parameter is renamed as a prediction.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No fitted free parameters are introduced: the input exponents theta, beta, t are hypotheses supplied by cited theorems, not fitted values. The paper's constructions (pushforward discriminant, T(pi), q_* Sur) are explicit definitions with no unexplained entities. All load-bearing background results are listed as axioms.

assumptions (5)

domain assumption [AO21, Theorem 1.1 and Corollary 1.2] establish the asymptotic for fibers over π for abelian T with no local restrictions; used as black box in Theorem 1.11.
Load-bearing input for fiber counts; the paper's Remark 6.2 notes the original had an error fixed in [AO23] and argues the no-local-restriction case is unaffected.
domain assumption [DW88] asymptotic for S3-extensions over a number field with local conditions, used in Theorem 3.1.
Provides the fiber count for S3 ≀ B.
domain assumption Class group torsion bounds: [BST+20] for 2-torsion, [LOS24] for average 2-torsion, [DW88] for 3-torsion averages, [LOWW21] for 3-torsion in 2-extensions.
Used to verify the input bounds (1.1) and (1.4) in the corollaries.
domain assumption Upper bounds for extension counts: [Alb20] for nilpotent groups with arbitrary invariant, [EV06], [Sch95], [Lem24].
Used to bound q_* Sur(Gk,G;X) in applications.
standard math Standard analytic number theory (Perron formula, contour integration, convexity bounds for Dedekind zeta functions) and class field theory.
Background tools used in Section 6.

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Cite this review

Pith. "Pith review of Inductive methods for counting number fields." pith.science (2026). https://pith.science/paper/PSMBZHPM

@misc{pith2026250118574,
  author       = {Pith},
  title        = {Pith review of: Inductive methods for counting number fields},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PSMBZHPM}},
  note         = {Machine review of arXiv:2501.18574}
}

abstract

We give a new method for counting extensions of a number field asymptotically by discriminant, which we employ to prove many new cases of Malle's Conjecture and counterexamples to Malle's Conjecture. We consider families of extensions whose Galois closure is a fixed permutation group $G$. Our method relies on having asymptotic counts for $T$-extensions for some normal subgroup $T$ of $G$, uniform bounds for the number of such $T$-extensions, and possibly weak bounds on the asymptotic number of $G/T$-extensions. However, we do not require that most $T$-extensions of a $G/T$-extension are $G$-extensions. Our new results use $T$ either abelian or $S_3^m$, though our framework is general.

Discussion (0). Continue with ORCID to comment.

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Reference graph

Works this paper leans on

14 extracted references · 11 canonical work pages · cited by 4 Pith papers

[11]

Pointwise Bound for $\ell$-torsion in Class Groups II: Nilpotent Extensions

[Wan20] Jiuya Wang. Pointwise Bound for ℓ-torsion in Class Groups II: Nilpotent Extensions. arXiv:2006.10295 [math] , June

work page Pith review arXiv 2006
[12]

Pointwise bound for ℓ-torsion in class groups: Elementary abelian extensions

[Wan21b] Jiuya Wang. Pointwise bound for ℓ-torsion in class groups: Elementary abelian extensions. Journal für die reine und angewandte Mathematik (Crelles Jou rnal), 2021(773):129–151, April

work page 2021
[1985]

Faithful Artin induction and the Chebotarev density theorem

[LOS24] Robert J. Lemke Oliver and Alexander Smith. Faithfu l Artin induction and the Chebotarev density theorem. (arXiv:2405.08383), May

work page Pith review arXiv
[1987]

Malle’s conjecture and Brauer groups of stacks

[LS24] Daniel Loughran and Tim Santens. Malle’s conjecture and Brauer groups of stacks. arXiv preprint arXiv:2412.04196,

arXiv
[1997]

[BF10] Karim Belabas and Étienne Fouvry

Computational algebra and number the- ory (London, 1993). [BF10] Karim Belabas and Étienne Fouvry. Discriminants cub iques et progressions arithmétiques. In- ternational Journal of Number Theory , 6(07):1491–1529,

work page 1993
[2006]

Ellenberg and Akshay Venkatesh

[EV07] Jordan S. Ellenberg and Akshay Venkatesh. Reﬂection principles and bounds for class group torsion. International Mathematics Research Notices , 2007,

work page 2007
[2009]

Asymptotics for number ﬁeld s and class groups

[Woo16] Melanie Matchett Wood. Asymptotics for number ﬁeld s and class groups. In Directions in Num- ber Theory: Proceedings of the 2014 WIN3 Workshop . Springer, New York, NY, 1st ed. 2016 edition edition, July

work page 2014
[2010]

Galois groups of random integer po lynomials and van der Waerden’s Conjec- ture

[Bha24] Manjul Bhargava. Galois groups of random integer po lynomials and van der Waerden’s Conjec- ture. Ann. of Math. , to appear, 2024+. [BHRD13] John N. Bray, Derek F. Holt, and Colva M. Roney-Doug al. The maximal subgroups of the low- dimensional ﬁnite classical groups , volume 407 of London Mathematical Society Lecture Note Series. Cambridge Univers...

work page 2024

Show all 14 references

[2016]

[Wri89] David J. Wright. Distribution of discriminants of a belian extensions. Proceedings of the London Mathematical Society, s3-58(1):17–50, 1989

1989
[2020]

G eometry-of-numbers methods over global ﬁelds I: Prehomogeneous vector spaces, Dec 2015

[BSW15] Manjul Bhargava, Arul Shankar, and Xiaoheng Wang. G eometry-of-numbers methods over global ﬁelds I: Prehomogeneous vector spaces, Dec 2015 . Preprint available at https://arxiv.org/abs/1512.03035. [BSW22] Manjul Bhargava, Arul Shankar, and Xiaoheng Wang. S quarefree va...

2015
[2021]

Power Savings for Counting (Twis ted) Abelian Extensions of Number Fields

[Alb24a] Brandon Alberts. Power Savings for Counting (Twis ted) Abelian Extensions of Number Fields. (arXiv:2402.03475), February

arXiv
[2022]

Counting nilpotent Galois extensions

[KM04] Jürgen Klüners and Gunter Malle. Counting nilpotent Galois extensions. Journal für die reine und angewandte Mathematik (Crelle’s Journal) , 2004(572):1–26,

2004
[2023]

Lemke Oliver, Jiuya Wang, and Melanie Mat chett Wood

[LOWW21] Robert J. Lemke Oliver, Jiuya Wang, and Melanie Mat chett Wood. The average size of 3-torsion in class groups of 2-extensions. (arXiv:2110.07712), October

arXiv
[2024]

Restricting the splitting types of a positive density set of places in number ﬁeld extensions

[Alb24b] Brandon Alberts. Restricting the splitting types of a positive density set of places in number ﬁeld extensions. Proceedings of the American Mathematical Society , 152(05):1907–1914,

1907

Pith tools

Reviewed August 9, 2026 · model on record in the stance chip above.

[1] [11]

Pointwise Bound for $\ell$-torsion in Class Groups II: Nilpotent Extensions

[Wan20] Jiuya Wang. Pointwise Bound for ℓ-torsion in Class Groups II: Nilpotent Extensions. arXiv:2006.10295 [math] , June

work page Pith review arXiv 2006

[2] [12]

Pointwise bound for ℓ-torsion in class groups: Elementary abelian extensions

[Wan21b] Jiuya Wang. Pointwise bound for ℓ-torsion in class groups: Elementary abelian extensions. Journal für die reine und angewandte Mathematik (Crelles Jou rnal), 2021(773):129–151, April

work page 2021

[3] [1985]

Faithful Artin induction and the Chebotarev density theorem

[LOS24] Robert J. Lemke Oliver and Alexander Smith. Faithfu l Artin induction and the Chebotarev density theorem. (arXiv:2405.08383), May

work page Pith review arXiv

[4] [1987]

Malle’s conjecture and Brauer groups of stacks

[LS24] Daniel Loughran and Tim Santens. Malle’s conjecture and Brauer groups of stacks. arXiv preprint arXiv:2412.04196,

arXiv

[5] [1997]

[BF10] Karim Belabas and Étienne Fouvry

Computational algebra and number the- ory (London, 1993). [BF10] Karim Belabas and Étienne Fouvry. Discriminants cub iques et progressions arithmétiques. In- ternational Journal of Number Theory , 6(07):1491–1529,

work page 1993

[6] [2006]

Ellenberg and Akshay Venkatesh

[EV07] Jordan S. Ellenberg and Akshay Venkatesh. Reﬂection principles and bounds for class group torsion. International Mathematics Research Notices , 2007,

work page 2007

[7] [2009]

Asymptotics for number ﬁeld s and class groups

[Woo16] Melanie Matchett Wood. Asymptotics for number ﬁeld s and class groups. In Directions in Num- ber Theory: Proceedings of the 2014 WIN3 Workshop . Springer, New York, NY, 1st ed. 2016 edition edition, July

work page 2014

[8] [2010]

Galois groups of random integer po lynomials and van der Waerden’s Conjec- ture

[Bha24] Manjul Bhargava. Galois groups of random integer po lynomials and van der Waerden’s Conjec- ture. Ann. of Math. , to appear, 2024+. [BHRD13] John N. Bray, Derek F. Holt, and Colva M. Roney-Doug al. The maximal subgroups of the low- dimensional ﬁnite classical groups , volume 407 of London Mathematical Society Lecture Note Series. Cambridge Univers...

work page 2024