Lower bounds on the $\ell$-rank of ideal class groups

Daniel E. Martin

REVIEW 1 major objections 5 minor 1 cited by

Lower bounds on the $\ell$-rank of ideal class groups

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Pith's one-line read For ℓ-divisible extensions, one prime with an ℓ-divisible ramification index below it forces a lower bound on the ℓ-rank of the class group.

desk verdict A genuinely improved ℓ-rank bound under a new group-theoretic hypothesis, with a sound proof and a real density consequence; the main limitation is the restrictiveness of the ℓ-divisibility condition. read the letter →

arxiv 2501.09865 v1 pith:SHOPRP5E submitted 2025-01-16 math.NT

Daniel E. Martin This is my paper · ORCID

classification math.NT MSC 11R2911R32

keywords idealclassgroupsℓ-rankramificationℓ-divisibilityfieldtowersnilpotentextensionsdihedralgenustheory

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 seeks to show that a single wildly ramified prime can force nontrivial $\ell$-torsion in the ideal class group of a number field, provided the Galois group of the extension satisfies a new combinatorial condition it calls $\ell$-divisibility. The main theorem bounds $\operatorname{rk}_\ell \mathrm{Cl}(K)$ from below by the number of primes of the base field over which at least one prime of $K$ has ramification index divisible by $\ell$, divided by a small group-theoretic constant, minus unit-rank and degree terms. Previous results required every prime above the counted prime to become an $\ell$th power; here only one such prime is needed. The author verifies the $\ell$-divisibility hypothesis for towers of Galois extensions, nilpotent extensions, and certain semidirect-product extensions, and uses the lower bound together with a classical class-field-tower criterion to prove new results on infinite class field towers, including a density statement for nilpotent extensions.

What carries the argument

The key machinery is the notion of $\ell$-divisibility for a group pair $G/H$ (Definition 1.3): for every element $\sigma\in G-H$ of order $\ell$, the pair must admit finitely many subgroups $H\le G_1,\ldots,G_\delta\le G$ such that some $G$-conjugate of $\sigma$ lies in $G_i$ and has its $G_i$-conjugacy class disjoint from $H$. This condition is what lets the proof turn a prime with a single $\ell$-divisible ramification index into a prime that becomes an $\ell$th power inside an intermediate field, unlocking the classical relative lower bound. The appendix shows the condition holds for Galois towers, nilpotent groups, and certain semidirect products; in the dihedral case $D_n/\langle s\rangle$, it is $2$-divisible exactly when $4\mid n$.

What would settle it

Take any $\ell$-divisible extension $K/F$ with $t_\ell(K/F)$ large enough to make the right-hand side of Theorem 1.4 positive, for example a dihedral quartic field with many ramified primes of type $\mathfrak{p}_1^2\mathfrak{p}_2$, compute $\operatorname{rk}_\ell \mathrm{Cl}(K)$ exactly, and compare it with the theorem's lower bound; a smaller rank would refute the theorem.

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Extended reading notes

Core claim

The paper's central claim, Theorem 1.4, is that for an $\ell$-divisible extension of number fields $K/F$, if $t_\ell(K/F)$ is positive then $$\operatorname{rk}_\ell \mathrm{Cl}(K) \ge \frac{t_\ell(K/F)}{\delta_\ell(K/F)} - \operatorname{rk}_\ell O_K^\times + \operatorname{rk}_\ell O_F^\times - e_\ell(K/F),$$ where $t_\ell(K/F)$ counts prime ideals of $F$ having at least one prime of $K$ above them with ramification index divisible by $\ell$, $\delta_\ell(K/F)$ is the minimal number of intermediate groups required by Definition 1.3, and $e_\ell(K/F)$ is the $\ell$-adic exponent of $[K:F]$. The proof converts each counted prime $p$ of $F$ into a prime $q$ in some intermediate field $F_i$ whose powers become $\ell$th powers in $K$, so that a relative version of the classical class-rank bound applies. The new feature is that the conversion works even when $p$ has several primes above it in $K$ and only one of them has ramification index divisible by $\ell$; previous bounds required all of them to do so.

Load-bearing premise

The entire bound rests on the $\ell$-divisibility condition: for every element of order $\ell$ that does not fix $K$, some conjugate of that element must have its whole conjugacy class inside an intermediate Galois group stay outside the subgroup of automorphisms fixing $K$; if this group-theoretic premise fails, the theorem gives nothing.

Editorial extensions

If this is right

For a tower $F=F_0\subseteq\cdots\subseteq F_n=K$ of Galois extensions, the lower bound becomes $t_\ell(K/F)/n - \operatorname{rk}_\ell O_K^\times + \operatorname{rk}_\ell O_F^\times - e_\ell(K/F)$, so ramification in early layers still contributes even when higher layers are unramified.
For nilpotent Galois groups, strong $\ell$-divisibility gives an analogous bound counted at primes of $K$ rather than $F$, with the constant $\Delta_\ell(K/F)$.
If the relative discriminant has sufficiently many prime factors, the lower bound crosses the classical class-field-tower criterion threshold, so the $\ell$-class field tower of $K$ is infinite.
Among nilpotent Galois extensions with fixed group over a number field, the proportion with finite class field towers tends to zero as the discriminant grows; the paper proves an explicit $O((\log\log x)^{e-1}/\log x)$ upper bound.
Dihedral extensions of degree divisible by $8$ gain $2$-rank contributions from primes that split as $\mathfrak{p}_1^2\mathfrak{p}_2$ or $\mathfrak{p}_1^2\mathfrak{p}_2\mathfrak{p}_3$, a situation where genus theory contributes nothing.

Reading between the lines

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

The theorem's value depends on how small $\delta_\ell(K/F)$ can be made; for a given small Galois group one can compute the optimal intermediate subgroups directly, and the appendix's catalogue is only a first step.
A natural computational test is to enumerate small dihedral quartic fields, compute the $2$-rank of their class groups, and see how often the new lower bound exceeds the genus-theory bound and by how much.
Because $\ell$-divisibility is preserved under composita with a common base field, the theorem can be iterated to construct families of extensions with arbitrarily large $\ell$-rank from staged ramification, suggesting quantitative lower bounds that grow with the discriminant.
The same mechanism—converting a prime with one $\ell$-divisible index into an $\ell$th power in an intermediate field—may transfer to narrow class groups or Selmer groups, though the paper only treats ideal class groups.

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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

Bottom line: this is a solid, honest paper that earns its main theorem. Martin replaces the all-ramification-indices-divisible-by-ℓ condition in Roquette–Zassenhaus and Connell–Sussman with an at-least-one condition, under a new group-theoretic hypothesis he calls ℓ-divisibility. That is a genuinely different mechanism, and it pays off: the nilpotent density result—almost all nilpotent G/1-extensions have infinite class field towers, conditional on Klüners–Malle—is new and does not reduce to earlier genus-theory or Roquette–Zassenhaus arguments.

What is actually new and good: Definitions 1.3 and 1.8 are new group-theoretic conditions, and the appendix proves them for a reasonable stock of examples—towers of Galois extensions, nilpotent groups, composita under a common base, and semidirect products including dihedral D_n with 4|n. Theorem 1.4 is the clean statement: t_ℓ(K/F) ramified primes, each with at least one index divisible by ℓ, force class group ℓ-rank at least t_ℓ/δ_ℓ minus unit ranks and the ℓ-adic degree exponent. The proof is structurally sound. The inertia-group step around (2.7)–(2.8) works: the Sylow argument correctly forces the intervening prime q to become an ℓth power in K, and Lemma 2.2 is a faithful adaptation of Connell–Sussman that drops the principal-ideal hypothesis they needed. The final Kummer-exponent bound is fine: the field generated by the relevant ℓth roots has ℓ-power degree dividing [K:F_{i0}]. Lemma 2.2's exact-sequence argument is correct.

Soft spots, in proportion. The ℓ-divisibility condition is restrictive and nonconstructive. It must be verified for each pair G/H, and the paper does not show it holds generally; the appendix works for specific families. That is a real limitation, but it is a hypothesis, not a gap—the paper says so honestly. Theorem 1.9's gain over Theorem 1.4 is heuristic; the paper labels it as such. The density theorem leans on Klüners–Malle and Malle's weak conjecture as external inputs; the citation pattern is appropriate, no self-citation, and the dependence is explicit. There are minor typos ('Lemma 2.3' instead of 'Proposition 2.3', and the 'a such σ' phrase in Lemma 4.6), but nothing that affects the mathematics.

Who it is for: number theorists working on class group bounds, genus theory, and class field towers. It deserves a serious referee; an editor should not desk reject it. My own verdict is accept after minor revisions, with the main request being a clearer discussion of how restrictive ℓ-divisibility is in practice.

Referee Report

1 major / 5 minor

Summary. The paper introduces a group-theoretic condition called ℓ-divisibility for Galois pairs G/H (Definition 1.3) and proves that for an ℓ-divisible extension K/F, each prime of F over which at least one prime of K has ℓ-divisible ramification index contributes at least 1/δℓ(K/F) to the ℓ-rank of the class group, up to unit-rank and ℓ-adic degree-error terms (Theorem 1.4). A stronger version, strong ℓ-divisibility (Definition 1.8), gives a bound supported on primes of K rather than of F (Theorem 1.9). The paper then combines Theorem 1.4 with the Golod–Shafarevich theorem to produce criteria for infinite class field towers (Corollaries 3.1 and 3.3) and an upper bound for the count of nilpotent extensions with finite class field towers (Theorem 3.6 and Corollary 1.15). The appendix verifies ℓ-divisibility and strong ℓ-divisibility for Galois towers, nilpotent groups, semidirect products, and composita, and gives explicit values for dihedral groups.

Significance. The main innovation is replacing the usual hypothesis that every ramification index at a prime is divisible by ℓ with the weaker condition that at least one is. This is a genuine strengthening, and the paper correctly identifies the group-theoretic obstruction with the example of cubic fields, where abundant two-prime splitting does not force 2-torsion. The proofs are detailed and mostly self-contained given standard inputs such as Connell–Sussman, Golod–Shafarevich, and Klüners–Malle; the appendix provides checkable criteria for the nonconstructive ℓ-divisibility hypothesis. There are no fitted constants, and the bounds depend only on explicit group-theoretic invariants. If Theorem 1.4 stands, it gives a new and fairly general mechanism for producing ℓ-torsion from ramification and yields a density statement for infinite class field towers in nilpotent families. The main caveat is that ℓ-divisibility is a strong premise that must be verified case by case, but the paper is transparent about this limitation.

major comments (1)

[4.2, proof of Theorem 4.8] The proof asserts that every term S_i of an arbitrary central series of the Sylow ℓ-subgroup S is a characteristic subgroup of G. This is false: central series terms of S need only be normal in S, and they need not be invariant under conjugation by G. For example, if G = C_p^2 ⋊ C_q with q acting irreducibly on C_p^2, then S = C_p^2 is the unique Sylow p-subgroup, but a line N in S gives a central series 1 < N < S for S while N is not normal in G. Since the proof subsequently forms quotients G/S_i and subgroups HS_i, an arbitrary central series is not usable. The argument can be repaired by choosing a central series of S consisting of G-invariant subgroups, for instance the upper central series of S, which preserves the stated bound on δℓ(G/H). This correction is needed because Theorem 4.8 underpins Corollary 4.9 and therefore the nilpotent density result in Theorem 3.6.

minor comments (5)

[3, proof of Corollary 3.1] The reference to “Lemma 2.3” should be to Proposition 2.3, and the expression “tℓ(K/Q)” should be “tℓ(K/F)”.
[4.2, Lemma 4.6] In the induction step, “the subgroup generated by a such σ” should read “the subgroup generated by all such σ”; as printed, the subgroup generated by a single element is not necessarily normal, and the proof of the induction step requires the subgroup generated by all elements of order ℓ^n.
[2, proof of Theorem 1.9] There are small typos: “There must be some σ” should be lowercase, and “pigeon-hole principal” should be “pigeon-hole principle”.
[3, proof of Theorem 3.6] The inequality “rkℓ O*_K ≤ rkℓ O*_K + 1 ≤ n” is not coherent as printed; it should presumably be rk O*_K + 1 ≤ d (or a similar bound using the degree), and should be corrected.
[4.3, after Theorem 4.10] The phrase “This has the affect” should be “This has the effect”.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: Theorem 1.4 is a genuine lower bound derived from a structural group-theoretic hypothesis, with no fitted parameters and no load-bearing self-citation.

full rationale

I walked the derivation chain of Theorem 1.4 and its use in Sections 3 and 4. The main proof combines Lemma 2.2, a relative version of the Connell–Sussman/Roquette–Zassenhaus inequality, with the group-theoretic condition of ℓ-divisibility. For each ramified prime counted by tℓ(K/F), the proof constructs, via inertia groups and the ℓ-divisibility hypothesis, a prime q in some intermediate field Fi0 such that qOK is an ℓth power; the pigeonhole principle then yields at least tℓ/δℓ distinct such primes in a single Fi0. The final Kummer-theoretic term is bounded by eℓ(K/Fi0) ≤ eℓ(K/F). Nothing here is fitted to class group data: δℓ(K/F) is defined purely in terms of the Galois group, tℓ counts prime ideals, and the unit and exponent terms are standard invariants. The conclusion does not appear as an input to Definition 1.3 or to any lemma. The appendix verifies ℓ-divisibility for towers, nilpotent groups, semidirect products, and composite fields by independent group-theoretic arguments; for example, Corollary 4.11 is proved directly from the structure of Dn. References to Roquette–Zassenhaus, Connell–Sussman, Golod–Shafarevich, and Klüners–Malle are external standard results, not self-citations, and are not used to smuggle in the target bound. The ℓ-divisibility hypothesis is restrictive and must be checked for each application, but that is a substantive premise rather than a circular step. I found no equation, definition, or fitted parameter that makes the stated rank lower bound true by construction.

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

The central claim is a theorem in algebraic number theory; it introduces no empirical data, no fitted parameters, and no new physical or mathematical objects. The ℓ-divisible and strongly ℓ-divisible definitions are structural invariants of finite groups, not invented entities.

assumptions (5)

standard math The Connell-Sussman genus bound (2.1) is correct.
Lemma 2.2 quotes [CS70, proof of Theorem 1] without re-deriving it; the main lower bound inherits this dependence.
standard math The Golod-Shafarevich criterion (Theorem 1.11) is valid.
Used in Corollary 3.1 and Theorem 3.6 to infer infinite class field towers from sufficiently large ℓ-rank, and in the contrapositive to bound class group ranks of subfields in Theorem 3.6.
standard math Nilpotent Galois extensions satisfy N(F,G/1,x) > c x^a (Kluners-Malle [KM04]).
Theorem 3.6 uses this to convert the upper count of finite-tower fields into a zero-density statement; Malle's discriminant parametrization D=s^{1/a}t is also used in the same proof.
standard math Landau's count of integers with k distinct prime factors.
Lemma 3.5 depends on Landau's theorem to estimate the number of possible discriminant norms up to x; the proof reproduces the standard argument.
domain assumption Unramified base change preserves unramifiedness: if M/K is unramified and L/K is finite, then ML/L is unramified.
Theorem 3.6 uses this standard fact to conclude that intermediate subfields of a finite-tower field have finite towers, so Golod-Shafarevich bounds their class group ranks.

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Cite this review

Pith. "Pith review of Lower bounds on the $\ell$-rank of ideal class groups." pith.science (2026). https://pith.science/paper/SHOPRP5E

@misc{pith2026250109865,
  author       = {Pith},
  title        = {Pith review of: Lower bounds on the $\ell$-rank of ideal class groups},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SHOPRP5E}},
  note         = {Machine review of arXiv:2501.09865}
}

abstract

For a prime number $\ell$ and an extension of number fields $K/F$, we prove new lower bounds on the $\ell$-rank of the ideal class group of $K$ based on prime ramification in $K/F$. Unlike related results from the literature, our bound is supported on prime ideals in $F$ over which at least one (rather than each) prime in $K$ has ramification index divisible by $\ell$. This bound holds with a proviso on the Galois group of the normal closure of $K/F$, which is satisfied by towers of Galois extensions, intermediate fields in nilpotent extensions, and intermediate fields in dihedral extensions of degree $8n$, to name a few. We also use our lower bound to prove a new density result on number fields with infinite class field towers.

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Reference graph

Works this paper leans on

9 extracted references · 9 canonical work pages · cited by 1 Pith paper

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Pith tools

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[1] [1]

Class number and ramiﬁca tion in number ﬁelds

[BR63] Armand Brumer and Michael Rosen. Class number and ramiﬁca tion in number ﬁelds. Nagoya Mathematical Journal 23 (1963), pp. 97–101. [Bru65] Armand Brumer. Ramiﬁcation and class towers of number ﬁe lds. Michi- gan Mathematical Journal 12.2 (1965), pp. 129–131. [BV15] Manjul Bhargava and Ila Varma. On the mean number of 2-to rsion ele- ments in the cl...

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A positive proportion of some quadratic number ﬁ elds with inﬁnite Hilbert 2-class ﬁeld tower

[Mou16] Ali Mouhib. A positive proportion of some quadratic number ﬁ elds with inﬁnite Hilbert 2-class ﬁeld tower. The Ramanujan Journal 40.2 (2016), pp. 405–412. [Neu13] J¨ urgen Neukirch. Algebraic number theory. Vol

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An explicit family of cubic number fields with large $2$-rank of the class group

[KK17] Yasuhiro Kishi and Toru Komatsu. Imaginary quadratic ﬁelds whose ideal class groups have 3-rank at least three. Journal of Number Theory 170 (2017), pp. 46–54. [KM04] J¨ urgen Kl¨ uners and Gunter Malle. Counting nilpotent Galois extensions. Journal f¨ ur die reine und angewandte Mathematik 572 (2004), pp. 1–26. [Kul16] Avinash Kulkarni. An explici...

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Class groups of dihedral extensions

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On the p-Rank of Class Groups of p-Extensions

[Liu24] Yuan Liu. On the p-Rank of Class Groups of p-Extensions. International Mathematics Research Notices 2024.6 (2024), pp. 5274–5325. [LYW21] Aaron Levin, Shengkuan Yan, and Luke Wiljanen. Quadratic ﬁelds with a class group of large 3-rank. Acta Arithmetica 197 (2021), pp. 275–292. [Mal02] Gunter Malle. On the distribution of Galois groups. Journal of...

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A class rank estimat e for algebraic number ﬁelds

[RZ69] Peter Roquette and Hans Zassenhaus. A class rank estimat e for algebraic number ﬁelds. Journal of the London Mathematical Society 1.1 (1969), pp. 31–38. [Shp08] Igor E. Shparlinski. Inﬁnite Hilbert class ﬁeld towers over cy clotomic ﬁelds. Glasgow Mathematical Journal 50.1 (2008), pp. 27–32. [Ten15] G´ erald Tenenbaum. Introduction to analytic and ...

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Clemson University, O-110 Martin Hall, 2020 Parkw ay Drive, Cl emson, SC Email address : dem6@clemson.edu 23

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[8] [2020]

A geometric a p- proach to large class groups: a survey, pp

Chap. A geometric a p- proach to large class groups: a survey, pp. 1–15. [GS64] Evgeniy Golod and Igor Shafarevich. On the class ﬁeld tower. Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya 28.2 (1964), pp. 261–

work page 1964