REVIEW 2 major objections 6 minor 12 references
Counterexamples for T\"urkelli's Modification on Malle's Conjecture
T0 review · 2 major / 6 minor · reviewed 2026-08-08 · deepseek-v4-flash
Pith's one-line read Türkelli's modification of Malle's conjecture fails for infinitely many wreath-product groups, because cyclotomic subfields that lift over function fields cannot lift over Q.
desk verdict Genuinely useful counterexamples to Türkelli's modification, but Theorem 1.3's b(G,Q) formula has an internal p=2 error that makes the stated formula false for an infinite family. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The argument runs on two machines. First, the φ-twisted cyclotomic action: for a projection π:G→G/N and a cyclotomic character φ, the group G(π,φ)=G×_{π,φ}Gal(Q(μ_d)/Q) acts on the minimal-exponent elements of Ker(π) by (x,y)·g = $x^{{-1}}$g^y x, and the number of orbits b(π,φ)=|S_min(Ker(π)_{inv})/G(π,φ)| is the predicted contribution of the subfamily. Second, the embedding problem for cyclotomic extensions: over function fields, the maximal constant field extension makes Gal(Q^cyc/Q) projective, so every such embedding problem is solvable; over number fields, the local-global principle of Theorems 4.5 and 4.6 decides which cyclotomic subfields Q(μ_n) can be embedded into a C_d-extension, and the local criteria at ℓ and ∞ reduce the admissible n. The inductive inequality from [AOWW25] then converts the solvable subfields into the true b-constant.
What would settle it
Enumerate all C_d-extensions F/Q for a concrete pair (ℓ,d), e.g., (3,4) or (5,8), compute b(C_ℓ,F)=[F∩Q(μ_ℓ):Q] for each, and check which cyclotomic subfields M_n⊂Q(μ_ℓ) can be embedded into a C_d-extension using the local criteria at ℓ and ∞ from Theorem 4.6; if the maximal embeddable n differs from the formula ∏_{i:r_i=s_i}$p_i^{{s_i}}$·2^s, or if the [AOWW25] inequality fails for a larger d, Theorem 1.3's b(G,Q) is wrong.
Extended reading notes
Core claim
The central discovery is that the b-constant in Malle's conjecture is not determined by the Galois group of the relevant cyclotomic extension alone: function fields and number fields with the same Gal(Q(μ_ℓ)/Q) can have different b-values. Concretely, Theorem 1.3 computes, for an odd prime ℓ and d = ∏ $p_i^{{r_i}}$ ≠ 2, the group G = C_ℓ ≀ C_d: Malle's original constant is b_M(G,Q) = 1, Türkelli's modified constant and the function-field constant are both b_T(G,Q) = b(G,F_q(t)) = ∏ $p_i^{{s_i}}$ with s_i = min(r_i, v_{p_i}(ℓ-1)), while the true number-field constant is b(G,Q) = ∏_{i: r_i = s_i} $p_i^{{s_i}}$ · 2^s, where s = v_2(ℓ-1) - 1 if v_2(d) > v_2(ℓ-1) and s = 0 otherwise. Thus there are groups with b(G,F_q(t)) > b(G,Q). The mechanism is that the maximal cyclotomic subfield that can actually be embedded into a C_d-extension over Q is smaller than over F_q(t), because the local-global principle for central embedding problems over Q forbids some lifts.
Load-bearing premise
The computed number-field constant b(G,Q) depends on the unpublished inductive inequality of [AOWW25, Cor. 1.6] and on the local-global principle for central embedding problems over Q (Theorems 4.5–4.6); if either gives way, the asserted values of b(G,Q) in Theorem 1.3 are unsupported.
Editorial extensions
If this is right
- Türkelli's modification is false over number fields for infinitely many wreath products C_ℓ ≀ C_d with d ≠ 2; the true b lies between Malle's and Türkelli's predictions in these cases.
- The b-constant is genuinely field-dependent: even when Gal(F_q(t)(μ_ℓ)/F_q(t)) ≃ Gal(Q(μ_ℓ)/Q), the function-field count carries a higher log-power than the number-field count.
- Counting by product of ramified primes does not repair Malle's conjecture: Klüners' groups C_ℓ ≀ C_{ℓ-1} (ℓ ≥ 5) and the nilpotent groups C_{ℓ^2} ≀ C_ℓ already violate the predicted b_M(G_rad,Q).
- The refined Malle's conjecture (Conjecture 6) predicts that splitting the count by liftable cyclotomic intersections restores a uniform asymptotic of the form X^{1/a} (log X)^{b(π,φ)-1}, with b(π,φ) given by orbit counts under the twisted action.
- If the refinement is right, the original Malle constant b(G,k) should be the maximum of b(π,φ) over pairs (π,φ) satisfying the lifting condition.
Reading between the lines
- The paper's reliance on an unpublished joint preprint [AOWW25] means the number-field side of Theorem 1.3 is conditional on that inductive method and its class-group uniformity bounds; if those bounds fail for some d, the strict inequality b(G,F_q(t)) > b(G,Q) might fail even if the group-theoretic constants stand.
- The local-global criterion suggests testable arithmetic: for groups like C_3 ≀ C_4, the non-liftable factors Q(μ_3) are precisely the ones with largest b(π,φ), so a direct enumeration of C_4-extensions over Q(μ_3)-containing fields could confirm b(G,Q) = b_M(G,Q).
- The same embedding obstruction likely produces discrepancies for other invariants (conductor, product of ramified primes) and for base number fields k with non-abelian cyclotomic Galois groups, since the projective-constant-field argument is unique to function fields.
- A full answer to the embedding problem for cyclotomic extensions (Question 1.2) would determine whether the refined conjecture's maximum is attained at the 'maximal' cyclotomic subfield or at smaller ones.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper studies Malle's conjecture on counting number fields by discriminant and its modifications. It presents counterexamples to Türkelli's modified conjecture (Conjecture 3), showing that the b-constant in the asymptotic can differ between number fields and function fields. The main theorem (Theorem 1.3) claims, for groups G = C_l ≀ C_d with d a product of prime powers, an explicit value for the true b-constant over Q, and shows it is smaller than both Türkelli's modified constant and the function-field constant in some cases. The paper also proves (Theorems 1.6 and 1.7) that Kümners' original counterexamples extend to counting by product of ramified primes, and proposes a refined Malle conjecture (Conjectures 6 and 7) that accounts for cyclotomic embedding problems. The proofs combine explicit group-theoretic orbit computations, an inductive argument over C_d-extensions, and local-global principles for central embedding problems.
Significance. If the main results were correct as stated, the paper would be significant: it would disprove Türkelli's modification, exhibit a number-field/function-field discrepancy in the b-constant, and introduce a plausible refined conjecture. The group-theoretic computations in Lemmas 3.1-3.3 are explicit and checkable, the function-field lower bound in Lemma 3.9 is derived from Wright's theorem with effective class-group bounds, and the paper is generally clearly written. However, the number-field formula in Theorem 1.3 contains an internal error for a family of examples at the prime 2, and the theorem's proof relies on an unpublished preprint with overlapping authorship. These issues are significant enough that the paper requires substantial revision before the results can be accepted as stated; nevertheless, the qualitative phenomenon that Türkelli's modification fails may still be salvageable after correcting the formula.
major comments (2)
- [Section 3.2, proof of Lemma 3.7] The computation of b(G,Q) in Theorem 1.3 is internally inconsistent at p=2 when val_2(d) > val_2(ell-1) and val_2(ell-1) >= 2. In the proof of Lemma 3.7, the maximal n is described by setting, for p=2 with r_i > u_i, t_i = s_i - 1 = u_i - 1. But the first condition quoted from Theorem 4.6 requires ell ≡ 1 mod p_i^{r_i} whenever t_i > 0, i.e. u_i >= r_i. For r_i > u_i this condition fails, so t_i must be 0. Concretely, for ell=5 and d=8, the theorem claims b(G,Q)=2, but no C_8-extension can contain Q(sqrt(5)): the tame inertia group of Q_5 has 2-part of order 4, and a cyclic character of order 8 would require an element of order 8 on (Z/5)^×, which has order 4. The correct value is b(G,Q)=1. Thus the factor 2^s in Theorem 1.3 is unjustified and the stated formula is wrong for an infinite family. The qualitative inequality b(G,F_q(t)) > b(G,Q) may still hold after correction, but the theorem as stated must be revised.
- [Theorem 1.3, number-field side] The equality b(G,Q) = max_{F/Q, Gal(F/Q)=C_d} b(C_ell,F) is imported from [AOWW25, Corollary 1.6], an unpublished preprint with overlapping authorship. Since this inductive inequality and its class-group uniformity hypothesis are load-bearing for the claimed value of b(G,Q), the theorem is conditional on external unpublished work. Please either provide a proof of the relevant corollary, state Theorem 1.3 explicitly as conditional on [AOWW25], or cite a published source containing the required result.
minor comments (6)
- [Abstract] There are typographical errors: 'Conj ecture' and 'ex tension' should be 'Conjecture' and 'extension'.
- [Remark 2.13] The tame relator is written as xyx^{-1} = y^p, but the local parameter is |v| elsewhere; please unify the notation and clarify that p denotes the residue characteristic.
- [Proof of Lemma 3.7] In the sentence 'It then follows that b(T, Q) is this particular n', the symbol b(T,Q) is undefined; it should presumably be b(G,Q) or b(C_ell,F).
- [Proof of Lemma 3.2] The phrase 'We first compute bM(Grad,Q). We first count the number of GQ-orbits' contains a duplicated 'We first'.
- [Notation] The letter Q is used both for a general global field and for the rationals, which can be confusing in statements such as Theorem 1.3; please reserve a separate notation for the rationals if possible.
- [Section 4.2] The assertion 'We do have X^1(Q, mu_{2^r}) = 0 for all r, since 2 has full decomposition group' is stated without a proof or reference; a citation would help the reader verify this nontrivial fact.
Circularity Check
Main number-field b-value is imported from a shared-author preprint; group-theoretic and function-field computations are independent.
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self citation load bearing
[Section 3.2 (Proof of Theorem 1.3, over Q), Eq. (3.15)]
"By [AOWW25, Corollary 1.6] for d > 2 and G = Cℓ ≀ Cd the inequality is satisfied as 1/2 + p/(d(p-1)) < ℓ/(ℓ-1), where p is the minimal prime divisor of d. Then we have b(G,Q) = max_{F/Q,Gal(F/Q)=C_d} b(Cℓ,F)."
The central number-field counterexample value b(G,Q) in Theorem 1.3 is not derived in this paper: the load-bearing equality (3.15) is imported from [AOWW25], an unpublished preprint coauthored by the present author. Everything after that step—computing b(Cℓ,F) via [Wri89] and maximizing using Theorem 4.6—is bookkeeping once (3.15) is granted. The stated formula b(G,Q)=∏ p_i^{s_i}·2^s therefore rests on a self-citation whose corollary is neither proved nor reproduced here; if that inductive inequality fails, the number-field half of Theorem 1.3 is unsupported. The function-field side and the Türkelli/Malle constant computations are independent, so the circularity is partial rather than total.
full rationale
The paper's group-theoretic computations of b_M and b_T (Lemma 3.1, Lemma 3.2, Lemma 3.3) are self-contained exercises in orbit counting under cyclotomic actions, and the function-field counting argument (Section 3.3) is carried out in the paper with explicit uniformity bounds. The only load-bearing import is the number-field equality b(G,Q)=max_{F/Q,Gal(F/Q)=C_d} b(Cℓ,F), which is taken verbatim from [AOWW25, Corollary 1.6], a preprint with overlapping authorship. That is a genuine self-citation dependency but not a definitional equivalence: the cited inequality is a substantive external theorem, and the paper does not merely rename its own inputs. A separate correctness concern, noted by the reviewer but not counted as circularity, is that the p=2 branch of Lemma 3.7 appears internally inconsistent with the local condition of Example 4.7; if correct, it would invalidate the stated formula in an infinite family but would not change the circularity assessment.
Assumptions & free parameters
assumptions (4)
- standard math Wright's theorem on abelian extensions over function fields (Theorem I.3 and Theorem 7.2 in [Wri89]) gives the weighted and local counts needed for b(C_ℓ,F).
- domain assumption [AOWW25, Corollary 1.6] reduces b(C_ℓ≀C_d,Q) to max_F b(C_ℓ,F) whenever 1/2 + p/(d(p-1)) < ℓ/(ℓ-1).
- standard math Local-global principle for central embedding problems over Q: X^2(Q,A)=0 for trivial modules A, so E(G_Q,φ,π) is solvable if and only if it is locally solvable.
- standard math The cyclotomic Galois group G_{F_q(t)}^{cyc} is isomorphic to \hat Z and is projective, while the number-field cyclotomic Galois group is not projective.
Cite this review
Pith. "Pith review of Counterexamples for T\"urkelli's Modification on Malle's Conjecture." pith.science (2026). https://pith.science/paper/MP2XKV4C
@misc{pith2026250204261,
author = {Pith},
title = {Pith review of: Counterexamples for T\"urkelli's Modification on Malle's Conjecture},
year = {2026},
howpublished = {\url{https://pith.science/paper/MP2XKV4C}},
note = {Machine review of arXiv:2502.04261}
}
abstract
We give counterexamples for the modification on Malle's Conjecture given by T\"urkelli. T\"urkelli's modification on Malle's conjecture is inspired by an analogue of Malle's conjecture over a function field. As a consequence, our counterexamples demonstrate that the $b$ constant can differ between function fields and number fields. We also show that Kl\"uners' counterexamples give counterexamples for a natural extension of Malle's conjecture to counting number fields by product of ramified primes. We then propose a refined version of Malle's conjecture which implies a new conjectural value for the constant $b$ for number fields.
Reference graph
Works this paper leans on
-
[5]
Malle's conjecture for fair counting functions
[KP23] Peter Koymans and Carlo Pagano. Malle’s conjecture f or fair counting functions. arXiv preprint arXiv:2309.04838 ,
-
[6]
Non-abelian cohen–lenstra heuristics in the presence of roots of unity
[Liu22] Yuan Liu. Non-abelian cohen–lenstra heuristics in the presence of roots of unity. arXiv preprint arXiv:2202.09471 ,
-
[12]
[SW23] Will Sawin and Melanie Matchett Wood. Conjectures fo r distributions of class groups of extensions of number fields containing roots of uni ty. arXiv preprint arXiv:2301.00791,
- [1999]
-
[2005]
Malle’s conjectur e for nonic heisenberg exten- sions
[FK21] Étienne Fouvry and Peter Koymans. Malle’s conjectur e for nonic heisenberg exten- sions. arXiv preprint arXiv:2102.09465 ,
-
[2008]
On a conjecture for $\ell$-torsion in class groups of number fields: from the perspective of moments
[PTBW19] Lillian B Pierce, Caroline L Turnage-Butterbaugh , and Melanie Matchett Wood. On a conjecture for ℓ-torsion in class groups of number fields: from the perspecti ve of moments. arXiv preprint arXiv:1902.02008 ,
work page Pith review arXiv 1902
-
[2012]
On Malle's conjecture for nilpotent groups, I
[KP21] P. Koymans and C. Pagano. On Malle’s conjecture for ni lpotent groups, I. arXiv: 2103.17223,
-
[2021]
Inductive methods for counting number fields
[AOWW25] Brandon Alberts, Robert J Lemke Oliver, Jiuya Wang , and Melanie Matchett Wood. Inductive methods for counting number fields. arXiv preprint arXiv:2501.18574 ,
Show all 12 references
-
[2022]
Homological stabili ty for hurwitz spaces and applications
[LL25] Aaron Landesman and Ishan Levy. Homological stabili ty for hurwitz spaces and applications. arXiv preprint arXiv:2503.03861 ,
-
[2023]
Ellenberg, T
[ETW17] J. Ellenberg, T. Tran, and C. Westerland. Fox-Neuwi rth-Fuks cells, quantum shuffle algebras and Malle’s conjecture for function field. arXiv:1701.04541,
-
[2024]
A predicted dis- tribution for Galois groups of maximal unramified extension s
[L WZB19] Yuan Liu, Melanie Matchett Wood, and David Zureick -Brown. A predicted dis- tribution for Galois groups of maximal unramified extension s. arXiv preprint arXiv:1907.05002,
1907 arXiv
-
[2025]
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 ,
Reviewed August 8, 2026 · model on record in the stance chip above.
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