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Improving the trivial bound for $\ell$-torsion in class groups

T0 review · 2 major / 4 minor · reviewed 2026-08-09 · deepseek-v4-flash

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Pith's one-line read For every number field and every ℓ≥2, the ℓ-torsion of the class group is o(sqrt(D_K)), an unconditional log-power saving.

desk verdict The balancing step flagged by the reader is actually correct once you plug in the definition of V_K; the paper's uniform log-power saving is real and deserves peer review. read the letter →

arxiv 2502.03464 v1 pith:VAXS35OH submitted 2025-02-05 math.NT

Robert J. Lemke Oliver , Asif Zaman This is my paper · ORCID
classification math.NT MSC 11R2911R42
keywords classgrouptorsionℓ-torsionnumberfieldstrivialboundDedekindzetaresidueArakelovdiscriminantboundsunconditionalsavings
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

Fix any number field $K$ and any integer $\ell \ge 2$. The paper proves that the $\ell$-torsion subgroup of the ideal class group satisfies $|\mathrm{Cl}_K[\ell]| = o_{[K:\mathbb{Q}],\ell}(D_K^{1/2})$, an unconditional log-power saving over the refined trivial bound; this is the first such general statement valid for all $K$ and all $\ell$. The quantitative form is Corollary 1: $|\mathrm{Cl}_K[\ell]| \ll D_K^{1/2} (\log D_K)^{-r_K+\rho_K-1}(\log\log D_K)^{3[K:\mathbb{Q}]/2}$. The argument is soft and uniform: it balances a lattice-counting lemma of [EV07], in the refined integral-ideal form of [HB24], against the size of the residue of the Dedekind zeta function at $s=1$. A small residue means few small non-inert unramified primes, and that scarcity itself forces a saving via the class number formula. A reader should care because it removes the last general barrier of the trivial bound for torsion in class groups, with no restriction on degree, Galois group, or ramification.

What carries the argument

The load-bearing object is Lemma 3, the lattice-counting lemma of [EV07] in the refined form of [HB24]. It asserts that $|\mathrm{Cl}_K[\ell]|$ is bounded above by $\kappa_K \sqrt{D_K}/M$, where $M$ counts unramified integral ideals $\mathfrak{n}$ of $K$ with squarefree norm at most $D_K^{(1-\eta)/(2\ell([K:\mathbb{Q}]-1))}$ and relatively prime to the different. The lemma is proved by bounding the volume of the Arakelov class group from above through the class number formula and bounding the volume of the quotient $P_\ell/P$ from below by $M$. The supporting machinery is the zeta-side estimate: a finite Euler product $H_K(s,x)$ constructed from the local factors of $\zeta_K$, Mellin inversion with a test function, and a convexity estimate for $\zeta_K(1/2+it)$ (Lemma 5). These ingredients convert the count of such ideals into an upper bound for $\kappa_K$ in terms of an unknown smooth scale $z$, and the proof then takes the supremum over the admissible range of $z$.

What would settle it

For a concrete sequence of number fields (for example real quadratic fields of growing discriminant), compute the quantity $M$ in Lemma 3 and compare $|\mathrm{Cl}_K[\ell]|$ with $\kappa_K D_K^{1/2}/M$. If for even one pair $(K,\ell)$ the torsion exceeds a fixed constant times that ratio, the volume lower bound behind Lemma 3 is false; reproducing the inequality in many cases would support it. Since the implied constants are in principle effective, this check is feasible for fixed small degree and fixed $\ell$.

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Extended reading notes

Core claim

On its own terms the paper establishes Theorem 1: if $|\mathrm{Cl}_K|$ is close to the largest size allowed by the refined trivial bound, parametrized by $V_K \ge [K:\mathbb{Q}]/\delta$, then the $\ell$-torsion is exceptionally small, $|\mathrm{Cl}_K[\ell]| \ll |\mathrm{Cl}_K|(\log D_K)^{-\delta V_K}$. Since $V_K$ is never larger than about $(\log D_K)(\log\log D_K)^{-3/2}$, this inverse relationship interpolates to the uniform saving of Corollary 1. The dichotomy underneath is that either $K$ has many small unramified prime ideals of degree one, in which case the [EV07]--[HB24] counting lemma saves directly, or it has few such primes, in which case the residue $\kappa_K$ of the Dedekind zeta function is small and the class number formula pulls the class group down. Balancing the two alternatives at the threshold $y = D_K^{1/(4\ell([K:\mathbb{Q}]-1))}$ gives the log-power saving. Theorem 2 is the conditional companion: under a subconvexity bound for $\zeta_K(1/2+it)$ with exponent saving, the same balance upgrades to full power savings $D_K^\Delta$ for $\Delta > 1/2 - 1/(2\ell([K:\mathbb{Q}]-1))$, with effective implied constants for all number fields, including those with a quadratic subfield.

Load-bearing premise

The proof assumes that the number of unramified integral ideals with squarefree norm up to $D_K^{(1-\eta)/(2\ell([K:\mathbb{Q}]-1))}$ is a valid lower bound for the volume of the quotient $P_\ell/P$ in the Arakelov class group; every saving flows through this count, and if that volume bound is weaker than claimed the estimate $|\mathrm{Cl}_K[\ell]| \ll \kappa_K D_K^{1/2}/M$ fails.

Editorial extensions

If this is right

  • Corollary 1 applies to every $K \ne \mathbb{Q}$ and every $\ell \ge 2$: $|\mathrm{Cl}_K[\ell]| \ll D_K^{1/2}(\log D_K)^{-r_K+\rho_K-1}(\log\log D_K)^{3[K:\mathbb{Q}]/2}$, saving a log power over the refined trivial bound.
  • When the class group is close to its maximal size ($V_K$ large), the saving is amplified: torsion drops by a factor $(\log D_K)^{-\delta V_K}$ relative to the class number.
  • Under the subconvexity hypothesis (1.9), Theorem 2 upgrades the saving to $|\mathrm{Cl}_K[\ell]| \ll D_K^\Delta$ for any $\Delta > 1/2 - 1/(2\ell([K:\mathbb{Q}]-1))$, and the implied constants can be made effective for all $K$, including fields with quadratic subfields.
  • The method imposes no restriction on the Galois group, degree, or ramification of $K$, so it covers sparse and high-degree cases where previous pointwise power savings did not reach.

Reading between the lines

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

  • Editorial inference: the same balance between counting ideals and bounding the zeta residue should transfer to average-family problems, where a family's splitting statistics of small primes determine how much of the saving survives on average.
  • Editorial inference: Remark 7 identifies norms with no small prime factors as the bottleneck; any sharper estimate for integers whose prime factors are all small would feed directly into a stronger unconditional exponent in Corollary 1.
  • Editorial inference: the effective saving for fields with a quadratic subfield in Theorem 2 suggests a general template for bypassing ineffective lower bounds on residues, which may apply to other problems where such ineffective bounds were the only input.
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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 claims the first unconditional saving over the trivial bound for ℓ-torsion in class groups that holds for every number field K and every ℓ ≥ 2. The method combines the Ellenberg–Venkatesh volume lemma with a new auxiliary function H_K(s,x) (Lemma 4), a residue estimate for the Dedekind zeta function, and a balancing argument. Corollary 1 states an explicit log-power saving, Theorem 1 gives an inverse relationship for class groups close to extremal size, and Theorem 2 derives conditional power savings from a subconvexity hypothesis. The proof is soft and builds on cited results of Landau, Silverman, Stark, Ellenberg–Venkatesh, and Heath-Brown.

Significance. If correct, Corollary 1 would be a notable breakthrough: it would give the first uniform unconditional saving over the trivial bound |Cl_K[ℓ]| ≪ D_K^{1/2+ε} for all number fields and all ℓ, with an explicit log-power. The argument is conceptually appealing, as it combines a small-residue principle with a many-small-primes principle. However, the proof rests on a false bound in Lemma 4, which is load-bearing for both Theorem 1 and Theorem 2. As written, the paper does not establish its main claims.

major comments (2)
  1. [Section 2, Lemma 4] The asserted upper bound |HK(1/2+it,x)| ≪ (e(log D_K)^{1/2}/log x)^C is false, and the proof given only yields the trivial bound |HK(1/2+it,x)| ≤ exp(O((log D_K)^{1/2}) + O(log log x)). The real part of the exponent in (2.8) can be positive; the absolute-value estimate on ∑(λ_K(p)-λ^♭_K(p))/p^{1/2+it} does not give decay in x. For a concrete counterexample, let K be a quadratic field whose discriminant D_K is the product of all primes up to Y, and take x ≥ D_K. By Kronecker's theorem, since the numbers log q for primes q|D_K are linearly independent over Q, there exists t such that q^{-1/2-it} is close to -q^{-1/2} for all q|D_K simultaneously. Then |HK(1/2+it,x)| ≈ ∏_{q|D_K}(1+q^{-1/2}) ≈ exp(c∑_{q|D_K} q^{-1/2}) ≈ exp(c' (log D_K)^{1/2}/log log D_K), which exceeds every power of log D_K. Consequently the error term in the Mellin integral for S(x) in Section 3 is not bounded as claimed, and the derivation of (3.4) fails. The same issue propagates to Theorem 2 through equation (4.4). Since Lemma 4 is used to control the dominant error term, this is a load-bearing error.
  2. [Section 3, balancing step] The balancing-step concern about an alleged exponent error does not land. The text derives |Cl_K| ≪ (log z)^{[K:Q]} D^{1/2}(log D)^{-r+ρ-1}(log log D)^{[K:Q]/2}. Combining this with the definition (1.7), V_K^{[K:Q]} D^{1/2}(log D)^{-r+ρ-1}(log log D)^{3[K:Q]/2} = |Cl_K|, gives V_K^{[K:Q]} (log log D)^{[K:Q]} ≪ (log z)^{[K:Q]}, and taking [K:Q]-th roots yields V_K log log D ≪ log z, exactly as stated. The suggested correction to V_K^{1/[K:Q]} would be needed only if the exponent [K:Q] applied to V_K alone rather than to the product V_K log log D.
minor comments (4)
  1. [Section 3, equation (3.1)] The statement D_K^2 ≤ x ≤ D_K^3 is false for quadratic fields: with [K:Q]=2, x = D_K^4. The lower bound is sufficient for the subsequent error absorption, but the displayed range should be corrected.
  2. [Section 2, Lemma 4 proof] The sentence "The last estimate is crude but sufficient" is misleading because the crude estimate does not imply the claimed upper bound; the bound asserted in the lemma is in fact false, as explained in the major comment above.
  3. [References] The reference "[A V23]" contains a typographical spacing error; it should read "[AV23]" or similar.
  4. [Remark 7] The notation D^{1/2ℓ([K:Q]-1)}_K is ambiguous; the exponent should be parenthesized as D^{1/(2ℓ([K:Q]-1))}_K.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the main bound is deduced from external results (Ellenberg–Venkatesh, Heath-Brown, Silverman, Landau, Stark) and self-contained analytic estimates; self-citations are not load-bearing.

full rationale

No circular step was found. The paper's central estimate is a deduction from cited external results: Lemma 3 is the Ellenberg–Venkatesh volume bound with Heath-Brown's refinement, Lemmas 4–6 are self-contained analytic estimates proved in the paper, and the final class-group bound follows by combining these with the class number formula, Landau's bound on the residue, and Silverman's regulator lower bound. The quantity V_K defined in (1.7) is a normalization of the class group size, not an input to the target; Theorem 1 expresses a conditional saving in terms of V_K, and Corollary 1 then follows from the unconditionally available upper bound on V_K. No parameter is fitted to data, no prediction is renamed from a fit, and no load-bearing premise is justified solely by a self-citation. The cited works by the authors ([LOTZ24], [TZ23], [LOS24]) appear only in the survey of related work and are not used in the proof. The reader's flagged concern about the balancing step in Section 3 is a possible exponent error in deriving z ≥ (log D_K)^{3δV_K} from V_K log log D_K ≪ log z; that is a question of mathematical soundness, not circularity, and does not affect the circularity score.

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

No constants are fitted to data; δ, η, and Δ are universal or hypothesis parameters, not fitted values. No new physical or mathematical entities are postulated; λ^♭_K and H_K are technical devices defined in the proof, not entities with independent evidence requirements.

assumptions (5)
  • domain assumption Ellenberg–Venkatesh volume lemma (Lemma 3), including Heath-Brown's integral-ideal variant
    Provides the core inequality |Cl_K[ℓ]| ≪ κ_K D_K^{1/2}/M; no proof is given in this paper.
  • domain assumption Silverman's regulator lower bound R_K ≫ (log D_K)^{r_K-ρ_K}
    Used with the class number formula to translate residue bounds into class number bounds; cited from [Sil84].
  • standard math Landau's residue upper bound κ_K ≪ (log D_K)^{[K:Q]-1}
    Background input for the refined trivial bound (1.5).
  • domain assumption Ineffective Brauer–Siegel lower bound for κ_K in Theorem 2 case (i)
    Provides κ_K ≫ D_K^{-δ/(4ℓ([K:Q]-1))}; it is a standard but ineffective theorem.
  • domain assumption Stark's effective lower bound for κ_K in Theorem 2 case (ii)
    Gives κ_K ≫ (log D_K)^{-1} for fields without a quadratic subfield; cited from [Sta74].

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Cite this review

Pith. "Pith review of Improving the trivial bound for $\ell$-torsion in class groups." pith.science (2026). https://pith.science/paper/VAXS35OH

@misc{pith2026250203464,
  author       = {Pith},
  title        = {Pith review of: Improving the trivial bound for $\ell$-torsion in class groups},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VAXS35OH}},
  note         = {Machine review of arXiv:2502.03464}
}
abstract

For any number field $K$ with $D_K=|\mathrm{Disc}(K)|$ and any integer $\ell \geq 2$, we improve over the commonly cited trivial bound $|\mathrm{Cl}_K[\ell]| \leq |\mathrm{Cl}_K| \ll_{[K:\mathbb{Q}],\varepsilon} D_K^{1/2+\varepsilon}$ on the $\ell$-torsion subgroup of the class group of $K$ by showing that $|\mathrm{Cl}_K[\ell]| = o_{[K:\mathbb{Q}],\ell}(D_K^{1/2})$. In fact, we obtain an explicit log-power saving. This is the first general unconditional saving over the trivial bound that holds for all $K$ and all $\ell$.

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Forward citations

Cited by 1 Pith paper

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

  1. A new pointwise bound for $3$-torsion of class groups

    math.NT 2025-05 conditional novelty 6.0 of 10

    A new combination of known techniques reduces the pointwise 3-torsion exponent from 1/3 to about 0.3193 and extends average ℓ-torsion bounds to real quadratic fields.

Reference graph

Works this paper leans on

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