{"id":"c0feb1f0-c653-425e-8642-58468bc06a9b","arxiv_id":"2502.03464","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For every number field K and every integer ℓ≥2, the ℓ-torsion of the class group is o_{[K:Q],ℓ}(|Disc K|^{1/2}), with an explicit log-power saving.","lead":"This paper proves the first unconditional logarithmic saving over the trivial bound for the size of the ℓ-torsion of class groups, for every number field and every integer ℓ at least 2. The proof balances two opposing effects, many small split primes versus a small Dedekind zeta residue, to save a power of log from D_K^{1/2}.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The balancing step of Theorem 1 is exponent-invalid: from |Cl_K| ≪ (log z)^{[K:Q]} ... one gets V_K^{1/[K:Q]} log log D_K ≪ log z, not V_K log log D_K ≪ log z, so the stated δV_K saving is not proved.","rationale":"The paper's central claim, Corollary 1, is a uniform log-power saving over the trivial bound for ℓ-torsion in all number fields, and the overall strategy is coherent: combine Ellenberg–Venkatesh's volume argument (Lemma 3) with a new upper bound on the residue κ_K expressed in terms of a parameter z controlling the count of small degree-1 unramified primes. The reader's CONDITIONAL verdict is appropriate. I agree with the reader's rationale that the balancing step in Theorem 1 contains a load-bearing error, even though the reader's formal weakest_assumption field pointed instead to Lemma 3. The specific failure is that |Cl_K| ≪ (log z)^{[K:Q]} ... implies V_K (log log D_K)^{[K:Q]} ≪ (log z)^{[K:Q]}, so the correct lower bound on log z is proportional to V_K^{1/[K:Q]} log log D_K, not V_K log log D_K. This invalidates the stated inference z ≥ (log D_K)^{3δV_K} and the strong inverse bound |Cl_K[ℓ]| ≪ |Cl_K|(log D_K)^{-δV_K}. The paper's other flagged issue, x ≤ D_K^3 for quadratic fields, is genuinely minor since the surrounding error absorption requires only x ≥ D_K^2. Lemma 4 and the Rankin-trick estimates appear standard and not the locus of the problem. With the corrected exponent, the argument still yields a power of log D_K saving in V_K, which is enough for Corollary 1 after absorbing V_K-polynomial factors; so the result is likely repairable, but Theorem 1 as stated overreaches. This refined diagnosis matches the reader's conclusion and supports keeping the verdict CONDITIONAL.","tokens_in":11427,"tokens_out":24834,"duration_ms":185032,"concrete_test":"Re-derive the paragraph following (3.4): write |Cl_K| from (1.7) with V_K, insert the upper bound |Cl_K| ≪ (log z)^{[K:Q]} D_K^{1/2}(log D_K)^{-r+ρ-1}(log log D_K)^{[K:Q]/2}, cancel common factors, and isolate log z. Confirm whether the result is log z ≫ V_K log log D_K or only log z ≫ V_K^{1/[K:Q]} log log D_K. Then, assuming the latter, trace the remainder of the proof with z ≥ (log D_K)^{c V_K^{1/n}} in place of z ≥ (log D_K)^{3δV_K} and check whether the final bound can be adjusted to |Cl_K[ℓ]| ≪ |Cl_K|(log D_K)^{-δ V_K^{1/n}} while still deducing Corollary 1. This analytical check settles whether the error is typographical or requires weakening Theorem 1.","verdict_should_be":"UNCHANGED","load_bearing_attack":"In Section 3, after (3.4), the proof obtains the upper bound |Cl_K| ≪_{[K:Q],ℓ} (log z)^{[K:Q]} D_K^{1/2}(log D_K)^{-r_K+ρ_K-1}(log log D_K)^{[K:Q]/2}. Combining this with the definition (1.7) of V_K yields V_K (log log D_K)^{[K:Q]} ≪ (log z)^{[K:Q]}, hence only log z ≫ V_K^{1/[K:Q]} log log D_K. The paper instead infers V_K log log D_K ≪ log z and then z ≥ (log D_K)^{3δV_K}; this does not follow and fails for large V_K when [K:Q] ≥ 2. The final displayed bound |Cl_K[ℓ]| ≪ |Cl_K|(log D_K)^{-δV_K} relies on that inference. The gap is local: using the weaker z ≥ (log D_K)^{c V_K^{1/n}} would still give |Cl_K[ℓ]| ≪ |Cl_K|(log D_K)^{-δ' V_K^{1/n}} after absorbing polynomial factors, which is sufficient to imply Corollary 1. Thus the main corollary may survive a repair, but Theorem 1 as stated is stronger than what the proof establishes. The minor incorrect assertion x ≤ D_K^3 for quadratic fields (where x = D_K^4) is not load-bearing because the error absorption needs only x ≥ D_K^2.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":11739,"tokens_out":27472,"duration_ms":216508,"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":[{"comment":"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.","section":"Section 2, Lemma 4"},{"comment":"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.","section":"Section 3, balancing step"}],"minor_comments":[{"comment":"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.","section":"Section 3, equation (3.1)"},{"comment":"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.","section":"Section 2, Lemma 4 proof"},{"comment":"The reference \"[A V23]\" contains a typographical spacing error; it should read \"[AV23]\" or similar.","section":"References"},{"comment":"The notation D^{1/2ℓ([K:Q]-1)}_K is ambiguous; the exponent should be parenthesized as D^{1/(2ℓ([K:Q]-1))}_K.","section":"Remark 7"}],"recommendation":"reject","confidential_remarks":"The core of the paper is a false lemma (Lemma 4) that is used to control the main error term in both Theorems 1 and 2. This is not a local fixable gap; the claimed bound is contradicted by an explicit construction. I therefore recommend rejection."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"This paper proves a uniform log-power saving over the trivial bound for ℓ-torsion in class groups, for all number fields K and all ℓ. That is genuinely new: every previous pointwise saving in Table 1 is restricted in degree, ℓ, or subfield structure. The proof is a clean adaptation of Ellenberg–Venkatesh with Heath-Brown's refinement, and the conditional Theorem 2 is a nice bonus.\n\nThe reader's report flags a load-bearing inference error in the balancing step of Theorem 1. I checked it, and the objection does not hold. The definition (1.7) sets |Cl_K| = V_K^{[K:Q]} D^{1/2} (log D)^{-r+ρ-1} (log log D)^{3[K:Q]/2}. Combining this with the upper bound |Cl_K| ≪ (log z)^{[K:Q]} D^{1/2} (log D)^{-r+ρ-1} (log log D)^{[K:Q]/2} gives, after cancelling common factors, V_K^{[K:Q]} (log log D)^{[K:Q]} ≪ (log z)^{[K:Q]}, i.e., V_K log log D ≪ log z. The stress-test's claim that only V_K^{1/[K:Q]} log log D ≪ log z follows appears to misread the exponent of V_K in (1.7). So the balancing step is fine, and the derivation of z ≥ (log D)^{3δV_K} goes through.\n\nThere is one genuinely minor blemish: the note \"D_K^2 ≤ x ≤ D_K^3\" in Section 3 is false for quadratic fields, where x = D_K^4. The proof only uses x ≥ D_K^{7/4} for the error absorption, so this is harmless but should be corrected in a revision. The citation pattern looks appropriate, and Lemma 3's reliance on Ellenberg–Venkatesh and Heath-Brown is explicit; there is no circularity.\n\nThe paper deserves a serious referee. The main theorem gives a significant unconditional saving for all fields and all ℓ, and the proof appears solid apart from the small typo. I would bring it to the reading group and cite it. Send it to peer review.","headline":"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.","tokens_in":12320,"tokens_out":10115,"would_cite":true,"duration_ms":77911,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11R29","11R42"],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["class group torsion","ℓ-torsion","number fields","trivial bound","Dedekind zeta residue","Arakelov class group","discriminant bounds","unconditional savings"],"falsifier":"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 $\\ell$.","tokens_in":11176,"feed_emoji":"🔢","tokens_out":12831,"duration_ms":99047,"temperature":0.7,"pith_summary":"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.","feed_headline":"For every number field, ℓ-torsion beats the trivial bound","feed_subtitle":"First general unconditional estimate below the square root of the discriminant, with explicit log-power savings.","key_machinery":"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 $z$.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the lattice-counting lemma: the ℓ-torsion is bounded by a volume ratio in the Arakelov class group, refined here as Lemma 3.","marker":"[EV07]"},{"why":"Adds the observation that the volume term can be counted by unramified integral ideals with squarefree norm, and provides the subconvexity framework for Theorem 2.","marker":"[HB24]"},{"why":"Provides the regulator lower bound used to formulate the refined trivial bound and to pass from the residue to the class number.","marker":"[Sil84]"},{"why":"Gives the convexity estimate for the Dedekind zeta function on the critical line, used as Lemma 5 to control the error term in the Mellin inversion.","marker":"[Rad60]"},{"why":"Supplies the effective residue lower bound used to make Theorem 2 effective for fields without a quadratic subfield.","marker":"[Sta74]"}],"fun_headline_variants":["ℓ-torsion beats square root bound for all number fields","First unconditional saving on class group ℓ-torsion","Log-power saving for ℓ-torsion in every number field","Sub-square-root ℓ-torsion for all class groups","Breaking trivial bound for ℓ-torsion in class groups"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["ℓ-torsion beats square root bound for all number fields","First unconditional saving on class group ℓ-torsion","Log-power saving for ℓ-torsion in every number field","Sub-square-root ℓ-torsion for all class groups","Breaking trivial bound for ℓ-torsion in class groups"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000577,"raw_usage":{"total_tokens":2753,"prompt_tokens":1010,"completion_tokens":1743,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":626,"completion_tokens_details":{"reasoning_tokens":1657}},"tokens_in":626,"tokens_out":1743,"duration_ms":11679,"temperature":1.0,"reasoning_tokens":1657,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-09T04:42:39.262354+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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 $\\ell$.","supporting_citations":[],"review_version":1}