{"id":"d9bf2c02-0d6f-4ef5-9211-89177aeb2c24","arxiv_id":"2411.13124","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A generalization of norm relations reduces class group computations for high-degree number fields to lower-degree subfields, with a polynomial-time proof under GRH and examples up to degree 105.","lead":"Mathematicians found a new way to compute class groups of very large number fields by breaking the problem into smaller subfields. It turns computations that previously took months into days for fields of degree more than 100.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 4.4's proof mistakenly asserts φ∘ψ = id although Prop. 2.14 yields only c·id; the claimed surjectivity on S-unit groups, and thus the unconditional polynomial-time conclusion of Theorem A, is unsupported.","rationale":"The reader's weakest assumption was the GRH/abstract mismatch, and the reader also noted an inconsistent equation in Theorem 4.4's proof. My stress-test focuses on that equation, which is the more fundamental flaw: it directly invalidates the correctness proof as written. However, the gap is plausibly repairable by invoking the saturation step, so the appropriate verdict remains CONDITIONAL, unchanged from the reader. The GRH mismatch is a related but distinct presentational issue: if GRH is not actually used, the paper should state an unconditional theorem; if it is used, Theorem A and the abstract must be qualified. I agree partially with the reader because they identified the GRH mismatch as the weakest assumption, whereas I see the false φ∘ψ = id identity as the load-bearing internal inconsistency. The concrete test is an analytical re-derivation that would settle whether the algorithm's correctness proof can be completed; it also clarifies the role of GRH.","tokens_in":20725,"tokens_out":20463,"duration_ms":199983,"concrete_test":"Re-derive the proof of Theorem 4.4 using the actual identity φ∘ψ = c·id from Prop. 2.14, and verify that the p-saturation steps in Algorithm 4.3 (performed for primes p dividing (n!)², with multiplicities) are sufficient to prove that the final V equals O^×_{K,S}. In particular, check that c divides |G|² (Theorem 2.16) and |G| ≤ n!, and that iterated p-th root extraction for each such prime kills the finite quotient O^×_{K,S}/V. If the saturation argument cannot be made to work, the theorem fails; if it works, state precisely where GRH is used, and if GRH is not used, remove it from Theorem 4.4 and from the abstract's unqualified polynomial-time claim.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The proof of the central Theorem 4.4 states: 'Since there is a generalised norm relation, we know that there exists an integer c, a surjective morphism ... and an injective morphism ..., such that φ∘ψ = id (by proposition 2.14).' This is incorrect: Proposition 2.14 and Definition 2.12 only guarantee the existence of ψ and φ with φ∘ψ = c·id for a positive integer c, and φ is not asserted to be surjective, only to have image of finite index. Consequently, Proposition 3.5 gives φ_M∘ψ_M = c·id, i.e. c·M(H) ⊆ im(φ_M), not a surjection φ_M. The proof then claims a surjective induced map on S-unit groups, which does not follow. Algorithm 4.3's saturation step may repair this, but the proof never supplies the necessary argument that p-saturation for primes dividing (n!)² converts finite-index inclusion into equality. As written, the correctness of Algorithm 4.3—and hence Theorem A's polynomial-time conclusion—is unproven even under GRH. Independently, Theorem 4.4 assumes GRH while the abstract and Theorem A do not, so the unconditional polynomial-time claim is also unsupported by the paper's own theorem.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces 'generalised norm relations' in the group algebra Q[G] for a finite group G, extending the classical norm relations of Biasse--Fieker--Hofmann--Page. It proves equivalent characterisations (Proposition 2.7), a field-theoretic criterion (Theorem 2.21), and a bound on the optimal coefficient (Theorem 2.16). The main algorithmic contribution is Algorithm 4.3, which claims to compute a basis of the S-unit group of a number field K from bases of S-unit groups of subfields K_i, with polynomial-time complexity; Theorem 4.4 states this under GRH. The paper also presents a heuristic class-group algorithm (Algorithm 4.6) and large computational examples, including a degree-105 field whose class group is computed in about 5 CPU days.","tokens_in":21002,"tokens_out":7653,"duration_ms":71868,"significance":"If the correctness gap identified below is repaired and the GRH qualification is stated consistently, this is a useful contribution to computational algebraic number theory. The generalised norm relation framework genuinely extends [2], the bound c(J,H) | |G|^2 improves the analogous |G|^3 bound for classical norm relations, and the examples (e.g. S5, A5, C7 x A5) show that the method can handle fields that are out of reach of direct computation. The paper includes explicit algorithms and reproducible computational experiments, which are valuable even where the complexity proof needs repair.","major_comments":[{"comment":"The proof of Theorem 4.4 states: 'Since there is a generalised norm relation, we know that there exists an integer c, a surjective morphism ... φ ... and an injective morphism ... ψ ..., such that φ∘ψ = id (by proposition 2.14).' This is not what Proposition 2.14 and Definition 2.12 deliver. They only guarantee φ∘ψ = c(J,H)·id for a positive integer c(J,H), with φ having image of finite index rather than being surjective. Consequently Proposition 3.5 gives φ_M∘ψ_M = c·id_{M(H)}, i.e. c·M(H) ⊆ im(φ_M), and the induced map on S-unit groups is not shown to be surjective. The correctness claim 'this proves the correctness' therefore does not follow as written. The missing step is to invoke Theorem 2.16 (c(J,H) divides |G|^2, and |G| ≤ n!) and to prove explicitly that the p-saturations in step 5 of Algorithm 4.3, performed for all primes dividing (n!)^2 with multiplicity, convert this finite-index inclusion into equality. This argument is absent from the proof. A related slip occurs at the end of the proof of Proposition 2.14, where the text writes 'Ψ ∘ Φ' when Definition 2.12 requires 'Φ ∘ Ψ'; with the correct order the conclusion is φ∘ψ = c·id, still not the identity.","section":"§4, Theorem 4.4 proof; Definition 2.12; Proposition 2.14; Proposition 3.5"},{"comment":"There is a statement-level inconsistency about GRH. The abstract and Theorem A advertise 'a polynomial time algorithm' without qualification, while Theorem 4.4 begins 'Assume the generalized Riemann Hypothesis (GRH). Then this algorithm is correct and its complexity is polynomial in the size of the input.' As written, the unconditional polynomial-time claim in Theorem A and the abstract is not supported by the paper's own theorem. The GRH hypothesis must be added to Theorem A and the abstract, or an unconditional proof must be supplied.","section":"Abstract, Introduction (Theorem A), §4 Theorem 4.4"}],"minor_comments":[{"comment":"The last line of the proof writes 'Ψ ∘ Φ' where the definition of c(J,H) and the surrounding argument require 'Φ ∘ Ψ'; this typo should be corrected to avoid confusion with the identity/c·id issue in Theorem 4.4.","section":"Proposition 2.14"},{"comment":"In the final sentence of the proof, the notation 'N_{C/M}(ι_L(x))' introduces a symbol M that has not been defined; the intended field should be named (apparently the compositum field C considered as an extension of L).","section":"Theorem 1.18"},{"comment":"There are several typographical slips, e.g. 'so me' in the abstract, 'a set a subgroups' in Definitions 2.2 and 2.4, and the OCR-style rendering 'C4×C4 2' in Example 6.1; these should be cleaned up.","section":"Abstract and various definitions"},{"comment":"Reference [7] is listed as 'GE.' with an incomplete author name; the full name and details should be provided.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"The central concern raised in the stress-test is genuine: the proof of Theorem 4.4 contains a load-bearing misstatement of Proposition 2.14, claiming a surjective φ and φ∘ψ = id where only φ∘ψ = c·id with finite-index image is available. The gap is very likely repairable by a saturation argument using Theorem 2.16, and the paper's main ideas remain valuable, so I recommend major revision rather than rejection. The GRH qualification mismatch between Theorem 4.4 and Theorem A/abstract should also be fixed. I would ask the editor to ensure the author addresses both issues explicitly in the revision, because the current text does not prove the main theorem as stated."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nThe short version: this paper has a genuinely new idea—generalized norm relations with an arbitrary subgroup H—and it backs it up with impressive computations, including a class group that went from a five-month computation to five days. But the proof of the main theorem has a gap, and the abstract overstates what is proved.\n\nWhat is actually new: replacing the norm element N_H (for trivial H) by N_H for any subgroup H, and showing that this buys you relations that do not come from classical norm relations in subgroups or quotients (their S4, S5, A5, C7×A5 examples). The compositum criterion (Theorem 2.21) is a useful practical tool, letting you check relations without computing Galois groups. The bound c(J,H) | |G|² improves the |G|³ bound for classical norm relations. The computational examples are the real payoff and look credible.\n\nThe soft spots are real but fixable. In the proof of Theorem 4.4, the author asserts that Proposition 2.14 gives φ∘ψ = id. It gives φ∘ψ = c·id. That matters: the induced map on S-unit groups is not necessarily surjective; its image only contains c-th powers. The algorithm's saturation step is clearly designed to handle this, and with the bound c | (n!)², the argument can be made: repeatedly adjoining p-th roots for all p | c turns finite-index inclusion into equality. But the proof as written does not supply that argument, so the correctness claim is unsupported as it stands. A referee should ask for this to be written out.\n\nThe second issue is presentation: Theorem 4.4 assumes GRH, while the abstract and Theorem A present the polynomial-time claim unqualified. The algorithm may in fact need GRH only for standard subroutines, but the statement should match the theorem. The paper also does not release code or data, which is a minor reproducibility setback given the computational claims.\n\nOn the citation pattern, the paper engages fairly with the prior norm-relation literature and the Mackey functor sources; no red flags there.\n\nWho should read this: anyone doing computational algebraic number theory, especially on class groups of high-degree fields. It deserves a serious referee: the idea is sound, the examples are strong, and the gap is patchable. I would not desk-reject it, but I would insist on a revised proof of Theorem 4.4 and a corrected abstract.","headline":"Genuinely new norm-relation generalization with impressive examples, but the main proof has a fixable gap and the abstract overstates the GRH qualification.","tokens_in":21499,"tokens_out":6560,"would_cite":true,"duration_ms":57568,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11R29","11R32","11Y40"],"pacs":[],"model":"deepseek-v4-flash","headline":"Class groups can be computed by reducing to lower-degree subfields via generalized norm relations.","keywords":["generalized norm relations","S-unit groups","class group computation","Mackey functors","Galois closure","compositums","Hecke algebras","number fields"],"falsifier":"Take the $C_7 \\times A_5$ field of Example 6.2, compute its S-units with Algorithm 4.3 and independently with a direct method; any discrepancy in the returned basis or in the final class group refutes the central claim. A cheaper test: verify the bound $c(\\mathcal{J},H) \\mid |G|^2$ on a large finite group with a known relation; one counterexample would refute Theorem 2.16 and with it the polynomial complexity.","tokens_in":20551,"feed_emoji":"🔢","tokens_out":5816,"duration_ms":52890,"temperature":0.7,"pith_summary":"This paper claims that the class group and S-unit group of a number field can be computed from the S-unit groups of smaller subfields whenever a certain algebraic identity, called a generalized norm relation, holds in the Galois group. The identity is a group-algebra equation $N_H = \\sum_i a_i N_{J_i} b_i$, and its existence is shown equivalent to a purely field-theoretic condition expressible with compositums. The author proves Theorem A: under the generalized Riemann hypothesis, there is an algorithm, polynomial in the input size, that takes such subfield unit bases and returns a basis of the S-units of the target field, hence its class group. This would matter because the cost of computing class groups by the standard subexponential method grows badly with degree and discriminant, and an inductive reduction to lower-degree fields circumvents that bottleneck. The paper also reports a concrete success: a degree-105 field whose class group was computed in about five days, where classical norm-relation methods did not finish in five months.","feed_headline":"Norm-relation induction computes big class groups from small subfields","feed_subtitle":"Under GRH, the method runs in polynomial time and solved a degree-105 case in five days.","key_machinery":"The central objects are generalized norm relations in the group algebra $\\mathbb{Q}[G]$: an equality $N_H = \\sum_{i=1}^{\\ell} a_i N_{J_i} b_i$, where $N_H = \\sum_{h\\in H} h$ is the norm element of a subgroup $H$ fixing the target field $K$, and the $J_i$ fix the auxiliary subfields. The mechanism that carries the argument is the transfer of this group-algebra identity to cohomological Mackey functors: by proposition 3.5, a relation with $\\varphi \\circ \\psi = d\\cdot \\mathrm{id}$ between permutation modules gives, for every cohomological Mackey functor $M$, maps between $M(J_i)$ and $M(H)$ whose composition is $d\\cdot \\mathrm{id}$. Applied to $M(H) = O_{\\tilde{K}^H,S}^{\\times}$, this produces the S-unit basis. The coefficient bound $c(\\mathcal{J},H) \\mid |G|^2$ makes the saturation step finite and polynomial.","core_discovery":"The central claim is that norm relations can be generalised from the trivial subgroup to an arbitrary subgroup $H$ of the Galois group $G$ of the Galois closure. Given subfields $K_i = \\tilde{K}^{J_i}$, a generalised norm relation $N_H = \\sum_i a_i N_{J_i} b_i$ in $\\mathbb{Q}[G]$ transfers to the S-unit groups: the Mackey functor $M(H) = O_{\\tilde{K}^H,S}^{\\times}$ admits maps $\\varphi_M$ and $\\psi_M$ with $\\varphi_M \\circ \\psi_M = d\\cdot \\mathrm{id}$, so a basis of the S-units of the $K_i$ yields a basis of the S-units of $K$. Equivalently, the relation exists iff there is a surjective $\\mathbb{Q}[G]$-module morphism from $\\bigoplus_i \\mathbb{Q}[G/J_i]$ to $\\mathbb{Q}[G/H]$, which is checked without computing Galois groups using the compositum action $C\\cdot x = N_{C/L}(\\iota_K(x))$. The optimal coefficient $c(\\mathcal{J},H)$ is proved to divide $|G|^2$, which yields the polynomial bound via saturation over primes dividing $(n!)^2$.","pith_inferences":["If the polynomial-time claim is right, computing class groups of large-degree fields could be redirected to searching for generalized norm relations in the Galois group rather than attacking the field directly; the author explicitly leaves the search problem open.","The compositum-action criterion suggests a possible algorithmic shortcut: one could try to find relations by linear algebra over $\\mathbb{Q}$ on the embeddings, without ever computing the Galois group, which may scale to fields beyond current Galois-closure limits.","Because the transfer works for any cohomological Mackey functor, the same induction could in principle compute other invariants such as unit groups of group rings or K-theory, though the paper only mentions S-units and class groups.","The bound $c(\\mathcal{J},H) \\mid |G|^2$ hints that the saturation over primes dividing $(n!)^2$ might be replaceable by a smaller set depending only on the group, which would improve constants."],"forward_implications":["If Theorem A holds, class groups of number fields admitting generalized norm relations can be computed in polynomial time in the input size under GRH, including the S-unit bases of subfields.","For Galois extensions, classical norm relations become a special case, so the method extends the reach of the prior norm-relation framework to non-Galois fields.","Fields with no classical norm relation in any quotient may still admit generalized norm relations; the paper exhibits $S_5$ and $C_7 \\times A_5$ examples with concrete speedups.","The optimal-denominator bound drops from $|G|^3$ to $|G|^2$, improving the provable complexity of the saturation step.","A practical variant, correct on termination, verified $\\mathrm{Cl}(K)=1$ for a degree-105 field in about 5 days CPU time, where classical methods did not finish in over 5 months."],"supporting_citations":[{"why":"Supplies the classical norm-relation framework and the saturation method (corollary 4.13) that the generalisation extends.","marker":"[2]"},{"why":"Supplies the cohomological Mackey functor induction (proposition 3.5) that transfers permutation-module relations to S-unit groups.","marker":"[3]"},{"why":"Supplies the G-functor and Hecke-operator isomorphisms (theorems 3.3 and 3.4) used to define the compositum action and the transfer maps.","marker":"[13]"},{"why":"Provides the classical character-relation ancestor showing group-ring relations between permutation characters yield arithmetic relations between subfields.","marker":"[4]"},{"why":"Gives the standard subexponential algorithm for class groups and regulators, the baseline that the inductive method aims to beat.","marker":"[5]"},{"why":"Supplies the approximation of the residue of the Dedekind zeta function at $s=1$, used in Algorithm 4.6 to verify the regulator-class-number product.","marker":"[1]"}],"fun_headline_variants":["Generalized norm relations power class group algorithm","Class group computation via subfield induction shortcuts","Induction on norm relations computes big class groups","Reducing class group degree via generalized norm relations","Subfield induction yields faster class group algorithm"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The polynomial-time guarantee rests on the generalized Riemann hypothesis, a standard unproven assumption about the distribution of prime numbers; if that hypothesis fails, the algorithm's run-time bound no longer follows.","fun_headline_variants_meta":{"raw":{"variants":["Generalized norm relations power class group algorithm","Class group computation via subfield induction shortcuts","Induction on norm relations computes big class groups","Reducing class group degree via generalized norm relations","Subfield induction yields faster class group algorithm"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000157,"raw_usage":{"total_tokens":1187,"prompt_tokens":875,"completion_tokens":312,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":491,"completion_tokens_details":{"reasoning_tokens":244}},"tokens_in":491,"tokens_out":312,"duration_ms":4159,"temperature":1.0,"reasoning_tokens":244,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T16:48:37.478329+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take the $C_7 \\times A_5$ field of Example 6.2, compute its S-units with Algorithm 4.3 and independently with a direct method; any discrepancy in the returned basis or in the final class group refutes the central claim. A cheaper test: verify the bound $c(\\mathcal{J},H) \\mid |G|^2$ on a large finite group with a known relation; one counterexample would refute Theorem 2.16 and with it the polynomial complexity.","supporting_citations":[{"cited_title":"Norm relations and computa- tional problems in number fields","cited_arxiv_id":null,"evidence_quote":"Supplies the classical norm-relation framework and the saturation method (corollary 4.13) that the generalisation extends."},{"cited_title":"Class group relations from Burnside ring idempotents","cited_arxiv_id":null,"evidence_quote":"Supplies the cohomological Mackey functor induction (proposition 3.5) that transfers permutation-module relations to S-unit groups."},{"cited_title":"On G-functors. II: Hecke operators and G-functors","cited_arxiv_id":null,"evidence_quote":"Supplies the G-functor and Hecke-operator isomorphisms (theorems 3.3 and 3.4) used to define the compositum action and the transfer maps."},{"cited_title":"Beziehungen zwischen Klassenzahlen von Teilk¨ orpern eines galoisschen K¨ orpers","cited_arxiv_id":null,"evidence_quote":"Provides the classical character-relation ancestor showing group-ring relations between permutation characters yield arithmetic relations between subfields."},{"cited_title":"A subexponential algorithm for the determination of class groups and regulators of algebraic number fields","cited_arxiv_id":null,"evidence_quote":"Gives the standard subexponential algorithm for class groups and regulators, the baseline that the inductive method aims to beat."},{"cited_title":"Computing the residue of the Dedekind zeta function","cited_arxiv_id":null,"evidence_quote":"Supplies the approximation of the residue of the Dedekind zeta function at $s=1$, used in Algorithm 4.6 to verify the regulator-class-number product."}],"review_version":1}