{"id":"e5b6a25c-b3c1-4a1d-95a8-0c54416ad0ca","arxiv_id":"2506.08865","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A complete classification of when a_p(f) ≡ x mod λ is governed by a congruence condition on p, in terms of the projective mod λ Galois image, extending Swinnerton-Dyer.","lead":"This paper identifies exactly when the p-th Fourier coefficient of a modular eigenform, taken modulo a prime, can be predicted by a congruence rule on the prime p itself. The classification is phrased in terms of the Galois representation attached to the form, which tells working number theorists when a hard coefficient condition can be replaced by a simple arithmetic rule on p.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"A5 row of Theorem 1.10 hinges on an unpublished finite search; the advertised complete classification is not independently auditable at this point.","rationale":"I read the full text and worked through the main proof. Proposition 2.1 is the dictionary translating congruence conditions on p into cosets of [G_λ,G_λ], and its 'only information determined by a congruence condition' step is valid once M is taken to be a multiple of the conductor of Z_λ; the proof is terse but fixable. Theorems 1.4–1.6 and the dihedral/PSL2/PGL2 rows of Theorem 1.10 are explicit and internally consistent. The genuine soft spot is the A5 exception set: the proof delegates the decisive computation to an unspecified finite search, with no code and no bound, and the table's (⋆) condition (ℓ=2,3,5,29 and specific trace classes) depends entirely on that search. This does not demonstrate a contradiction, so a conditional verdict is appropriate; it is a reproducibility and auditability gap, not a known error. I therefore agree with the reader's CONDITIONAL verdict but not with the identification of Proposition 2.1 as the weakest assumption, hence 'partial' agreement.","tokens_in":23454,"tokens_out":31052,"duration_ms":366224,"concrete_test":"Independently reproduce the A5 search: for primes ℓ up to an explicit bound (compute the bound from the condition that T∩αT is empty; include at least ℓ≤100), enumerate all conjugacy classes of A5 subgroups in PGL2(F_ℓ), choose an odd lift to GL2(F_ℓ) (determinant character containing -1), compute the trace set of each coset of [G,G], and check whether every class in F_ℓ^× is missed by some coset. Compare with the exception set in Theorem 1.10, especially ℓ=29 and x∈{±2,±5}, and publish the code and the bound.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The weakest load-bearing point is not Proposition 2.1 (which is sound after enlarging M to a multiple of the conductor of Z_λ) but the A5 case in the proof of Theorem 1.10. The text reduces the determination of the exception set (⋆) to 'a finite computer search' after asserting that 'it is easy to see that T∩αT≠∅ for only finitely many ℓ', but provides no code, no bound on ℓ, and no description of the enumeration of A5 subgroups or their lifts to GL2. Since Theorem 1.10 is the complete classification advertised in the abstract, an error in this search would change the semi-λ-abelian exceptions for ℓ=2,3,5,29 (or add others) and would propagate into the 'Yes unless (⋆)' entry for A5. The surrounding group theory (Theorems 1.4–1.6 and the dihedral/PSL2/PGL2 rows) is explicit and checkable; the A5 row is the one place where the published argument delegates the decisive computation without artifacts.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies, for a normalised cuspidal eigenform f and a prime λ of its coefficient field, when the congruence a_p(f) ≡ x mod λ can be described by a congruence condition on the prime p. It introduces the notions of weakly λ-abelian, semi-λ-abelian, and λ-abelian forms, and classifies these properties in terms of the projective image of the associated mod λ Galois representation. The main results are Theorem 1.4 (total λ-abelianness is equivalent to Borel image), Theorems 1.5 and 1.6 (weakly, respectively fully, λ-abelian for a nonzero class forces x=0 and dihedral projective image with specified restrictions on n and ℓ), Propositions 1.8 and 1.9 (explicit moduli and Legendre-symbol criteria in the dihedral cases), and Theorem 1.10, a table giving for every possible projective image whether f is semi-λ-abelian on F_λ^×, whether it is semi-λ-abelian for 0, and the density c_λ of primes with a_p(f) ≡ 0. The paper closes with several worked examples coming from weight-2 newforms attached to rational elliptic curves.","tokens_in":23510,"tokens_out":12939,"duration_ms":155763,"significance":"If the main results are correct, the paper gives a satisfying and quite general extension of Swinnerton-Dyer's classical congruence analysis, reducing the question to group theory in a transparent way. The strengths of the manuscript are its explicit, elementary matrix arguments for Theorems 1.4–1.6, the conductor bounds in Proposition 1.8, the clean Legendre-symbol criterion in Proposition 1.9, and the rich set of elliptic-curve examples that illustrate each row of the classification. The proof of the A5 row of Theorem 1.10, however, delegates the decisive finite check to an unpublished computation, which currently prevents the advertised complete classification from being independently auditable.","major_comments":[{"comment":"The A5 case is not fully proved in the manuscript. After reducing the exceptional classes T to the condition that T ∩ αT ≠ ∅ for a scalar α of multiplicative order 4, the text states that this occurs for only finitely many ℓ and that the problem reduces to a finite computer search, but it gives no bound on ℓ, no description of how A5 subgroups or their lifts to GL_2(F_λ) are enumerated, and no code or output. Since Theorem 1.10 is the advertised complete classification and the resulting condition (⋆) determines the A5 row for ℓ = 2, 3, 5, 29, an error in this search would change the classification. This gap should be closed by a reproducible computation, for example an explicit finite bound on ℓ with a complete case check, or by attaching the code and its output.","section":"§2.6, proof of Theorem 1.10 (A5 row)"}],"minor_comments":[{"comment":"The title contains a typo: 'modλV alues' should read 'mod λ values'.","section":"Title"},{"comment":"The sentence 'By Lemma 2.4, the commutator subgroup [G_λ, G_λ] will therefore contain a diagonalisable matrix A ≠ 1' refers to a nonexistent Lemma 2.4; the intended reference appears to be Proposition 2.4.","section":"§2.3, proof of Theorem 2.11"},{"comment":"The reference [Zyw15] lists the same arXiv identifier as [CMM24] (arXiv:2412.01803), which is not the Zywina paper on possible images of mod ℓ representations of elliptic curves; the correct URL or citation should be supplied.","section":"References"},{"comment":"In the Borel row of the table, the quantity d appears in '0 or 1/d for d | q ± 1' but is not defined in the theorem statement; it is only later identified in the proof as the order of the character ψ_1/ψ_2, and this definition should be moved into the statement.","section":"Theorem 1.10, table"},{"comment":"The proof of Proposition 2.1 would be easier to check if it stated explicitly that the modulus M may need to be enlarged to a multiple of the conductor of the maximal abelian subextension Z_λ, since the initial definition of weak λ-abelianness allows an arbitrary M.","section":"Proposition 2.1"}],"recommendation":"major_revision","confidential_remarks":"The only substantive obstacle is the undocumented A5 finite search in the proof of Theorem 1.10; the rest of the classification appears sound and the examples are well chosen. I would be willing to support acceptance once the authors provide a complete, reproducible verification of the A5 row and fix the minor issues listed above."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here's my take. The paper generalizes Swinnerton-Dyer's [SD88] dichotomy from level-one rational eigenforms to arbitrary level, nebentypus, and coefficient field, and adds the full semi-λ-abelian table with densities. Theorems 1.4–1.6 and Propositions 1.8–1.9 are proven by explicit matrix arguments that I can follow, and Section 3 genuinely illustrates the claims rather than just citing them. That is real content.\n\nThe soft spot is exactly where your stress-test puts it: the A5 row of Theorem 1.10 relies on a finite computer search that is not described. No code, no bound on ℓ, no description of how A5 subgroups of PGL2(F_ℓ) are enumerated. Since that row is part of the advertised complete classification, an error would change the exception set and propagate into the table. This is the one load-bearing gap. I'd cap soundness at 6 for it, as your reader did. The rest of the proof structure checks out: the Borel, dihedral, PSL2/PGL2, A4, and S4 arguments are explicit enough for a referee to verify in an afternoon.\n\nMinor points: several trace-set identifications are deferred as 'routine computations', the Section 3 tables ship no computational artifacts, and the Zywina reference carries the wrong arXiv identifier (it duplicates [CMM24]'s). Each is easily addressable.\n\nI disagree slightly with the stress-test's worry about Proposition 2.1. That dictionary is sound: the Kronecker-Weber/class field theory direction is standard, and the conductor issue is handled by enlarging M. The paper is honest about what is a Lemma and what is a Proposition.\n\nWho this is for: number theorists working on mod λ Galois representations, and computational people interested in elliptic curves and congruences. It deserves a serious referee. The referee should demand that the A5 search either be fully specified with bounds and code, or that the statement be marked as dependent on a computational search. As printed, it is close but not fully auditable. I'd engage with it.","headline":"Solid generalization of Swinnerton-Dyer's dichotomy, except the A5 row of the complete table rests on an underspecified finite search.","tokens_in":24218,"tokens_out":3356,"would_cite":true,"duration_ms":35391,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11F33","11F80","11G05","11R45"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper classifies exactly when the condition a_p(f) ≡ x mod λ for a classical modular eigenform is determined by a congruence on the prime p, in terms of the projective image of the mod λ Galois representation.","keywords":["modular forms","Fourier coefficients","congruences","Galois representations","elliptic curves","dihedral groups","trace of Frobenius","Chebotarev density theorem"],"falsifier":"Take a weight-2 newform whose projective mod λ image is PSL_2(F_λ) with #F_λ > 3 and find a nonzero f-proper class x such that the set of primes with a_p(f) ≡ x mod λ is a union of residue classes modulo some M; the classification predicts this is impossible, so one such example would refute it. A computationally easier check is to test Theorem 1.5: any weakly λ-abelian nonzero class must have Borel image, so scanning small levels and primes λ for an implication 'p ≡ r mod M implies a_p(f) ≡ x mod λ' with x ≠ 0 and non-Borel image would settle it immediately.","tokens_in":23097,"feed_emoji":"🧮","tokens_out":9654,"duration_ms":107505,"temperature":0.7,"pith_summary":"The paper asks a sharp arithmetic question: when can the condition that the p-th Fourier coefficient a_p(f) of a normalized cuspidal eigenform is congruent to a fixed value x modulo λ be rewritten as a congruence condition on the prime p itself? It answers the question completely in terms of the projective image of the mod λ Galois representation attached to f. The classification says that a nonzero residue class x can be governed by a congruence on p only when that image is contained in a Borel (upper-triangular) subgroup, making the form totally λ-abelian. The remaining λ-abelian phenomena are concentrated at x = 0 and occur exactly for dihedral projective images D_n with n = 2 or odd n ≥ 3 and ℓ odd, with weak variants for all dihedral n > 1 with ℓ ∤ n. A final table settles the weaker semi-λ-abelian notion for every possible projective image, including the densities of primes with a_p(f) ≡ 0 mod λ.","feed_headline":"Fourier coefficients follow prime congruences only in two cases","feed_subtitle":"The classification says nonzero values need Borel images, while vanishing is congruence-governed exactly in dihedral cases.","key_machinery":"The load-bearing object is the mod λ Galois representation ρ_{f,λ} and its trace map, together with the commutator subgroup [G_λ, G_λ] of its image. Proposition 2.1 shows that the only information a congruence condition on p can carry is which coset of [G_λ, G_λ] contains Frobenius at p, because congruence conditions correspond exactly to splitting in abelian extensions by Kronecker-Weber and class field theory. Since a_p(f) mod λ equals the trace of the Frobenius element, the classification reduces to asking on which cosets of the commutator subgroup the trace is constant, or zero, or never equal to a given value. Two elementary facts do most of the work: in PGL_2, traceless elements are exactly involutions (Lemma 2.7), and the commutator subgroup of a dihedral group D_n is the rotation subgroup generated by $r^{2}$ (Lemma 2.13). The subgroup classification of PGL_2(k) is invoked only for the semi-λ-abelian table.","core_discovery":"The central claim is that the relation between the congruence a_p(f) ≡ x mod λ and a congruence on p is fully classified by the projective image Ḡ_λ of ρ_{f,λ} in PGL_2(F_λ). f is totally λ-abelian exactly when G_λ is conjugate into the Borel subgroup of GL_2(F_{$λ^{2}$}) (Theorem 1.4). If f is weakly λ-abelian for some f-proper x but not totally λ-abelian, then x = 0 and this happens exactly when Ḡ_λ ≅ D_n is dihedral with n > 1 and ℓ ∤ n (Theorem 1.5); for the stronger λ-abelian property, the same conclusion holds with x = 0 and Ḡ_λ ≅ D_n where n = 2 or n is odd, ℓ is odd, and ℓ ∤ n (Theorem 1.6). Theorem 1.10 extends the analysis to semi-λ-abelian forms: for each possible projective image—Borel, PGL_2(k), PSL_2(k), dihedral, A_4, S_4, A_5—it records whether f is semi-λ-abelian for all nonzero f-proper classes, whether it is semi-λ-abelian for 0, and the density c_λ of primes with a_p(f) ≡ 0 mod λ.","pith_inferences":["The same commutator-coset dictionary should apply to Hilbert modular forms and to any 2-dimensional Galois representation with a trace-compatible coefficient field; the argument is essentially representation-theoretic and does not use the classical level structure except through the group-theoretic setup.","For rational elliptic curves, this classification sharpens the classical fact that supersingular primes are often hard to describe: they admit a prime-congruence description exactly in the dihedral and Borel image cases, so outside those cases no modulus M can capture a_p = 0.","The densities c_λ in Theorem 1.10 give a practical statistical test: compare the observed proportion of primes with a_p(f) ≡ 0 mod λ against the table; a persistent mismatch would indicate either a misidentified projective image or a counterexample to the classification.","One could test the sharpness of Proposition 1.8 by computing the minimal modulus M for dihedral examples and checking whether it ever exceeds the stated divisor; the proposition claims the modulus is always controlled by rad(Nℓ) and the exponent of the Galois image, which is a concrete finite verification."],"forward_implications":["For any classical normalized cuspidal eigenform, a nonzero residue class x can be λ-abelian only if the mod λ Galois image is conjugate into the Borel subgroup; every non-Borel λ-abelian congruence rule is a rule for vanishing.","Vanishing a_p(f) ≡ 0 mod λ is governed by an exact congruence on p exactly when the projective image is dihedral D_n with n = 2 or odd n ≥ 3, ℓ odd and ℓ ∤ n, with explicit moduli M constructed from rad(Nℓ) and the exponent of the semisimplified image.","In the Borel case the entire vector of a_p(f) mod λ is determined by p modulo an explicit divisor of rad(Nℓ) · gcd(2 exp(im(ρ_λ^{ss,f})), Nℓ), so computations become finite checks of a single residue class.","For images PGL_2(F_λ) or PSL_2(F_λ) with more than 3 elements, no residue class at all is semi-λ-abelian: neither the coefficient value nor its vanishing can be implied by or imply a congruence on p.","For exceptional projective images A_4, S_4 and A_5, the nonzero classes are semi-λ-abelian except for the listed determinant exceptions, while the density of vanishing primes is 1/4, 3/8 and 1/4 for odd ℓ, so the classification gives sharp density predictions in every possible case."],"supporting_citations":[{"why":"Attaches the λ-adic Galois representation to f, so that a_p(f) mod λ becomes the trace of Frobenius.","marker":"[Del71]"},{"why":"Provides the earlier classification of coefficient congruences for level-one rational eigenforms that this paper extends to general level and nebentypus.","marker":"[SD88]"},{"why":"Supplies the explicit subgroup classification and trace computations for PGL_2(k) used in the dihedral arguments and the semi-λ-abelian table.","marker":"[Fab23]"},{"why":"Shows non-CM eigenforms have only finitely many primes λ with non-surjective Galois image, bounding when weakly λ-abelian behavior can occur.","marker":"[Rib85]"},{"why":"Provides the local Kronecker-Weber conductor bounds used to produce the explicit moduli M in Proposition 1.8.","marker":"[Ste02]"}],"fun_headline_variants":["Nonzero Fourier congruences force Borel images","Zero Fourier congruences require dihedral images","Prime congruences for Fourier coefficients: two image types","Fourier mod lambda: Borel for nonzero, dihedral for zero","Fourier congruences classified by Galois image shape"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is Proposition 2.1: a congruence condition on p can determine exactly the coset of the commutator subgroup of the Galois image that Frobenius at p occupies, and conversely each such coset is expressible by a congruence on p; if level structure, nebentypus, or ramification broke this dictionary for some eigenform, the classification would describe a different relation than the one it claims to classify.","fun_headline_variants_meta":{"raw":{"variants":["Nonzero Fourier congruences force Borel images","Zero Fourier congruences require dihedral images","Prime congruences for Fourier coefficients: two image types","Fourier mod lambda: Borel for nonzero, dihedral for zero","Fourier congruences classified by Galois image shape"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000249,"raw_usage":{"total_tokens":1584,"prompt_tokens":1015,"completion_tokens":569,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":631,"completion_tokens_details":{"reasoning_tokens":504}},"tokens_in":631,"tokens_out":569,"duration_ms":6424,"temperature":1.0,"reasoning_tokens":504,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T05:01:38.955832+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a weight-2 newform whose projective mod λ image is PSL_2(F_λ) with #F_λ > 3 and find a nonzero f-proper class x such that the set of primes with a_p(f) ≡ x mod λ is a union of residue classes modulo some M; the classification predicts this is impossible, so one such example would refute it. A computationally easier check is to test Theorem 1.5: any weakly λ-abelian nonzero class must have Borel image, so scanning small levels and primes λ for an implication 'p ≡ r mod M implies a_p(f) ≡ x mod λ' with x ≠ 0 and non-Borel image would settle it immediately.","supporting_citations":[],"review_version":1}