{"id":"0631227a-230c-4e01-8269-a7fcc9cf9d7e","arxiv_id":"2604.02723","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":6.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"New K4 and K5 eta-quotient families express Hecke eigenform coefficients via finite-field periods and produce identities with Appell series F1^p and F2^p.","lead":"The paper constructs two explicit families of eta-quotients called K4 and K5 from hypergeometric series using Jacobi theta functions and cubic analogues. These constructions express Fourier coefficients of certain weight-2 and weight-4 Hecke eigenforms as finite-field period functions and yield new identities with finite-field Appell series.","discovery_kind":"new_method","skeptic_critique":{"model":"grok-4.3","headline":"Modularity and transformation properties of the newly constructed K4 and K5 eta-quotient families under the relevant congruence subgroups remain the least-secured step linking them to Hecke eigenform coefficients.","rationale":"The reader's weakest assumption correctly isolates the unverified modularity step for K4/K5. With full text now available, the same step remains load-bearing because the paper's novelty consists precisely in asserting that these particular eta-quotients produce the finite-field period expressions; no independent machine-checked verification or numerical table is referenced in the abstract, so a direct coefficient comparison is the minimal concrete check that would either confirm or refute the link without requiring re-derivation of the entire hypergeometric background.","tokens_in":1732,"tokens_out":422,"duration_ms":31418,"concrete_test":"Select the lowest-level weight-2 Hecke eigenform treated in the paper (e.g., the one attached to the first elliptic curve in the list); compute the first 20 coefficients of the K4 eta-quotient expansion directly from its definition in terms of Jacobi theta functions, then compare term-by-term with the known Fourier coefficients of that eigenform. Agreement to machine precision for all terms confirms the modularity link; discrepancy >1 in any coefficient falsifies the claim for that form.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The central claim requires that the K4 (built from weight-1/2 Jacobi theta functions) and K5 (cubic analogues) eta-quotients satisfy exact transformation laws and produce q-expansions whose coefficients equal the finite-field period expressions for the target weight-2 and weight-4 Hecke eigenforms. If the level, character, or weight of these eta-quotients deviates from that of the eigenforms (or if the hypergeometric-to-eta identification in the construction step contains an implicit normalization error), the claimed identities relating a_n to F1^p and F2^p special values fail to hold. This is the single assumption whose failure would invalidate the explicit modularity statements.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The paper constructs two explicit families of eta-quotients, denoted K4 (from weight-1/2 Jacobi theta functions) and K5 (from cubic analogues), motivated by the Explicit Hypergeometric Modularity Method of Allen et al. It claims that these families allow the Fourier coefficients of certain weight-2 and weight-4 Hecke eigenforms to be expressed in terms of finite-field period functions, yielding new identities that relate those coefficients to special values of the finite-field Appell series F1^p and F2^p.","tokens_in":1870,"tokens_out":560,"duration_ms":20801,"significance":"If the constructions and identifications hold, the work supplies explicit eta-quotient realizations that make the hypergeometric modularity statements concrete and computable. This could facilitate direct evaluation of eigenform coefficients via finite-field periods and produce verifiable identities between modular forms and Appell series over finite fields, extending the EHMM framework with explicit, potentially machine-checkable links.","major_comments":[{"comment":"The transformation laws and modularity of the newly defined K4 and K5 eta-quotients under the relevant congruence subgroups (presumably stated in the construction sections) are asserted but not accompanied by explicit verification of the slash-operator action or character; without this, the claimed equality between the q-expansion coefficients and the finite-field period expressions for the target Hecke eigenforms cannot be confirmed.","section":"Construction of K4 and K5 (likely §2–3)"},{"comment":"No explicit formulas for the K4 or K5 q-expansions, no numerical checks against known eigenform coefficients, and no sample identities relating a_n to F1^p or F2^p special values are supplied in the abstract or visible text; the central claim therefore rests on an unverified identification step.","section":"Application to Hecke eigenforms (likely §4)"}],"minor_comments":[{"comment":"Notation for the finite-field Appell series F1^p and F2^p should be defined at first use with a reference to the precise normalization employed.","section":"Introduction"},{"comment":"The precise levels and characters of the target Hecke eigenforms should be stated explicitly when the K4/K5 constructions are introduced.","section":"Main constructions"}],"recommendation":"major_revision","confidential_remarks":"The manuscript appears to be an extension of prior EHMM work; the citation pattern to Allen et al. is appropriate, but the absence of any computational verification or explicit formulas in the visible text raises a scope-fit concern for a number-theory journal that typically expects concrete, checkable identities."},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for the careful reading and constructive suggestions. We address each major comment below and have revised the manuscript accordingly to make the verifications and applications more explicit.","responses":[{"response":"We agree that the explicit verification of the slash-operator action and character was not sufficiently detailed. The modularity of K4 and K5 follows from the known transformation properties of the underlying weight-1/2 Jacobi theta functions (for K4) and their cubic analogues (for K5), but we have now added, in the revised Sections 2 and 3, the full computation of the action under a set of generators for the relevant congruence subgroups (including explicit matrix representatives and the resulting multiplier system/character). These calculations confirm the claimed modularity and thereby justify the identification of the q-expansion coefficients with the finite-field period functions.","revision_made":"yes","referee_comment":"[Construction of K4 and K5 (likely §2–3)] The transformation laws and modularity of the newly defined K4 and K5 eta-quotients under the relevant congruence subgroups (presumably stated in the construction sections) are asserted but not accompanied by explicit verification of the slash-operator action or character; without this, the claimed equality between the q-expansion coefficients and the finite-field period expressions for the target Hecke eigenforms cannot be confirmed."},{"response":"The abstract is necessarily concise and does not contain explicit expansions or numerical examples. The full manuscript derives the q-expansions of K4 and K5 from their eta-quotient definitions in Sections 2–3 and obtains the identities with F1^p and F2^p in Section 4. To address the concern directly, we have added (i) the first several terms of the q-expansions of both families, (ii) explicit numerical checks matching the resulting coefficients against known tables for specific weight-2 and weight-4 Hecke eigenforms (e.g., the form attached to the elliptic curve of conductor 37), and (iii) concrete sample identities such as a_p = F_1^p(…) for small primes p. These additions render the identification step verifiable by direct computation.","revision_made":"partial","referee_comment":"[Application to Hecke eigenforms (likely §4)] No explicit formulas for the K4 or K5 q-expansions, no numerical checks against known eigenform coefficients, and no sample identities relating a_n to F1^p or F2^p special values are supplied in the abstract or visible text; the central claim therefore rests on an unverified identification step."}],"tokens_in":1349,"tokens_out":555,"duration_ms":31481,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"Maity and Barman define two new eta-quotient families, K4 from weight-1/2 Jacobi theta functions and K5 from their cubic analogues. They use these to write the Fourier coefficients of some weight-two and weight-four Hecke eigenforms in terms of finite-field period functions and then produce identities that relate those coefficients to special values of the finite-field Appell series F1^p and F2^p. The work is framed as an explicit extension of the EHMM framework from Allen et al., with the new families supplying the concrete link rather than just invoking the general method.","headline":"Maity and Barman define two new eta-quotient families K4 and K5 that give explicit hypergeometric expressions for coefficients of certain weight-2 and weight-4 Hecke eigenforms plus identities with finite-field Appell series, but the transformation properties of those families are the part that needs the closest check.","tokens_in":2376,"tokens_out":229,"would_cite":false,"duration_ms":35366,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":{"model":"grok-4.3","evidence":[{"relation":"unclear","rs_module":"IndisputableMonolith/Foundation/RealityFromDistinction.lean","rs_theorem":"reality_from_one_distinction","paper_passage":"We construct two explicit families of eta-quotients, which we call the K4 and K5 functions, from the hypergeometric background... express the Fourier coefficients of certain Hecke eigenforms of weight two and four in terms of finite field period functions."},{"relation":"unclear","rs_module":"IndisputableMonolith/Cost/FunctionalEquation.lean","rs_theorem":"washburn_uniqueness_aczel","paper_passage":"K4(r)(τ) := η(2τ)^{24r−8} η(τ)^{−24r+16} ... K5(r)(τ) := η(3τ)^{12r−2} η(τ)^{−12r+6}"}],"headline":"Number-theoretic constructions of eta-quotients and hypergeometric modularity for Hecke eigenforms bear no structural relation to RS forcing from distinction.","alignment":"orthogonal","rationale":"The paper develops explicit families K4 and K5 of eta-quotients from hypergeometric data HDK4(r) and HDK5(r), proves identities linking their Fourier coefficients to finite-field period functions P(HD;1;p) and Appell series F1^p/F2^p, and invokes the EHMM of Allen et al. to obtain Hecke eigenforms. None of its central objects (rising factorials, Jacobi sums, Gross-Koblitz, Euler integrals, Schwarz maps to Hauptmoduls for Γ0(2)/Γ0(3), or Galois families of eta-quotients) appear in the RS chain. RS theorems such as reality_from_one_distinction, Jcost uniqueness via Aczél, AlexanderDuality (D=3), ArithmeticFromLogic (LogicNat ≃ Nat), or AlphaCoordinateFixation have no counterpart here; the paper is a conventional calculation inside arithmetic geometry and modular forms.","tokens_in":56543,"confidence":"high","tokens_out":462,"duration_ms":16588,"cache_read_input_tokens":128,"cache_creation_input_tokens":0},"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"Certain Hecke eigenforms of weights two and four have their Fourier coefficients expressed in terms of finite field period functions using new eta-quotient families.","keywords":["eta-quotients","Hecke eigenforms","hypergeometric modularity","finite field period functions","Appell series","modular forms","weight two","weight four"],"falsifier":"A direct calculation for one of the Hecke eigenforms where the Fourier coefficient at a prime does not agree with the value computed from the corresponding finite field period function.","tokens_in":2600,"feed_emoji":"📐","tokens_out":697,"duration_ms":46026,"temperature":0.7,"pith_summary":"The paper builds two families of eta-quotients called the K4 and K5 functions from hypergeometric data, using weight 1/2 Jacobi theta functions and cubic analogues. These families then allow the Fourier coefficients of specific Hecke eigenforms in weights two and four to be written as finite field period functions. As a result, the work produces new identities that relate those coefficients directly to special values of the finite field Appell series F1^p and F2^p. A reader would care because this gives concrete, explicit ways to connect classical modular forms with hypergeometric objects defined over finite fields.","feed_headline":"Eta quotients link Hecke forms to finite field periods","feed_subtitle":"K4 and K5 families give explicit expressions for coefficients of weight two and four eigenforms and new Appell series identities.","key_machinery":"The K4 and K5 eta-quotient families constructed using weight 1/2 Jacobi theta functions and cubic analogues, serving as the explicit link from hypergeometric modularity to the Fourier coefficients of the eigenforms.","core_discovery":"Using constructions of the K4 and K5 eta-quotient families from the hypergeometric background and the theory of weight 1/2 Jacobi theta functions and their cubic analogues, the Fourier coefficients of certain Hecke eigenforms of weight two and four are expressed in terms of finite field period functions, yielding new identities relating the Fourier coefficients of modular forms to special values of the finite field Appell series F1^p and F2^p.","pith_inferences":["These expressions may enable more efficient computation of Fourier coefficients using algorithms from finite fields.","The method could be adapted to other classes of modular forms or Galois representations in higher dimensions.","Verifying the identities numerically for several small primes would test their validity in concrete cases.","Links to broader theories of hypergeometric functions in finite characteristic might be uncovered."],"forward_implications":["The Fourier coefficients of certain weight two Hecke eigenforms equal finite field period functions.","The Fourier coefficients of certain weight four Hecke eigenforms equal finite field period functions.","Identities are obtained that relate modular form coefficients to special values of F1^p and F2^p.","The Explicit Hypergeometric Modularity Method produces explicit eta-quotient representations for these forms."],"fun_headline_variants":["K4 and K5 eta quotients express eigenform coefficients as finite field periods","Hypergeometric background gives explicit modularity for weight two four Hecke forms","New identities relate eigenform coefficients to finite field Appell series","Explicit modularity of Hecke eigenforms via K4 and K5 eta quotients"],"cache_read_input_tokens":64,"weakest_assumption_plain":"The K4 and K5 eta-quotient families satisfy the precise transformation properties and modularity conditions needed to match the Fourier coefficients of the target Hecke eigenforms.","fun_headline_variants_meta":{"raw":{"variants":["K4 and K5 eta quotients express eigenform coefficients as finite field periods","Hypergeometric background gives explicit modularity for weight two four Hecke forms","New identities relate eigenform coefficients to finite field Appell series","Explicit modularity of Hecke eigenforms via K4 and K5 eta quotients"]},"model":"grok-4.3","cost_usd":0.009939,"raw_usage":{"total_tokens":4396,"prompt_tokens":626,"num_sources_used":0,"completion_tokens":80,"cost_in_usd_ticks":99387000,"prompt_tokens_details":{"text_tokens":626,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":3690,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":626,"tokens_out":80,"duration_ms":37190,"temperature":1.0,"reasoning_tokens":3690,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-05-13T19:05:28.241681+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"A direct calculation for one of the Hecke eigenforms where the Fourier coefficient at a prime does not agree with the value computed from the corresponding finite field period function.","supporting_citations":[],"review_version":1}