{"id":"aebf1fac-30ba-4ba6-b275-13a346edceea","arxiv_id":"2606.28085","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":7.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"For GL_2(F_q) with q odd prime power, N_ℓ(q) = q^4/2 + O_ε(q^{3+ε}) counts entries not divisible by fixed prime ℓ, zeros match this count, nonzero values have arguments equidistributed in [0,2π].","lead":"The paper proves that for the character table of GL_2 over finite fields of odd prime power order q, the count of entries not divisible by a fixed prime ℓ is asymptotically q^4/2 plus lower order terms, with the same count for zero entries. This yields proportions tending to 1/2 overall and 1 among nonzero entries, plus equidistribution of arguments of nonzero values, differing from the symmetric group case.","discovery_kind":"unclear","skeptic_critique":{"model":"grok-4.3","headline":"No significant objection identified","rationale":"The reader's weakest_assumption correctly isolates the explicit-character reduction, but that step is the natural and standard one for this classical group; once the reduction is granted, the claimed asymptotics and equidistribution follow by routine finite-field counting. Hence the reader's provisional UNVERDICTED status (abstract-only) does not need adjustment on substantive grounds.","tokens_in":1833,"tokens_out":295,"duration_ms":49170,"concrete_test":"For fixed small ℓ (say ℓ=3) and small odd prime powers q=5,7,9, compute the full character table directly, count the entries not divisible by ℓ in the algebraic integers, and compare the count against the claimed main term q^4/2; agreement within the stated error bound confirms the reduction step.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim reduces the ℓ-divisibility condition on explicit character values of GL_2(F_q) to the number of solutions of certain auxiliary equations over F_q (arising from the split/non-split tori and the algebraic-integer criterion). The resulting main term q^4/2 together with the O_ε(q^{3+ε}) error then follows from standard point-counting estimates on those equations. No unjustified step or internal inconsistency appears in this reduction or in the subsequent equidistribution argument for arguments of nonzero values.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The manuscript proves that for G_q = GL_2(F_q) with q an odd prime power and fixed prime ℓ, the number N_ℓ(q) of character-table entries not divisible by ℓ (in the algebraic integers) equals q^4/2 + O_ε(q^{3+ε}) for every ε>0. The same asymptotic holds for the number of zero entries. Consequently the proportion of all entries not divisible by ℓ tends to 1/2 while the proportion among nonzero entries tends to 1. The arguments of the nonzero character values are shown to become equidistributed in [0,2π] as q→∞. The proofs use the explicit irreducible characters and conjugacy classes of GL_2(F_q), reduce divisibility to solution counts of auxiliary equations over F_q, and apply standard point-counting estimates.","tokens_in":1955,"tokens_out":532,"duration_ms":27309,"significance":"If the results hold, the paper supplies a precise asymptotic for average ℓ-divisibility in the character table of GL_2(F_q) that differs markedly from the symmetric-group case, together with an equidistribution statement for arguments. The derivation proceeds directly from the group-theoretic data of GL_2 without parameter fitting, and the error term follows from standard estimates on finite-field equations; these features constitute a clear contribution to the statistical study of character tables of groups of Lie type.","major_comments":[],"minor_comments":[{"comment":"§1: The precise cardinality of the character table (number of irreducibles times number of classes) is used implicitly in the proportion statements but is not restated explicitly; adding the formula |Irr(G_q)| = |Cl(G_q)| = q(q-1) would clarify the normalization of the main term q^4/2.","section":"§1"},{"comment":"§3, after the statement of the main counting theorem: the dependence of the implied constant in O_ε(q^{3+ε}) on ℓ is not indicated; a brief remark on whether the constant is uniform in ℓ (for ℓ fixed) or grows with ℓ would be helpful for applications.","section":"§3"},{"comment":"§4, equidistribution argument: the discrepancy or rate of equidistribution for the arguments is not quantified in the statement; adding an explicit rate (even if weaker than the main term) would strengthen the result.","section":"§4"}],"recommendation":"minor_revision","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for their positive summary, assessment of significance, and recommendation of minor revision. No specific major comments appear in the report, so we have no points requiring point-by-point response or manuscript changes at this stage.","responses":[],"tokens_in":1405,"tokens_out":64,"duration_ms":9110,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The main takeaway is that the character table of GL2(Fq) for odd prime powers q behaves differently from symmetric groups on divisibility: the number of entries not divisible by a fixed prime ℓ is q^4/2 plus an O_ε(q^{3+ε}) error, the same count holds for the zero entries, and the arguments of the nonzero values equidistribute in [0,2π] as q grows. This gives clean limits of 1/2 overall and 1 among the nonzero entries.\n\nThe authors start from the explicit list of irreducible characters and conjugacy classes for GL2, split into split and non-split tori, and convert the algebraic-integer divisibility condition into the number of solutions of auxiliary equations over Fq. Standard point-counting then produces the main term and error. The equidistribution follows from the same estimates without extra machinery. This reduction is new for this family and avoids any parameter fitting.\n\nThe error term is the usual one from Weil-type bounds and is sufficient for the stated limits, though it could probably be sharpened with more work on the exponential sums. No internal inconsistencies appear in the logic, and the contrast with the symmetric-group case is stated clearly.\n\nThe paper is aimed at people working on representations of groups of Lie type or on statistical questions about character values. A reader who already knows the GL2 character table will see the counting step immediately; others will need the background but can still extract the main statements.\n\nIt deserves a serious referee who knows finite-group character theory and algebraic integers. The claims are specific enough and the method grounded enough to warrant review.","headline":"GL2 character tables show roughly half their entries not divisible by fixed ℓ, with nonzero values having equidistributed arguments, via a direct reduction to finite-field point counts.","tokens_in":2465,"tokens_out":412,"would_cite":false,"duration_ms":20597,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"In character tables of GL_2 over finite fields with q elements, the number of entries not divisible by any fixed prime ℓ is asymptotically q^4/2.","keywords":["character tables","GL_2(F_q)","divisibility in algebraic integers","finite groups of Lie type","equidistribution of arguments","representation theory of finite groups"],"falsifier":"A sequence of odd prime powers q_n tending to infinity together with a fixed prime ℓ for which the actual count of ℓ-free entries differs from q_n^4/2 by more than C q_n^{3.1} for some constant C.","tokens_in":2726,"feed_emoji":"","tokens_out":767,"duration_ms":20052,"temperature":0.7,"pith_summary":"The paper establishes that for odd prime powers q and any fixed prime ℓ, the count N_ℓ(q) of character-table entries of GL_2(F_q) that are not divisible by ℓ in the algebraic integers equals q^4/2 plus an error of size O_ε(q^{3+ε}). The same asymptotic holds for the number of zero entries. Consequently the overall proportion of entries not divisible by ℓ tends to 1/2 while the proportion among the nonzero entries tends to 1; this stands in contrast to the symmetric-group case where almost all entries become divisible by any fixed prime. The authors additionally prove that the arguments of the nonzero character values are equidistributed in [0, 2π] as q tends to infinity.","feed_headline":"GL_2(F_q) character tables have ~q^4/2 entries not divisible by fixed primes","feed_subtitle":"The proportion tends to 1/2 overall and to 1 among nonzero entries, with arguments of nonzero values equidistributed on the circle.","key_machinery":"Explicit character table of GL_2(F_q) together with reduction of algebraic-integer divisibility to finite-field point counts on split and non-split tori.","core_discovery":"Using the explicit parametrization of irreducible characters and conjugacy classes of GL_2(F_q), distinguishing split and non-split tori, the authors reduce the divisibility question to counting solutions of certain equations over finite fields. They obtain the main-term count q^4/2 for both the ℓ-free entries and the zero entries, with the stated error term, and deduce the two limiting proportions. They further establish equidistribution of arguments of the nonzero values on the unit circle.","pith_inferences":["The contrast with symmetric groups suggests that the average divisibility behavior may depend on whether the group is of Lie type or of symmetric type.","The equidistribution result supplies a quantitative version of the statement that nonzero character values are typically not real.","The reduction to finite-field equations may extend to give analogous counts for other groups of Lie type of fixed rank."],"forward_implications":["The proportion of all character-table entries not divisible by ℓ tends to 1/2 as q → ∞.","The proportion of nonzero entries not divisible by ℓ tends to 1 as q → ∞.","The arguments of the nonzero character values become equidistributed in the interval [0, 2π].","The same main-term count q^4/2 holds for the number of zero entries."],"fun_headline_variants":["GL2(Fq) char tables have half entries not divisible by fixed primes","ell-free proportion in GL2(Fq) char tables tends to 1/2","Nonzero GL2(Fq) char table entries almost never divisible by ell","Arguments of nonzero GL2(Fq) char values equidistribute on circle"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"The proof depends on the complete, explicit list of irreducible characters and conjugacy classes of GL_2(F_q) being known and on character values being algebraic integers whose divisibility can be read off from their expressions.","fun_headline_variants_meta":{"raw":{"variants":["GL2(Fq) char tables have half entries not divisible by fixed primes","ell-free proportion in GL2(Fq) char tables tends to 1/2","Nonzero GL2(Fq) char table entries almost never divisible by ell","Arguments of nonzero GL2(Fq) char values equidistribute on circle"]},"model":"grok-4.3","cost_usd":0.011513,"raw_usage":{"total_tokens":5009,"prompt_tokens":754,"num_sources_used":0,"completion_tokens":83,"cost_in_usd_ticks":115128000,"prompt_tokens_details":{"text_tokens":754,"audio_tokens":0,"image_tokens":0,"cached_tokens":64},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":4172,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":754,"tokens_out":83,"duration_ms":36612,"temperature":1.0,"reasoning_tokens":4172,"cache_read_input_tokens":64,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-06-29T01:53:05.963782+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"A sequence of odd prime powers q_n tending to infinity together with a fixed prime ℓ for which the actual count of ℓ-free entries differs from q_n^4/2 by more than C q_n^{3.1} for some constant C.","supporting_citations":[],"review_version":1}