{"id":"37dd8b95-4374-42dd-a7d2-2027433297e5","arxiv_id":"2411.18419","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For many families of Hecke operators, the second coefficient of the characteristic polynomial never repeats, and this non-repetition yields a conditional uniqueness theorem for Fourier coefficients of modular forms.","lead":"This paper studies the second coefficients of characteristic polynomials of Hecke operators on spaces of modular forms, and proves that these coefficients do not repeat in several natural families. The main results are used to give a new conditional proof that modular forms of level one are determined by their 2nd or 4th Fourier coefficients.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 1.1 and 1.2 depend on the unproven numerical bounds (3.10); without a rigorous derivation of the constants 4.862 and 8.447, the large-N thresholds do not follow.","rationale":"The reader's conditional verdict is centered on the unproven numerical bounds (3.10) and reliance on the GitHub repository [9]. My read agrees: the decisive soft spot is (3.10). Theorems 1.1 and 1.2 are proved by bounding a difference of a2 values by a positive factor times a term that is negative only because E(N) is bounded using (3.10). The constants 4.862 and 8.447 are asserted without proof, and the 'smaller constants' claim is tied to an unpinned repository. Everything downstream—the universal thresholds N >= 3,392,663 and N >= 332,427, and hence the reduction to finitely many computer checks—depends on these constants. A wrong constant would not necessarily destroy the theorem, but it would change the thresholds and could make the computer verification miss a range. This is a missing proof, not an internal inconsistency, so the appropriate verdict remains conditional. I also note the paper's finite checks are only described, but I do not see a deeper flaw in the analytic bounding once (3.10) is granted.","tokens_in":18498,"tokens_out":9668,"duration_ms":77967,"concrete_test":"Independently re-derive the bounds (3.10) from [8, Lemma 2.4] with all constants tracked, and recompute the E(N) bound in (3.9) and the resulting thresholds. As a sanity check, compute M1 = max_{n<=10^7} 2^omega(n)/n^(1/4) and M2 = max_{n<=10^7} sigma_0(n)/n^(1/4). If M1 > 4.862 or M2 > 8.447, or if the analytic derivation yields larger constants, then the thresholds in Theorems 1.1 and 1.2 must be increased; if the increased thresholds lie outside the range checked by the code in [9], the theorems are not proved for the intervening N.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The pivotal step is (3.10): the paper states 2^omega(N) <= 4.862 N^(1/4) and sigma_0(N) <= 8.447 N^(1/4), claiming they follow by 'an identical argument as in [8, Lemma 2.4]' with 'smaller constants' from the GitHub repository [9]. These constants are substituted into E(N) in (3.9) to obtain E(N) < 11/8 for N >= 3,392,663, which is exactly the inequality making the difference (3.8) negative for all k. Theorem 1.2 uses the same bounds to get E(N) < 15/192 for N >= 332,427. The paper contains no proof of (3.10), and the 'smaller constants' claim is tied to an unpinned repository without a commit hash. If the true suprema of 2^omega(N)/N^(1/4) or sigma_0(N)/N^(1/4) exceed these constants, the thresholds grow and the finite computer verification described in [9] may no longer cover the transitional range. This is a missing proof, not an internal contradiction, but it is load-bearing because without (3.10) the strict-decrease/no-repeat claims are not established.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the second coefficient a2(T_m(N,2k)) of the characteristic polynomial of the Hecke operator T_m(N,2k) and proves non-repetition results in three aspects: vertical (weight aspect, Theorems 1.1 and 1.2), horizontal (level one, m=2 vs m=3, Theorem 1.3), and level aspect (m=2, prime levels, k≥58, Theorem 1.4). As an application, Theorem 1.5 extends Vilardi and Xue's result on distinguishing normalized Hecke eigenforms of level one by their m-th Fourier coefficient to m=2 and m=4, conditional on the irreducibility of the relevant Hecke characteristic polynomials. The proofs combine the Eichler-Selberg trace formula, a dimension formula, elementary bounds on divisor functions, and finite computer verification for ranges below explicit thresholds.","tokens_in":18698,"tokens_out":10720,"duration_ms":76870,"significance":"If the results hold, the non-repetition statements for a2 are new and go beyond the known trace non-repetition result of Chiriac and Jorza. The paper gives explicit, quantitative thresholds and a clear framework based on a2 rather than the trace, which the authors argue generalizes more easily to arbitrary m. Theorem 1.5 provides a conditional extension of the eigenform-distinguishing theorem from m=2,3 to m=2,4. However, two load-bearing issues currently prevent acceptance: the numerical bounds in (3.10) are asserted without proof, and an algebraic step in the proof of Theorem 1.2 appears to contain a sign/direction error. These issues affect the validity of the thresholds on which the finite verifications rest.","major_comments":[{"comment":"The bounds 2^omega(N) <= 4.862 N^(1/4) and sigma_0(N) <= 8.447 N^(1/4) are asserted without proof, attributed to [8, Lemma 2.4] and [2, Lemma 4.2] with 'smaller constants' from the GitHub repository [9]. These constants are load-bearing: they are substituted into E(N) in (3.9) to obtain E(N) < 11/8 for N >= 3,392,663, which makes the difference (3.8) negative, and they are used again in Section 4 to obtain the threshold N >= 332,427. The paper needs to provide a rigorous derivation of these exact constants, or a precise reference stating them, rather than an appeal to 'an identical argument.' Without this, the large-N part of Theorems 1.1 and 1.2 is incomplete.","section":"§3, Eq. (3.10)"},{"comment":"The displayed chain of inequalities in (4.3) contains an algebraic error. The second inequality claims that the left-hand side is at least (2k+1)/2 * [ (10k+5)/192 - 16/12 |E_{k+1}| - 1/12 |E_k| - (1/3)E_k^2 ], but multiplying the bracket by (2k+1)/2 gives negative coefficients -(2/3)(2k+1)|E_{k+1}| and -(1/24)(2k+1)|E_k|, which are smaller in magnitude than the corresponding negative terms on the left. Since these terms are negative, the inequality has the wrong direction for large |E_{k+1}|. The correct coefficients should be 8/3 and 1/6, respectively, to match after multiplication. As written, the derived bound for E(N) in (4.7) is too small, and the threshold N >= 332,427 is not justified. Please correct the algebra and recompute the threshold and the finite verification range.","section":"§4, Eq. (4.3)"},{"comment":"The proofs of Theorems 1.1, 1.2, 1.3, and 1.5 rely on finite computer checks whose results are recorded only in the GitHub repository [9], which is cited without a commit hash or a versioned archive. Because these checks cover the transitional ranges below the thresholds, their correctness is as load-bearing as the analytic bounds. The paper should specify the exact ranges verified (for example, the values of k_N for each N), include the code as an ancillary file or a permanent archival DOI, and pin the repository version. This is necessary for reproducibility and for the proof to be complete.","section":"§3, §4, §5, §7 (computer verification, [9])"}],"minor_comments":[{"comment":"The title page has a typo: 'POL YNOMIALS' should be 'POLYNOMIALS'.","section":"Title"},{"comment":"In the sentence before the computer-verification step, 'This verifies that a2(T2(N, 2k)) is strictly increasing for k >= k_N' should refer to a2(T4(N, 2k)), not T2.","section":"§4, end of proof of Theorem 1.2"},{"comment":"In the displayed equation for a2(T2(p,2k)), the term 'TrT2(p, 2k))^2' has an extra parenthesis; it should be '(TrT2(p, 2k))^2'.","section":"§6, proof of Theorem 1.4"},{"comment":"The notation 'Egv_2k' is used without an explicit definition; it should be defined as the set (or multiset) of eigenvalues of T_m(1,2k), counted with multiplicity.","section":"§7"},{"comment":"Reference [9] should include a version identifier (commit hash) or an archival DOI, since it is used as a load-bearing part of the proofs.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"The paper is within the scope of the journal and the main idea is promising, but the proof of Theorem 1.2 contains an algebraic error that affects the central claim, and the large-N estimates rest on unproved numerical constants. The authors should also be encouraged to make the repository code and the exact verification ranges permanently available. Given the self-citations to prior and forthcoming papers for key auxiliary bounds, the editor may wish to ensure that those bounds are independently verified or that the relevant arguments are included in this manuscript."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThe paper does something genuinely new: it proves non-repetition for the second coefficient a2 of Hecke polynomials in the weight, level, and index aspects, and it extends the Vilardi–Xue eigenform-distinguishing result to m = 2, 4 conditionally. Theorems 1.1 and 1.2 are the main events. The proofs are concrete: Eichler–Selberg trace formula, explicit bounds on every term, then finite computer checks. That is the right way to do this kind of extremal problem. The results appear new, and the asymptotic formula (4.8) nicely explains why square and non-square m behave differently.\n\nThe soft spots are real but not fatal. The large-N thresholds in Theorems 1.1 and 1.2 rest on the numerical bounds (3.10), 2^ω(N) ≤ 4.862 N^{1/4} and σ0(N) ≤ 8.447 N^{1/4}, which the paper does not prove. The text says 'an identical argument as in [8, Lemma 2.4]' with 'smaller constants' from a GitHub repository with no commit hash. Those constants are load-bearing: if they are wrong, the thresholds grow and the finite verifications in [9] may not cover the transitional range. The authors cite their own earlier work for several auxiliary estimates; that is legitimate if the cited results are correct, but it makes the paper not self-contained.\n\nThere are also small presentation slips—e.g., in the proof of Theorem 1.2 the text says 'This verifies that a2(T2(N, 2k)) is strictly increasing' where it should be T4, and Lemma 5.3 has a repeated bound with a typo ('2|P2k(1,4)|' twice). These are cosmetic.\n\nThe proof of Theorem 1.3 is clean and independent of the suspect constants. Theorem 1.4 appears solid. Theorem 1.5's conditional argument is straightforward once the non-repetition is in hand.\n\nVerdict: conditional, exactly as the reader said. The core ideas are good, the argument is structured properly, and I would bet the theorems are true. The missing derivation of (3.10) should be supplied before publication; ideally the authors include a proof of those elementary bounds in the paper, or at least pin down the repository with a commit hash and a verification script.\n\nThis paper deserves a serious referee. The referee's main job is to check the elementary but load-bearing inequalities in (3.10) and the computer code. If those check out, the paper is publishable as is.\n\nI'd bring it to a reading group focused on modular forms and Hecke operators; otherwise it is a specialized but honest piece.","headline":"New non-repetition results for a2 of Hecke polynomials, built on explicit trace bounds; the main theorems are plausible but hinge on unproven numerical constants and an unpinned computer verification.","tokens_in":19301,"tokens_out":2426,"would_cite":true,"duration_ms":19731,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11F25","11F72","11F11"],"pacs":[],"model":"deepseek-v4-flash","headline":"Hecke polynomial coefficients never repeat as the weight grows","keywords":["Hecke operator","Hecke polynomial","second coefficient","Eichler-Selberg trace formula","non-repetition","Hecke eigenforms","Fourier coefficients","level one modular forms"],"falsifier":"Compute $a_2(T_2(N,2k))$ for an odd $N$ and a range of $k$ where $s(N,2k)\\ge 2$: a single repeated value, or one pair with $k_1<k_2$ and $a_2(T_2(N,2k_1))\\le a_2(T_2(N,2k_2))$, would falsify Theorem 1.1. Alternatively, exhibit an odd $N$ with $2^{\\omega(N)}>4.862\\,N^{1/4}$ or $\\sigma_0(N)>8.447\\,N^{1/4}$, since the large-$N$ reduction depends on those inequalities.","tokens_in":18251,"feed_emoji":"🔢","tokens_out":6429,"duration_ms":46235,"temperature":0.7,"pith_summary":"This paper studies the second coefficient $a_2$ of the characteristic polynomial of the $m$-th Hecke operator $T_m(N,2k)$ on cusp forms of weight $2k$ and level $N$. Its central claim is that this coefficient does not take repeated values as $k$ varies: for any fixed odd level $N$, $a_2(T_2(N,2k))$ is strictly decreasing in $k$, and $a_2(T_4(N,2k))$ never repeats. It also proves a horizontal comparison at level one, $a_2(T_3(1,2k))<a_2(T_2(1,2k))$ for $k\\ge 12$, $k\\ne 13$, and a level-aspect inequality for prime levels when $k\\ge 58$. These non-repetition results are then applied to show that normalized Hecke eigenforms of level one are equal exactly when their second or fourth Fourier coefficients agree, assuming the relevant Hecke characteristic polynomials are irreducible. A sympathetic reader would care because this makes a quantitative, coefficient-level statement about the rigidity of Hecke data that had previously been known only for traces or under stronger conjectures.","feed_headline":"Second Hecke coefficients never repeat as weight grows","feed_subtitle":"For fixed odd level, the second coefficient never repeats; eigenforms are distinguishable by one Fourier coefficient.","key_machinery":"The load-bearing identity is the elementary relation $a_2(T_m(N,2k)) = \\frac12\\big((\\operatorname{Tr} T_m(N,2k))^2 - \\sum_{d\\mid m} d^{2k-1}\\operatorname{Tr} T_{m^2/d^2}(N,2k)\\big)$, which follows from the Hecke operator composition formula. The paper combines this with the Eichler–Selberg trace formula for the traces, and with the dimension formula for $s(N,2k)$, to express the difference $a_2(T_m(N,2k+2))-a_2(T_m(N,2k))$ as a main term plus error terms $\\theta_i(N)$ built from $2^{\\omega(N)}$, $\\sigma_0(N)$, and $\\psi(N)$. Explicit numerical bounds $2^{\\omega(N)}\\le 4.862\\,N^{1/4}$ and $\\sigma_0(N)\\le 8.447\\,N^{1/4}$ are used to show that the error term $E(N)$ is smaller than the main term for all $N$ beyond a finite threshold; the finitely many remaining pairs $(N,k)$ are checked by computer. For the horizontal and level aspects, the same machinery is specialized to level one and to prime levels, using Hurwitz class numbers and sharper bounds on $P_{2k}(t,m)$.","core_discovery":"The paper's central discovery is that the second coefficient $a_2$ of the Hecke polynomial is injective in the weight aspect in the two cases $m=2$ and $m=4$, for every fixed odd level $N$. For $m=2$ it proves the stronger monotonicity: $a_2(T_2(N,2k))$ strictly decreases as $k$ increases. For $m=4$ it proves non-repetition without monotonicity for small weights; the values are eventually increasing. The same coefficient also separates parameters in the other directions: at level one, $a_2(T_3(1,2k))<a_2(T_2(1,2k))$ for all $k\\ge 12$, $k\\ne 13$, and for fixed weight $k\\ge 58$ the value $a_2(T_2(p,2k))$ is strictly larger for larger odd primes $p$. The distinguishing application is that two normalized Hecke eigenforms of level one coincide if and only if their $m$-th Fourier coefficients coincide for $m=2$ or $m=4$, conditional on irreducibility of the relevant Hecke characteristic polynomials.","pith_inferences":["If the asymptotic formula (4.8) holds uniformly enough, the distinction between square and non-square $m$ suggests a general pattern: $a_2(T_m(N,2k))$ is eventually monotone decreasing for non-square $m$ and eventually increasing for square $m$, which would give a proof of the paper's Conjecture 4.1 for every $m$ with the same finite-check structure.","The same coefficient-level method should extend to the even-indexed coefficients $a_{2j}(T_m)$ once explicit asymptotic bounds are known; the non-repetition of these coefficients would yield distinguishing statements for higher Fourier coefficients of eigenforms without needing non-repetition of traces.","A testable extension is to compute $a_2(T_m(1,2k))$ for $m=5,6,7,\\ldots$ at small $k$ and check that no repetitions occur once $k$ passes the computed threshold, which would give numerical evidence for the paper's conjecture in the horizontal aspect as well."],"forward_implications":["For every fixed odd level $N$, the value $a_2(T_2(N,2k))$ is attained exactly once as $k$ ranges over weights with $s(N,2k)\\ge 2$; the same holds for $a_2(T_4(N,2k))$.","At level one, $a_2$ distinguishes $T_2$ from $T_3$ for all $k\\ge 12$, $k\\ne 13$: $a_2(T_3(1,2k))<a_2(T_2(1,2k))$.","For fixed weight $k\\ge 58$, different odd prime levels give different values of $a_2(T_2(p,2k))$, ordered by the size of the prime.","Assuming irreducibility of Hecke characteristic polynomials, normalized level-one Hecke eigenforms are determined by their second or fourth Fourier coefficient; this verifies the case $m=2,4$ of the relevant conjecture under that assumption.","The same proof strategy applies to any fixed $m\\ge 2$: for each $m$, $a_2(T_m(N,2k))$ is non-repeating for all sufficiently large $k+N$, with only finitely many computer checks remaining."],"supporting_citations":[{"why":"Supplies the expression for $a_2$ in terms of traces and the trace bounds used throughout Sections 3 and 4.","marker":"[2]"},{"why":"Provides the Eichler–Selberg trace formula that converts the trace terms into computable main terms and errors.","marker":"[5]"},{"why":"Gives the lemma whose argument yields the explicit bound $2^{\\omega(N)}\\le 4.862\\,N^{1/4}$ used for the large-$N$ reduction.","marker":"[8]"},{"why":"Contains the computer code verifying the finitely many remaining cases in the proofs of Theorems 1.1, 1.2, and 1.3.","marker":"[9]"},{"why":"Provides the detailed bounding steps and the asymptotic formula for $a_2(T_m(N,2k))$ used in the $m=4$ case and in the discussion for general $m$.","marker":"[10]"},{"why":"The result on distinguishing level-one eigenforms by the second Fourier coefficient that Theorem 1.5 extends to $m=2,4$.","marker":"[14]"},{"why":"States the conjecture on uniqueness of Fourier coefficients whose $m=2,4$ cases are verified here under the irreducibility assumption.","marker":"[15]"},{"why":"Supplies the level-one trace formula with Hurwitz class numbers used in Section 5 for the comparison $a_2(T_3)<a_2(T_2)$.","marker":"[16]"}],"fun_headline_variants":["Hecke second coefficient injective in weight aspect","One Fourier coefficient separates Hecke eigenforms","All aspects: second Hecke coefficient never repeats","Hecke eigenforms told apart by one coefficient"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof stands on two numerical pillars: the explicit bounds $2^{\\omega(N)}\\le 4.862\\,N^{1/4}$ and $\\sigma_0(N)\\le 8.447\\,N^{1/4}$, which are invoked from the literature and not proved here, and the correctness of the computer verifications in the accompanying repository for the finitely many remaining cases.","fun_headline_variants_meta":{"raw":{"variants":["Hecke second coefficient injective in weight aspect","One Fourier coefficient separates Hecke eigenforms","All aspects: second Hecke coefficient never repeats","Hecke eigenforms told apart by one coefficient"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001339,"raw_usage":{"total_tokens":5420,"prompt_tokens":901,"completion_tokens":4519,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":517,"completion_tokens_details":{"reasoning_tokens":4461}},"tokens_in":517,"tokens_out":4519,"duration_ms":27078,"temperature":1.0,"reasoning_tokens":4461,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T11:13:37.105333+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute $a_2(T_2(N,2k))$ for an odd $N$ and a range of $k$ where $s(N,2k)\\ge 2$: a single repeated value, or one pair with $k_1<k_2$ and $a_2(T_2(N,2k_1))\\le a_2(T_2(N,2k_2))$, would falsify Theorem 1.1. Alternatively, exhibit an odd $N$ with $2^{\\omega(N)}>4.862\\,N^{1/4}$ or $\\sigma_0(N)>8.447\\,N^{1/4}$, since the large-$N$ reduction depends on those inequalities.","supporting_citations":[{"cited_title":"Nonvanishing of second coefficients of Hecke polynomials","cited_arxiv_id":null,"evidence_quote":"Supplies the expression for $a_2$ in terms of traces and the trace bounds used throughout Sections 3 and 4."},{"cited_title":"Traces of Hecke operators, volume 133 of Mathematical Surveys and Mono- graphs","cited_arxiv_id":null,"evidence_quote":"Provides the Eichler–Selberg trace formula that converts the trace terms into computable main terms and errors."},{"cited_title":"Newspaces with nebentypus: An explicit dimension formula and classification of trivial newspaces","cited_arxiv_id":null,"evidence_quote":"Gives the lemma whose argument yields the explicit bound $2^{\\omega(N)}\\le 4.862\\,N^{1/4}$ used for the large-$N$ reduction."},{"cited_title":"Non-repetition a2 coefficient","cited_arxiv_id":null,"evidence_quote":"Contains the computer code verifying the finitely many remaining cases in the proofs of Theorems 1.1, 1.2, and 1.3."},{"cited_title":"Signs of the second coefficients of Hecke polynomials","cited_arxiv_id":null,"evidence_quote":"Provides the detailed bounding steps and the asymptotic formula for $a_2(T_m(N,2k))$ used in the $m=4$ case and in the discussion for general $m$."},{"cited_title":"Distinguishing eigenforms of level one","cited_arxiv_id":null,"evidence_quote":"The result on distinguishing level-one eigenforms by the second Fourier coefficient that Theorem 1.5 extends to $m=2,4$."},{"cited_title":"Uniqueness of Fourier coefficients of eigenforms","cited_arxiv_id":null,"evidence_quote":"States the conjecture on uniqueness of Fourier coefficients whose $m=2,4$ cases are verified here under the irreducibility assumption."},{"cited_title":"Traces des op´ erateurs de Hecke","cited_arxiv_id":null,"evidence_quote":"Supplies the level-one trace formula with Hurwitz class numbers used in Section 5 for the comparison $a_2(T_3)<a_2(T_2)$."}],"review_version":1}