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REVIEW 3 major objections 5 minor 16 references

Non-repetition of second coefficients of Hecke polynomials

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Pith's one-line read Hecke polynomial coefficients never repeat as the weight grows

desk verdict 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. read the letter →

arxiv 2411.18419 v2 pith:ITBNYSXF submitted 2024-11-27 math.NT

Archer Clayton , Helen Dai , Tianyu Ni , Erick Ross , Hui Xue , Jake Zummo This is my paper · ORCID
classification math.NT MSC 11F2511F7211F11
keywords HeckeoperatorpolynomialsecondcoefficientEichler-Selbergtraceformulanon-repetitioneigenformsFouriercoefficientslevelonemodularforms
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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))

What carries the argument

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 $P_{2k}(t,m)$.

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • 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.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

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.

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 (3)
  1. [§3, Eq. (3.10)] 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.
  2. [§4, Eq. (4.3)] 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.
  3. [§3, §4, §5, §7 (computer verification, [9])] 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.
minor comments (5)
  1. [Title] The title page has a typo: 'POL YNOMIALS' should be 'POLYNOMIALS'.
  2. [§4, end of proof of Theorem 1.2] 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.
  3. [§6, proof of Theorem 1.4] 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'.
  4. [§7] 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.
  5. [References] 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.

Circularity Check

0 steps flagged · score 2.0 of 10

No circular reduction found; the core non-repetition theorems are derived from trace formulas and auxiliary numerical bounds, though several load-bearing estimates are delegated to overlapping self-citations and an unproven repository-backed inequality.

full rationale

The derivation is not circular. Proposition 2.1 is an algebraic identity expressing a2 as a polynomial in Hecke traces; Proposition 2.2 (Eichler-Selberg trace formula) and Proposition 2.3 (dimension formula) are standard external results. Theorem 1.1 reduces monotonicity to the explicit inequality (3.8), whose large-N negativity depends on the numerical bounds (3.10) for 2^{omega(N)} and sigma_0(N). Those bounds are auxiliary estimates about divisor-counting functions; they neither assert nor assume any non-repetition of a2, so their use is not a reduction of the conclusion to itself. Theorems 1.2, 1.3, and 1.4 follow the same pattern of explicit trace bounds plus finite computation, and Theorem 1.5 uses Theorems 1.1 and 1.2 together with an explicit irreducibility hypothesis rather than presupposing the equality statement it proves. I nonetheless flag a presentation gap: (3.10) is stated as following from [8, Lemma 2.4] / [2, Lemma 4.2] with constants in the co-author's GitHub repository [9], and the finite computer verifications for the remaining N and k are also delegated to [9]. These are load-bearing, but they are not circular because none of the cited inputs contains the non-repetition conclusion. The central claims are derived, not assumed, so the circularity score is low; I set 2 to reflect the unproven auxiliary constants and the heavy overlap of supporting citations, not to indicate a circular reduction.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The central claim does not introduce new entities or data-fitted parameters. It depends on standard trace formulas and dimension formulas, plus a set of explicit numerical bounds and computational checks that are sourced from the authors' own prior and forthcoming work and a public GitHub repository. The two most load-bearing imported items are the constants in (3.10) and the correctness of the finite computer checks.

assumptions (6)
  • standard math Eichler-Selberg trace formula for Tr T_m(N,2k) (Proposition 2.2)
    Used throughout to compute traces of Hecke operators; standard theorem from Knightly-Li and Serre, cited as [5] and [13].
  • standard math Dimension formula for s(N,2k) (Proposition 2.3)
    Cited from Cohen-Stromberg [3, Corollary 7.4.3]; used repeatedly for the dimension of the cusp form space.
  • domain assumption Numerical bounds 2^omega(N) <= 4.862 N^(1/4) and sigma0(N) <= 8.447 N^(1/4) in (3.10)
    Attributed to [8, Lemma 2.4] and [2, Lemma 4.2] with smaller constants, but not proven in this paper. These bounds determine the large-N thresholds in Theorems 1.1 and 1.2.
  • domain assumption External trace estimates from [2] and [10]
    Sections 3, 4, and 6 use specific bounds on Tr T2, Tr T4, and Tr T16 taken from the authors' previous and forthcoming works; the proofs of those bounds are not reproduced here.
  • ad hoc to paper Correctness of the computer verification code in [9]
    Theorems 1.1, 1.2, and 1.3 reduce finitely many cases to computations; the paper assumes the public GitHub repository performs these checks correctly.
  • domain assumption Irreducibility of Hecke characteristic polynomials in Theorem 1.5
    The eigenform-distinguishing theorem is explicitly conditional on the characteristic polynomials of the relevant Hecke operators being irreducible, as stated in the theorem.

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Pith. "Pith review of Non-repetition of second coefficients of Hecke polynomials." pith.science (2026). https://pith.science/paper/ITBNYSXF

@misc{pith2026241118419,
  author       = {Pith},
  title        = {Pith review of: Non-repetition of second coefficients of Hecke polynomials},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ITBNYSXF}},
  note         = {Machine review of arXiv:2411.18419}
}
abstract

Let $T_m(N,2k)$ denote the $m$-th Hecke operator on the space $S_{2k}(\Gamma_0(N))$ of cuspidal modular forms of weight $2k$ and level $N$. In this paper, we study the non-repetition of the second coefficient of the characteristic polynomial of $T_m(N,2k)$. We obtain results in the horizontal aspect (where $m$ varies), the vertical aspect (where $k$ varies), and the level aspect (where $N$ varies). Finally, we use these non-repetition results to extend a result of Vilardi and Xue on distinguishing Hecke eigenforms.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

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