REVIEW 3 major objections 6 minor 1 cited by
Murmurations using Petersson trace formula
T0 review · 3 major / 6 minor · reviewed 2026-08-06 · deepseek-v4-flash
Pith's one-line read The paper proves a weight-aspect murmuration for modular forms with an averaging window as narrow as K^{1/3+ε}, using the Petersson trace formula.
desk verdict A real step forward for the trace-formula approach to murmurations, but the advertised range M >> K^{1/3+eps} rests on an unproved weakening of Li's expansion, so the headline result is conditional; the paper still deserves refereeing. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The central objects are the Petersson trace formula, which expresses the average of Hecke eigenvalues over an orthogonal basis of Hecke eigenforms as a delta symbol plus a sum of Kloosterman sums weighted by J-Bessel functions; Proposition 2, which evaluates the sum of Bessel functions over even weights as oscillatory integrals V1 and V2, so that the sign factor i^k selects the appropriate oscillation; and the stationary-phase expansion (17) of V2, which after Mellin-transform and contour-shift produces the main term from a single residue at s=1. A decorrelation lemma for sums of Kloosterman sums over primes, valid under GRH for Dirichlet L-functions, suppresses the off-diagonal contributions.
What would settle it
Numerically test the ratio (4) for M = $K^{{1/3+0.01}}$ across a growing range of K, comparing the deviation from $K^{{-1}}$/(√B+√A) with the paper's predicted error size; if the deviation fails to shrink, the unproved extension of the stationary-phase expansion is the likely culprit.
Extended reading notes
Core claim
On its own terms, the paper establishes that the weight-aspect murmuration ratio follows from the Petersson trace formula: after the sign of the functional equation cancels the Bessel phase in the trace formula, the weighted average is governed by an oscillatory integral whose asymptotic expansion yields the main term for the numerator, while the denominator is computed by the diagonal term of the trace formula, producing the ratio $K^{{-1}}$/(√B+√A). The proof requires no GRH for the L-functions of the modular forms themselves, only GRH for Dirichlet L-functions, because the smooth averaging over weights replaces the sharp cutoff used in earlier work. The improvement in the window width, from a much larger power of K to M >> $K^{{1/3+ε}}$, is the concrete advance over earlier results.
Load-bearing premise
Everything rests on the claim that the cited stationary-phase expansion stays valid when the averaging window is as narrow as $K^{{1/3+ε}}$, even though the original source only proved it for windows down to $K^{{3/8+ε}}$.
Editorial extensions
If this is right
- The murmuration ratio for the weight aspect is now known for families as small as M ≍ K^{1/3+ε}, roughly a K^{2/3} reduction in window width compared to the previous best, so the phenomenon is visible on much smaller slices of the spectrum.
- The use of a relative trace formula opens the door to deriving murmurations for automorphic forms on higher-rank groups, where the Selberg trace formula is unwieldy or unavailable.
- No GRH for the L-functions of the forms themselves is needed once the weight summation is smooth, removing a hypothesis that earlier sharp-cutoff results required.
- The constants in the main term arise from a single residue of an L-series L(s) = Σ μ(c)^2/(φ(c) c^s), tying the shape of the murmuration to elementary Euler factors rather than to the fine statistics of the family.
Reading between the lines
- If the unproved narrowing of the averaging window is verified, the same machinery should push toward even narrower windows, and the boundary of validity would indicate the natural scale at which individual forms' coefficients decorrelate from their functional-equation signs.
- The relative trace formula route suggests a complementary level-aspect proof, where the same sign-cancellation mechanism could recover or improve the existing level-aspect murmuration results.
- Because the main term comes from a single Euler-product residue, the ratio K^{-1}/(√B+√A) may be a universal first-order shape for weight-aspect murmurations across different weightings, with the weighting affecting only lower-order terms.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper proves a weight-aspect murmuration phenomenon for holomorphic modular forms of level 1. Assuming GRH for Dirichlet L-functions, it establishes that the ratio of the sign-weighted to unweighted sums of Hecke eigenvalues, with a Gaussian weight of width M centered at weight K, is asymptotic to K^{-1}/(√B+√A) for K^{1/3+ε} ≪ M ≪ K^{1-ε}. The proof uses the Petersson trace formula, a Bessel-function summation identity (Proposition 2), a Kloosterman decorrelation lemma under GRH, and an L-series residue computation. The advertised range M ≫ K^{1/3+ε} improves on the earlier K^{5/6+ε} range of Bober et al. and is presented as the first relative-trace-formula proof of murmurations.
Significance. If the central technical gap is repaired, this is a substantial contribution: it gives the sharpest known weight-window for weight-aspect murmurations and demonstrates that the trace-formula method is flexible enough to recover and improve results obtained by the Selberg trace formula. The paper is also methodologically clean: it uses no numerical fitting, contains no circular argument, and reduces the needed hypotheses to GRH for Dirichlet L-functions. The final main term is explicit and matches the expected density. However, the advertised improvement over previous work rests on an unproved weakening of a cited expansion of Li, so the significance is conditional on that point being resolved.
major comments (3)
- [§3.3, Eq. (17)] The proof of Theorem 1 relies on expansion (17) being valid for M ≫ K^{1/3+ε}, while the cited source [6, Proposition A.3 (and Proposition 5.1)] states the condition M ≫ K^{3/8+ε}. The sentence 'examining the proof shows that only the condition K^{1/3+ε} ≤ M is necessary' is not a proof, and no derivation or reference is supplied. This is load-bearing because the error bound in (19) and the Mellin residue computation in (20) are evaluated in the smaller range. If the weakening is not justified, Theorem 1 is only established for M ≫ K^{3/8+ε}. Please provide a complete proof of the weakened condition or state Theorem 1 with the stronger condition.
- [§3.3.2, error term Σ3] The text derives the condition K^{1/3+2/(3(4+3L2))} ≪ M and then states that this is ensured by the assumption M ≥ K^{1/3+ε}. This is not true for an arbitrary choice of L2: one must choose L2 so that 2/(3(4+3L2)) < ε. As written, the proof does not establish the claimed range unless L2 is explicitly taken to depend on ε. This is fixable, but it should be stated in the proof.
- [§3.3.2, c-sum restriction] The claim that the c-sum in (19) is restricted to 4π√x/(100K) ≤ c ≤ 400π√x/K, hence to a constant range, is not proved. The validity interval stated for (17) gives 1/(100K) ≤ |x| ≤ 100K, and with |x| = 4π√x/c this only implies c ≤ 400πK√x, which is as large as O(K²), not O(1). If V2 is indeed negligible outside [K/100, 100K], the precise statement from [6, Proposition A.3] should be quoted or proved. This matters because after the change of variables in §3.4 the error terms contain a factor φ(c); a non-constant range for c would invalidate the bound O(K^{2+L2}M^{-(3L2+3)}).
minor comments (6)
- [§2, Proposition 1, Eq. (7)] The right-hand side of (7) uses \widehat h(0), but the proof computes the contribution using \widehat W(0) ≍ M/K and obtains M K²|E|. As written, (7) appears to be off by a factor of K; please clarify the normalization or replace \widehat h(0) by \widehat W(0).
- [§3.3, Eq. (17)] The notation V2^*(x) is used before it is defined. Please define it explicitly, for example as the one-sided integral appearing in the displayed formula preceding (17).
- [§3.1, Eq. (11)] The integral representation (11) is used for real Bessel order ℓ ≥ 1; it would be helpful to add a parenthetical remark that the relevant Schläfli-type representation is valid for this range of ℓ.
- [§3.4] The sentence 'we just need to look at a small interval x ∈ [−M^ε, M^ε]' should specify that this ε is a fixed small constant and how it relates to the ε in the theorem statement.
- [Throughout] There are several typographical issues: 'paremeters' in Theorem 1, 'Propostion' in the citation [3, Proposition 2.1], and some inconsistent spacing in set definitions such as 'E =[A,B]'.
- [References] Reference [4] (Iwaniec and Sarnak) is listed but not cited in the body of the paper; please cite it where relevant or remove it from the bibliography.
Circularity Check
No circularity found: the derivation is self-contained relative to external lemmas; the only concern is an unproved weakening of Li's expansion condition, which is a rigor gap, not a circular step.
full rationale
The claimed derivation chain is non-circular. Theorem 1 is obtained by evaluating the numerator (2) via the Petersson trace formula (8), the weight-summation identity of Proposition 2 (proved in the paper from the Bessel integral representation and Poisson summation), Lemma 1 (a Kloosterman decorrelation stated under GRH for Dirichlet L-functions), and Li's power-series expansion (17) cited from [6, Proposition A.3]; the denominator is evaluated in Proposition 1 using the same Petersson formula and the prime number theorem. None of these inputs is defined in terms of the target ratio, none is fitted to the data being predicted, and the cited results are external (Iwaniec-Luo-Sarnak, Li, Sarnak) rather than self-citations. The only load-bearing assumption that is not fully justified is in Section 3.3: the authors assert that Li's condition M >= K^{3/8+epsilon} can be weakened to M >= K^{1/3+epsilon} by 'examining the proof', with no derivation supplied. If that weakening fails, Theorem 1 would only be established for M >= K^{3/8+epsilon}, and even in the claimed range there is the further technical condition K^{1/3 + 2/(3(4+3L2))} << M noted in Section 3.3.2. This is a correctness or rigor gap, not a circular step: it does not reduce the theorem to its own input, and no self-citation is load-bearing. Hence the circularity score is 0.
Assumptions & free parameters
assumptions (4)
- domain assumption Generalized Riemann Hypothesis for Dirichlet L-functions
- standard math Petersson trace formula for GL(2) level one
- standard math Li's expansion for the Bessel sum S_a(x) ([6, Proposition A.3]) in its original range M >> K^{3/8+eps}
- ad hoc to paper Weakened version of Li's expansion valid for M >> K^{1/3+eps}
Cite this review
Pith. "Pith review of Murmurations using Petersson trace formula." pith.science (2026). https://pith.science/paper/APOURETO
@misc{pith2026250711418,
author = {Pith},
title = {Pith review of: Murmurations using Petersson trace formula},
year = {2026},
howpublished = {\url{https://pith.science/paper/APOURETO}},
note = {Machine review of arXiv:2507.11418}
}
read the original abstract
We prove the murmuration phenomenon, which is a correlation between signs of functional equations and Fourier coefficients, in the case of modular forms in the weight aspect. We in particular improve the range of visibility of murmurations compared to previous results. This is the first approach to the murmuration phenomenon using a relative trace formula, showing its robustness.
Forward citations
Cited by 1 Pith paper
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Joint level-weight murmurations: prime averaging and the cubic pointwise range
Unconditionally, joint averages of root-number-weighted prime traces of squarefree-level holomorphic newforms converge to an explicit atomic measure on rational squares for every K = X^ρ with 0 < ρ < 1.
Reference graph
Works this paper leans on
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[1]
R., Lee, M., and Lowry-Duda, D
Bober, J., Booker, A. R., Lee, M., and Lowry-Duda, D. Murmurations of modular forms in the weight aspect, Oct. 2023. arXiv:2310.07746 [math]
arXiv 2023
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[2]
Murmurations of Elliptic Curves
He, Y.-H., Lee, K.-H., Oliver, T., and Pozdnyakov, A. Murmurations of Elliptic Curves. Experimental Mathe- matics 0, 0 (Oct. 2023), 1–13. Publisher: Taylor & Francis _eprint: https://doi.org/10.1080/10586458.2024.2382361
arXiv 2023
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[3]
Publications mathématiques de l’IHÉS 91, 1 (2000), 55–131
Iwaniec, H., Luo, W., and Sarnak, P.Low lying zeros of families of L-functions. Publications mathématiques de l’IHÉS 91, 1 (2000), 55–131
work page 2000
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[4]
Perspectives on the Analytic Theory of L-functions
Iwaniec, H., and Sarnak, P. Perspectives on the Analytic Theory of L-functions. Geom. Funct. Anal (2000), 705–741
work page 2000
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[5]
Murmurations of Dirichlet characters
Lee, K.-H., Oliver, T., and Pozdnyakov, A. Murmurations of Dirichlet characters. International Mathematics Research Notices 2025, 1 (Jan. 2025), rnae277. arXiv:2307.00256 [math]
work page Pith review arXiv 2025
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[6]
Li, X. Bounds for $\rGL(3) \times \rGL(2)$ $\rL$-functions and $\rGL(3)$ $\rL$-functions.Annals of Mathematics 173, 1 (2011), 301–336
work page 2011
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[7]
On murmurations and root numbers
Sarnak, P. On murmurations and root numbers. Letter (2023)
work page 2023
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[8]
W., and Templier, N.Families of L-functions and their symmetries
Sarnak, P., Shin, S. W., and Templier, N.Families of L-functions and their symmetries. In Families of automor- phic forms and the trace formula , Simons Symp. Springer, 2016, pp. 531–578
work page 2016
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[9]
Murmurations, Oct
Zubrilina, N. Murmurations, Oct. 2023. arXiv:2310.07681 [math]. School of mathematics (Zhuhai) Zhuhai Campus, Sun Y at-Sen University Tangjiawan, Zhuhai, Guangdong, 519082, China (PRC) Email address: kuanchi3@mail.sysu.edu.cn University of Lille CNRS, UMR 8524 — Laboratoire Pa...
2023 arXiv
Reviewed August 6, 2026 · model on record in the stance chip above.
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