On Murmurations and Trace Formulas

David Lowry-Duda

REVIEW 2 major objections 4 minor 28 references

On Murmurations and Trace Formulas

T0 review · 2 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read Murmuration patterns in L-functions can be proven systematically with trace formulas, and four new families are within reach.

desk verdict A useful, honest survey of trace-formula proofs of murmurations; no new theorems, and the §5 conjectures hinge on a weight-removal step the paper itself admits is unclear. read the letter →

arxiv 2506.01640 v1 pith:ATYRNS74 submitted 2025-06-02 math.NT

David Lowry-Duda This is my paper

classification math.NT MSC 11F6611F72

keywords murmurationstraceformulasL-functionsrootnumbersMaassformsmodularone-leveldensityKuznetsovformula

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

Murmurations are statistical correlations between the coefficients of L-functions and their root numbers within families ordered by conductor, a pattern first noticed in elliptic-curve data. The paper's central claim is that these phenomena are not scattered accidents: nearly every proved murmuration starts from a trace formula, and the same machinery should prove several new ones. Specifically, it argues that Kuznetsov's trace formula should yield an arithmetically weighted Maass-form murmuration, that Petersson and Eichler-Selberg methods should extend the weight-aspect and level-1 results to general level, and that the template applies to symmetric square lifts and to half-integral weight forms whose Dirichlet series lack Euler products. The unifying assertion is that 'using a reasonable trace formula and performing reasonable averages will probably prove some murmuration phenomena.'

What carries the argument

The central machinery is a small collection of trace formulas—Petersson, Kuznetsov, and Eichler-Selberg, together with the explicit Selberg trace formula for Hecke operators. Trace formulas rewrite spectrally averaged sums of Hecke eigenvalues as explicit sums over Kloosterman sums or class numbers, so that setting one slot to 1 and the other to a prime $p$ exposes the correlation that a murmuration measures. The load-bearing distinction is arithmetic normalization: denominators like $\|f\|^2$, proportional to $L(1,\operatorname{Sym}^2 f)$, appear naturally in the Petersson and Kuznetsov formulas, giving arithmetically weighted variants, while the Eichler-Selberg and Selberg routes can produce the unweighted average that matches observed data.

What would settle it

A concrete check is to compute both the arithmetically weighted average and the unweighted average for level-1 holomorphic cuspforms of increasing weight and compare their ratio; if this ratio does not approach a nonzero constant as $X$ grows, the weighted-to-unweighted translation on which the paper's program relies fails.

Watch

Extended reading notes

Core claim

The paper's core discovery is an architecture rather than a new theorem: murmuration behavior follows from a trace formula after averaging in every non-fixed aspect. It reads the earlier one-level density theorem for level-1 cuspforms as implicitly containing an arithmetically weighted murmuration through the Petersson formula, with weights $L(1,\operatorname{Sym}^2 f)^{-1}$ coming from Petersson norms, and it reads the more recent proof for weight-aspect cuspforms as using an Eichler-Selberg type formula to reach the unweighted shape that matches experiments. The same division of labour is expected for Maass forms: the Kuznetsov trace formula gives arithmetically weighted correlations, while the explicit Selberg trace formula gives unweighted ones. The four sketched families are Maass forms with arithmetic weights, general-level holomorphic and Maass forms, symmetric square lifts (using $\lambda_f(p^2)$ through a GL(2) trace formula), and half-integral weight forms where a Petersson formula still applies even though the attached Dirichlet series have no Euler product.

Load-bearing premise

The load-bearing premise is that a result that weights each form by an arithmetic factor such as $1/L(1,\operatorname{Sym}^2 f)$ can be converted into the unweighted average seen in data; the paper says outright that this conversion is not apparent.

Editorial extensions

If this is right

An arithmetically weighted Maass-form murmuration follows from the Kuznetsov trace formula at level 1, with a density qualitatively similar to the unweighted Maass result.
The weight-aspect and level-1 holomorphic results should extend to arbitrary level, with the density unchanged except for local factors at primes dividing the level.
Murmurations of symmetric square lifts of holomorphic modular forms can be studied through $\lambda_f(p^2)$ using a GL(2) trace formula, even though all such root numbers are trivial.
Half-integral weight forms admit a murmuration from the Petersson trace formula without needing an Euler product or Riemann hypothesis for the attached Dirichlet series; GRH for Dirichlet L-functions and a smooth cutoff should suffice.
If the approach is right, the practical bottleneck for proving new murmurations is the existence of a sufficiently explicit trace formula for the chosen family, not the phenomenon itself.
The paper's overall position is that trace formulas are the systematic route to proving murmurations, converting a data-driven phenomenon into analytic number theory.

Reading between the lines

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

A natural numerical test of the whole programme is to compute both sides of the weighted-to-unweighted comparison for level-1 cuspforms; if the ratio of the weighted average to the unweighted average tends to a nonzero constant, the open translation step is likely removable.
The exact equality of the level-1 Maass and holomorphic unweighted densities suggests a broader invariance: the same density, decorated only by local factors, may govern all GL(2) automorphic families once the arithmetic weights are stripped off.
The half-integral weight example, if proven, would show that murmurations are not an epiphenomenon of Riemann-hypothesis-type zero statistics, since the relevant Dirichlet series have zeros off the critical line.
If the program succeeds for general level, it would give a precise prediction for experimentalists: the murmuration density for newforms of level $N$ should match the level-1 density times local factors at primes dividing $N$.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

You should know this one is a survey, not a research paper. It has no new theorems or derivations, but it is a clear and honest map of the trace-formula approach to murmurations, and it is upfront about where the program is conjectural. I found the survey portions accurate and well written. The descriptions of ILS00, BBLLD25, BLLD+24, Zubrilina, Wang, Martin, and LOP25 are faithful, and the observation that all proven murmurations so far come from trace formulas is useful and, as far as I know, correct. The paper also highlights an interesting fact: the density function for weight-aspect holomorphic forms and eigenvalue-aspect Maass forms coincide, which is not obvious and worth stating.

The soft spot is exactly the one the stress-test note flags. The §5 conjectures are plausible but unproven, and the load-bearing premise is that arithmetic weights L(1,Sym^2 f)^{-1} coming from Petersson or Kuznetsov can be removed to recover standard unweighted murmurations. The paper itself concedes in §3 that 'it's not apparent how to translate an arithmetically weighted murmuration to a non-arithmetically weighted normalization.' So the four sketched families in §5 might end up proving only weighted versions of murmurations, not the observed unweighted phenomenon. This is not an internal inconsistency, and the paper's hedging is appropriate—it says 'probably' and 'should be possible'—but it does mean the central programmatic claim is a conjecture about a missing technique, not a consequence of existing derivations.

I would not call this a flaw of the survey; it is an honest limitation. The paper does not overclaim, and it openly identifies the key obstacle. If anything, I would have liked a bit more discussion of why one might hope the weight removal is possible, but that may be too much to ask from an expository note.

Who gets value from this? Anyone entering the murmurations literature, or a researcher wanting a concise summary of which trace formulas have been used and what remains open. It is not a breakthrough, but it is a well-organized synthesis with accurate citations. I would bring it to a reading group if the group is working in arithmetic statistics; otherwise it is more of a reference.

My recommendation: this deserves a serious referee, but as a survey, not as a research paper. A referee should check the technical summaries for accuracy and pressure-test whether the conjectures in §5 are as close to proven as the prose suggests—they are not, but they are honestly labeled. I would accept it for peer review with the expectation of minor revision, mostly to sharpen the language around the weight-removal problem.

Referee Report

2 major / 4 minor

Summary. This note surveys murmuration phenomena for automorphic L-functions through the lens of trace formulas. Section 2 recalls the general definition of a murmuration density (Eq. (2)). Section 3 explains how the ILS00 one-level density proof yields an arithmetically weighted murmuration (Eq. (6)) for level-1 holomorphic forms, and contrasts it with the unweighted result (Eq. (7)) proved via Eichler--Selberg in BBLLD25. Section 4 inventories other trace-formula-based proofs (Dirichlet characters, fixed-weight varying-level forms, imaginary quadratic fields, Maass forms) and stresses that no trace formula covers sparse families such as elliptic curves. Section 5 proposes four likely extensions: arithmetically weighted Maass-form murmurations via Kuznetsov, general level generalizations, symmetric-square murmurations, and half-integral weight forms without a Riemann hypothesis. The paper explicitly concedes the main technical obstacle: converting arithmetically weighted trace-formula statements into unweighted murmurations of the standard shape.

Significance. If the program in Section 5 is realized, the results would establish that murmuration phenomena are systematically accessible to classical analytic number theory across several new GL(2)-type families and even some non-RH Dirichlet series. The note is valuable as a concise survey of the existing proved cases and as an explicit research map; it correctly identifies the weighted-to-unweighted conversion as the central unresolved step. Its descriptions of published results are faithful, and its language is appropriately hedged. At the same time, no new theorem is proved, and the forward-looking claims are sketches rather than derivations; the paper's lasting value depends on the conjectures being realized in follow-up work.

major comments (2)

[§3 and §5.1–5.2] The load-bearing premise of the §5 program is that an arithmetically weighted trace-formula statement can be converted into an unweighted murmuration of the form (2), but the manuscript itself states in §3 that 'it's not apparent how to translate an arithmetically weighted murmuration to a non-arithmetically weighted normalization.' Equation (6) is weighted by L(1,Sym^2 f)^{-1}, and the Kuznetsov proposal in §5.1 reintroduces the analogous weight ||mu_j||^2 as L(1,Sym^2 mu_j); since these special values fluctuate and are correlated with the family, the weighted and unweighted averages need not share the same main term. The paper should either state explicitly that the §5 predictions are conditional on solving this translation problem, or provide a worked example in which the weight is removed. As written, the sentence 'Practice shows that using a reasonable trace formula and performing reasonable averages will probably prove some murmuration phenomena' (§5) is a conjecture about a missing technique, not a consequence of the displayed derivations.
[§5.1, displayed Kuznetsov formula] The displayed Kuznetsov formula in §5.1 omits the continuous-spectrum contribution on the spectral side; the formula as written equates the cuspidal sum plus \hat{b}(0) with the geometric side, while the standard formula (e.g. [IK04, Theorem 16.3]) contains an integral over the Eisenstein spectrum. If the author intends to absorb the continuous spectrum into \hat{b}(0), this should be explained; otherwise the proposed analysis of the Kuznetsov formula is incomplete exactly where it is used to predict Maass-form murmurations.

minor comments (4)

[§3, Eq. (6)] The sentence after (6) says the only difference between (6) and the standard shape (2) is the arithmetic weights, but (6) also contains a factor \sqrt{p} multiplying \lambda_f(p) in the numerator; the manuscript should clarify which coefficient convention is intended, since the unweighted result (7) from BBLLD25 uses \lambda_f(p) without this factor.
[§5.2] The assertion that the resulting murmuration densities in the level-N generalizations are 'essentially the same, except for some local factors coming from primes dividing the level' should be made more precise, since the local factors are exactly the part that requires new work.
[§5.3] The statement lambda_{Sym^2(f)}(p) = lambda_f(p^2) is only true for primes unramified in the symmetric square lift and away from the level; the manuscript should state this qualification.
[References] The reference [HLOa24] contains a typo in the author list ('Alexey Pozdnyakov and' should be 'Alexey Pozdnyakov').

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the paper is a conditional survey anchored in external theorems and contains no fitted-input predictions.

full rationale

This paper is a survey and research announcement rather than a derivation chain. The two proven murmuration theorems it highlights, [BBLLD25] and [BLLD+24], are prior independent results by the author and collaborators; they are cited as established theorems, not derived inside this note. Equation (6) is explicitly attributed to the proofs in [ILS00], an external source, and the surrounding discussion presents it as an unpacking of that work rather than a new prediction. The §5 program is explicitly heuristic, with the paper stating that 'using a reasonable trace formula and performing reasonable averages will probably prove some murmuration phenomena,' and each proposed family is accompanied by caveats about missing technical steps. The most important limitation is the conceded difficulty in §3 that 'it's not apparent how to translate an arithmetically weighted murmuration to a non-arithmetically weighted normalization'; this is an open technical gap, not a circular reduction, because the arithmetically weighted and unweighted statements are not equated by construction. No fitted parameters appear, and no equation in the paper reduces to its own input by definition. Self-citations occur, but they point to externally established theorems with stated assumptions and are not used to forbid alternatives or to smuggle in an unproved ansatz. Accordingly, the appropriate finding is no significant circularity.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

The paper introduces no free parameters or invented entities. Its claims rest on external mathematical facts: the validity of the cited trace formulas, the correctness of the cited murmuration theorems, and the GRH assumption appearing in the quoted theorems.

assumptions (3)

domain assumption GRH is assumed in the quoted one-level density and murmuration theorems.
§3 quotes Theorems from [ILS00] and [BBLLD25] that are conditional on GRH; the survey inherits this assumption.
standard math The explicit trace formulas (Petersson, Kuznetsov, Eichler-Selberg, Selberg-Strömbergsson) are valid as quoted.
§4 writes down these formulas from [Pet32], [IK04], [Chi22], [Str16]; they are established or adopted without re-derivation.
standard math The cited murmuration results are correct.
The survey's narrative assumes the correctness of [BBLLD25], [BLLD+24], [Zub23], [LOP25], [Wan25], and [Mar23].

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Cite this review

Pith. "Pith review of On Murmurations and Trace Formulas." pith.science (2026). https://pith.science/paper/ATYRNS74

@misc{pith2026250601640,
  author       = {Pith},
  title        = {Pith review of: On Murmurations and Trace Formulas},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ATYRNS74}},
  note         = {Machine review of arXiv:2506.01640}
}

abstract

In recent work with Bober, Booker, Lee, Seymour-Howell, and Zubrilina, we proved murmuration behavior for Maass forms in the eigenvalue aspect and for modular forms in the weight aspect. Both used an approach based on the Selberg trace formula. But different trace formulas, including those due to Kuznetsov or Petersson, offer different variations. We examine murmurations from the perspective of different trace formulas and outline several families of $L$-functions where one can likely prove additional murmuration behavior.

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Reference graph

Works this paper leans on

28 extracted references · 19 canonical work pages

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [1]

Booker, Min Lee, and David Lowry-Duda

Jonathan Bober, Andrew R. Booker, Min Lee, and David Lowry-Duda. Murmurations of modular forms in the weight aspect. To appear in Algebra and Number Theory , 2025. https://arxiv.org/abs/2310.07746

arXiv 2025

[2] [2]

Booker, Min Lee, David Lowry-Duda, Andrei Seymour-Howell, and Nina Zubrilina

Andrew R. Booker, Min Lee, David Lowry-Duda, Andrei Seymour-Howell, and Nina Zubrilina. Murmurations of M aass forms, 2024

work page 2024

[3] [3]

Farmer, and Martin R

Brian Conrey, David W. Farmer, and Martin R. Zirnbauer. Autocorrelation of ratios of L -functions. Commun. Number Theory Phys. , 2(3):593--636, 2008

work page 2008

[4] [4]

Twist-minimal trace formula for holomorphic cusp forms

Kieran Child. Twist-minimal trace formula for holomorphic cusp forms. Res. Number Theory , 8(1):Paper No. 11, 27, 2022

work page 2022

[5] [5]

Murmurations and ratios conjectures

Alex Cowan. Murmurations and ratios conjectures. 2024. http://arxiv.org/abs/2408.12723v3

arXiv 2024

[6] [6]

W. Duke, J. B. Friedlander, and H. Iwaniec. The subconvexity problem for A rtin L -functions. Invent. Math. , 149(3):489--577, 2002. http://dx.doi.org/10.1007/s002220200223

doi:10.1007/s002220200223 2002

[7] [7]

Murmurations of elliptic curves

Yang-Hui He, Kyu-Hwan Lee, Thomas Oliver, and Alexey Pozdnyakov and. Murmurations of elliptic curves. Experimental Mathematics , 0(0):1--13, 2024

work page 2024

[8] [8]

Analytic number theory , volume 53 of American Mathematical Society Colloquium Publications

Henryk Iwaniec and Emmanuel Kowalski. Analytic number theory , volume 53 of American Mathematical Society Colloquium Publications . American Mathematical Society, Providence, RI, 2004

2004