Murmurations in the depth aspect

Catinca Mujdei; Claire Burrin; Hsin-Yi Yang; Min Lee; Vivian Kuperberg

REVIEW 2 major objections 2 minor 1 cited by

Murmurations in the depth aspect

T0 review · 2 major / 2 minor · reviewed 2026-07-13 · grok-4.5

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Pith's one-line read The murmuration density for Hecke modular forms of fixed weight and prime-power level is computed explicitly as the power of the prime tends to infinity.

desk verdict Clean abstract claim of an explicit depth-aspect murmuration density for prime-power levels, but the supplied body is too corrupted to verify the argument. read the letter →

arxiv 2603.25564 v2 pith:E7YLBFUI submitted 2026-03-26 math.NT

Claire Burrin , Vivian Kuperberg , Min Lee , Catinca Mujdei , Hsin-Yi Yang This is my paper · ORCID

classification math.NT MSC 11F1111F3011F6611M41

keywords murmurationsHeckeformsmodulardepthaspectprime-powerleveldensityfunctionL-functionsspectralaverages

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 the oscillatory patterns that appear when you average arithmetic data (such as Hecke eigenvalues) over families of modular forms or L-functions and then look at the averages as a continuous function of a scaling parameter. This paper establishes that such a limiting density exists, and writes it down in closed form, for the family of Hecke forms of fixed weight k whose levels are pure powers of a fixed odd prime. The computation is carried out in the depth aspect: the level N = ℓ^a grows by letting the exponent a tend to infinity while the prime ℓ stays fixed. A sympathetic reader cares because an explicit density turns a mysterious numerical oscillation into a concrete, checkable analytic object that can be compared with other families and used to predict averages at large depth.

What carries the argument

The passage from discrete averages of Hecke eigenvalues to a continuous limiting density, carried by a Petersson/trace-formula identity (or equivalent spectral average) that remains controllable uniformly in the depth aspect a → ∞.

What would settle it

Compute the empirical average of the relevant Hecke eigenvalues (or root-number-weighted coefficients) for forms of level ℓ^a with successively larger a and check whether the resulting histogram converges to the closed-form density claimed in the paper.

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Extended reading notes

Core claim

For the family of Hecke forms of weight k and level N = ℓ^a (ℓ a fixed odd prime), the murmuration density function exists in the limit a → ∞ and is given by an explicit closed-form expression obtained by averaging the relevant Hecke eigenvalues (or spectral sums) and passing to a continuous density.

Load-bearing premise

The analytic estimates that convert averages of Hecke data into a limiting density stay valid uniformly as the exponent a goes to infinity for a fixed odd prime.

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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

The one thing to know is that this paper claims an explicit murmuration density for Hecke forms of fixed weight k and level N = ℓ^a (ℓ fixed odd prime, a → ∞). That is a natural next case in the murmurations program, not a new framework, and the abstract states it cleanly.

What is new is the depth-aspect specialization itself: previous murmuration densities have been worked out in other aspects, and this fills in the prime-power level family with a → ∞. From the readable fragments one can see the usual ingredients—averages of Hecke data, spectral sums, and a limiting oscillatory density—so the setup looks standard for the subfield. The claim is external-benchmark style rather than circular, and there are no free parameters or invented objects in the abstract.

The soft spot is severe and practical: the full-text extract we have is almost entirely garbled (encoding garbage, mixed non-NT material, unreadable formulae). I cannot inspect the main theorem statement, the precise range of a, or the error-term control that would justify uniformity as a → ∞. That analytic uniformity is load-bearing; without it the density is only a claim. This is not a detected mathematical error—it is an inability to check the paper at all from the supplied source. Confidence therefore stays low.

Who it is for: specialists already working on murmurations or averages of modular forms in the depth aspect. They will want the closed form if the derivation holds. It is not for a general number-theory audience or for anyone expecting a conceptual breakthrough.

I would send it to peer review once a clean manuscript is available. The abstract is sharp enough and the problem is real enough that a referee who can read the proofs should decide. Until then I would not cite it or bring the garbled extract to reading group. If the clean version checks out, it is solid subfield progress; right now we simply cannot tell.

Referee Report

2 major / 2 minor

Summary. The paper claims to compute the murmuration density function for the family of Hecke forms of fixed weight k and prime-power level N=ℓ^a, where ℓ is a fixed odd prime and a o∞. From the abstract and the few readable fragments, the intended result is an explicit limiting averaged oscillatory density for Hecke data (eigenvalues or related spectral sums) in the depth aspect, obtained via standard analytic tools such as Petersson/trace formulae. The supplied full-text extract is almost entirely corrupted by encoding garbage, mixed non-NT material (including a cs.CV arXiv stamp), and unreadable formulae, so the precise main theorem, error estimates, and derivation cannot be inspected.

Significance. If the claimed density is correctly derived with uniform error control as a o∞, the result would be a genuine contribution to the emerging literature on murmurations of modular forms, extending known conductor-aspect phenomena to the depth aspect for prime-power levels. Explicit closed-form densities of this type are of interest for understanding average oscillatory behaviour of Hecke eigenvalues and for comparisons with other families. The abstract states a clean computational claim of the sort that is typically valued in analytic number theory; however, because the body is unreadable, no machine-checked proofs, reproducible code, or fully checkable error estimates can be credited on the present extract.

major comments (2)

The full-text extract is severely corrupted (replacement characters throughout, unreadable formulae, and intrusion of unrelated cs.CV material with arXiv stamp 2603.25565). Consequently the main theorem statement, the precise range of a, and the depth-aspect error estimates cannot be verified. The central claim that a limiting murmuration density exists and is given by a closed form therefore remains uncheckable on the supplied manuscript.
From the abstract and readable fragments, the argument must pass from averages of Hecke eigenvalues (via Petersson or trace formulae) to a limiting density as a o∞ for fixed odd prime ℓ. Uniform control of the error terms in this depth aspect is load-bearing; without a readable derivation or explicit error hypotheses, it is impossible to confirm that the claimed density is justified.

minor comments (2)

Even the readable fragments contain mixed or garbled section headings and formula scraps that prevent assessment of notation consistency or reference completeness.
The abstract is clear, but the body must be re-supplied in a clean, properly encoded form before any further technical review is possible.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found: claimed murmuration density is an asymptotic computation, not a quantity forced by its own inputs.

full rationale

The paper's central claim (abstract) is that the murmuration density for Hecke forms of fixed weight k and prime-power level N=ℓ^a (ℓ fixed odd prime, a→∞) is computed explicitly as a limiting averaged oscillatory density. That is a standard external-benchmark style of analytic-number-theory claim: one averages Hecke data (via Petersson/trace formulae or related spectral sums) and extracts a limiting density. Nothing in the readable abstract or fragments indicates that the density is defined in terms of itself, fitted to a subset of the same data and then re-presented as a prediction, or forced by a uniqueness theorem or ansatz that only the authors supply. The supplied full-text extract is heavily encoding-corrupted, so individual equations cannot be inspected; under the hard rule that circularity may be asserted only when a specific reduction can be quoted, no circular step can be exhibited. Residual dependence on prior definitions of murmurations is ordinary citation, not load-bearing circularity. Score 0; steps empty.

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

The paper is a pure-math asymptotic computation. It inherits standard modular-forms and analytic-number-theory background (Hecke theory, Petersson inner products / trace formulas, known murmuration setup) and does not appear to introduce free fitted constants or new physical entities. Because the body is unreadable, the ledger lists only what the abstract and field context force; finer ad-hoc analytic hypotheses may exist in the proofs.

assumptions (3)

standard math Standard theory of Hecke eigenforms of weight k and level N=ℓ^a, including existence of a basis of newforms and Hecke eigenvalues.
Background for any average over such forms; invoked by the family definition in the abstract.
domain assumption Prior definition and existence framework for murmuration density functions in other aspects (level/conductor aspects) as developed in the recent murmuration literature.
The paper computes 'the' murmuration density in a new aspect, so it relies on the established notion of that density from prior work.
domain assumption Analytic estimates (trace formula / Petersson formula / spectral large sieve type bounds) remain strong enough as a→∞ with ℓ fixed odd prime to pass from finite averages to a limiting density.
Load-bearing for any depth-aspect limit; not checkable from the corrupted body.

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Cite this review

Pith. "Pith review of Murmurations in the depth aspect." pith.science (2026). https://pith.science/paper/E7YLBFUI

@misc{pith2026260325564,
  author       = {Pith},
  title        = {Pith review of: Murmurations in the depth aspect},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/E7YLBFUI}},
  note         = {Machine review of arXiv:2603.25564}
}

abstract

We compute the murmuration density function for the family of Hecke forms of weight $k$ and prime power level $N=\ell^a$, with $\ell$ a fixed odd prime and $a\to \infty$.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Joint level-weight murmurations: prime averaging and the cubic pointwise range
math.NT 2026-07 conditional novelty 8.0 of 10

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.

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Reviewed July 13, 2026 · model on record in the stance chip above.