The fourth moment of holomorphic Hecke cusp forms in shorter intervals

Jinghai Liu

REVIEW 2 major objections 5 minor 16 references

The fourth moment of holomorphic Hecke cusp forms in shorter intervals

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Pith's one-line read The paper proves that, for any fixed $0<c\le 1/4$, the average fourth moment of holomorphic Hecke cusp forms over a weight window of length $K^{3/4+c}$ is $6/\pi + O(K^{-\delta})$, matching the random wave prediction and improving the…

desk verdict Genuine short-interval improvement claimed, but the step that buys the new interval, Lemma 5.8, is a one-line citation that does not obviously transfer, so the main theorem is not established as written. read the letter →

arxiv 2501.10971 v1 pith:JULQVKWS submitted 2025-01-19 math.NT

Jinghai Liu This is my paper · ORCID

classification math.NT MSC 11F1111F3011F6611F67

keywords holomorphicHeckecuspformsfourthmomentrandomwaveconjectureshortintervalsinweightautomorphicL-functionsKloostermansumsBesselfunctionsspectralaverages

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

The paper shows that the average fourth moment of holomorphic Hecke cusp forms of weight near $K$ is $6/\pi + O(K^{-\delta})$ when the average is taken over a short window of weights of length $H = K^{3/4+c}$, for any fixed $0 < c \le 1/4$. This matches the value predicted by the random wave conjecture for these arithmetic eigenfunctions and improves on the earlier result that required a window of length comparable to $K$. The proof reduces the fourth moment to central values of automorphic $L$-functions and then estimates the resulting sums of Bessel functions and Kloosterman sums. The limit of the admissible window length is set by one technical estimate for the third error term.

What carries the argument

The load-bearing identity is the triple-product formula expressing $|\langle F^2, G\rangle|^2$, for $F$ of weight $k$ and $G$ of weight $2k$, as $\frac{\pi^3}{2(2k-1)}\frac{L(1/2,g)\,L(1/2,\mathrm{sym}^2 f\times g)}{L(1,\mathrm{sym}^2 f)^2 L(1,\mathrm{sym}^2 g)}$. Combined with the decomposition $\|F\|_4^4 = \sum_{g\in B_{2k}}|\langle F^2,G\rangle|^2$, it turns the fourth moment into average central values of degree-8 $L$-functions. The average over the Hecke basis is then evaluated with the trace formula, which replaces the spectral sum by Kloosterman sums weighted by products of the Bessel functions $J_{k-1}$ and $J_{2k-1}$. The main term $6/\pi$ emerges from the diagonal contribution, and the off-diagonal pieces are bounded using Weil's bound for Kloosterman sums together with asymptotic lemmas for weighted averages of Bessel functions; the hardest of these is the third error term of Lemma 5.8.

What would settle it

Work out the Poisson-summation and stationary-phase analysis of Lemma 5.8 for the specific weight $w'$ supported on $(1,(K+H)/K)$ with $H = K^{3/4+c}$; if the resulting phase function lacks the required non-stationary decay, the estimate $E_3 \ll K^{-1/4+\epsilon} K/H$ fails, and the claimed $O(K^{-\delta})$ error term in Theorem 1.2 would not follow.

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Extended reading notes

Core claim

With $\|F\|_2 = 1$, Theorem 1.2 states that for $H = K^{3/4+c}$ the smoothed average $\frac{2}{HW}\sum_{k\equiv 0 \bmod 2} w((k-K)/H) \frac{12}{k}\sum_{f\in B_k}\|F\|_4^4$ equals $6/\pi + O(K^{-\delta})$ for some $0 < \delta \le c$. Here $w$ is a smooth non-negative function supported on $(0,1)$, $W$ is its integral, and $B_k$ is the Hecke basis of holomorphic cusp forms of weight $k$. This is the fourth moment averaged over weights in $[K,K+H]$, and the constant $6/\pi$ is the random wave prediction. The result is obtained by the same overall route as the full-interval theorem: the fourth moment is written as a sum of products of central $L$-values, the average over forms is converted to Kloosterman sums, and the error terms are bounded by negative powers of $K$.

Load-bearing premise

The whole improvement rests on assuming that the bound for the third error term, which was proved for the full-interval weight, transfers unchanged to the new smooth weight supported on $(1,(K+H)/K)$ with $H = K^{3/4+c}$, even though the paper's own remark says the stationary-phase route behind that bound does not plainly apply.

Editorial extensions

If this is right

The random wave prediction for holomorphic Hecke cusp forms holds on average over weight windows of length $K^{3/4+c}$, which is shorter than the full dyadic range by a factor of $K^{1/4-c}$.
The proof shows that the admissible window length is controlled by the quality of the Bessel-Kloosterman estimates; in particular, the first two error terms are already amenable to $H = K^{1/2+c}$, so the bottleneck is the third error term.
For any fixed $c \le 1/4$, the error term in the average is $O(K^{-\delta})$, so the result gives a genuine power saving rather than a slowly decaying error.
The theorem extends the known fourth-moment average from intervals of length comparable to $K$ down to $K^{3/4+c}$, a substantial shrinking of the averaging window.

Reading between the lines

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

Because the first two error terms already work for $H = K^{1/2+c}$, a repair of the third error term would plausibly shorten the admissible window to $K^{1/2+c}$; that stronger statement is not established in the paper.
The key oscillatory integral behind Lemma 5.8 could be checked numerically for moderate $K$ and $c=1/4$; a failure of the expected decay would indicate that the improvement over the full-interval result depends on a missing argument.
The same reduction-plus-Bessel-estimate structure may carry over to short-interval averages of other moments or to Maass forms, provided the corresponding analogue of the third error-term estimate is available.

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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

Short version: the paper has a real new statement but the step that buys the new interval, Lemma 5.8, is a one-line citation to Khan's paper with a weight that doesn't satisfy the cited lemma's hypotheses, and the author's own Remark 5.9 concedes the standard method fails. As written, Theorem 1.2 is not established.

What's new and good: The statement is new: Khan's [9] averages over weights in [K, 2K], so H is about K, while here H = K^{3/4+c}. The main term derivation through Watson's formula, Petersson's trace formula, and the residue computation producing 6/pi is clean and standard. The author is honest: Remarks 1.3, 5.3, 5.7, and 5.9 identify exactly what controls H. Lemma 5.2 and Remark 5.3 show some independent technique.

Soft spots: Lemma 5.8 is load-bearing. The proof is the single line "By [9, Lemma 3.5]" applied to a new smooth weight w'(k/K) supported on (1, (K+H)/K), where H/K = K^{-1/4+c}. The cited lemma is for h supported on (1,2) with bounded derivatives; the new weight has derivatives of size (K/H)^j, and oscillatory Bessel estimates are sensitive to that. Remark 5.9 says after Poisson summation the index properties are "very poor", which undercuts the stationary-phase route behind the cited lemma. Without a verification that the transfer works, (5.31) is not available and no improvement over Khan follows. This is not a fatal arithmetical error but a missing proof at the exact spot that decides the theorem.

Also the notation hurts: E1 and E2 are used both as the two pieces of ET in (5.7) and as two of the three pieces of E2 in (5.10); Lemma 5.4 is titled "For part E1" when it is actually for the second piece of the split. The reader has to reverse-engineer which bound covers which range; that is fixable but it is real friction.

The diagonal and the main term are fine, and the overall strategy is coherent. The paper has no fitted parameters and no self-citations; the constant 6/pi comes out of the diagonal.

Verdict: As written it is not publishable, but the gap is specific and probably addressable. This is exactly the kind of paper a referee should see: the intended audience, analytic number theorists working on moments of L-functions, would benefit from a complete proof of Lemma 5.8. Send it to peer review; if the lemma can be made rigorous, the short-interval improvement is a solid, if narrow, result.

Referee Report

2 major / 5 minor

Summary. The paper claims Theorem 1.2: for fixed 0<c≤1/4 and H=K^{3/4+c}, with the normalization ||F||_2=1, the smoothed average over even weights k∈[K,K+H] of (2/(HW))Σ_k w((k-K)/H)(12/k)Σ_{f∈B_k}||F||_4^4 equals 6/π+O(K^{-δ}) for some 0<δ≤c. The route is standard: Watson's formula reduces the fourth moment to averaged central values of L(1/2,g)L(1/2,sym^2 f×g); the Petersson trace formula produces a diagonal term plus off-diagonal terms; the diagonal gives the constant 6/π. The new ingredient is to run this argument over a short weight interval of length H=K^{3/4+c}, shorter than Khan's H≈K. The paper splits the main off-diagonal term into pieces and bounds them in Lemmas 5.2, 5.4, 5.6 and 5.8. The decisive piece is Lemma 5.8, whose proof is a one-line appeal to Khan's Lemma 3.5 after changing the weight to w'(k/K) supported on (1,(K+H)/K).

Significance. If Theorem 1.2 is correct, it is a genuine improvement: it would confirm the random-wave prediction on average over weight windows of length K^{3/4+c}, shrinking Khan's previous K-length window by a power of K. The main term is parameter-free and emerges cleanly from Watson's formula and the diagonal of the Petersson trace formula, with no fitted constants; the paper is also refreshingly honest in Remarks 1.3 and 5.9, which identify Lemma 5.8 as the exact source of the admissible range of H. However, the main term and the E1/E2 bounds do not by themselves deliver the theorem; the entire shorter-interval gain rests on the E3 bound in Lemma 5.8, and that bound is not proved at the new scale. The paper's significance is therefore conditional on a missing technical argument.

major comments (2)

[§5, Lemma 5.8 and Remark 5.9] The proof of Lemma 5.8 consists of defining w' supported on (1,(K+H)/K) and then asserting E3 << K^{-1/4+ε} K/H << K^{-δ} 'By [9, Lemma 3.5]'. This does not verify that Khan's Lemma 3.5 applies to the new weight. Khan's lemma is proved for a fixed-scale smooth weight h(k/K) supported on (1,2) with bounded derivatives, whereas w'(k/K) is supported on an interval of length H/K = K^{-1/4+c} and has derivatives of size K^{j(1/4-c)}. Oscillatory Bessel-sum estimates are sensitive to exactly this derivative growth, and the stationary-phase or Poisson-summation method behind [9, Lemma 3.5] must be rerun at the new scale before (5.31) is available. Remark 5.9 explicitly concedes that for this weight the index properties after Poisson summation are 'very poor' and power saving is difficult. Since Remark 1.3 states that the range of H in Theorem 1.2 is determined exactly by Lemma 5.8, this is the load-bearing step: without a proof of (5.31), Theorem 1.2 does not improve on Khan's H≈K result. This is not a routine citation; it is a missing proof.
[§5, Lemma 5.2] The proof of the first part of (5.14) cites Lemma 3.1, but the subsequent bound appears to use only |J|≤1 together with Weil's bound and divisor estimates; if so, the reference to Lemma 3.1 is misleading. In the second part, the claimed decay by partial integration via (3.8) would require derivative bounds of size K^{-j} for a function supported on [K,2K], while w((k-K)/H) has support length H and derivatives of size H^{-j}=K^{-j(3/4+c)}; the stated justification is therefore not directly applicable. The desired decay in that range is more readily obtained from (3.2), since the Bessel argument is O(K^ε), but the proof should say which estimate is being used and where.

minor comments (5)

[§5, (5.7)-(5.10)] The notation E1,E2 is recycled: E1 and E2 are defined in (5.7) as the two terms of ET, and then (5.10) redefines E2 = E1 + E2 + E3 with new E1 and E2. Lemmas 5.2, 5.4 and 5.6 therefore refer ambiguously to 'E1' and 'E2'. Please rename the three subsums in (5.10), for example D1,D2,D3, throughout Lemmas 5.2-5.8.
[§5, Lemma 5.4 heading] Lemma 5.4 is titled 'For part E1', but in the context of the decomposition (5.10) it appears to concern one of the new subsums of the old E2; the heading should be updated to match the renamed decomposition.
[§3, Lemmas 3.4 and 3.5] The support conventions for the weight function are not harmonized: Lemma 3.4 states that h is supported on (0,1), while Lemma 3.5 says 'h as in Lemma 3.4' but its proof defines functions for u∈(1,2) and applies Lemma 3.1 to functions supported on [K,2K]. Please state the exact support and derivative conditions for each lemma.
[§5, Remark 5.3] There is a typo: 'Possion summation formula' should be 'Poisson summation formula'.
[§5, Remark 5.5] Remark 5.5 asserts without proof that Lemma 5.4 can be made valid for H=K^{1/2+c} by the same method as Remark 5.3. Since this range is not used in Theorem 1.2, either give the argument or delete the remark to avoid an unverified claim.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the main term comes from Watson's formula and the diagonal of the trace formula, and the contested short-interval bound is an external lemma transfer, not an equivalent input.

full rationale

I walked the derivation chain and found no step that reduces, by the paper's own equations or by self-citation, to its own inputs. The constant 6/pi in Theorem 1.2 is produced by Watson's formula, the identity (4.4), and the diagonal term of the Petersson trace formula, all external and explicit; it is not fitted or defined in terms of the target average. No parameter is fitted to a subset of the data and then renamed a prediction. There are no self-citations: all load-bearing references (Watson [14], Khan [9], Iwaniec [7], Iwaniec–Kowalski [8]) are external. The passage the skeptic highlights, Lemma 5.8, is indeed load-bearing and is justified by a one-line appeal to Khan [9, Lemma 3.5] with a different weight function; the paper itself flags in Remark 5.9 that after Poisson summation the index properties are 'very poor' and in Remark 1.3 that the admissible H is decided exactly by Lemma 5.8. This is an explicit limitation: if the transfer of Khan's bound fails, the improvement over Khan's H ~ K result is not established. But that is a soundness or correctness gap, not circularity, because the cited bound is independent external evidence rather than an input equivalent to the theorem. Per the hard rules, a derivation gap is scored under correctness risk, not circularity. I therefore assign score 0.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The paper introduces no new objects and no fitted constants: the main term 6/pi is fixed by Watson's formula and the diagonal of Petersson. What the central claim rests on is a stack of quoted analytic number theory lemmas from [7], [8], [9] and [14], plus the asserted transfer in Lemma 5.8 of Khan's E3 bound to the narrow weight window, which is the load-bearing and least supported input.

assumptions (5)

domain assumption Watson's Rankin-Selberg formula (2.12) expressing |(F^2, G)|^2 as a ratio of central L-values
Imported from [14, Theorem 3]; it is the bridge from the fourth moment to L-functions and is unproved here.
domain assumption Petersson trace formula (Lemma 2.1)
Cited from [8, Proposition 14.5]; splits the fourth moment into the diagonal main term and off-diagonal Kloosterman sums.
domain assumption Approximate functional equations (Lemma 2.2) and the truncated inverse symmetric-square L-value (Lemma 2.3)
Quoted from [9, Lemmas 1.4 and 1.5]; they make the central L-values finite sums with controllable tails.
domain assumption Bessel averaging lemmas 3.1, 3.3, 3.4 and 3.5
Quoted from [7, Lemma 5.8] and [9, Lemmas 1.6, 1.8, 1.9]; the oscillatory analysis of the weight sums is not re-derived.
ad hoc to paper Transfer of [9, Lemma 3.5] to the narrow weight window w'((k-K)/H) in Lemma 5.8
This is the decisive saving E3 << K^(-1/4+epsilon) * K/H that fixes H = K^(3/4+c). Remark 5.9 concedes the new weight gives very poor index properties under Poisson summation, and the hypotheses of the cited lemma are not verified.

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Pith. "Pith review of The fourth moment of holomorphic Hecke cusp forms in shorter intervals." pith.science (2026). https://pith.science/paper/JULQVKWS

@misc{pith2026250110971,
  author       = {Pith},
  title        = {Pith review of: The fourth moment of holomorphic Hecke cusp forms in shorter intervals},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/JULQVKWS}},
  note         = {Machine review of arXiv:2501.10971}
}

abstract

Let $0<c\le 1/4$ be fixed. For $H = K^{\frac{3}{4}+ c}$, we find the average value of the fourth moment of holomorphic Hecke cusp forms of weight varies within $[K,K+H]$, improving a previous result of Khan.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

16 extracted references · 10 canonical work pages

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Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.

[1] [9]

On the fourth moment of holomorphic Hecke cusp forms

Khan, R., “On the fourth moment of holomorphic Hecke cusp forms.” Ramanujan J. 34, no. 1 , 83–107, (2014) . https://doi.org/10.1007/s11139-013-9505-z

work page doi:10.1007/s11139-013-9505-z 2014

[2] [1]

Regular and irregular semiclassical wavefunctions

Berry, M. V., “Regular and irregular semiclassical wavefunctions.” J. Phys. A 10 , no. 12, 2083–2091, (1977). http://stacks.iop.org/0305-4470/10/2083

work page 1977

[3] [2]

Distribution of mass of holomorphic cusp forms

Blomer, V., Khan, R., Young, M., “Distribution of mass of holomorphic cusp forms.” Duke Math. J.162, no.14, 2609–2644, (2013). https://doi.org/10.1215/00127094-2380967

doi:10.1215/00127094-2380967 2013

[4] [3]

On the fourth moment of Hecke-Maass forms and the random wave conjecture

Buttcane, J., Khan, R., “On the fourth moment of Hecke-Maass forms and the random wave conjecture.” Compos. Math.153, no.7, 1479–1511, (2017). https://doi.org/10.1112/S0010437X17007199

work page doi:10.1112/s0010437x17007199 2017

[5] [4]

Automorphic Forms and L-Functions for the Group GL(n,R)

Goldfeld, D., “Automorphic Forms and L-Functions for the Group GL(n,R).” Cambridge Studies in Advanced Mathematics, vol. 99. Cambridge University Press, Cambridge (2006). With an appendix by Kevin A. Broughan. https://doi.org/10.1017/CBO9780511542923

doi:10.1017/cbo9780511542923 2006

[6] [5]

Coefficients of Maass forms and the Siegel zero

Hoffstein, J., Lockhart, P., “Coefficients of Maass forms and the Siegel zero.” With an appen- dix by Dorian Goldfeld, Hoffstein and Daniel Lieman. Ann. of Math. (2) 140 , no. 1, 161–181, (1994). https://doi.org/10.2307/2118543

work page doi:10.2307/2118543 1994

[7] [6]

Mass equidistribution for Hecke eigenforms

Holowinsky, R., Soundararajan, K., “Mass equidistribution for Hecke eigenforms.” Ann. Math. 172(2), 1517-1528, (2010). http://doi.org/10.4007/annals.2010.172.1517

work page doi:10.4007/annals.2010.172.1517 2010

[8] [7]

Topics in Classical Automorphic Forms

Iwaniec, H., “Topics in Classical Automorphic Forms.” Graduate Studies in Mathematics, vol. 17. Am. Math. Soc., Providence, (1997). https://doi.org/10.1090/gsm/017

doi:10.1090/gsm/017 1997