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Spectral fourth moments of Hecke--Maa{\ss} cusp forms

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Pith's one-line read For Heegner points, the spectral fourth moment of Hecke–Maass cusp forms is essentially T^{2+ε}.

desk verdict Strong, likely-correct paper proving the optimal spectral fourth moment at Heegner points, but the proof as written omits a load-bearing local integral bound and needs revision before it is complete. read the letter →

arxiv 2607.13518 v1 pith:NWIBU5NG submitted 2026-07-15 math.NT

Edgar Assing , Peter Humphries This is my paper · ORCID
classification math.NT MSC 11F7211F6711F66
keywords Hecke–MaasscuspformsHeegnerpointsspectralfourthmomentWaldspurgerformulaLindelöfonaverageshiftedconvolutionsumoftwosquaresDiophantineexponents
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

On a dyadic spectral interval of length T, the family of Hecke–Maass cusp forms contains about T^2 forms, so the trivial bound for the fourth moment of their values at a fixed point is T^3, while the Lindelöf-on-average prediction is T^{2+ε}. This paper proves that prediction for every Heegner point: for fixed level q and discriminant D, the sum of |φ(z0)|^4 over cusp forms with spectral parameter in [T,2T] is O_{q,D,ε}(T^{2+ε}). The proof runs Waldspurger's formula through an approximate functional equation and the spectral large sieve, reducing the problem to a fourth moment of central L-values. The result also yields unconditional bounds for the shifted convolution error term for r(n), and shows that the pointwise Diophantine exponent at a Heegner point is equal to 1.

What carries the argument

The load-bearing tool is Waldspurger's formula in the form of Lemma 2.4, which converts |φ(z0)|^2 at a Heegner point into a sum over class-group characters of the central values L(1/2, π⊗σ_ξ), divided by L(1, π, ad). Once the pointwise value is expressed through L-values, an approximate functional equation (Lemma 2.2) and the spectral large sieve (Lemma 2.1) bound the fourth moment. Independently, Theorem 1.10 provides a second path to the key inequality (7) via the pretrace formula and a lattice-point count, where the Heegner-point structure reduces the counting to representations by a positive-definite binary quadratic form; this path avoids Waldspurger.

What would settle it

Compute or numerically estimate the local integrals I_{S,D}(g·φ_A, ξ_A) for a fixed ramified prime p|qD by direct local computation, and check whether they remain O_{q,D}(1) as T→∞ and as ξ ranges over Cl_D; if any of them grows with T or with the conductor of ξ, the proof of Lemma 2.4 collapses. A direct numerical test: for small level q and discriminant D, compare |φ(z0)|² with (1/|Cl_D|) Σ_ξ L(1/2,π⊗σ_ξ)/L(1,π,ad) for several forms φ in the dyadic interval and see whether the ratio stays bounded as T varies.

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Extended reading notes

Core claim

The paper's central claim is Theorem 1.2: for any fixed integer q≥1, negative fundamental discriminant D<0, and Heegner point z0 of discriminant D and level q, the spectral fourth moment Σ_{φ∈B(q), T≤t_φ≤2T} |φ(z0)|^4 is bounded by O_{q,D,ε}(T^{2+ε}). This is the essentially optimal, best-possible bound for this family, confirming Chamizo's Conjecture 1.1 for all Heegner points. The paper derives two consequences: Theorem 1.3, an unconditional estimate for the error term in the correlation sum Σ_{n≤N} r(n)r(n+m) with piecewise power savings and a mean-square bound in m, and Corollary 1.7, the equality κ(z0)=1 of the pointwise Diophantine exponent for Heegner points on the modular surface.

Load-bearing premise

The argument rests on Lemma 2.4, whose proof requires the local toric-period/Whittaker integrals I_{S,D}(g·φ_A, ξ_A) at the ramified places to be bounded by O_{q,D}(1); the paper asserts this follows by trivial estimation but omits the details, and this local bound is essential for converting |φ(z0)|^2 into a sum of central L-values.

Editorial extensions

If this is right

  • Chamizo's conjecture is verified for all Heegner points, giving the Lindelöf-on-average size for this spectral family.
  • The shifted-convolution estimate for r(n) becomes unconditional, with error-term exponents matching the best known conditional results over a wide range of shifts, plus a power-saving mean-square bound in m.
  • The pointwise Diophantine exponent κ(z0) equals 1 for every Heegner point on the modular surface, matching the trivial lower bound.
  • The density-type inequality (7) is established for Heegner points without any use of the Ramanujan–Petersson conjecture.
  • The proof method treats the spectral aspect for fixed level q, and the same Waldspurger-plus-large-sieve structure is ready to be reused for related families of automorphic L-functions.

Reading between the lines

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

  • The proof of Lemma 2.4 omits the details of the local integral estimate I_{S,D}(g·φ_A, ξ_A) ≪_{q,D} 1; making this estimate explicit would turn the main theorem's implied constant into an effective one and would indicate whether the method extends to non-Heegner points.
  • Since the lattice-point counting in Proposition 4.3 only uses that z0 is a root of a binary quadratic form, a similar pretrace argument should apply to any quadratic irrational point, not only Heegner points of a fixed order, potentially giving (7) for all CM points in line with the random-wave heuristic.
  • The same combination of Waldspurger's formula and the spectral large sieve, applied at the second-moment level, may yield or improve subconvexity bounds for L(1/2, π⊗σ_ξ) at Heegner points, because the large sieve supplies extra averaging over π.
  • If the local integral bound in Lemma 2.4 were ever shown to fail for some ramified prime, the present proof of Theorem 1.2 would collapse even though the theorem itself might still be true; checking this numerically for level q>1 is a feasible falsification test.
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Editorial analysis

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

Referee Report

3 major / 6 minor

Summary. The paper proves Theorem 1.2: for fixed q, a Heegner point z0 of level q and fundamental discriminant D<0, and T≥1, the spectral fourth moment Σ_{φ∈B(q), T≤tφ≤2T} |φ(z0)|^4 is O_{q,D,ε}(T^{2+ε}). This confirms Chamizo's Conjecture 1.1 for Heegner points. The proof combines Waldspurger's formula (Lemma 2.4) to bound |φ(z0)|^2 by a sum of central values L(1/2, π⊗σξ)/L(1,π,ad), an approximate functional equation with π-independent weight (Lemma 2.2), and a spectral large sieve (Lemma 2.1). Applications are given to shifted convolution sums of r(n) (Theorem 1.3) and to pointwise Diophantine exponents (Corollary 1.7). A second, more classical route to the density hypothesis (7) is developed in Theorem 1.10 using the pretrace formula and a counting argument.

Significance. If correct, the paper settles a folklore conjecture (Conjecture 1.1 of Chamizo) for the natural family of Heegner points, giving the Lindelöf-on-average bound for a fourth moment that had previously been inaccessible. The applications are substantive: Theorem 1.3 removes the conditional Ramanujan–Petersson hypothesis from earlier shifted-convolution estimates, and Corollary 1.7 proves the optimal pointwise Diophantine exponent κ(z0)=1 for Heegner points. The proof is non-circular: it does not assume Conjecture 1.1 and uses standard, published tools. A genuine strength is the self-contained proof of Theorem 1.10 via the pretrace formula, which provides an independent verification of the density hypothesis needed for the Diophantine application, and the detailed counting arguments in Propositions 4.2 and 4.3. The main weakness is that two key auxiliary lemmas (Lemma 2.4 and the non-cuspidal part of Lemma 2.2) are only sketched or deferred, and Lemma 2.1 is stated in a false unrestricted form. These are fixable, but they currently leave the central proof incomplete.

major comments (3)
  1. [§2.4, Lemma 2.4] The inequality I_{S,D}(g·φ_A, ξ_A) ≪_{q,D} 1 is asserted with 'We omit the details.' This is load-bearing: Lemma 2.4 converts the point evaluation |φ(z0)|^2 into a sum of central L-values, and if any local factor (particularly the archimedean matrix-coefficient factor on the compact torus T(R), or the non-archimedean factors at split/ramified primes dividing qD) grew with t_π, the final T^{2+ε} bound in Theorem 1.2 would fail. The citations to [BBK, Lemma 10.3] and [MW] do not by themselves establish the needed uniform bound in the present rescaling and ramification set S={∞}∪{p|qD}. Please supply the calculation, or give precise statements with the required uniformity.
  2. [§2.2, Lemma 2.2] The approximate functional equation is proved only for cuspidal σξ, with the non-cuspidal (genus-character) case dismissed as 'straightforward since the L-function factors.' But Theorem 1.2 sums over all ξ∈Cl_D, including real characters for which σξ is an Eisenstein representation. This case is used and is load-bearing. The factorization should be written explicitly (or a reference given that covers Eisenstein σξ with the same π-independent weight).
  3. [§2.1, Lemma 2.1] As stated, the left-hand side sums over all π with c(π)|q with no spectral restriction, so the inequality is false: the number of terms grows like T² (and in fact the full sum over all t_π diverges), while the right-hand side is independent of T. The proof invokes (12), which has the restriction |t_φ|≤T. The intended truncated version (with |t_π|≤T, or T≤t_π≤2T) is standard and is what the proof of Theorem 1.2 actually uses, but the lemma statement and the application in §3 must be corrected.
minor comments (6)
  1. [§1.1] In the sentence preceding Conjecture 1.1, the text reads 'for all φ∈B(q) and.' — the phrase is incomplete; it should say 'for all z0∈Γ0(q)\H'.
  2. [§2.1, proof of Lemma 2.1] The displayed sum on the right has a typo: 'X_{N<n≤N}' should be 'X_{N<n≤2N}'.
  3. [§1.2, proof of Theorem 1.3] The final sentence says 'so that the desired results follow from Theorem 1.3'; this should refer to Theorem 1.2.
  4. [§2.4, Example 2.3] The notation Q^+_{1,-4} is used without definition; presumably it is the subset of positive-definite forms. Please define.
  5. [§1.3, Corollary 1.7] There is a mismatched parenthesis in 'For a Heegner point z0∈SL2(Z)\H)'.
  6. [References] Reference [Sard] is formatted with an arXiv id followed by '. (2019)' which is inconsistent with the other entries; please clean up.

Circularity Check

1 steps flagged · score 1.0 of 10

No load-bearing circularity: the fourth-moment bound follows from Waldspurger's formula, the spectral large sieve, and approximate functional equations; only a non-load-bearing self-reference typo appears in the application section.

  1. other [Section 1.2, proof of Theorem 1.3]
    "Both of these points z0 are Heegner points of discriminant D=−4, so that the desired results follow from Theorem 1.3."

    Read literally, this sentence justifies the theorem being proved (Theorem 1.3) by citing Theorem 1.3 itself, which would be circular. The surrounding text shows the intended citation is Theorem 1.2, which has just been established for the two Heegner points q=1,z0=i and q=2,z0=(-1+i)/2 of discriminant D=-4. So this is a typographical self-reference in an application, not a load-bearing assumption of the main derivation.

full rationale

The central claim (Theorem 1.2) is not circular. Its proof applies Lemma 2.4 (Waldspurger's formula converting |φ(z0)|^2 into a positive sum of central L-values), Cauchy-Schwarz, the standard bounds L(1,π,ad)≪T^ε, the approximate functional equation (Lemma 2.2), and the spectral large sieve (Lemma 2.1). None of these inputs assumes Chamizo's Conjecture 1.1 or the target T^{2+ε} bound; the sieve supplies the T^2+N loss that yields the result after optimizing N≤T^{2+ε}. Lemma 2.4 depends on an omitted verification of the local toric-period bound I_{S,D}(g·φ_A,ξ_A)≪1; this is a completeness/correctness gap (if the bound failed the L-value upper bound on |φ(z0)|^2 could degrade), but it is not a circular reduction because the bound is not the target result. The self-citations in the paper ([Hu] in a footnote for (8); [HK] in Remark 4.9 as an alternative route) are auxiliary; the needed estimate (8) is cited to [Ch99]/[Ch22], and Theorem 1.10 is proved independently by the pretrace formula and quadratic-form counting, explicitly circumventing Waldspurger. The only in-text self-reference is the proof of Theorem 1.3 ending with 'follow from Theorem 1.3' instead of Theorem 1.2; this is a typo, non-load-bearing, and does not affect the main theorem. Overall: no significant circularity; score reflects only this minor textual self-reference.

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

No fitted numerical parameters appear. The central claim rests entirely on external theorems in automorphic forms plus one internally asserted but unproved local estimate.

assumptions (5)
  • domain assumption Waldspurger's formula and the toric period formula bound central L-values by toric periods (Lemma 2.4)
    Used to convert |φ(z0)|^2 into a sum of L(1/2,π⊗σξ)/L(1,π,ad). External deep theorem; the paper cites [Wa], [BBK], [MW].
  • domain assumption Spectral large sieve / Kuznetsov formula bounds (Lemma 2.1 and (8))
    External results from Drappeau and Chamizo/Kuznetsov; control sums of Hecke eigenvalues over the spectral family.
  • ad hoc to paper Local integral estimate I_{S,D}(g·φ_A, ξ_A) ≪_{q,D} 1
    Needed for Lemma 2.4; the paper states 'We omit the details' instead of proving it. This is a load-bearing unproved step.
  • ad hoc to paper Approximate functional equation for L(1/2,π⊗σξ) with π-independent weight (Lemma 2.2), including non-cuspidal σξ
    Lemma 2.2 says modifications for non-cuspidal ξ are 'straightforward' but does not carry them out.
  • domain assumption Hoffstein–Lockhart / Li bounds L(1,π,ad) ≍ T^{±ε}
    Used in (11) to absorb L(1,π,ad) in the fourth-moment sum; external theorem.

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Pith. "Pith review of Spectral fourth moments of Hecke--Maa{\ss} cusp forms." pith.science (2026). https://pith.science/paper/NWIBU5NG

@misc{pith2026260713518,
  author       = {Pith},
  title        = {Pith review of: Spectral fourth moments of Hecke--Maa\ss cusp forms},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NWIBU5NG}},
  note         = {Machine review of arXiv:2607.13518}
}
abstract

In this note, we establish essentially optimal bounds for certain spectral moments of automorphic forms for $\mathrm{GL}(2)$. More precisely, we consider the family of Hecke--Maa{\ss} cusp forms with spectral parameter in a dyadic interval and study the fourth moment of these forms evaluated at a Heegner point. We additionally present applications of our main result to the shifted convolution problem involving the sum of two squares function $r(n)$ as well as to pointwise Diophantine exponents.

Figures

Figures reproduced from arXiv: 2607.13518 by the authors.

Figure 1
Figure 1. We compare different exponents β(α) arising in the esti￾mate E(N, Nα ) ≪ϵ Nβ(α)+ϵ , where α ∈ [0, 2]. The exponent in red is conditional on the Ramanujan–Petersson conjecture and originates from (3). We display the exponents arising from (5) in blue, which we establish unconditionally in Theorem 1.3. Finally, we show the exponents from (4) in green. In the latter cases, we have used the current record θ = 7 64 . m c… view at source ↗

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