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REVIEW 3 major objections 5 minor 25 references

Quadratic forms of modular forms

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Pith's one-line read Assuming GRH, products of Hecke eigenforms decorrelate unless their pair sets match.

desk verdict Worth reading and worth refereeing, but the completion of the main conjecture rests on two unproved estimates delegated to an unpublished preprint; that gap must be closed before the paper is accepted. read the letter →

arxiv 2507.21951 v3 pith:PFKS7ZBP submitted 2025-07-29 math.NT

Shenghao Hua This is my paper · ORCID
classification math.NT MSC 11F1211F3011F66
keywords modularformsquadraticHeckeeigenformsdecorrelationL-functionsmomentsℓ^pnormsGRH
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 tries to establish a conditional decorrelation law for products of holomorphic Hecke eigenforms. Concretely, for $L^2$-normalized eigenforms of even weights with $k_1+k_2=k_3+k_4$, the inner product $\langle f_1f_2,f_3f_4\rangle$ should be $1+\delta_{f_1=f_2}$ when the two sets $\{f_1,f_2\}$ and $\{f_3,f_4\}$ coincide, and $o(1)$ otherwise. Under the Generalized Riemann Hypothesis, the Generalized Ramanujan Conjecture in one case, a non-overlap condition for four distinct forms, and analytic continuation of certain triple-product $L$-functions, the paper proves this for every quadruple with at least two distinct forms. It also proves conditional upper bounds for the $\ell^p$ norm of a quadratic form over a Hecke basis, establishing that the coefficients are not uniformly small. The interest is that this converts the inner-product question into a moment problem for central $L$-values and finishes the remaining cases of a natural conjecture.

What carries the argument

The argument runs on moment estimates for central values of automorphic $L$-functions. A triple-product formula (from [24]) rewrites each coefficient $\langle fg,h\rangle$ in terms of $L(1/2,f\times g\times h)$, $L(1/2,h)$ and $L(1/2,\operatorname{sym}^2 f\times h)$, divided by $L(1,\operatorname{sym}^2\cdot)$ factors that are known to grow slowly under GRH. Proposition 1.10 supplies the two averaged moment bounds (1.6) and (1.7), proved with the method of [22]: a Dirichlet-polynomial approximation of $\log L(1/2,\cdot)$, an orthogonality formula for Hecke eigenvalues over the basis, and a contour-integral step that controls prime sums by $\log\log\log k$. The remaining cases of Theorem 1.2 rest on two fractional-moment estimates quoted from [14].

What would settle it

Compute the two fractional moments displayed in §2.2 for increasing weights and check whether they decay like powers of $\log k$; a failure of the claimed $\log^{-3/8}$ or $\log^{-1/4}$ decay is a direct contradiction. Alternatively, search for a pair of distinct quadruples with equal weight sum whose inner product differs from the predicted diagonal or zero value by a nonvanishing constant.

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Extended reading notes

Core claim

The central discovery is the conditional validity of Conjecture 1.1 for all non-identical quadruples. The previously known cases are the shared-pair configuration $\{f_1,f_2\}=\{f_3,f_4\}\neq\{f_1\}$ and the double-diagonal configuration $f_1=f_2\neq f_3=f_4$; the new work handles the remaining cases, where $\{f_1,f_2\}\neq\{f_3,f_4\}$ and $f_1\neq f_2$ or $f_3\neq f_4$, using two fractional-moment estimates. Along the way, Theorem 1.4 gives, for any $p>0$, the bound $\|Q\|_{\ell^p,H_k}\ll k^{1/p-1/2}(\log^{-(2-p)/4+(p+1)\varepsilon}k+\log^{-(2-p)/8+(p+1)\varepsilon}k)$ for a quadratic form $Q$ in Hecke eigenforms; if all diagonal coefficients vanish, the exponent of the logarithm improves and the $\ell^p$ norm is $o(1)$ for every $p>2$. The paper also proves that no sparse Hecke-basis vector with a bounded number of nonzero coordinates can represent a generic quadratic form, because such a representation would force a coefficient larger than the $\ell^p$ bound permits.

Load-bearing premise

The proof depends on two moment estimates quoted without derivation from an unpublished preprint, and on the still-open analytic continuation of certain triple-product L-functions; if either fails, the completion of the conjecture collapses.

Editorial extensions

If this is right

  • Conjecture 1.1 is conditionally true for every quadruple with at least two distinct forms; the only unresolved case is the classical $L^4$-norm problem for a single form.
  • If two quadratic forms share no diagonal terms, their inner product is asymptotically the sum of the products of their common off-diagonal coefficients (Corollary 1.8).
  • For a quadratic form with zero diagonal, the $\ell^p$ norm over the Hecke basis is $o(1)$ for $p>2$; hence most coordinates are small but a few are of larger-than-average size.
  • No quadratic form of the type considered can be written as a linear combination of at most $L$ Hecke eigenforms once the weight is large enough (Theorem 1.6).

Reading between the lines

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

  • A likely route to removing the conditional hypotheses is to prove the two fractional-moment estimates in §2.2 directly; they are finite averages over a Hecke basis and may be checkable numerically at moderate weights.
  • The same moment method should extend to congruence subgroups, half-integral weight forms, or Hilbert modular forms, where analogues of the triple-product formula exist; the paper does not pursue these.
  • The distinction between the configurations $f_1=f_2, f_3=f_4$ and $f_1=f_3, f_2=f_4$ suggests that any conjecture for products of three or more forms will need a new mechanism, because products of eigenforms are not eigenforms.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. The paper studies quadratic forms in spaces of holomorphic cusp forms for SL2(Z). Its main claim, Theorem 1.2, is that Conjecture 1.1 holds conditionally for all quadruples of L2-normalized Hecke eigenforms with at least two distinct forms: assuming GRH (plus GRC when the unordered pairs coincide), a distinctness condition f1×f2 not isomorphic to f3×f4 when all four forms are distinct, and analytic continuation of certain triple-product L-functions involving symmetric squares, the inner product ⟨f1f2, f3f4⟩ is δ_{{f1,f2}={f3,f4}}(1+δ_{f1=f2}) + o(1) as k1+k2→∞. The proof is via Soundararajan's method: Proposition 1.10 gives conditional fractional-moment estimates (1.6) and (1.7), and Watson's formula converts these into ℓp bounds for expansion coefficients in a Hecke basis. Theorem 1.4 states a conditional upper bound for the ℓp norm of such quadratic forms, and Theorems 1.6 and 1.11 draw consequences about the distribution of the coefficients. The detailed parts of the proof are the Petersson-based high-moment bound (Lemma 2.2), the Perron/Littlewood estimate for prime sums (Lemma 2.4), and the Soundararajan-type tail bound (Lemma 2.5), all in Section 2.

Significance. If the central claims are correct, the paper completes a natural conjecture on decorrelation of products of Hecke eigenforms and connects it with a mixed L4-norm problem. The statement is genuinely quantitative: the moment exponents in Proposition 1.10 are derived from Gaussian integrals and Dirichlet-polynomial estimates, not fitted to data, and there are no free parameters. The paper also gives a clean formulation of an ℓp norm on Hecke-basis coefficients and derives structural information about the size distribution of those coefficients. The proofs of Lemmas 2.2, 2.4, and 2.5 are presented in detail and follow established templates. However, the two fractional-moment estimates that carry the remaining cases of Theorem 1.2 are not proved in the manuscript; they are asserted to follow from an unpublished preprint. This makes the completeness of the proof of Conjecture 1.1 conditional on material that is not available in the paper. The analytic continuation of L(s,sym²f×sym²g×sym²h) is also a genuinely open input, though it is stated as an assumption.

major comments (3)
  1. [§2.2, proof of Theorem 1.2] The two fractional-moment displays after 'Let f3 ≠ f4' are asserted to 'follow directly from the proof of [14, Proposition 5.1]' with no derivation. These estimates are the only argument for the remaining cases f3≠f4 and f1≠f2 with f3≠f4, and [14] is an unpublished preprint. The manuscript must supply the derivation, including the variance computation for the mixed Dirichlet polynomial, the treatment of three half-power L-factors, and the diagonal and overlap terms controlled by f1×f2 not isomorphic to f3×f4. As written, Theorem 1.2 is not established for those cases; if a derivation is not supplied, the two estimates should be stated as explicit assumptions rather than as consequences.
  2. [Lemma 2.4 and Proposition 1.10(1.7)] The proof of (2.6)–(2.12) uses the analytic continuation of L(s,sym²f×sym²g×sym²h) through Perron's formula and then applies a Littlewood bound (2.13). The statement of Lemma 2.4 does not impose a distinctness condition on f,g,h, while the proof asserts sym²f, sym²g, sym²h are pairwise non-isomorphic via [20], and (2.19) later requires f and g distinct. If sym²f and sym²g coincide, the sum in (2.7) has a main term of size log log x, not O(log log log(k1+k2)). Since Proposition 1.10(1.7) and hence Theorem 1.9 depend on this lemma, the paper should state explicitly the exact distinctness hypotheses on f,g,h and explain how the exceptional cases are handled.
  3. [§2.2, case #{f1,f2,f3,f4}=3] The strong-multiplicity-one argument in the paragraph 'When f1≠f2, f3≠f4, and #{...}=3' is difficult to follow and appears to reverse the implication. If f1=f3 and f2≠f4, the assumption f1×f2 ≁ f3×f4 is automatic, and the Rankin–Selberg convolution (sym²f1⊞1)×(f2×f4) has no pole at s=1, so the displayed partial sum should be o(X), not ≫X. The text seems to be describing what would happen if the overlap f1×f2 ∼ f3×f4 occurred. This needs to be rewritten clearly, because it is part of the verification that no overlap contribution appears in the fractional moment.
minor comments (5)
  1. [Lemma 2.4, proof] In the sentence 'we know that sym² f, sym² g, sym² g are self-dual cusp forms', the third entry should be sym² h.
  2. [Equation (2.19)] The displayed formula is ambiguous: it should be l² log(log x/log y) if that is the intended expression. As printed, l² log log x / log y can be misread as l² log(log x)/log y, which is not the identity used later.
  3. [Theorem 1.9 and Proposition 1.10] The statement of (1.7) does not exclude f=g, but the proof of Lemma 2.4 and (2.19) requires distinctness of the forms. If f=g, the moment in (1.7) reduces to a moment of L(1/2,sym²f×h), which is covered by (1.6) with one exponent zero; this should be said explicitly, or the theorem statement should include f≠g.
  4. [Theorem 1.11] The notation 'A=[1,A]' is confusing; it appears to mean the interval [1,A] but is written as a set containing the constant A. Also, the proof of Theorem 1.11 is only a sketch and should state more precisely how the upper bound from Theorem 1.4 implies the existence of a coordinate larger than the average order.
  5. [Throughout] The paper relies heavily on the unpublished preprint [14] both for Proposition 5.1 and for the two cases of Theorem 1.2 quoted at the start of Section 2.2. Since these results are load-bearing, the manuscript should explicitly list which statements are proved here and which are quoted from [14].

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the derivation is a conditional, parameter-free moment computation; the only serious weakness is reliance on unpublished preprint [14], which is a verifiability gap rather than a circular reduction.

full rationale

The paper's derivation chain is genuinely conditional and does not reduce its conclusions to its inputs by construction. Conjecture 1.1 is attacked through Watson's formula, which converts inner products of products of eigenforms into central values of L-functions, and the consequent moment estimates (1.6) and (1.7) are proved by Soundararajan's method with Gaussian-integral computations (for example, the shift mu(k1+k2)=(-1/2+epsilon)l log log(k1+k2) and the variance sigma(k1+k2)^2=l^2 log log(k1+k2)). No parameter is fitted to the target o(1) or to the logarithmic decay rates; those rates emerge from variance and Gaussian identities. Theorems 1.4 and 1.6 follow from Theorem 1.9 by Minkowski's inequality and a Fourier-coefficient argument, neither of which presupposes the conclusion. The manuscript does, however, contain a load-bearing external dependence: (1.6) is asserted to be a special case of [14, Proposition 5.1], and the two fractional-moment estimates completing Theorem 1.2 in Section 2.2 are asserted to 'follow directly from the proof of [14, Proposition 5.1]' without derivation. This is a serious completeness and verifiability issue, because [14] is an unpublished preprint and the estimates are the only argument for the remaining cases of Theorem 1.2. But it is not circularity: the estimates are not shown to be equivalent to the paper's assumptions, nor are they fitted to the theorem's conclusion, and no equation in the paper reduces a prediction to an input. Thus the honest finding is no significant circularity, with the caveat that the proof is not self-contained.

Assumptions & free parameters 0 free parameters · 8 assumptions · 0 invented entities

The central claims rest on GRH for several families of L-functions, two analytic-continuation hypotheses (including an open degree-27 triple symmetric-square product), GRC in one case, the distinctness condition f₁×f₂≁f₃×f₄, and on the unproved §2.2 moment estimates delegated to a collaborator's preprint. No free parameters are fitted to data; all exponents in the moment bounds are computed from Gaussian-integral heuristics. No new entities are introduced.

assumptions (8)
  • domain assumption GRH for L(s,h), L(s,f×g×h), L(s,sym²f), L(s,sym²g), L(s,sym²h), L(s,sym²f×sym²g), L(s,sym²f×sym²h), L(s,sym²g×sym²h), L(s,sym²f×sym²g×sym²h) as listed in Theorem 1.9.
    All ℓ^p bounds and decorrelation statements are conditional on these; invoked throughout §2 and §2.2.
  • domain assumption Analytic continuation of L(s, sym²f×sym²g×sym²h), the degree-27 triple product of three GL(3) symmetric-square forms.
    Open in general; used in Lemma 2.4 via Perron's formula to bound the prime sums (2.6)-(2.12), which feed Lemma 2.5 and Proposition 1.10.
  • domain assumption Analytic continuation of triple-product L-functions involving symmetric squares in the mixed cases of Theorem 1.2(iii).
    Theorem 1.2 hypothesis for cases #{f₁,f₂}=2 or #{f₃,f₄}=2 with unequal pairs.
  • domain assumption Generalized Ramanujan Conjecture when {f₁,f₂}={f₃,f₄}≠{f₁}.
    Theorem 1.2(i), inherited from Huang's [14] treatment of that case.
  • domain assumption f₁×f₂ ≁ f₃×f₄ when #{f₁,f₂,f₃,f₄}=4.
    Theorem 1.2(ii); needed to avoid overlap in the fractional moments. Stated explicitly, likely automatic but unproved.
  • domain assumption The two mixed fractional-moment estimates in §2.2 hold as stated (asserted to follow from [14, Prop 5.1]).
    These estimates are the unproved input for Theorem 1.2's remaining cases.
  • standard math Watson's Rankin triple product formula and the nonnegativity of central L-values (Lapid's theorem).
    Used in the proof of Theorem 1.9 to convert L-value moments into ⟨fg,h⟩ and ⟨f²,h⟩ bounds.
  • standard math Lau-Wu bounds (log log k)^{-1} ≪ L(1,sym²φ) ≪ (log log k)^3 under GRH.
    Cited as [17, Theorem 3]; used to absorb denominator factors in Theorem 1.9 and the proof of Theorem 1.6.

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Cite this review

Pith. "Pith review of Quadratic forms of modular forms." pith.science (2026). https://pith.science/paper/PFKS7ZBP

@misc{pith2026250721951,
  author       = {Pith},
  title        = {Pith review of: Quadratic forms of modular forms},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PFKS7ZBP}},
  note         = {Machine review of arXiv:2507.21951}
}
abstract

In this paper, we study quadratic forms in spaces of holomorphic cusp forms. We show, conditionally, that when two quadratic forms in Hecke eigenforms share no common diagonal terms, their inner product is expected to converge to the sum of the products of their common off-diagonal coefficients. This phenomenon could be interpreted as a mixed $L^4$-norm problem. We also define the $\ell^p$-norm of a holomorphic cusp form via its expansion with respect to an orthonormal Hecke basis. We then establish a conditional upper bound for the $\ell^p$-norm, and deduce that the coefficients of quadratic forms of holomorphic cusp forms in the Hecke basis are not uniformly small, being dominated by small-amplitude components. This behavior is consistent with the expected distribution of orthogonal families of $L$-functions.

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