{"id":"3f8828a2-79da-490f-82e6-f4baa6550168","arxiv_id":"2507.21951","paper_version":3,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Under GRH and analytic continuation hypotheses, the paper completes a decorrelation conjecture for products of Hecke eigenforms and proves new ℓ^p-norm bounds for quadratic forms in the Hecke basis.","lead":"This paper studies quadratic forms built from Hecke eigenforms, proving conditional results on when their inner products decorrelate. It also introduces ℓ^p norms of cusp forms in the Hecke basis and bounds them under the Riemann Hypothesis.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 1.2's remaining cases rest on two fractional-moment estimates in §2.2 that are asserted to follow from unpublished [14, Prop 5.1] with no derivation; until those estimates are proven, the completion of Conjecture 1.1 is not established.","rationale":"The paper's central claim is Theorem 1.2, the conditional completion of Conjecture 1.1. The theorem explicitly depends on the two fractional-moment estimates in §2.2, and those estimates are not proven in the manuscript: they are delegated to an unpublished preprint by a collaborator. The reader's weakest_assumption identifies exactly this gap, and the manuscript itself states the delegation, so this is not a manufactured concern. I considered the other flagged issues: the Minkowski inequality for 0<p<1 is mis-invoked but the error is absorbable by the finite N,B constants; the analytic continuation of triple-product L-functions is a genuine assumption but it is explicitly stated, not an internal gap; the lower-bound step in Theorem 1.6 has an apparent size error but the intended contradiction can be repaired using the standard GRH lower bound L(1,sym²φ) ≫ (log log k)^{-1}. None of these affects the main completion of Conjecture 1.1 as directly as the unproven §2.2 estimates. A verdict of CONDITIONAL is therefore appropriate: the framework and the surrounding arguments are coherent, but the two estimates that carry the remaining cases must either be proved in the paper or shown to be immediate special cases of a publicly available proof of [14, Proposition 5.1]. If the requested derivation were supplied and the exponents verified, the central claim would be substantially reinforced.","tokens_in":12859,"tokens_out":15183,"duration_ms":165742,"concrete_test":"Obtain the full proof of [14, Proposition 5.1] or have the author write out the two §2.2 estimates from first principles. Specifically, re-run the Soundararajan variance computation for the polynomial P(h)=Σ_{p≤x} (λ_{f1}(p)λ_{f2}(p)+λ_{f3}(p)λ_{f4}(p)) p^{-1/2-1/log x}(1−log p/log x) and check that the Gaussian integral yields exactly log^{-1/4+2ε}(k1+k2); if any cross-term survives under f1×f2≁f3×f4, the exponent worsens and Theorem 1.2's remaining cases are unproved. For the first estimate, recompute with P(h) incorporating λ_{sym²f1}(p)+λ_{f3×f4}(p)+λ_h(p), and verify the diagonal terms in the Petersson sieve reproduce log^{-3/8+2ε} k1.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Conjecture 1.1 is completed in Theorem 1.2 only by two displays in §2.2: (1/k1)Σ_{h∈H_{2k1}} L(1/2,h)^{1/2}L(1/2,sym²f1×h)^{1/2}L(1/2,f3×f4×h)^{1/2} ≪ log^{-3/8+2ε} k1, and the analogous two-factor estimate with L(1/2,f1×f2×h)^{1/2}L(1/2,f3×f4×h)^{1/2} ≪ log^{-1/4+2ε}(k1+k2). The text says both 'follow directly from the proof of [14, Proposition 5.1]', where [14] is an unpublished preprint. These estimates are the only argument for the cases f3≠f4 and f1≠f2, f3≠f4 of Theorem 1.2; if either estimate is false or has a weaker logarithmic exponent, the o(1) conclusion can fail. No derivation is given of the variance computation for the mixed Dirichlet polynomial, of the treatment of two half-power L-factors, or of the prime-sum estimates involving λ_{sym²f3}(p)λ_{sym²f4}(p)λ_{sym²h}(p). The surrounding lemmas (2.2–2.5) are proven only for the single f×g×h moment; the transition to products of two different fractional L-functions is not automatic and requires checking diagonal and overlap terms in the Petersson sieve. This is an omitted proof, not merely an unproved hypothesis: GRH and analytic continuation are explicitly assumed, but the derivation of the two estimates from those assumptions is not supplied.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":1872,"tokens_out":2595,"duration_ms":207217,"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":[{"comment":"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.","section":"§2.2, proof of Theorem 1.2"},{"comment":"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.","section":"Lemma 2.4 and Proposition 1.10(1.7)"},{"comment":"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.","section":"§2.2, case #{f1,f2,f3,f4}=3"}],"minor_comments":[{"comment":"In the sentence 'we know that sym² f, sym² g, sym² g are self-dual cusp forms', the third entry should be sym² h.","section":"Lemma 2.4, proof"},{"comment":"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.","section":"Equation (2.19)"},{"comment":"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.","section":"Theorem 1.9 and Proposition 1.10"},{"comment":"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.","section":"Theorem 1.11"},{"comment":"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].","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"The central issue is the unpublished status of [14] and the two unproved fractional-moment estimates in Section 2.2. If these estimates cannot be supplied, the author should restate Theorem 1.2 with them as hypotheses; otherwise the claim of completing Conjecture 1.1 is stronger than what is proved. The editor may also wish to verify that [14] is publicly available and that the results quoted from it are stated correctly."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"I'd say this paper is genuinely worth engaging with, but it has one load-bearing gap that needs to be named plainly. The main result, Theorem 1.2, completes Conjecture 1.1 in the mixed cases only because of two fractional-moment estimates in §2.2, and those are asserted to follow from the proof of [14, Proposition 5.1] with no derivation. That's not a small omission: the cases f3≠f4 and f1≠f2, f3≠f4 collapse if either estimate has a weaker logarithmic exponent or an extra overlap term. The lemmas in §2.1 are proven for a single f×g×h moment; moving to products of two fractional L-functions requires checking the Petersson sieve diagonal and the off-diagonal contributions, and that work is simply not shown. Since [14] is an unpublished preprint, the reader cannot verify the claim by looking it up. This is the first thing I'd ask the author to fix.\n\nWhat the paper does well: the ℓ^p-norm framework is new, and Theorem 1.4 plus the applications (Theorems 1.6 and 1.11) give a clean way to see that Hecke-basis coefficients of quadratic forms cannot all be uniformly small. The proof of Proposition 1.10 in §2.1 is detailed and follows standard Soundararajan machinery correctly—Watson's formula, the Perron/Littlewood bounds, the tail estimates via Lemma 2.5. The mixed moment (1.7) is a real contribution, and the dependence on GRH and analytic continuation is spelled out honestly. No data fitting or circularity here; the exponents come from actual Gaussian integrals.\n\nSmaller issues: the proof of Theorem 1.6 contains a line where the displayed lower bound gives |c_{r0}| ≫ |a| L^{-1} (log log k)^{-1}, not the claimed ≫ |a| L log^ε k; the intended contradiction with Theorem 1.4 can probably be repaired, but the exponent as written is wrong. Corollary 1.8 needs a pairwise-distinctness condition on the f_i; without it the RHS is off by constant factors. And Minkowski's inequality is invoked for 0<p<1, where it fails; the error is absorbable, but it should be flagged or avoided. None of these are fatal; the §2.2 gap is the only real obstacle.\n\nWho is this for? Analytic number theorists working on L-functions, moments, and the distribution of automorphic forms. It deserves a serious referee, but the referee should demand a full derivation of the two §2.2 estimates before recommending acceptance. I'd send it out.","headline":"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.","tokens_in":13861,"tokens_out":1928,"would_cite":false,"duration_ms":23045,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11F12","11F30","11F66"],"pacs":[],"model":"deepseek-v4-flash","headline":"Assuming GRH, products of Hecke eigenforms decorrelate unless their pair sets match.","keywords":["modular forms","quadratic forms","Hecke eigenforms","decorrelation","L-functions","moments","ℓ^p norms","GRH"],"falsifier":"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.","tokens_in":12375,"feed_emoji":"🧮","tokens_out":12145,"duration_ms":125567,"temperature":0.7,"pith_summary":"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.","feed_headline":"Modular-form products decorrelate unless pairs match","feed_subtitle":"Conditional proof completes a conjecture on inner products of quadratic forms","key_machinery":"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].","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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)."],"supporting_citations":[{"why":"Supplies the two previously known cases of Conjecture 1.1 and the proposition whose proof is quoted for the remaining fractional-moment estimates.","marker":"[14]"},{"why":"Provides the moment method for L-functions used to prove Proposition 1.10.","marker":"[22]"},{"why":"Triple-product formula expressing $\\langle fg,h\\rangle$ through central L-values, the bridge from the inner-product problem to moments.","marker":"[24]"},{"why":"Orthogonality formula for Hecke eigenvalues averaged over the basis, used in Lemma 2.2.","marker":"[21]"},{"why":"Explicit bounds for L-functions on the critical line used in Lemma 2.3 to bound $\\log L(1/2,f\\times g\\times h)$.","marker":"[5]"},{"why":"Nonnegativity of central L-values, needed to apply the triple-product formula.","marker":"[16]"},{"why":"Bounds for $L(1,\\operatorname{sym}^2\\phi)$ under GRH, used to normalize the moment estimates.","marker":"[17]"},{"why":"Shows distinct symmetric-square lifts on $\\mathrm{GL}(3)$ are non-isomorphic, used in Lemma 2.4.","marker":"[20]"},{"why":"Establishes that the symmetric-square lift of a holomorphic eigenform is an automorphic cusp form on $\\mathrm{GL}(3)$.","marker":"[9]"}],"fun_headline_variants":["Quadratic forms decorrelate unless diagonal terms match","Modular form quadratic forms: no correlation without shared diagonals","Conditional proof: Hecke eigenform products decorrelate unless pairs align","Cusp form quadratic forms: decorrelation away from common diagonal terms","Matching diagonal entries tie quadratic forms in Hecke basis"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Quadratic forms decorrelate unless diagonal terms match","Modular form quadratic forms: no correlation without shared diagonals","Conditional proof: Hecke eigenform products decorrelate unless pairs align","Cusp form quadratic forms: decorrelation away from common diagonal terms","Matching diagonal entries tie quadratic forms in Hecke basis"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000457,"raw_usage":{"total_tokens":2303,"prompt_tokens":964,"completion_tokens":1339,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":580,"completion_tokens_details":{"reasoning_tokens":1252}},"tokens_in":580,"tokens_out":1339,"duration_ms":11432,"temperature":1.0,"reasoning_tokens":1252,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T12:19:03.568790+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Moments of the Riemann zeta function.Ann","cited_arxiv_id":null,"evidence_quote":"Provides the moment method for L-functions used to prove Proposition 1.10."},{"cited_title":"Rankin Triple Products and Quantum Chaos","cited_arxiv_id":"0810.0425","evidence_quote":"Triple-product formula expressing $\\langle fg,h\\rangle$ through central L-values, the bridge from the inner-product problem to moments."},{"cited_title":"Lower bounds for moments ofL-functions: symplectic and orthogonal examples.Multiple Dirichlet series, automorphic forms, and analytic number theory,293– 303, Proc","cited_arxiv_id":null,"evidence_quote":"Orthogonality formula for Hecke eigenvalues averaged over the basis, used in Lemma 2.2."},{"cited_title":"Explicit upper bounds forL-functions on the critical line.Proc","cited_arxiv_id":null,"evidence_quote":"Explicit bounds for L-functions on the critical line used in Lemma 2.3 to bound $\\log L(1/2,f\\times g\\times h)$."},{"cited_title":"On the nonnegativity of Rankin-SelbergL-functions at the center of symmetry.Int","cited_arxiv_id":null,"evidence_quote":"Nonnegativity of central L-values, needed to apply the triple-product formula."},{"cited_title":"A density theorem on automorphicL-functions and some applications.Trans","cited_arxiv_id":null,"evidence_quote":"Bounds for $L(1,\\operatorname{sym}^2\\phi)$ under GRH, used to normalize the moment estimates."},{"cited_title":"An exercise concerning the selfdual cusp forms on GL(3).Indian J","cited_arxiv_id":null,"evidence_quote":"Shows distinct symmetric-square lifts on $\\mathrm{GL}(3)$ are non-isomorphic, used in Lemma 2.4."},{"cited_title":"A relation between automorphic representations of GL(2) and GL(3)","cited_arxiv_id":null,"evidence_quote":"Establishes that the symmetric-square lift of a holomorphic eigenform is an automorphic cusp form on $\\mathrm{GL}(3)$."}],"review_version":1}