{"id":"e2e094ea-5af1-42ce-8878-f2800a0ed1e5","arxiv_id":"2607.13518","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For Heegner points, the spectral fourth moment of Hecke–Maass cusp forms in a dyadic interval is O(T^{2+ε}), matching the Lindelöf-on-average prediction.","lead":"This paper proves an essentially optimal bound for the fourth moment of Hecke–Maass cusp forms at Heegner points, settling a 1999 conjecture of Chamizo in this special case. It also makes two applications: an unconditional error term for shifted sums of the two-squares function and optimal pointwise Diophantine exponents for Heegner points.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lemma 2.4's asserted bound on local toric-period integrals is unproved; the main theorem depends on it, so the proof as written is incomplete.","rationale":"The paper's central result, Theorem 1.2, is a fourth-moment bound for Hecke–Maass forms at Heegner points. The proof follows the standard automorphic-template: Waldspurger's formula (Lemma 2.4), an approximate functional equation (Lemma 2.2), and the spectral large sieve (Lemma 2.1). The reader identified Lemma 2.4's local-integral bound as the weakest assumption; I agree. Without a rigorous proof of I_{S,D}(g·φ_A, ξ_A) ≪_{q,D} 1, the connection between |φ(z0)|² and L-values is not established. The gap is real, but it is likely fillable: the archimedean torus is compact, so a unitary matrix coefficient is bounded pointwise by 1 and the integral over T(R) is bounded by the torus volume; the non-archimedean factors are finite sums of normalized matrix coefficients, which should be O_p(1) with fixed conductor. The authors' reference to [BBK, Lemma 10.3] and [MW] suggests they know the details but chose to omit them. This supports a CONDITIONAL verdict rather than REJECT. The secondary issue of Lemma 2.1's missing spectral cutoff is a typo, not a substantive flaw. No other hidden step appears more fragile: the Cauchy–Schwarz reductions, the application of the large sieve, and the counting arguments in Section 4 are standard and sufficiently detailed. Thus, the reader's assessment is accurate and no verdict change is needed.","tokens_in":19081,"tokens_out":13501,"duration_ms":134800,"concrete_test":"Write out the local integrals I_{S,D}(g·φ_A, ξ_A) explicitly for each place in S, using [MW, §§3–4] and [BBK, Lemma 10.3]. For the archimedean place, express I_∞ as an integral over the compact torus T(R)≅S^1 of a normalized matrix coefficient and verify |I_∞| ≤ vol(T(R)) ≪_D 1, independent of t_π. For each p|qD, verify that the local integral is a finite sum of normalized matrix coefficients, each bounded by O_p(1), independent of t_π. If any factor has t_π-dependence, Lemma 2.4 fails and Theorem 1.2 must be weakened accordingly.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The proof of Theorem 1.2 rests on Lemma 2.4, which converts the point evaluation |φ(z0)|² into a sum of central L-values. That lemma requires the local integral bound I_{S,D}(g·φ_A, ξ_A) ≪_{q,D} 1. The paper says this can be achieved by trivial estimation or by citing [BBK, Lemma 10.3]/[MW], but the details are omitted. This is not a cosmetic omission: if any local factor grows with the spectral parameter t_π, Lemma 2.4 would be false and the final T^{2+ε} bound would degrade to T^{2+A+ε} for some A>0. In particular, the archimedean factor must be checked for t_π-independence; while T(R) is compact for D<0, the matrix coefficient of a Maass form of large t_π could in principle be unbounded on that torus, and the non-archimedean factors at split primes require a uniform bound on sums of matrix coefficients. The authors point to existing literature, but the application is not demonstrated. Thus the central claim is conditional on a missing verification. (Secondary: Lemma 2.1 as stated omits the spectral restriction |t_π|≤T; without it the displayed inequality is false. The intended truncated version is standard and is what the proof of Theorem 1.2 actually uses, so this is fixable and not the primary obstacle.)","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":19540,"tokens_out":8552,"duration_ms":82858,"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":[{"comment":"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.","section":"§2.4, Lemma 2.4"},{"comment":"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).","section":"§2.2, Lemma 2.2"},{"comment":"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.","section":"§2.1, Lemma 2.1"}],"minor_comments":[{"comment":"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'.","section":"§1.1"},{"comment":"The displayed sum on the right has a typo: 'X_{N<n≤N}' should be 'X_{N<n≤2N}'.","section":"§2.1, proof of Lemma 2.1"},{"comment":"The final sentence says 'so that the desired results follow from Theorem 1.3'; this should refer to Theorem 1.2.","section":"§1.2, proof of Theorem 1.3"},{"comment":"The notation Q^+_{1,-4} is used without definition; presumably it is the subset of positive-definite forms. Please define.","section":"§2.4, Example 2.3"},{"comment":"There is a mismatched parenthesis in 'For a Heegner point z0∈SL2(Z)\\H)'.","section":"§1.3, Corollary 1.7"},{"comment":"Reference [Sard] is formatted with an arXiv id followed by '. (2019)' which is inconsistent with the other entries; please clean up.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"The main theorem is very likely correct, and the omitted local computations in Lemma 2.4 and Lemma 2.2 are standard for experts. The authors should be encouraged to supply full details, especially because the paper's central claim depends on them. The false unrestricted statement of Lemma 2.1 is a minor but necessary correction. The paper is a good fit for a leading number theory journal once these points are addressed."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear [Colleague],\n\nShort version: Assing and Humphries have the right result — the T^{2+ε} fourth moment at Heegner points — and the overall strategy is sound, but one load-bearing step is asserted rather than proved. The paper as written is not complete; it is likely fixable.\n\nWhat's new: Theorem 1.2 is a genuinely new unconditional bound in the spectral aspect, matching Chamizo's Conjecture 1.1 for Heegner points. The prior state of the art was conditional (Chamizo) or level-aspect (Khayutin–Nelson–Steiner). The proof combines Waldspurger's formula, the approximate functional equation, and the spectral large sieve in a clean way. The applications to shifted r(n) correlations (Theorem 1.3) and pointwise Diophantine exponents (Corollary 1.7) are natural and work if the main theorem holds. I especially like the alternative proof of Theorem 1.10 via the pretrace formula and a quaternary quadratic-form count; that part is self-contained and convincing.\n\nThe soft spot is Lemma 2.4. To convert point evaluation into central L-values, the authors need I_{S,D}(g·φ_A, ξ_A) ≪ 1 uniformly in the spectral parameter. They write \"We omit the details.\" That is a real gap, not a rhetorical one. If the archimedean or non-archimedean local factors had even a small power of t_π growth, the final exponent would degrade from T^{2+ε} to something larger. The cited sources ([BBK, Lemma 10.3], [MW]) look like they should cover this, and I suspect a specialist can fill the gap in a few pages, but the proof as written is incomplete. A referee cannot verify Theorem 1.2 without either a direct proof of this bound or a precise identification of where it appears in the literature.\n\nThere is also a smaller issue: Lemma 2.1 is stated with no restriction on t_π, which is false as written; the proof of Theorem 1.2 needs the version with T ≤ t_π ≤ 2T, which is standard and follows from the displayed (12). That is a fixable typo, not a conceptual problem. Lemma 2.2 similarly defers \"straightforward\" modifications for non-cuspidal ξ; I would not flag that separately.\n\nIf I had to bet, the main theorem is true and the omissions are repairable. But \"likely true\" is not the same as \"proved\". The paper deserves a serious referee; it should go out for review with a request to fill in Lemma 2.4.\n\nBest,\n[You]","headline":"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.","tokens_in":19900,"tokens_out":4365,"would_cite":true,"duration_ms":41877,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11F72","11F67","11F66"],"pacs":[],"model":"deepseek-v4-flash","headline":"For Heegner points, the spectral fourth moment of Hecke–Maass cusp forms is essentially T^{2+ε}.","keywords":["Hecke–Maass cusp forms","Heegner points","spectral fourth moment","Waldspurger formula","Lindelöf on average","shifted convolution","sum of two squares","Diophantine exponents"],"falsifier":"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.","tokens_in":19016,"feed_emoji":"🔢","tokens_out":9298,"duration_ms":85153,"temperature":0.7,"pith_summary":"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.","feed_headline":"Spectral fourth moment at Heegner points is essentially T^2","feed_subtitle":"Waldspurger plus the spectral large sieve confirms Chamizo's conjecture for Heegner points, with news for r(n) and Diophantine exponents.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"fun_headline_variants":["Fourth moment at Heegner points is essentially T^2","Chamizo's conjecture confirmed for Heegner points","Sharp spectral moments yield new r(n) correlation bounds","Optimal Hecke–Maass moment: T^2, with Diophantine payoff","Essentially optimal fourth moment: T^2 at Heegner points"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Fourth moment at Heegner points is essentially T^2","Chamizo's conjecture confirmed for Heegner points","Sharp spectral moments yield new r(n) correlation bounds","Optimal Hecke–Maass moment: T^2, with Diophantine payoff","Essentially optimal fourth moment: T^2 at Heegner points"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000794,"raw_usage":{"total_tokens":3296,"prompt_tokens":673,"completion_tokens":2623,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":417,"completion_tokens_details":{"reasoning_tokens":2532}},"tokens_in":417,"tokens_out":2623,"duration_ms":19809,"temperature":1.0,"reasoning_tokens":2532,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-02T04:56:21.077916+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[],"review_version":1}