{"id":"e298931a-9b48-4303-80bc-e235f6f43ed3","arxiv_id":"2508.16813","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"For random cusp forms, the expected sup norm on compact sets is of order sqrt(log k), and globally it is of order k^{1/4} times a logarithm.","lead":"A new random model for weight-k cusp forms on the modular surface shows that, on any fixed compact piece, the expected maximum height grows like the square root of log k. On the whole fundamental domain, the expected maximum is much larger, growing like k^{1/4} up to a log factor.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Proof of Proposition 3.1 evaluates a singular main term at z=w; the lower bound in Theorem 1.1(i) rests on this step.","rationale":"The reader's weakest_assumption focused on the global lower bound, specifically on the existence of a high-variance point z_k and the transfer E[sup] ≥ E[|h_k(z_k)|]. That step is actually sound: the point z_k exists (take y_k=(k−1)/(4π)), and the inequality is trivial. The reader also noticed the '2iy/(z−z)=1' typo but dismissed it as harmless. However, this typo is a symptom of a deeper problem: Theorem 3.3 is applied at z=w where its leading term is singular. This is directly load-bearing because Proposition 3.1's variance lower bound is the basis for the compact-domain lower bound E[M_k] ≫ √log k, a principal claim. The other major claims (global upper/lower bounds, concentration) rely on Theorem 3.4's cusp analysis, which is independent and appears solid. Therefore the weakest point in the paper is the diagonal evaluation in the bulk, not the global lower bound. The verdict remains CONDITIONAL because the gap may be fixable by clarifying the kernel's definition, but the proof as written is incomplete.","tokens_in":22743,"tokens_out":20567,"duration_ms":214879,"concrete_test":"Re-derive R_k(z,z) for a bulk point z (e.g., z=2i) and small even k (e.g., k=12) directly from the sum ∑_γ ℓ_γ(z,z)^k using the manuscript's definition ℓ_γ(z,w)=√(yv)b_γ(z,−w), and compare with the finite left side of (3.8). If the sum diverges or disagrees, check whether replacing −w by \\bar{w} in the definition yields the stated constant in (3.6). As a second check, consult [25, Theorem 2.15] to confirm whether the reproducing kernel should be B_k(z,\\bar{w}) and whether the formula for ℓ_γ should contain z−\\bar{w} rather than z−w.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The proof of Proposition 3.1 sets z=w in Theorem 3.3, whose main term is 2(2i√(yv)/(z−w))^k. At z=w this term is undefined, and the text 'noting 2iy/(z−z)=1' is not a valid substitution since z−z=0. The natural correction z−\\bar{z}=2iy does not match the denominator (z−w). This indicates a likely sign/bar error in the definition of ℓ_γ (possibly R_k(z,w) should involve B_k(z,\\bar{w}) rather than B_k(z,−w)). The identity (3.8) linking the covariance to R_k is stated without proof, and Proposition 3.1's variance asymptotics (3.6), used for the compact-domain lower bound E[M_k] ≫ √log k in Theorem 1.1(i), depends on this questionable diagonal evaluation. If the singular main term cannot be interpreted as a limit, the lower bound lacks a valid proof.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces a random weight-k cusp form ensemble for SL2(Z) by taking coefficients uniformly on the unit sphere in an orthonormal basis of the Petersson-normed space S_k, and writes h_k(z)=y^{k/2}g_k(z). The main results are: (i) for any fixed compact subdomain K of the modular fundamental domain, the expected supremum of |h_k| over K is of order sqrt(log k), with exponential concentration about the median; (ii) for the global supremum over the whole fundamental domain, the expected value satisfies k^{1/4} << E[M_k^g] << k^{1/4} sqrt(log k), again with exponential concentration, at a scale k^{1/2} in the exponent. The proofs proceed by analyzing the covariance kernel r_k(z,w) through the Bergman kernel, deriving diagonal variance estimates on the bulk and near the cusp, then using exact spherical L^p moments, Lévy concentration, and L^p-to-sup comparison arguments.","tokens_in":22948,"tokens_out":19735,"duration_ms":236331,"significance":"If the results are correct, the paper settles the true order of the expected supremum for random cusp forms on compacta, matching the square-root-logarithmic behavior known for random waves and supporting the conjectured sup-norm bounds for Hecke forms. The global result, with the cusp contribution k^{1/4} up to a logarithmic factor, is a new and natural analogue of Xia's bounds for Hecke forms. The covariance-kernel estimates, in particular the near-cusp analysis, are of independent interest. The exposition is largely transparent, with explicit error terms and no fitted parameters. However, one load-bearing diagonal evaluation is not justified as written and must be fixed before the compact lower bound is established.","major_comments":[{"comment":"The proof of Proposition 3.1 sets z=w in Theorem 3.3, whose main term is 2(2i sqrt(yv)/(z-w))^k. At z=w this term is undefined. The text 'noting 2iy/(z-z)=1' is not a valid substitution, since z-z=0. The identity (3.8) then does not yield the stated diagonal variance (3.6). This is load-bearing: the lower bound in Theorem 1.1(i) uses (3.6) through (5.1)-(5.3). The natural correction is to replace w by \\bar{w} in the definition of R_k and in the main term of Theorem 3.3, so that the gamma-I contribution is 2i sqrt(yv)/(z-\\bar{w}) and at z=w it equals 1. Then R_k(z,z)/2 -> 1 and (3.6) follows. The subsequent estimates in the proof of Theorem 3.3 appear to go through with this replacement because they only depend on |w| and |z-w|. As submitted, however, Proposition 3.1 has no valid proof, and (3.8)-(3.9) need to be rechecked with the corrected conjugation.","section":"§3.2, Eq. (3.7), Theorem 3.3, and Proof of Proposition 3.1"}],"minor_comments":[{"comment":"The function f(t) is defined as t^{k-1} e^{-4\\pi t} 1_{(0,\\infty)}(t), but the Fourier transform is computed with e^{-4\\pi y t}. The definition should be f(t)=t^{k-1} e^{-4\\pi y t}. In the line after (3.21), 'e^{-4\\pi n y}' appears to be a typo for 'e^{-4\\pi y m}'.","section":"Lemma 3.5"},{"comment":"In the choice of z_k, the text says 'x_k \\in (0,1) arbitrary', but the fundamental domain requires |x_k| \\le 1/2. The intended statement is probably x_k \\in (-1/2,1/2), and one may simply set x_k=0. The existence of y_k is fine, e.g. y_k=(k-1)/(4\\pi).","section":"§5.1, lower bound of Theorem 1.3(i)"},{"comment":"The title of §5 reads 'Proof of Theorem 1.1(i.) and Theorem 1.3(ii.)', but the global lower bound is Theorem 1.3(i). Also, Theorem 3.3 is referred to as 'Theorem 3.3(i.)' without a labelled part (i).","section":"Section title and theorem labels"},{"comment":"The equality in law G_k(z) = ζ · g_k(z) is stated for each fixed z, which is sufficient for the paper's use, but since G_k is a field the notation could suggest equality of processes. Clarifying 'pointwise in z' would be helpful.","section":"§2.2.2, Eq. (2.7)"}],"recommendation":"major_revision","confidential_remarks":"The central issue is almost certainly a conjugation/bar typo: replacing w by \\bar{w} in (3.7) and Theorem 3.3 repairs the diagonal evaluation and seems consistent with all later estimates. If the authors confirm this correction and re-derive (3.8)-(3.9) accordingly, the paper should be acceptable. The global lower bound and the concentration results do not depend on the problematic diagonal evaluation, so I do not see a circularity or novelty problem. The manuscript fits the journal's scope."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"This paper earns its keep. It determines the true order of magnitude of the expected supremum of random weight-k cusp forms on compacta: E[M_k] ≍ sqrt(log k), matching the random-wave heuristic. The global expected supremum is pinned to k^{1/4} up to a sqrt(log k) factor, with concentration around the median in both regimes. The proof strategy is coherent: covariance kernel analysis via the Bergman kernel, L^p estimates, and a Gaussian comparison for the global lower bound. The off-diagonal Bergman asymptotics (Theorem 3.3) and the near-cusp analysis (Theorem 3.4) are genuinely new and will be useful beyond this paper.\n\nThe soft spot is a cluster of typos around the diagonal evaluation. In the proof of Proposition 3.1, \"noting 2iy/(z−z)=1\" is evidently meant to be \"2iy/(z−\\bar{z})=1\". More substantially, Theorem 3.3 states the main term as 2(2i√(yv)/(z−w))^k, which is singular at z=w; the proof derives this from the chosen definition of ℓ_γ. The natural correction is to replace z−w by z−\\bar{w} throughout the statement and the proof's leading-term computation. With that correction, R_k(z,z) = 2 + O(...), which is exactly what Proposition 3.1 needs. As printed, the proof of the compact-domain lower bound is missing a valid evaluation step. I do not think this is a real gap: the corrected formula is standard Bergman kernel asymptotics, and the rest of the argument is independent. But a referee should insist the authors state the theorem correctly and show the diagonal limit carefully.\n\nEverything else holds up. Proposition 3.2's variance bounds near the cusp are careful, and the global lower bound's choice of y_k with (k−1)/(4πy_k) integer is valid (take y_k = (k−1)/(4π)). The concentration arguments are standard Lévy plus L^p interpolation, executed cleanly. The citation pattern is honest; the unproven sqrt(log k) gap between the global upper and lower bounds is acknowledged.\n\nWho is this for? Anyone working on sup norms of automorphic forms, random waves, or Bergman kernel asymptotics. It deserves a serious referee. Recommend acceptance pending the typo fix.","headline":"Resolves the order of magnitude for random cusp form suprema; main gap is a fixable sign/bar typo in the Bergman kernel diagonal evaluation.","tokens_in":23519,"tokens_out":10426,"would_cite":true,"duration_ms":98195,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11F11","60G60"],"pacs":[],"model":"deepseek-v4-flash","headline":"The expected maximum of a random weight-k cusp form is of order √(log k) on fixed compact sets and of order k^{1/4} (up to a logarithm) over the whole modular surface, with exponential concentration in both regimes.","keywords":["random cusp forms","sup norm","Bergman kernel","covariance kernel","exponential concentration","modular group","expected supremum","large weight"],"falsifier":"Take the model literally and simulate it exactly: for k = 24, 48, …, 400, draw coefficients uniformly on the unit sphere of C^N and evaluate y^{k/2}|Σ a_j f_j(z)| on a fine grid over the fundamental domain up to height y = k/2. The global claim predicts the sample mean of the maximum stays between c k^{1/4} and C k^{1/4}√(log k) with absolute constants; a mean falling below c k^{1/4} at growing k would kill the single-resonant-point lower bound. Separately, compute the exact diagonal variance y^k Σ|f_j(iy)|²: the predicted sharp peaks of size ≍√k at heights y = (k−1)/(4πn) and O(1) values else","tokens_in":22629,"feed_emoji":"🎲","tokens_out":19502,"duration_ms":195586,"temperature":0.7,"pith_summary":"Take a weight-k cusp form for the modular group — a holomorphic function on the upper half-plane that transforms with a fixed power law and decays at the cusp — and draw its coefficients in an orthonormal basis uniformly from the unit sphere. The paper determines, up to constants, how large the invariant function y^{k/2}|g_k(z)| gets. On any fixed compact piece of the modular surface the expected maximum is of order √(log k), with the whole distribution concentrated around its median on the scale of a constant; over the entire surface the expected maximum jumps to order k^{1/4} (up to a √(log k) factor) and is attained near height y ≈ k/(4π), close to the cusp. The interest is that the compact rate matches the conjectured k^ε sup-norm bounds for individual Hecke forms, while the global rate reproduces, for a generic random form, the k^{1/4±ε} scale that is provable only on average for the hardest deterministic forms.","feed_headline":"Random cusp forms peak at √log k on compacta, k^1/4 globally","feed_subtitle":"On the whole modular surface the peak grows like k^1/4 up to a log factor; both maxima concentrate tightly.","key_machinery":"The covariance kernel r_k(z,w) of the invariant random field h_k(z)=y^{k/2}g_k(z), a constant multiple of the weight-k Bergman kernel R_k(z,w)=Σ_{γ∈SL2(Z)} ℓ_γ(z,w)^k, the reproducing kernel of the cusp-form space. The argument asymptotically evaluates this kernel: away from elliptic points and the cusp, γ=±I dominates, giving R_k(z,w)=2(2i√(yv)/(z−w))^k plus exponential error; near the cusp, Γ_∞ dominates and Poisson summation turns the kernel into 2(4πy)^k/Γ(k) Σ m^{k−1}e^{−4πmy}, analyzed by Laplace's method. These variance estimates feed exact L^p expectations (p≈log k) for the upper bounds and the spherical concentration inequality for the exponential tails; the global lower bound rests","core_discovery":"Random cusp-form maxima are governed, to leading order, by the diagonal of the covariance kernel. In the bulk of the modular surface the variance of h_k = y^{k/2}g_k is nearly constant — (k−1)/(4πN) plus exponentially small errors — and exact L^p formulas with p ≈ log k give E[M_k] ≍ √(log k) on every fixed compact set. Near the cusp the variance fluctuates arithmetically: about 1 at low heights, but ≍ √k at heights y with (k−1)/(4πy) near an integer, in particular y ≈ k/(4π). One such resonant point already has expected modulus ≫ k^{1/4} (via the Gaussian identity Z = ζ·h_k), forcing the global expected supremum to k^{1/4}, while the same machinery gives k^{1/4}√(log k) from above. Concentr","pith_inferences":["Because Theorem 3.3 pins down the full off-diagonal covariance to leading order (a |z−w|^{−k} decay in the bulk), a Rice-type computation becomes a routine follow-up that the paper does not carry out: the two-point correlation function of zeros of g_k in the bulk should match that of the Gaussian field with the same covariance kernel.","The variance peaks at heights y ≈ (k−1)/(4πn) have relative width about 1/√k, so the set of 'large-variance' heights occupies a sparse, arithmetically defined set; my reading is that the global maximum is a deterministic-resonance phenomenon — conditioned on being near such a peak, the local maximum should already be of order k^{1/4}, which Monte Carlo simulation can test directly.","The methods allow the compact domain to grow to height C√k while keeping the √(log k) rate; this suggests a threshold phenomenon the paper does not address — the expected supremum should begin climbing from √(log k) toward k^{1/4} only for heights substantially beyond √k, and locating that transition is a concrete open test."],"forward_implications":["On any fixed compact set the random sup norm is almost surely O(√(log k)) (independent draws across k, Borel–Cantelli), in line with the conjectured k^ε sup-norm bound for Hecke forms.","Globally the random sup norm is almost surely O(k^{1/4}√(log k)), and with high probability the maximum sits near height y ≈ k/(4π), the location of maximal variance.","The compact-domain estimates extend to domains growing polynomially with k, up to height C√k, with the √(log k) rate unchanged.","The results transfer to Gaussian coefficients and to real coefficients; the spherical ensemble is linked to its Gaussian counterpart by a chi-distributed factor ζ with mean 1+O(1/N).","The paper's heuristic analysis of the high-variance 'island' near y ≈ k/(4π) suggests the true global order is exactly k^{1/4}, i.e. the √(log k) gap in the proved bounds is likely an artifact of the method."],"supporting_citations":[{"why":"Supplies the reproducing-kernel identity expressing the Bergman kernel of S_k, which yields the covariance kernel formula (3.9), and the global sup-norm bound k^{3/4} used in the global concentration argument.","marker":"[25]"},{"why":"Contributes the strategy of expanding the Bergman kernel as a sum over SL2(Z) and isolating dominant terms, refined here with uniform exponential error terms.","marker":"[8]"},{"why":"Provides the hyperbolic lattice-point counting bound that controls the non-identity terms in the Bergman kernel expansion.","marker":"[14]"},{"why":"The Gamma-integral identity used with Poisson summation to evaluate the cuspidal part of the kernel as the series 2(4πy)^k/Γ(k) Σ m^{k−1}e^{−4πmy}.","marker":"[13]"},{"why":"The compact sup-norm bound sup_K y^{k/2}|f| ≪_K k^{1/2}‖f‖₂, used to set the Lipschitz constant for the concentration inequality and the L^p interpolation on compacta.","marker":"[21]"},{"why":"The Hecke-form benchmark k^{1/4±ε} for global sup norms that Theorem 1.3 is compared against and matches up to a logarithm.","marker":"[26]"},{"why":"The template for the method: concentration inequalities for random linear combinations plus L^p-norm estimates to control expected suprema, adapted here to modular forms.","marker":"[7]"},{"why":"States the standard concentration inequality for Lipschitz functions on the sphere, used to prove exponential tails and the median–mean comparisons.","marker":"[17]"}],"fun_headline_variants":["Cusp form max: √log k locally, k^1/4 globally","Random cusp forms spike to k^1/4 at cusp","Supremum: √log k compacta, k^1/4 globally","Cusp forms: compact peak √log k, cusp peak k^1/4"],"cache_read_input_tokens":2688,"weakest_assumption_plain":"The global k^{1/4} lower bound rests on one carefully placed point z_k at height y_k ≈ k/(4π) where (k−1)/(4πy_k) is an exact integer: the paper needs the pointwise variance there to be genuinely ≫√k and the one-point expectation E|h_k(z_k)| to inherit the Gaussian lower bound through the comparison Z = ζ·h_k. If either the variance estimate at that single point or that Gaussian transfer fails, the k^{1/4} lower bound has no other support.","fun_headline_variants_meta":{"raw":{"variants":["Cusp form max: √log k locally, k^1/4 globally","Random cusp forms spike to k^1/4 at cusp","Supremum: √log k compacta, k^1/4 globally","Cusp forms: compact peak √log k, cusp peak k^1/4"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000626,"raw_usage":{"total_tokens":2702,"prompt_tokens":685,"completion_tokens":2017,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":429,"completion_tokens_details":{"reasoning_tokens":1930}},"tokens_in":429,"tokens_out":2017,"duration_ms":15947,"temperature":1.0,"reasoning_tokens":1930,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-05T17:07:48.471747+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take the model literally and simulate it exactly: for k = 24, 48, …, 400, draw coefficients uniformly on the unit sphere of C^N and evaluate y^{k/2}|Σ a_j f_j(z)| on a fine grid over the fundamental domain up to height y = k/2. The global claim predicts the sample mean of the maximum stays between c k^{1/4} and C k^{1/4}√(log k) with absolute constants; a mean falling below c k^{1/4} at growing k would kill the single-resonant-point lower bound. Separately, compute the exact diagonal variance y^k Σ|f_j(iy)|²: the predicted sharp peaks of size ≍√k at heights y = (k−1)/(4πn) and O(1) values else","supporting_citations":[{"cited_title":"Uniform bounds on sup norms of holomorphic forms of real weight","cited_arxiv_id":null,"evidence_quote":"Supplies the reproducing-kernel identity expressing the Bergman kernel of S_k, which yields the covariance kernel formula (3.9), and the global sup-norm bound k^{3/4} used in the global concentration argument."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Contributes the strategy of expanding the Bergman kernel as a sum over SL2(Z) and isolating dominant terms, refined here with uniform exponential error terms."},{"cited_title":"Spectral Methods of Automorphic Forms","cited_arxiv_id":null,"evidence_quote":"Provides the hyperbolic lattice-point counting bound that controls the non-identity terms in the Bergman kernel expansion."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"The Gamma-integral identity used with Poisson summation to evaluate the cuspidal part of the kernel as the series 2(4πy)^k/Γ(k) Σ m^{k−1}e^{−4πmy}."},{"cited_title":"On the asymptotic distribution of zeros of modular forms","cited_arxiv_id":null,"evidence_quote":"The compact sup-norm bound sup_K y^{k/2}|f| ≪_K k^{1/2}‖f‖₂, used to set the Lipschitz constant for the concentration inequality and the L^p interpolation on compacta."},{"cited_title":"On L∞ norms of holomorphic cusp forms","cited_arxiv_id":null,"evidence_quote":"The Hecke-form benchmark k^{1/4±ε} for global sup norms that Theorem 1.3 is compared against and matches up to a logarithm."},{"cited_title":"and Lebeau, G","cited_arxiv_id":null,"evidence_quote":"The template for the method: concentration inequalities for random linear combinations plus L^p-norm estimates to control expected suprema, adapted here to modular forms."},{"cited_title":"The concentration of measure phenomenon","cited_arxiv_id":null,"evidence_quote":"States the standard concentration inequality for Lipschitz functions on the sphere, used to prove exponential tails and the median–mean comparisons."}],"review_version":1}