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On the supremum of random cusp forms

T0 review · 1 major / 4 minor · reviewed 2026-08-05 · deepseek-v4-flash

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Pith's one-line read 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.

desk verdict Resolves the order of magnitude for random cusp form suprema; main gap is a fixable sign/bar typo in the Bergman kernel diagonal evaluation. read the letter →

arxiv 2508.16813 v1 pith:2QYDDGX7 submitted 2025-08-22 math.NT math.CAmath.PR

Bingrong Huang , Stephen Lester , Igor Wigman , Nadav Yesha This is my paper · ORCID
classification math.NTmath.CAmath.PR MSC 11F1160G60
keywords randomcuspformssupnormBergmankernelcovarianceexponentialconcentrationmodulargroupexpectedsupremumlargeweight
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

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.

What carries the argument

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

What would settle it

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

Watch

Extended reading notes

Core claim

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

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

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

  • 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.
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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

1 major / 4 minor

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.

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 (1)
  1. [§3.2, Eq. (3.7), Theorem 3.3, and Proof of Proposition 3.1] 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.
minor comments (4)
  1. [Lemma 3.5] 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}'.
  2. [§5.1, lower bound of Theorem 1.3(i)] 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).
  3. [Section title and theorem labels] 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).
  4. [§2.2.2, Eq. (2.7)] 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.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: covariance estimates are derived from the Bergman kernel and lattice-point bounds; no fitted input is renamed as a prediction and no load-bearing self-citation chain appears. One flagged proof gap in Proposition 3.1 is a formal diagonal-evaluation issue, not a circular reduction.

full rationale

The derivation is self-contained against external benchmarks. The random ensemble is defined by uniform spherical coefficients (1.1)-(1.2); the covariance kernel r_k(z,w) is computed from the Petersson inner product and the Bergman kernel via (3.3), (3.7)-(3.9), with (3.8) taken from Steiner [25, Thm 2.15]—an external, independent source, not a self-citation. The variance estimates in Propositions 3.1 and 3.2 follow from the Bergman-kernel analysis of Theorem 3.3 (lattice-point bounds, hyperbolic counting, Poisson summation) and Theorem 3.4 through Lemmas 3.5-3.6, not from fitting. The upper bounds for E[M_k] and E[M^g_k] use L^p-norm estimates and known sup-norm bounds of Rudnick and Steiner; the compact lower bound uses the variance lower bound (3.6); the global lower bound uses an explicit point z_k with (k-1)/(4π y_k) ∈ Z, the Gaussian comparison (5.6)-(5.8), and the elementary inequality E[sup] ≥ E[|h_k(z_k)|]. No step equates a target result to an input by construction, and there are no author self-citations in the reference list. The only manuscript passage worth flagging is in the proof of Proposition 3.1: 'noting 2iy/(z−z)=1' — formally z−z=0. This is a proof gap or typo (likely the Bergman-kernel variable should be ar z so that 2iy/(z−ar z)=1), and it affects the rigor of the diagonal evaluation underlying the lower bound in Theorem 1.1(i). It is not a circularity: the singular main term is not a fitted parameter, and the target result is not assumed. Thus the circularity score is low, with the caveat that the Proposition 3.1 diagonal step needs correction.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The paper relies on standard results in modular forms, Bergman kernel theory, and concentration of measure. No free parameters are fitted to data. No new entities are postulated.

assumptions (6)
  • domain assumption Sup norm bound on compacta: sup_{z∈K} y^{k/2}|f(z)| ≪_K k^{1/2} ||f||_2 for all f ∈ S_k.
    Used to give the Lipschitz constant of the sup norm on the sphere, yielding concentration around the median (Theorem 1.1(ii)). Cited from Rudnick [21, Proposition A.1].
  • domain assumption Global sup norm bound: sup_{z∈H^2} y^{k/2}|f(z)| ≪ k^{3/4} ||f||_2.
    Used for global concentration (Theorem 1.3(ii)) and in Lemma 5.1. Cited from Steiner [25, Corollary 1.4].
  • standard math Reproducing property of the Bergman kernel, formula (3.8): Σ y^{k/2} f_j(z) v^{k/2} f_j(w) = (k-1)/(4π) R_k(z,w)/2.
    Connects the covariance kernel to the Bergman kernel. Cited from Steiner [25, Theorem 2.15].
  • standard math Lévy's inequality on the unit sphere S^{2N-1}.
    Used to prove exponential concentration of ||h_k||_p and ||h_k||_∞ around their medians (Lemmas 4.3 and proofs in §4.3).
  • domain assumption Uniform distribution of coefficients on the unit sphere and the resulting law of |a_1|.
    Defines the random model (1.1) and underlies Lemma 4.5, the exact formula for E[||h_k||_p^p].
  • standard math Hyperbolic lattice point bound: #{ γ ∈ SL2(Z): u(z, γz) ≤ X } ≪ y X.
    Used in the proof of Theorem 3.3 to bound the Bergman kernel error term. Cited from Iwaniec [14, Corollary 2.12].

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Pith. "Pith review of On the supremum of random cusp forms." pith.science (2026). https://pith.science/paper/2QYDDGX7

@misc{pith2026250816813,
  author       = {Pith},
  title        = {Pith review of: On the supremum of random cusp forms},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2QYDDGX7}},
  note         = {Machine review of arXiv:2508.16813}
}
abstract

A random ensemble of cusp forms for the full modular group is introduced. For a weight-$k$ cusp form, restricted to a compact subdomain of the modular surface, the true order of magnitude of its expected supremum is determined to be $\asymp \sqrt{\log{k}}$, in line with the conjectured bounds. Additionally, the exponential concentration of the supremum around its median is established. Contrary to the compact case, it is shown that the global expected supremum, which is attained around the cusp, grows like $k^{1/4}$, up to a logarithmic factor.

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