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REVIEW 1 major objections 6 minor 17 references

Joint equidistribution of newforms

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Pith's one-line read Newform masses jointly equidistribute assuming GRH

desk verdict A serious, well-structured conditional theorem with a genuinely new conjecture; the proof is dense and one key L(1)-control lemma needs close checking, but this deserves full peer review. read the letter →

arxiv 2509.01602 v1 pith:TZBD2X4J submitted 2025-09-01 math.NT

Asbj{o}rn Christian Nordentoft , Radu Toma This is my paper · ORCID
classification math.NT MSC 11F6711F7211M26
keywords jointequidistributionnewformsquantumuniqueergodicityHeckecorrespondencefractionalmomentsofL-functionsWatson-IchinoformulaGeneralizedRiemannHypothesisarithmetichyperbolicsurfaces
topics The Riemann Hypothesis
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 proposes Conjecture A: as the level q grows, the mass of a Hecke–Maaß newform on the cover Y_q, pushed forward through the natural embedding into the product Y_1 × Y_1 of a fixed arithmetic hyperbolic surface with itself, becomes equidistributed with respect to the uniform product measure. The authors prove this conjecture, conditionally on the Generalized Riemann Hypothesis, for compact surfaces attached to division quaternion algebras, with an effective rate (log q)^{-1/4+ε}. The significance is that this single statement combines two usually separate phenomena: quantum unique ergodicity for the eigenfunction's mass and equidistribution of Hecke points in the level aspect. If correct, it immediately implies the Kowalski–Michel–VanderKam conjecture for Maaß newforms in the compact case.

What carries the argument

The load-bearing mechanism is the Watson–Ichino formula, an exact identity expressing the integral of three automorphic forms as a ratio of completed L-functions; it converts the spectral expansion of each Weyl sum into a sum over newforms φ of |L(1/2, F⊗F⊗φ) L(1/2, f1⊗f2⊗φ)|^{1/2}. The proof bounds this fractional moment by approximating log L-values with short Dirichlet polynomials (Soundararajan–Chandee under GRH), then applying high-moment estimates for Hecke eigenvalues from the Bruggeman–Kuznetsov formula, organized by a Gaussian random model for the joint distribution of the two log-L processes.

What would settle it

For a fixed division quaternion algebra (e.g. the smallest discriminant), choose f1 = f2 a non-constant Hecke–Maaß form on Y1, and for an increasing sequence of prime levels q compute the fractional-moment sum in Theorem 4.9 over the finite set of newforms φ with t_φ ≤ 100, evaluating L(1/2, F⊗F⊗φ) and L(1/2, f1⊗f2⊗φ) directly; if the sum grows faster than C_ε q (log q)^{-1/4+ε} for a universal C_ε, the theorem is false. More directly, numerical computation of the inner product ⟨|F_q|², f1·l_q f2⟩_q for the lowest-eigenvalue newform F_q that does not tend to 0 as q grows would refute Conjectur

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Extended reading notes

Core claim

The central result is Theorem 1.1: for B a division quaternion algebra, along prime levels q, Conjecture A holds under GRH with an effective rate (log q)^{-1/4+ε} for every ε>0. Concretely, the pushforward measures (ι_q)_* |F_q|² μ_q converge weakly to μ_1 ⊗ μ_1, and the proof exhibits explicit polynomial control in the spectral parameters of the test functions. The mechanism is spectral: Weyl sums for the pair (f1, f2) are expanded over newforms φ on Y_q, each term is converted by the Watson–Ichino formula into a ratio of triple-product L-functions, and the problem becomes a fractional moment estimate for L(1/2, F⊗F⊗φ) L(1/2, f1⊗f2⊗φ). The authors establish that this fractional moment decay

Load-bearing premise

The proof assumes the Generalized Riemann Hypothesis for every L-function it uses—in particular the triple products L(s, F⊗F⊗φ) and L(s, f1⊗f2⊗φ) and their functorial lifts of degree up to 12—and it also uses the numerical bound 7/64 towards the Ramanujan conjecture; if either premise gave way, the estimates producing the log q decay would not close.

Editorial extensions

If this is right

  • Conjecture A implies the Kowalski–Michel–VanderKam conjecture for Maaß newforms in the compact case, since projection onto the first factor recovers the one-sided pushforward.
  • The proof yields an effective rate of equidistribution, (log q)^{-1/4+ε}, rather than merely a qualitative convergence statement.
  • Under the weaker Generalized Lindelöf Hypothesis, the same equidistribution holds whenever the newform's Laplace eigenvalue grows like q^ε, illustrating the 'equidistribution in stages' principle.
  • The fractional moment bound is stronger for f1≠f2 (saving (log q)^{-3/8}) than for f1=f2 (saving (log q)^{-1/4}), reflecting negative correlation of the two L-functions in the diagonal case.
  • The argument requires automorphy and analytic control of L-functions of degree up to 12, including the Kim–Sarnak bound 7/64, so the range of validity is tied to the available Ramanujan-type bounds.

Reading between the lines

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

  • The Gaussian model in Section 5 suggests a central limit theorem for the family of log-L values; one could test numerically for small q whether the correlation between log L(1/2,F⊗F⊗φ) and log L(1/2,f1⊗f2⊗φ) matches the predicted dependence, especially in the diagonal case f1=f2.
  • The same fractional-moment machinery may apply to the non-compact case Y_0(q) once the Eisenstein contribution is controlled; the authors note they already bound the cuspidal part there, so a natural next step is to extend the theorem to the split algebra.
  • The analogy drawn with the Mixing Conjecture hints at a family of joint-equidistribution results where the two factors are different correspondences; the ratio D/q² and the eigenvalue t_F play parallel roles, which could guide a unified conjecture.
  • Because the proof uses GRH for degree up to 12 L-functions, a numerical check of the functional equations for small conductors could identify where the Ramanujan-bound hypothesis is genuinely needed.
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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 / 6 minor

Summary. The paper introduces Conjecture A for joint equidistribution of newforms on compact arithmetic hyperbolic surfaces: for a sequence of Hecke-Maass newforms F_q on covers Y_q of a fixed quaternion surface Y_1, the pushforward of |F_q|^2 under the Hecke correspondence iota_q: Y_q -> Y_1 x Y_1 should converge to the uniform product measure. The main theorem proves this under GRH for division quaternion algebras and prime levels q, with the effective rate (log q)^(-1/4+epsilon). The proof is spectral: Weyl sums are expanded in a basis of Y_q, the old spectrum is sieved out, and the remaining newform contribution is expressed through the Watson-Ichino formula as a fractional moment of triple product L-functions. The fractional moment is then bounded using Soundararajan-Chandee's GRH upper bounds for central L-values and high moments obtained from the Bruggeman-Kuznetsov formula.

Significance. If correct, this is a substantial advance. It establishes the natural combination of quantum unique ergodicity and Hecke-point equidistribution in the level aspect, conditional on GRH, and it implies the Kowalski-Michel-VanderKam conjecture for Maass newforms in the compact case. The paper is unusually explicit about the structural constants: the Watson-Ichino local constants are written out in Proposition 3.6, the old-spectrum sieve is computed in Lemma 4.4, and the main technical bound is reduced to four clearly stated estimates (Lemmas 8.5-8.8). A further strength is that the mean and variance in the fractional-moment argument are computed from Euler products and Hecke relations rather than fitted to the target answer; the Kim-Sarnak bound 7/64 < 1/8 is used in an essential and transparent way.

major comments (1)
  1. [§8.1, Lemma 8.2] The proof of Lemma 8.2 is not correct as written, and the lemma is load-bearing for Theorem 4.9 through Corollary 8.3. The displayed manipulation after 'completing the square' contains a sign error: -(δ y+2)^2/(δ^2-1)+y^2 equals (y^2+4δ y+4)/(1-δ^2), not (δ^2-1)^{-1}(-y^2+4δ y+4). More importantly, the claimed bound 'bounded above by 3/(δ^2-1)' is false already for the value of δ needed in the paper: take δ=1/4 and λ_F(p)=λ_f1(p)=λ_f2(p)=1, which respects the Kim-Sarnak bounds; the coefficient then equals about 2.32, which is larger than 3/(δ^2-1)=-3.2. The auxiliary factor also involves log(L1+L2) for a sum of L-functions, which is not an Euler product, so its p-th coefficient is not the sum of the individual log coefficients as silently assumed. The lemma may still be recoverable by a cruder bound of the form O_δ(1+p^{7/16}) and by using the convergence of ∑_p p^{-9/16}, but the curren
minor comments (6)
  1. [Abstract] Typo: 'let ι_q be embedding' should be 'let ι_q be the embedding'.
  2. [§4.3, after Corollary 4.5] The passage from the compact surface Y_q to the noncompact Y_0(qD) is quite terse: the norms in the compact orthonormal basis are probability-normalized, while the Kuznetsov formula in Theorem 6.1 uses the standard normalization. The displayed inequality before Theorem 4.9 presumably absorbs the volume factors into the q^{-1} and the weight h(i s_φ)/L(1,sym^2 φ), but this is not shown. Please include the normalization calculation.
  3. [§4.2, Proposition 4.6] In the statement of Proposition 4.6, 'f1(q.f2)' should be 'f1 · (l_q f2)'.
  4. [§8.1.1, Lemma 8.2] The phrase 'p-th Dirichlet coefficient' is ambiguous when fractional powers and quotients of L-functions are involved; the proof appears to work with the coefficient of the logarithm. Please specify this explicitly and justify the manipulation involving log(L1+L2).
  5. [Typesetting] The word 'Ackowledgements' on page 1 is misspelled.
  6. [§8.4, final integration] The sentence 'the last three integrals can be bounded by O_ε(1)' is correct only because the exponents are negative for q large; it would help to display the negative exponent explicitly, since the range of V depends on log log q and Δ.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the main result is a conditional proof from GRH whose mean and variance inputs are computed, not fitted.

full rationale

The paper's central claims are conditional: assuming GRH (Conjecture B), the authors prove Conjecture A with an effective rate. The target measure (ι_q)_*|F|^2 μ_q is not used as an input; the proof spectrally expands the Weyl sums via Watson–Ichino and reduces to bounding fractional moments of genuine L-functions. The mean and variance entering the random model are derived, not fitted: μ_q = −3/2 log log q or −2 log log q follows from Euler products and Mertens' theorems, and the variance constants 3 and 6 follow from the Hecke relations in Lemma 3.5. The final rates (log q)^{−3/8} and (log q)^{−1/4} are the Gaussian expectation E(exp(X/2)) computed from those parameters (Section 5). No parameter is fitted to the quantity being predicted; in particular Theorem 4.9 is an upper bound for an average of L-values, not a restatement of the conjectured equidistribution. Lemma 8.2, flagged by the skeptical note, is an internal analytic estimate bounding ratios of L(1)-values via Corollary 6.6; even if its algebra were incomplete or erroneous, that would be a correctness defect rather than circularity. Citations to prior work (Soundararajan, Chandee, Blomer–Brumley, Lester–Radziwiłł) are external and not self-referential; the authors do not rely on a uniqueness theorem or ansatz from their own prior work. The derivation chain is therefore self-contained apart from standard external theorems (GRH, Kim–Sarnak, Watson–Ichino, Kuznetsov), and no step is equivalent to its inputs by construction.

Assumptions & free parameters 3 free parameters · 7 assumptions · 0 invented entities

No free parameters are fitted to data: the mean and variance of log ℒ(φ) are derived from Euler products, Hecke relations (Lemma 3.5) and Mertens' theorems, and the listed 'parameters' are bookkeeping choices that do not affect the final rate. The axioms are either explicitly assumed (GRH, compactness) or standard imported theorems cited to the original sources. No new objects (particles, forces, dimensions, conserved quantities) are postulated; the only new item, Conjecture A, is a statement about existing objects and needs no independent physical evidence.

free parameters (3)
  • bookkeeping scale Δ = log log log q = any slowly growing function works
    Introduced in Section 8.2 to split the ranges of V in the fractional-moment argument; the proof only needs Δ→∞ with Δ ≤ (log log q)^δ for small δ, so the choice is a convenience and does not affect the output rate.
  • spectral cutoff at t_φ ≤ 100 = 100 (any fixed constant works)
    The truncation of the spectral expansion in (1.1) and Theorem 4.9 is justified by the decay of the archimedean L-factor (Lemma 3.8); the number 100 is arbitrary, not fitted to data.
  • implied constants A and B in truncations = exist by the stated inequalities
    A in x = exp(9A log q / (εV)) is the implied constant of the GRH-level bound (8.14); B is the constant in Chandee's inequality (6.4). All later estimates permit dependence on these, so they are structural rather than tuned.
assumptions (7)
  • domain assumption GRH for the automorphic L-functions appearing, in particular the triple products and their functorial lifts up to degree 12
    Conjecture B (Section 3.1) is assumed in Theorem 1.1 and used in Proposition 6.9, Corollaries 6.6-6.8, and throughout Section 8; without it the effective equidistribution proof does not go through.
  • standard math Kim-Sarnak bound towards Ramanujan: |α_π(p,j)| ≤ p^{7/64}
    Cited to [Kim03, Appendix 2]; used critically in Lemma 3.8, Proposition 6.9 and Section 8.3, with the authors noting in Section 1.2.1 that 7/64 < 1/8 is crucial.
  • standard math Automorphy and holomorphy of sym² and sym⁴ of GL(2) cusp forms (Gelbart-Jacquet, Kim)
    Cited to [GJ78] and [Kim03]; used in Lemmas 3.1 and 3.3 to define the Euler products and the s=1 factors in the mean and variance.
  • standard math Watson-Ichino triple product formula with explicit local constants
    Imported as Proposition 3.6 from [Ich08], [Woo12] and [Nel11]; it is the bridge from inner products to L-functions used in Proposition 4.2 and Corollary 4.5.
  • standard math Bruggeman-Kuznetsov formula for square-free level with positive multiplier W_φ
    Theorem 6.1 is imported from [Iwa02] and [HK20, Lemma A.9]; it underlies the high-moment estimates of Propositions 7.1 and 7.2.
  • standard math Soundararajan-Chandee upper bound for log|L(1/2, π)| under GRH
    Proposition 6.9 extends [Sou09] and [Cha09, Thm. 2.1] with the (1-ε)μ and larger-x variant described in Remark 6.10; the core inequality (6.6) is imported.
  • domain assumption Compactness of Y_q for division algebras (purely discrete spectrum)
    Section 4 restricts to non-split B; the spectral expansion over an orthonormal basis of Y_q is discrete, which is essential to the fractional-moment argument (Remark 1.2).

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Pith. "Pith review of Joint equidistribution of newforms." pith.science (2026). https://pith.science/paper/TZBD2X4J

@misc{pith2026250901602,
  author       = {Pith},
  title        = {Pith review of: Joint equidistribution of newforms},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TZBD2X4J}},
  note         = {Machine review of arXiv:2509.01602}
}
abstract

Let $Y_1$ be a compact arithmetic hyperbolic surface associated to a maximal quaternion order, let $Y_q$ be a cover associated to an Eichler suborder of prime level $q$, and let $\iota_q$ be embedding of $Y_q$ as the Hecke correspondence into $Y_1 \times Y_1$. Let $\mu_1$ and $\mu_q$ be the invariant probability measures on $Y_1$ and $Y_q$, respectively. If $F$ is a newform on $Y_q$, we conjecture that the pushforward measure $(\iota_q)_\ast(\lvert F \rvert^2 \mu_q)$ converges weakly to the uniform measure $\mu_1 \times \mu_1$, as $q$ tends to infinity. We prove this conjecture with an effective rate of equidistribution, assuming GRH.

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