{"id":"2a873cf1-af03-49b0-aa44-1296018f772d","arxiv_id":"2509.01602","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"Under GRH, a prime-level newform's mass pushed through the Hecke correspondence equidistributes on Y1 × Y1 at rate (log q)^{-1/4+ε}, confirming the authors' new joint conjecture.","lead":"On a class of curved surfaces built from arithmetic data, the authors conjecture that a special wave's mass spreads uniformly across a two-copy product surface as a size parameter grows. They prove this with an explicit convergence rate, assuming the Generalized Riemann Hypothesis, one of the deepest open conjectures in mathematics.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 4.9's rate depends on the unexpanded coefficient inequality in Lemma 8.2; a sign error there would destroy the (log q)^{-1/4} saving.","rationale":"The reader's weakest assumption was GRH and the Kim–Sarnak bound; those are external hypotheses explicitly stated by the paper. My stress-test found a different, internal soft spot: Lemma 8.2, whose proof is too compressed for the load it carries. The lemma is essential to neutralize the potentially large L(1)-values that otherwise enter the exponent in (8.5). If Lemma 8.2 is correct, the argument likely goes through; if not, the main rate fails. The paper has no formal verification, and this algebraic step is exactly the kind that can hide a sign error. I do not claim the lemma is false—only that it is the most load-bearing unverified step and should be checked independently. I therefore recommend a CONDITIONAL acceptance, pending verification of Lemma 8.2 (or a fully expanded derivation). This does not contradict the reader's ACCEPT outright, but adds a concrete condition that should be satisfied before full confidence.","tokens_in":46038,"tokens_out":19392,"duration_ms":202206,"concrete_test":"Explicitly recompute the logarithmic Dirichlet coefficient of ℒ_2(s)^δ / ℒ_F(s) at a prime p for both cases f1=f2 and f1≠f2. Verify, with a symbolic algebra system, that after adding the auxiliary factor's exponent 1/(δ^2−1) the coefficient is ≤ 3/(δ^2−1) for all primes p and all 0<δ<1, using only the Hecke relations and the Kim–Sarnak bound |λ(p)|≤2p^{7/64}. In particular, test the displayed completing-the-square identity numerically for random primes and for p=2,3,5. If the inequality fails for any p, Lemma 8.2 collapses and Theorem 4.9 does not follow; if it holds for all p, the concern is resolved.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The central conditional theorem reduces, via Corollary 8.3, to the fractional-moment estimate (8.5). The passage from (8.5) to the claimed bound in Theorem 4.9 relies on Lemma 8.2, which asserts that (ℒ_2(1))^δ / ℒ_F(1) ≪ (log log q)^{O(1)} for any 0<δ<1. This lemma is the only mechanism controlling the L(1) factors L(sym^2 F), L(sym^4 F), L(sym^2 f1⊗sym^2 f2), etc., which under GRH alone can be as large as exp((log q)^{7/8}) by Corollary 6.8. If those factors entered the exponent uncontrolled, the claimed (log q)^{-1/4+ε} rate would be swamped. The proof of Lemma 8.2 is a single paragraph: it asserts that after completing the square, the p-th Dirichlet coefficient of ℒ_2(s)^δ/ℒ_F(s) times an auxiliary factor to the power 1/(δ^2−1) is bounded above by 3/(δ^2−1) ≤ 0, then invokes Corollary 6.6. No intermediate algebra is shown. A sign error, a missing factor, or a failure of the bound for some primes would make Corollary 6.6 inapplicable, and the main term of the fractional moment would be uncontrolled. This is an internal correctness risk distinct from the external GRH assumption, and it is not flagged in the reader's verdict.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":46295,"tokens_out":38588,"duration_ms":395069,"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":[{"comment":"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","section":"§8.1, Lemma 8.2"}],"minor_comments":[{"comment":"Typo: 'let ι_q be embedding' should be 'let ι_q be the embedding'.","section":"Abstract"},{"comment":"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.","section":"§4.3, after Corollary 4.5"},{"comment":"In the statement of Proposition 4.6, 'f1(q.f2)' should be 'f1 · (l_q f2)'.","section":"§4.2, Proposition 4.6"},{"comment":"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).","section":"§8.1.1, Lemma 8.2"},{"comment":"The word 'Ackowledgements' on page 1 is misspelled.","section":"Typesetting"},{"comment":"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.","section":"§8.4, final integration"}],"recommendation":"major_revision","confidential_remarks":"The paper is well written and the overall strategy is convincing. The main obstacle is the proof of Lemma 8.2, which is localized but load-bearing; the stress-test concern about the unexpanded algebra in that lemma is legitimate. I believe the lemma's conclusion is likely true for the range of δ needed in the application (δ ≈ 1/4), and a corrected argument should be a local fix rather than a change of strategy. I therefore recommend major revision, not rejection."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this is a real advance. The paper states a new conjecture—combining quantum unique ergodicity with Hecke-point equidistribution in the level aspect—and proves it under GRH for compact quaternion levels with an effective rate. If Theorem 1.1 stands, it resolves a natural strengthening of the Kowalski–Michel–VanderKam conjecture and goes beyond anything in the cited literature. The reader's accept verdict is right.\n\nWhat is actually new: Conjecture A is not in KMV, Nelson, or Michel–Venkatesh; those address a single projection or point masses. The proof reduces the problem to a fractional moment with explicit mean and variance that come from the Euler product and Hecke relations, not from fitted parameters. The paper is also honest about its limitations: compactness, prime levels, GRH, and the Kim–Sarnak bound are all flagged. That transparency is worth credit.\n\nSoft spots, in proportion: the main fragility is the stack of deep imports—GRH, Watson–Ichino, Kuznetsov, Kim functoriality, Soundararajan–Chandee. That is declared, not hidden. The internal step I would send a referee to is Lemma 8.2. The stress-test note is right that the one-paragraph algebra is compressed: the p-th coefficient inequality is asserted after a completing-square step, and a sign error there would destroy the (log q)^{-1/4} saving. I do not think it is obviously wrong—the structure is standard and the claim is plausible—but the displayed algebra is not fully expanded, and everything downstream depends on it. That is a referee's task, not a desk-reject reason. The Eisenstein gap and the prime-level restriction are disclosed, so they should not be counted as hidden flaws.\n\nWho this is for: analytic number theorists working on QUE, Hecke correspondences, or fractional moments. It deserves a serious referee. I would send it out and ask the referee specifically to expand Lemma 8.2 and double-check the ranges in Lemmas 8.5–8.8. My own verdict is conditional-accept unless that lemma turns out to have a genuine error.","headline":"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.","tokens_in":602,"tokens_out":769,"would_cite":true,"duration_ms":64968,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11F67","11F72","11M26"],"pacs":[],"model":"deepseek-v4-flash","headline":"Newform masses jointly equidistribute assuming GRH","keywords":["joint equidistribution","newforms","quantum unique ergodicity","Hecke correspondence","fractional moments of L-functions","Watson-Ichino formula","Generalized Riemann Hypothesis","arithmetic hyperbolic surfaces"],"falsifier":"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","tokens_in":45778,"feed_emoji":"📐","tokens_out":11140,"duration_ms":116331,"temperature":0.7,"pith_summary":"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.","feed_headline":"Newform masses jointly equidistribute assuming GRH","feed_subtitle":"The mass of a Hecke–Maaß newform, pushed to Y1×Y1, converges to the uniform product measure at rate (log q)^{-1/4+ε}.","key_machinery":"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.","core_discovery":"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","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"supplies the Watson–Ichino formula that converts spectral inner products into ratios of triple-product L-functions.","marker":"[Wat02]"},{"why":"gives the general trilinear-form theorem underlying Proposition 3.6's explicit local constants.","marker":"[Ich08]"},{"why":"provides the short-Euler-product upper bound for log |L(1/2,π)| under GRH used in the central-value approximation.","marker":"[Sou09]"},{"why":"extends Soundararajan's inequality to general L-functions; Proposition 6.9 starts from Chandee's bound.","marker":"[Cha09]"},{"why":"blueprint for fractional-moment bounds under GRH; its Kuznetsov-based high-moment argument is adapted in Propositions 7.1–7.2.","marker":"[BB24]"},{"why":"first use of this fractional-moment technique in an equidistribution problem; supplies the range-splitting template.","marker":"[LR20]"},{"why":"provides automorphy of sym^4 and the Kim–Sarnak bound 7/64, both used throughout to control local factors.","marker":"[Kim03]"},{"why":"source of the Bruggeman–Kuznetsov formula and the Hecke relations used in the high-moment estimates.","marker":"[Iwa02]"},{"why":"Hoffstein–Lockhart bounds for L(1,sym²) control the normalizing factors in the Watson–Ichino application.","marker":"[HL94]"}],"fun_headline_variants":["Newform masses jointly equidistribute under GRH","Effective equidistribution of newform masses on product surfaces","Newforms' masses spread uniformly on doubled surface","Joint equidistribution of newforms with rate under GRH","Newform mass converges to uniform product measure assuming GRH"],"cache_read_input_tokens":2688,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Newform masses jointly equidistribute under GRH","Effective equidistribution of newform masses on product surfaces","Newforms' masses spread uniformly on doubled surface","Joint equidistribution of newforms with rate under GRH","Newform mass converges to uniform product measure assuming GRH"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000554,"raw_usage":{"total_tokens":2472,"prompt_tokens":737,"completion_tokens":1735,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":481,"completion_tokens_details":{"reasoning_tokens":1658}},"tokens_in":481,"tokens_out":1735,"duration_ms":12973,"temperature":1.0,"reasoning_tokens":1658,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-05T12:23:47.625952+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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","supporting_citations":[],"review_version":1}