{"id":"ddc101d4-eeb6-4093-9413-71a9747f086c","arxiv_id":"1908.06348","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Low-lying zero densities for Hecke-Maass L-functions over class-number-one imaginary quadratic fields match orthogonal random matrix ensembles in the level and eigenvalue aspects.","lead":"This paper proves that the lowest zeros of L-functions attached to Hecke-Maass forms over class-number-one imaginary quadratic fields follow the orthogonal random matrix ensembles, in both the level aspect and the eigenvalue aspect. An appendix adds a simpler proof for Maass forms over the rationals and new results separating even and odd forms.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Corollary 3.7 appears to overclaim: H_{T,M}(z) is not negligible in the transition range |Im z|≈T, yet Lemma 3.8 uses it to discard the c<√(n1n2) part of the Kloosterman sum; (3.14) lacks a proof for that range.","rationale":"The reader's identified assumption h_F = 1 is a real scope restriction: the standing assumption at the start of Section 2 is load-bearing and the abstract omits it. However, that is a limitation of the statement rather than a gap in the proof for the cases where h_F = 1. By contrast, the transition-range issue around Corollary 3.7 is a potential gap inside the proof of the eigenvalue-aspect theorems. The paper itself flags the non-negligibility of H in the transition range in Remark 3.6 and the introduction, which makes the unqualified statement of Corollary 3.7 surprising. I read this as an imprecise or missing hypothesis in Corollary 3.7 and an omitted estimate in Lemma 3.8, rather than as a deliberate or fraudulent claim. Because the missing argument concerns the central eigenvalue-aspect results, the paper should not be accepted as complete until the c < √(n1n2) range is correctly bounded. This is consistent with a CONDITIONAL verdict, though for a different reason than the reader's h_F = 1 concern.","tokens_in":26590,"tokens_out":41683,"duration_ms":393237,"concrete_test":"Evaluate H_{T,M}(iT) numerically or by stationary phase for a concrete choice, e.g., M = T^{3/4}, h(t) = e^{-t^2}, T = 10^6, using the representation (3.2)-(3.3). If the value is ≍ MT rather than O(T^{-A}), Corollary 3.7 is false as stated. Then insert this size of H into the c < √N(n1n2) part of the Kloosterman sum in (3.14) and check whether that portion is indeed o(MT^2); if not, the proof of Lemma 3.8 is incomplete and the eigenvalue-aspect theorems need a new argument.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"In the proof of Lemma 3.8, the bound (3.14) is obtained by treating N(c) ≥ √N(n1n2) with Lemma 3.2 and then dismissing the complementary range N(c) < √N(n1n2), where |z| > 1, via Corollary 3.7. As printed, Corollary 3.7 says H_{T,M}(z) = O_A(T^{-A}) for 1 < |z| ≤ T. This conflicts with Lemma 3.5(2) and Remark 3.6: in the transition range |Im z| ≈ T, H_{T,M} is not negligible. For example, Remark 3.6 gives J_{it}(iy) ≈ 2π/√(4π²y²−t²) for y ≈ T and t ≈ T, so H_{T,M}(iT) is of size MT, not T^{-A}. Thus the c < √N(n1n2) contribution with |z| in the transition range is not controlled by the written proof. If Corollary 3.7 was intended to include an extra hypothesis such as |Im z| ≤ cT/(2π), then the missing complementary range must be handled separately; the proof of (3.14) does not do this. Since (3.14) is the estimate used in §5.2 and §5.3, Theorems 1.6 and 1.7 are not fully established as written. The level-aspect Theorem 1.4, which uses the fixed-h bound (3.13), is not affected by this issue.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves Katz--Sarnak density results for low-lying zeros of L-functions attached to Hecke--Maass cusp forms over imaginary quadratic fields, under the standing assumption that the class number h_F = 1. Theorem 1.4 establishes that the 1-level density in the level aspect, with square-free level q and a fixed spectral test weight h, converges to the orthogonal density W_1(O) for Fourier support in (-3/2, 3/2). Theorems 1.6 and 1.7 treat the Laplace-eigenvalue aspect at full level with a localizing weight h_{T,M}: the 1-level density is claimed to converge to W_1(SO(even)) with support in (-1,1), and the 2-level density to W_2(SO(even)) with support in (-1/2,1/2), the latter using that all signs of the functional equation are +1. The proofs combine the explicit formula, the Kuznetsov trace formula for GL(2) over the three-dimensional hyperbolic space, and an analysis of the associated Bessel integrals. An appendix revisits the SL_2(Z) Maass-form case and includes results for even and odd forms conditional on the Riemann hypothesis.","tokens_in":26837,"tokens_out":11445,"duration_ms":107951,"significance":"If the main theorems are valid, the paper extends the Iwaniec--Luo--Sarnak and Alpoge--Miller framework to Maass forms over imaginary quadratic fields and provides an eigenvalue-aspect determination of SO(even) symmetry via the 2-level density. The target densities are taken from the Katz--Sarnak and Miller literature rather than fitted, and the trace-formula and explicit-formula reductions are standard. The level-aspect theorem appears to be supported by the written proof. The eigenvalue-aspect theorems, however, depend on a Bessel-integral estimate whose proof contains a false statement; as printed, Theorems 1.6 and 1.7 are not fully established.","major_comments":[{"comment":"Corollary 3.7 is false as stated. It asserts H_{T,M}(z) = O_A(T^{-A}) for 1 < |z| ≤ T with no restriction on arg z, but Lemma 3.5(2) and Remark 3.6 show that the transition range |Im z| ≈ T is not negligible. For example, for z = iT one has |z| = T, and the formula in Remark 3.6 gives J_{it}(iT) ≈ 2π (4π^2 T^2 - t^2)^{-1/2} for t ≈ T; integrating against h_{T,M}(t)t^2 dt produces a contribution of size MT, not T^{-A}. Lemma 3.5(1) does not imply Corollary 3.7 because it requires |Im z| ≤ cT/(2π). This invalid statement is used in the proof of (3.14) in Lemma 3.8 to discard the range N(c) < sqrt(N(n_1 n_2)); that range contains values with |Im z| ≈ T, and also values with |z| > T, which Corollary 3.7 does not cover at all. Since (3.14) is the estimate applied in §5.2 and §5.3, Theorems 1.6 and 1.7 are not established as written. Theorem 1.4, whose proof uses the fixed-h estimate (3.13), is not affected by this issue.","section":"§3.2 (Corollary 3.7), §3.3 (Lemma 3.8), §5.2--5.3"}],"minor_comments":[{"comment":"The abstract and the informal statement of the main results omit the standing assumption h_F = 1, which is introduced at the beginning of Section 2 and is used throughout the paper (Hecke operators, Γ_0(q), spectral weights, the functional equation). The abstract should either state this hypothesis explicitly or the theorems should be proved for all imaginary quadratic fields.","section":"Abstract and §2"},{"comment":"Corollary 3.7 is stated without proof and without any hint of the hypotheses needed to derive it from Lemma 3.5(1). If the intended statement required restrictions such as |Im z| ≤ cT/(2π) and |Re z| ≤ T M^{1-ε}, those restrictions must be stated and the complementary transition range must be handled separately.","section":"§3.2, Corollary 3.7"},{"comment":"In the proof of (3.14), the split between the two ranges of N(c) is not written explicitly; the displayed formula jumps from a sum over N(c) ≥ sqrt(N(n_1 n_2)) to an error term 1/T^A. The reader should be told exactly which estimate controls each range of c, including the range where |z| > T.","section":"§3.3, Lemma 3.8"}],"recommendation":"major_revision","confidential_remarks":"The paper has a genuine result in Theorem 1.4, but the advertised eigenvalue-aspect results depend on a Bessel-integral statement that contradicts the authors' own Remark 3.6. If the t-aspect theorems cannot be repaired within the scope of the revision, the authors should consider withdrawing or substantially restating Theorems 1.6 and 1.7, or restricting the support and weights so that the missing transition range is genuinely negligible. The h_F = 1 assumption should also be made prominent."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here's my read on arXiv:1908.06348. The genuinely new pieces are Theorem 1.4 in the level aspect (support up to 3/2, matching the orthogonal density O) and the appendix, which gives a cleaner proof of the Alpoge–Miller result for SL2(Z) Maass forms and detects even vs odd forms under RH for Dirichlet L-functions. The appendix also correctly flags a gap in Alpoge–Miller's Eisenstein bound. Those parts look solid and are worth citing.\n\nThe soft spot is the eigenvalue-aspect proof. The stress-test note is right: Corollary 3.7 claims H_{T,M}(z) is negligible for 1 < |z| ≤ T, but in the transition range |Im z| ≈ T the Bessel integral is not negligible. For z near iT, J_{it}(iy) has a square-root singularity and H_{T,M} is roughly T^{3/2} in that range, not T^{-A}. That invalidates the use of Corollary 3.7 in Lemma 3.8(3.14) to discard the c < √(n1n2) contribution.\n\nThere is also a separate arithmetic issue in §5.2. The bound (3.14) gives the prime sum as ≤ (M/T) ∑_{p≤T^{4/v1}} p^{-1/4} log p ≈ M T^{3/v1-1}. For support in (−1,1), i.e. v1 < 1, the exponent 3/v1−1 is strictly greater than 2, so this is not o(MT^2) as the text claims. The assertion \"which is op(MT^2)\" is false for every v1 < 1. So the proof of Theorem 1.6 does not close, and Theorem 1.7 inherits the problem.\n\nThe abstract also omits the standing assumption h_F = 1; the body is explicit about it, so that is a minor overclaim rather than a mathematical gap.\n\nOn balance: Theorem 1.4 and the appendix are likely correct and useful. The eigenvalue-aspect theorems may be true, but the written proof has a load-bearing gap that a referee should check carefully—both the Bessel transition range and the prime-sum bound. This deserves peer review, but major revision is likely.","headline":"Level-aspect result and appendix are worth a look, but the eigenvalue-aspect proof has a real gap; Theorems 1.6 and 1.7 are not established as written.","tokens_in":27477,"tokens_out":23960,"would_cite":false,"duration_ms":202528,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11M50"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that low-lying zeros of L-functions for Hecke–Maass forms over imaginary quadratic fields follow the random-matrix orthogonal symmetry prediction in both the level and eigenvalue aspects, for test functions with…","keywords":["low-lying zeros","Maass forms","imaginary quadratic fields","n-level density","Kuznetsov trace formula","Katz-Sarnak heuristics","orthogonal symmetry","Bessel integrals"],"falsifier":"For the class-number-one field F=Q(i), compute the averaged one-level density (1.7) for a sequence of square-free levels with growing norm, using the explicit formula and tabulated Hecke eigenvalues. Theorem 1.4 predicts convergence to ∫φ W1(O) for every even Schwartz function with Fourier transform supported in (−3/2, 3/2); a deviation for any such function would falsify the main level-aspect result.","tokens_in":26286,"feed_emoji":"📐","tokens_out":10001,"duration_ms":94402,"temperature":0.7,"pith_summary":"The paper aims to verify the Katz–Sarnak prediction that low-lying zeros of L-functions are distributed like eigenvalues of random matrices, for the previously untreated family of Hecke–Maass forms over an imaginary quadratic field. In the level aspect, it proves that the averaged one-level density converges to the orthogonal density W1(O) for square-free levels when the test function's Fourier transform is supported in (−3/2, 3/2). In the eigenvalue aspect, it proves that the averaged one- and two-level densities converge to the SO(even) densities, with supports in (−1, 1) and (−1/2, 1/2) respectively. An appendix gives a simpler proof of the analogous real-field result and treats even and odd Maass forms on SL2(Z). If correct, the paper establishes orthogonal symmetry for these families in the stated support ranges and shows that the two-level density can distinguish the orthogonal flavor.","feed_headline":"Low-lying zeros of L-functions match random-matrix type","feed_subtitle":"For Maass forms over imaginary quadratic fields, the zeros near the central point follow the orthogonal prediction.","key_machinery":"The argument is carried by the spectral Kuznetsov trace formula for GL2 over an imaginary quadratic field (Proposition 2.1), combined with the explicit formula that turns averaged one- and two-level densities into weighted sums over prime ideals. A Möbius-inversion step (Lemma 2.2) passes from full-level averages to newform averages and introduces Kloosterman–Bessel terms; the decisive estimates are bounds on those terms (Lemmas 3.8 and A.4–A.9), obtained by stationary-phase analysis of Bessel integrals. In the eigenvalue aspect the weight function h_T,M(t) = h((t−T)/M) + h((t+T)/M) localizes spectral sums to conductors near T, and a Bessel-integral analysis shows the Kloosterman terms are negligible in the stated support ranges.","core_discovery":"The central discovery is Theorem 1.4 together with Theorems 1.6 and 1.7: for Hecke–Maass newforms over a fixed imaginary quadratic field of class number 1, the averaged one-level density tends to the orthogonal density W1(O) as N(q) tends to infinity, and in the full-level eigenvalue aspect the one- and two-level densities tend to the SO(even) densities. The level-aspect result holds for Fourier transforms supported in (−3/2, 3/2), the eigenvalue-aspect results for supports in (−1, 1) and (−1/2, 1/2), with the two-level computation separating SO(even) from the other orthogonal ensembles. The appendix also proves a variant of the corresponding statement for Maass forms over Q, with support up to (−1−µ, 1+µ), conditional on the Riemann hypothesis.","pith_inferences":["Editorial inference: because the class-number-1 condition appears only inside the paper and not in the abstract, a reader of the abstract alone could reasonably believe the theorem covers all imaginary quadratic fields; the proved statement is narrower.","Editorial inference: the (−3/2, 3/2) support barrier in the level aspect is tied to the Weil-bound estimate for Kloosterman sums, so pushing beyond it would likely require a new treatment of the Bessel integrals rather than a routine sharpening.","Editorial inference: the same Kuznetsov-plus-explicit-formula template should yield analogous statements for holomorphic modular forms over imaginary quadratic fields, since the trace formula and explicit formula have such analogues."],"forward_implications":["For any imaginary quadratic field of class number 1 and any square-free level ideal, the averaged one-level density of low-lying zeros has the orthogonal random-matrix limit when the Fourier transform is supported in (−3/2, 3/2).","In the full-level eigenvalue aspect, the symmetry type is SO(even) rather than merely O: the two-level density computation identifies the orthogonal flavor.","The same methods give a simpler proof of the known real-field result for SL2(Z) Maass forms and prove SO(even) and SO(odd) density statements for even and odd Maass families, conditional on the Riemann hypothesis.","The stated support restrictions are part of the theorem: the verification is partial, not an unconditional statement for all test functions."],"supporting_citations":[{"why":"Supplies the Katz–Sarnak density conjecture and the orthogonal ensemble definitions that the paper's main theorems verify.","marker":"[KS2]"},{"why":"Provides the explicit formula and the newform-averaging and Möbius-inversion machinery used to express averaged densities as prime sums.","marker":"[ILS]"},{"why":"Supplies the spectral Kuznetsov trace formula for GL2 over imaginary quadratic fields, the main identity behind the level-aspect argument.","marker":"[LG]"},{"why":"Introduces the space of weight functions H(S,N) and the sum formula whose constants are adopted.","marker":"[BM1]"},{"why":"Gives the Γ0(q) setup over GL2(O) and the trace-formula constants used in Proposition 2.1.","marker":"[Ven]"},{"why":"Provides the Kim–Sarnak bound on Hecke eigenvalues that controls the tails of the explicit formula and the prime sums.","marker":"[BB]"},{"why":"Provides the weight function h_T,M and the Bessel-integral analysis that makes the eigenvalue-aspect proof short.","marker":"[Li]"},{"why":"Shows that the two-level density distinguishes the three orthogonal ensembles for arbitrarily small support, justifying the conclusion of Theorem 1.7.","marker":"[Mil2]"}],"fun_headline_variants":["Maass zero densities obey orthogonal law","Level-aspect zeros match orthogonal prediction","Eigenvalue aspect: SO(even) for Maass forms","Imaginary quadratic Maass zeros: orthogonal type","Random matrix type for Maass L-functions"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof assumes the imaginary quadratic field has class number 1—that is, every ideal in its ring of integers is principal—and the theorems are not proved for fields with more than one ideal class.","fun_headline_variants_meta":{"raw":{"variants":["Maass zero densities obey orthogonal law","Level-aspect zeros match orthogonal prediction","Eigenvalue aspect: SO(even) for Maass forms","Imaginary quadratic Maass zeros: orthogonal type","Random matrix type for Maass L-functions"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000926,"raw_usage":{"total_tokens":3920,"prompt_tokens":847,"completion_tokens":3073,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":463,"completion_tokens_details":{"reasoning_tokens":3001}},"tokens_in":463,"tokens_out":3073,"duration_ms":23183,"temperature":1.0,"reasoning_tokens":3001,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T12:49:10.473750+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For the class-number-one field F=Q(i), compute the averaged one-level density (1.7) for a sequence of square-free levels with growing norm, using the explicit formula and tabulated Hecke eigenvalues. Theorem 1.4 predicts convergence to ∫φ W1(O) for every even Schwartz function with Fourier transform supported in (−3/2, 3/2); a deviation for any such function would falsify the main level-aspect result.","supporting_citations":[],"review_version":1}