Low-lying zeros of $L$-functions for Maass forms over imaginary quadratic fields

Sheng-chi Liu; Zhi Qi

REVIEW 1 major objections 3 minor 30 references

Low-lying zeros of $L$-functions for Maass forms over imaginary quadratic fields

T0 review · 1 major / 3 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read 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…

desk verdict 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. read the letter →

arxiv 1908.06348 v2 pith:A72V42E6 submitted 2019-08-17 math.NT

Sheng-Chi Liu , Zhi Qi This is my paper · ORCID

classification math.NT MSC 11M50

keywords low-lyingzerosMaassformsimaginaryquadraticfieldsn-leveldensityKuznetsovtraceformulaKatz-SarnakheuristicsorthogonalsymmetryBesselintegrals

topics The Riemann Hypothesis

open problems 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 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.

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

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.

Reading between the lines

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

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.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the 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.

The 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.

There 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.

The 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.

On 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.

Referee Report

1 major / 3 minor

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.

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 (1)

[§3.2 (Corollary 3.7), §3.3 (Lemma 3.8), §5.2--5.3] 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.

minor comments (3)

[Abstract and §2] 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.
[§3.2, Corollary 3.7] 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.
[§3.3, Lemma 3.8] 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.

Circularity Check

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No circularity: the target densities are external Katz–Sarnak objects and the proof derives them from the explicit formula and Kuznetsov trace formula.

full rationale

The paper's central claims (Theorems 1.4, 1.6, 1.7) are statements that certain averaged 1- and 2-level densities tend to W1(O), W1(SO(even)), and W2(SO(even)). These densities and their Fourier transforms are quoted from Katz–Sarnak and Miller, not defined in terms of the paper's own quantities. The derivations proceed by the explicit formula (Lemma 4.1), the spectral Kuznetsov trace formula (Proposition 2.1 and its corollaries), and estimates for the resulting Kloosterman–Bessel and Eisenstein terms. The main constants, such as phi-hat(0)+1/2 phi(0), emerge from the diagonal term and Gamma-asymptotics rather than being fitted or imposed. The paper's use of prior work by the authors (e.g., [Qi1], [Qi2] for Bessel integral representations and bounds) is for standard auxiliary estimates that are stated with hypotheses and are not premises equivalent to the conclusions; no 'uniqueness theorem' or ansatz is imported from self-citations to force the symmetry type. The standing assumption h_F = 1 restricts the scope of the theorems but is not a circular input. A reviewer concern that Corollary 3.7 may overclaim in the transition range |Im z| approximately T is a correctness gap in a technical estimate, not a case of a prediction reducing to its input by construction. No fitted parameter is renamed as a prediction, and the target densities are not recovered from data fitted in the paper. Therefore no significant circularity is present.

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

The paper rests on several heavy external theorems, chiefly the Kuznetsov trace formula for Γ0(q) over imaginary quadratic fields, the Kim-Sarnak bound, Weil's bound for Kloosterman sums, and a Landau lower bound for the Dedekind zeta function on the 1-line. No new entities or fitted constants are introduced. The appendix adds the Riemann hypothesis for classical Dirichlet L-functions as a conditional assumption.

assumptions (7)

standard math Kuznetsov trace formula for Γ0(q) over imaginary quadratic fields (Proposition 2.1, quoted from [LG, Theorem 11.3.3], [BM1], [Ven])
Central tool that converts spectral sums into Kloosterman and Eisenstein terms; accepted in the literature though the primary source is a PhD thesis.
standard math Kim-Sarnak bound |α_f(p)|, |β_f(p)| ≤ N(p)^{7/64} (equation (2.5), from [BB])
Used to bound the prime power terms P2 and ν ≥ 3 in the explicit formula; this is a theorem, not an assumption.
standard math Weil bound for Kloosterman sums S(n1,n2;c) << N(n1,n2,cd^{-1})^{1/2} N(c)^{1/2+ε} (equation (2.2))
Used throughout to estimate Kloosterman-Bessel terms.
standard math Landau lower bound |ζF(1+2it)| >> 1/log(|t|+3) (footnote 4)
Estimates for the Eisenstein contribution Ξ^★_q rely on this uniform lower bound.
standard math Stationary phase lemma (Lemma 2.6, improvement of [BKY, Lemma 8.1])
Used to show exponential integrals are negligible away from stationary points; stated without proof in the paper.
domain assumption Class number one hypothesis hF = 1 (Section 2)
Restricts the theorem to the nine imaginary quadratic fields with class number one; the abstract omits this restriction, which is a specific weakness of the paper as written.
domain assumption Riemann hypothesis for classical Dirichlet L-functions (Appendix A)
Assumed for Theorems A.1 and A.3 to bound Eisenstein terms and to use the averaged Kloosterman sum estimate; does not affect Theorems 1.4, 1.6, and 1.7.

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Pith. "Pith review of Low-lying zeros of $L$-functions for Maass forms over imaginary quadratic fields." pith.science (2026). https://pith.science/paper/A72V42E6

@misc{pith2026190806348,
  author       = {Pith},
  title        = {Pith review of: Low-lying zeros of $L$-functions for Maass forms over imaginary quadratic fields},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/A72V42E6}},
  note         = {Machine review of arXiv:1908.06348}
}

abstract

We study the $1$- or $2$-level density of families of $L$-functions for Hecke--Maass forms over an imaginary quadratic field $F$. For test functions whose Fourier transform is supported in $\left(-\frac 32, \frac 32\right)$, we prove that the $1$-level density for Hecke--Maass forms over $F$ of square-free level $\mathfrak{q}$, as $\mathrm{N}(\mathfrak{q})$ tends to infinity, agrees with that of the orthogonal random matrix ensembles. For Hecke--Maass forms over $F$ of full level, we prove similar statements for the $1$- and $2$-level densities, as the Laplace eigenvalues tends to infinity.

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Works this paper leans on

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

Alpoge, N

L. Alpoge, N. Amersi, G. Iyer, O. Lazarev, S. J. Miller, and L. Zhang. Maass waveforms and low-lying zeros. Analytic Number Theory , pages 19--55. Springer, 2015

work page 2015

[2] [2]

A Bessel delta-method and exponential sums for GL(2)

K. Aggarwal, R. Holowinsky, Y. Lin, and Z. Qi. A B essel delta-method and exponential sums for GL(2) . preprint, arXiv:1906.05485 , 2019

work page Pith review arXiv 1906

[3] [3]

Alpoge and S

L. Alpoge and S. J. Miller. Low-lying zeros of M aass form L -functions. Int. Math. Res. Not. IMRN , (10):2678--2701, 2015

2015

[4] [4]

C. B. Balogh. Asymptotic expansions of the modified B essel function of the third kind of imaginary order. SIAM J. Appl. Math. , 15:1315--1323, 1967

work page 1967

[5] [5]

Blomer and F

V. Blomer and F. Brumley. On the R amanujan conjecture over number fields. Ann. of Math. (2) , 174(1):581--605, 2011

work page 2011

[6] [6]

Barrett, F

O. Barrett, F. W. K. Firk, S. J. Miller, and C. Turnage-Butterbaugh. From quantum systems to L -functions: pair correlation statistics and beyond. Open Problems in Mathematics , pages 123--171. Springer, 2016

work page 2016

[7] [7]

Blomer, R

V. Blomer, R. Khan, and M. P. Young. Distribution of mass of holomorphic cusp forms. Duke Math. J. , 162(14):2609--2644, 2013

2013

[8] [8]

R. W. Bruggeman and R. J. Miatello. Sum formula for SL _2 over a number field and S elberg type estimate for exceptional eigenvalues. Geom. Funct. Anal. , 8(4):627--655, 1998

work page 1998