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Low-Lying Zeros of $L$-functions of Ad\'elic Hilbert Modular Forms and their Convolutions

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Pith's one-line read Under GRH, low-lying zeros of Hilbert modular form L-functions over any totally real field follow the orthogonal density, with Fourier support 3/2; the Rankin–Selberg family follows the symplectic density.

desk verdict Substantial conditional advance in low-lying zeros for Hilbert modular forms; the flagged bottleneck (7.1) is actually covered by the stated hypotheses. read the letter →

arxiv 2412.03034 v2 pith:RM6VNNRH submitted 2024-12-04 math.NT

Alia Hamieh , Peng-Jie Wong This is my paper · ORCID
classification math.NT MSC 11F4111F6711F3011F1111F1211N75
keywords HilbertmodularformsRankin-Selbergconvolutions1-leveldensitylow-lyingzerosKatz-SarnakconjectureautomorphicL-functionscentralvaluessymplecticsymmetry
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

Under the generalised Riemann hypothesis, this paper proves instances of the Katz–Sarnak density conjecture for $L$-functions over arbitrary totally real number fields. For primitive Hilbert modular forms of square-free level, the averaged $1$-level density of low-lying zeros is the orthogonal density $W^{(O)}(x)=1+\tfrac12\delta_0(x)$, and the Fourier support of the test function can reach $u=\tfrac32$ when the level dominates the weight. For Rankin–Selberg convolutions $L(s,f\times g)$ with $g$ fixed, the averaged density is the symplectic density $W^{(Sp)}(x)=1-\tfrac{\sin(2\pi x)}{2\pi x}$, with support $u=\tfrac34$ in the level aspect, provided the class number of $F$ is odd or $g$ is non-dihedral. As applications, the paper bounds the average order of $L(s,f)$ at $s=\tfrac12$ and obtains a positive proportion of non-vanishing for the convolution family at the central point. The point of these statements is that they pin down the predicted symmetry type in settings where earlier work required class number one or parallel weights, and they give explicit support limits rather than an abstract convergence.

What carries the argument

The carrying mechanism is the explicit formula of Proposition 2.1, combined with the Petersson trace formula for Hilbert modular forms (Proposition 4.1). The explicit formula rewrites $D(f;\varphi)$ as a conductor term, a term $-\delta_f\varphi(0)/2$, and a sum over prime ideals of $C_f(\mathfrak p)\widehat\varphi(\log N(\mathfrak p)/\log R)/N(\mathfrak p)^{1/2}$. Averaging over the family, the Petersson trace formula turns the prime sum into a diagonal term plus a Kloosterman–Bessel sum; the diagonal produces the density $W^{(O)}$ or $W^{(Sp)}$, while the off-diagonal part is split as $\Delta'_{k,n}+\Delta^\infty_{k,n}$ and bounded using Weil's bound for Kloosterman sums and GRH-controlled sums over $\operatorname{Sym}^2$ coefficients. For convolutions, the main term contains $\delta_{f\times g}$, the order of the pole of $L(s,\operatorname{Sym}^2(f\times g))$ at $s=1$; Theorem 3.1 computes $\delta_{f\times g}$ in all cases, and the averaged asymptotic (7.1), proved when the class number is odd or $g$ is non-dihedral, makes the averaged main term equal to $\int\varphi W^{(Sp)}$.

What would settle it

Work in a totally real field $F$ of even narrow class number, fix a dihedral $g\in\Pi_{k'}(\mathfrak n')$, and compute the average of $\delta_{f\times g}$ over $f\in\Pi_k(\mathfrak n)$ as $N(\mathfrak n)\to\infty$ using the cases in Theorem 3.1. If the proportion of dihedral $f$ that are twist-equivalent to $g$ is not $o(1)$, the average exceeds $1$, so the symplectic density in Theorem 1.2 fails; the paper notes that its argument gives no control in this case.

Watch

Extended reading notes

Core claim

The central discovery is Theorem 1.1 and Theorem 1.2. Let $F$ be a totally real field, $k\in(2\mathbb N)^n$, $\mathfrak n$ a square-free integral ideal, and let $\Pi_k(\mathfrak n)$ be the set of primitive forms of weight $k$ and level $\mathfrak n$. Theorem 1.1 asserts that, under GRH for $\zeta_F(s)$, $L(s,f)$, and $L(s,\operatorname{Sym}^2 f)$, $$\frac{1}{|\Pi_k(\mathfrak n)|}\sum_{f\in\Pi_k(\mathfrak n)}D(f;\varphi)\sim\int_{-\infty}^{\infty}\varphi(x)$W^{{(O)}}$(x)\,dx$$ whenever $\widehat\varphi$ is supported in $(-u,u)$ with $u=\frac32\frac{\log N(\mathfrak n)}{\log(N(\mathfrak n)N(k)^2)}+\frac43\frac{\log N(k)}{\log(N(\mathfrak n)N(k)^2)}$. Theorem 1.2 asserts the symplectic law for the averaged $D(f\times g;\varphi)$ over $f\in\Pi_k(\mathfrak n)$ with $g$ fixed and $(\mathfrak n,\mathfrak n')=1$, with $u=\frac34\frac{\log N(\mathfrak n)}{\log(N(\mathfrak n)N(k)^2)}+\frac23\frac{\log N(k)}{\log(N(\mathfrak n)N(k)^2)}$, under GRH for $L(s,f\times g)$ and $L(s,\operatorname{Sym}^2(f\times g))$, and either odd class number or $g$ non-dihedral. The proof also produces a complete table of pole orders $\delta_{f\times g}$ of $L(s,\operatorname{Sym}^2(f\times g))$ at $s=1$ (Theorem 3.1), which is what selects one density law rather than the other.

Load-bearing premise

The central results are conditional on the generalised Riemann hypothesis, and the Rankin–Selberg density additionally depends on the averaged asymptotic (7.1), $\frac1{|\Pi_k(\mathfrak n)|}\sum_{f\in\Pi_k(\mathfrak n)}\delta_{f\times g}\to1$, which the paper proves only when the class number of $F$ is odd or the fixed $g$ is non-dihedral.

Editorial extensions

If this is right

  • The orthogonal symmetry type of the Katz–Sarnak conjecture is confirmed for primitive Hilbert modular forms over any totally real field, with Fourier support up to $3/2$ in the level aspect.
  • For a fixed $g$, the family of Rankin–Selberg $L$-functions $L(s,f\times g)$ has the symplectic symmetry type with explicit support $3/4$ in the level aspect, under the stated conditions.
  • The average order of vanishing of $L(s,f)$ at the central point is bounded: $\limsup\sum_{m\ge1}mP_m(\mathfrak n)\le \frac1u+\frac12$.
  • The lower bound $\liminf Q_0(\mathfrak n)\ge \frac54-\frac{1}{2u}$ gives a positive proportion of non-vanishing for $L(\tfrac12,f\times g)$ whenever $u>\frac25$, and the theorem's level-aspect support $u=\frac34$ satisfies this.

Reading between the lines

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

  • If the averaged asymptotic (7.1) could be proved without the odd-class-number or non-dihedral restriction, Theorem 1.2 would extend to every fixed $g$; the missing input is a bound showing that dihedral forms twist-equivalent to $g$ make up $o(|\Pi_k(\mathfrak n)|)$ of the family.
  • The pole-order table of Theorem 3.1 is independently usable: exact orders of $L(s,\operatorname{Sym}^2(f\times g))$ at $s=1$ feed directly into any density or moment computation where a symmetric-square pole enters the explicit formula.
  • For the Hilbert family itself the paper's remark shows that Fourier support beyond $u=2$ is the threshold for a positive proportion of non-vanishing of $L(s,f)$; a stronger treatment of the off-diagonal Kloosterman sums, not the explicit formula, is the concrete step needed to cross it.
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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

3 major / 5 minor

Summary. The paper establishes conditional 1-level density results for low-lying zeros of L-functions attached to primitive Hilbert modular forms over arbitrary totally real fields, and for their Rankin-Selberg convolutions with a fixed form. Theorem 1.1 proves the orthogonal symmetry type for the family of Hilbert modular forms with explicit Fourier support u = (3/2 log N(n) + 4/3 log N(k)) / log(N(n)N(k)^2), thereby breaking the (-1,1) support barrier in the level aspect. Theorem 1.2 proves the symplectic symmetry type for the family f x g with a fixed g, under the condition that the class number is odd or g is non-dihedral. Theorem 1.4 derives applications to the average order of L(s,f) at the central point and to a positive proportion of nonvanishing of Rankin-Selberg L-functions. The proofs use the explicit formula, the Petersson trace formula for Hilbert modular forms, and a detailed calculation of the pole order delta_{f x g} of L(s, Sym^2(f x g)) at s=1.

Significance. If the results are correct, they provide a substantial generalization of the Iwaniec-Luo-Sarnak framework to Hilbert modular forms, removing the narrow-class-number-one and parallel-weight restrictions and giving explicit, non-trivial support for the test functions. The pole-order classification in Theorem 3.1 is also a useful contribution. The paper is careful in stating GRH hypotheses and in isolating the averaged pole-order asymptotic (7.1) as the main technical bottleneck. However, the proof as written contains a load-bearing error in the support computation of Section 6, and the hypotheses of Theorem 1.2 do not exactly match those used in Proposition 4.3. These issues require substantive repair rather than mere editing.

major comments (3)
  1. [§6, Eq. (6.6)-(6.7)] The displayed support condition (6.7) does not follow from (6.6) with Y=X^2 and X=(N(n)N(k))^eta. The first error term in (6.6) is bounded by (1/log R) X^{5/2-delta} R^u N(n)^{-1/2+delta+epsilon} N(k)^{delta-1/3}; substituting X=(N(n)N(k))^eta and requiring this term to be o(N(n)N(k)) gives, up to negligible epsilon and eta terms, u < [ (3/2 - eta(5/2-delta) - delta) log(N(n)N(k)) - (1/6) log N(k) ] / log(N(n)N(k)^2). The sign before delta is essential: in the weight aspect this yields u < 2/3 - (delta/2)(1+eta)+o(1), and in the level aspect u < 3/2 - delta - eta(5/2-delta)+o(1). The manuscript's displayed formula has +delta and therefore permits the advertised end-point supports u=3/2 and u=2/3, but those supports are not supplied by the estimates in the text. Since Theorem 1.1 depends on this bound, the proof needs repair.
  2. [§4.5, Prop. 4.3; §7.2] Theorem 1.2 assumes GRH only for zeta_F, L(s,f x g), and L(s, Sym^2(f x g)). Proposition 4.3, however, is derived from Proposition 2.1 using equation (3.1), whose error term requires control of L(s, ∧^2(f x g)) = L(s, Ad(f)) L(s, Ad(g)) as well; the sentence before (3.1) and the statement of Proposition 4.3 both make this explicit. As the paper stands, the hypotheses of Theorem 1.2 do not justify the delta_{f x g} term with the stated error O(log log / log R). Please either add GRH for L(s, Ad(f)) and L(s, Ad(g)) to the theorem, or prove the averaged second-moment asymptotic under the stated hypotheses.
  3. [§7.1] In the odd-class-number branch, the proof of (7.1) concludes that no f in Pi_k(n) is dihedral from the absence of nontrivial quadratic class-group characters. This inference is valid only if the self-twist character of a dihedral form of square-free level and trivial central character is necessarily unramified. The quoted result [14] is said to imply conductor 1 for twist characters, but the connection to the self-twist character of a dihedral form is not shown; the conductor of the quadratic character attached to K/F is generally the discriminant of K/F and is not visibly trivial. Because (7.1) is an essential input for Theorem 1.2 in this branch, please supply a complete proof of this local/global fact or modify the statement.
minor comments (5)
  1. [§7.2] The final displayed support condition has a '≤' where the derivation requires strict inequality for an o(N(n)N(k)) term; both this condition and (6.7) should be reformatted so that the bracketed numerator terms are unambiguous.
  2. [§8] In the proof of Theorem 1.4(ii), the index n is reused in the expression 'sum_{n>=1} Q_{2n}(n)'; a different summation index should be used to avoid confusion.
  3. [Theorem 3.1, case (b)] The phrase 'By a dimension consideration' is not an argument; please give a precise reference or a short proof for the order of the pole in the case where exactly one of f and g has property P.
  4. [§7.1] The sentence 'the number of g in Pi_k(n) such that g = f ⊗ chi ...' uses the symbol g for a variable in the same family as the fixed g of Theorem 1.2; please rename this variable to avoid ambiguity.
  5. [Throughout] There are TeX-editing artifacts such as 'Ad\'elic' and 'suppress l/suppress l' in reference [11] that should be cleaned before publication.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the 1-level density formulas are derived from the explicit formula and the Petersson trace formula, and the cited self-result [22] is a non-circular technical input.

full rationale

The derivation chain is self-contained in the relevant sense. Theorem 1.1 is obtained by combining the explicit formula of Proposition 2.1 (derived from the standard Weil explicit formula and GRH) with the Petersson trace formula and the auxiliary estimates in Section 5; no fitted parameter is renamed as a prediction. Theorem 1.2 follows from Proposition 4.3 and the averaged asymptotic (7.1), which is proved under exactly the hypotheses of the theorem: odd class number or non-dihedral g. In the odd-class-number case the absence of dihedral forms is justified by the conductor restriction of Ramakrishnan-Yang [14] together with the fact that a quadratic class-group character is trivial when the class number is odd; in the non-dihedral-g case the needed δ_{f×g}=1 is obtained directly from Theorem 3.1's non-twist-equivalent cases, which do not depend on the self-citation. The one self-citation, [22], is used in Theorem 3.1 for the pole order of L(s,Ad(π)×Ad(π)) when π is dihedral; this concerns a general automorphic L-function computation, not the target Katz-Sarnak density statement, and it is not needed for the δ-values that drive the main conditional theorems. Thus there is no step where a prediction reduces by construction to an input, no fitted constant is called a prediction, and no load-bearing premise rests solely on an unverified self-citation.

Assumptions & free parameters 2 free parameters · 5 assumptions · 0 invented entities

The central claim rests on the generalized Riemann hypothesis (explicitly assumed), standard analytic properties of automorphic L-functions, the Petersson trace formula, and prior results on twist-equivalence and pole orders. No new entities are introduced.

free parameters (2)
  • δ
    A fixed real number in (0,1/2) used in the J-Bessel function estimates (e.g., equations (4.5), (6.3)) and in the error term of Proposition 5.5. It is chosen within an allowed range and does not appear in the final theorems.
  • η
    A sufficiently small positive parameter used to set X=(N(n)N(k))^η and Y=X^2 in Sections 6-7 to balance error terms. It is chosen ad hoc for the proof and is optimized away.
assumptions (5)
  • domain assumption Generalized Riemann hypothesis for ζ_F(s), L(s,f), L(s,Sym²f) (Theorem 1.1) and additionally for L(s,f×g) and L(s,Sym²(f×g)) (Theorem 1.2).
    Explicitly assumed in Theorems 1.1, 1.2, and 1.4 to obtain the explicit formulas and the asymptotic for Λ_f(p²). If GRH fails, the proofs do not go through.
  • domain assumption Ramanujan-Petersson conjecture for the L-functions in the general formalism (Section 2.1).
    Section 2.1: 'we will further assume the Ramanujan-Petersson conjecture (denoted RPC) for L(s,f) as follows...' This is used to bound the m≥3 terms in the explicit formula. For Hilbert modular forms, the needed bound is known by Blasius [2], but the general assumption is stated.
  • standard math Analytic continuation and functional equation for L(s,f) and L(s,f×g) (Sections 4.4, 4.5).
    Based on known results for automorphic representations on GL(2) and Rankin-Selberg theory, cited to [10] and standard references. These properties are essential for applying the explicit formula.
  • standard math Petersson trace formula for Hilbert modular forms (Proposition 4.1).
    Due to Trotabas [20, Theorems 5.5 and 6.3], used in Sections 5-7 to average Fourier coefficients. This is a prior established theorem.
  • standard math Pole-order computations for L(s, Ad(π)×Ad(π)) and cuspidality criteria from [21] and [22].
    Used in the proof of Theorem 3.1 to compute δ_{f×g}. [22] is a self-citation (Wong), but published in Math. Res. Lett. 2022. These are technical inputs, not the target result.

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Pith. "Pith review of Low-Lying Zeros of $L$-functions of Ad\'elic Hilbert Modular Forms and their Convolutions." pith.science (2026). https://pith.science/paper/RM6VNNRH

@misc{pith2026241203034,
  author       = {Pith},
  title        = {Pith review of: Low-Lying Zeros of $L$-functions of Ad\'elic Hilbert Modular Forms and their Convolutions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RM6VNNRH}},
  note         = {Machine review of arXiv:2412.03034}
}
abstract

In this article, we study the density conjecture of Katz and Sarnak for $L$-functions of ad\'elic Hilbert modular forms and their convolutions. In particular, under the generalised Riemann hypothesis, we establish several instances supporting the conjecture and extending the works of Iwaniec-Luo-Sarnak and many others. For applications, we obtain an upper bound for the average order of $L$-functions of Hilbert modular forms at $s=\frac{1}{2}$ as well as a positive proportion of non-vanishing of certain Rankin-Selberg $L$-functions.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Unitary $n$-correlations with restricted support in random matrix theory

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Reference graph

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