On a Divisor Modular Form and a Theta Lift

Andreas Mono; Johann Stumpenhusen; Larry Rolen

REVIEW 2 major objections 3 minor 1 cited by

On a Divisor Modular Form and a Theta Lift

T0 review · 2 major / 3 minor · reviewed 2026-08-05 · deepseek-v4-flash

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Pith's one-line read A theta lift sends half-integral weight cusp forms to the weak Maass form ω_{k+1,D}, which encodes the divisor modular form of f_{k,D}.

desk verdict Nice construction, but Theorem 1.3 has a missing conjugate and is false as stated; the fix (kernel at −\bar z) likely saves it. read the letter →

arxiv 2509.01378 v1 pith:CUZ35VEC submitted 2025-09-01 math.NT

Andreas Mono , Larry Rolen , Johann Stumpenhusen This is my paper

classification math.NT MSC 11F1111E1611F27

keywords IndefinitethetaseriesWeakMaassformsliftsDivisormodularHyperbolicPoincaréHalf-integralweightPlusspaceBinaryquadratic

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 sets up a new theta lift in the theory of modular forms. Its central object is a function ω_{k+1,D}(z) built by summing, over all integral binary quadratic forms of discriminant D, the quantity Q_z/Q(z,1)^{k+1}; this sum has no poles on the upper half-plane and is a weak Maass form of weight 2k+2 with eigenvalue 2k. The authors prove that the generating function Λ_k(τ,z) of these functions, taken over positive discriminants D, transforms like a half-integral weight modular form in the plus space, exactly as the classical kernel that generates the hyperbolic Poincaré series f_{k,D} does. They then compute the Petersson inner product of Λ_k with the plus-space Poincaré series of index D and obtain ω_{k+1,D} up to an explicit gamma constant. Because ω_{k+1,D} also appears in a formula for the divisor modular form of f_{k,D}, the lift connects half-integral weight cusp forms directly to the zeros and poles of that series.

What carries the argument

The engine of the argument is the new function ω_{k+1,D}(z)=∑_{Q∈Q_D} Q_z/Q(z,1)^{k+1}, where Q_z=(a|z|^2+bx+c)/y encodes the hyperbolic geodesic attached to Q. Each summand is a weak Maass form piece: the function is modular of weight 2k+2, has eigenvalue 2k under the hyperbolic Laplacian, and splits into a holomorphic derivative part plus y^{-1}f_{k,D}(z). The generating function Λ_k(τ,z)=∑_{D>0} D^{k−1/2}ω_{k+1,D}(z)e^{2πiDτ} is shown, via a standard indefinite theta series criterion, to transform like a weight k+1/2 modular form in the plus space. The Petersson coefficient formula then turns a Fourier coefficient evaluation into the closed-form lift.

What would settle it

Take the function p(a,b,c) defined in Section 4 and compute its second mixed partial derivatives in a neighborhood that crosses the cone b^2 = 4ac; a discontinuity there would invalidate the application of the theta-series theorem. An independent high-precision evaluation of (E − Δ/4π)p − (k−1)p at random parameter triples would verify or refute the paper's computer algebra check.

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Extended reading notes

Core claim

The paper's main result is an exact evaluation: the Petersson inner product of the plus-space Poincaré series P^+_{k+1/2,D} with Λ_k(·,−z) equals Γ(k−1/2)/(6(4π)^{k−1/2}) ω_{k+1,D}(z). Since the P^+ series generate the plus space, this is a theta lift realizing ω_{k+1,D} as the image of the index-D series. The supporting results are modularity of Λ_k in weight k+1/2 for Γ0(4) and a formula expressing the divisor modular form of f_{k,D} as k/(2π) ω_{k+1,D}/f_{k,D} plus k/6 E_2^*. The lift therefore connects half-integral weight cusp forms to the zeros and poles of the hyperbolic Poincaré series.

Load-bearing premise

The argument treats a computer algebra check and a smoothness claim on a three-dimensional auxiliary function as established; the modularity theorem collapses if the check or the smoothness is wrong.

Editorial extensions

If this is right

Because the plus-space Poincaré series generate the entire plus space, the theta lift extends by linearity to a map defined on all half-integral weight cusp forms satisfying the plus-space condition.
Theorem 1.1(iii) gives an explicit formula for the divisor modular form of f_{k,D}: normalized by its first nonzero coefficient, it equals k/(2π) ω_{k+1,D}/f_{k,D} plus k/6 E_2^*.
Since Λ_k has the same modularity type as the classical kernel Ω_k, modularity-based constructions that work for Ω_k—such as coefficient extraction and Petersson inner products—have direct counterparts for Λ_k.
The lift pairs half-integral weight cusp forms with real-analytic modular objects that carry divisor data, making the zeros and poles of the hyperbolic Poincaré series accessible through theta-lift methods.

Reading between the lines

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

Editorial inference: extending the lift to the full plus space by linearity yields a natural map from half-integral weight cusp forms to a space spanned by the ω_{k+1,D}, potentially giving a new analog of the classical correspondence between half-integral and integral weight forms.
Editorial inference: the explicit divisor formula suggests that coefficients of ω_{k+1,D} could be translated into information about vanishing behavior of f_{k,D}; this could be tested numerically by comparing the lift values with known divisor data for small k and D.
Editorial inference: the same construction applied to negative discriminants, where f_{k,D} has poles at CM points rather than cusps, might produce a hyperbolic analog with singularities at CM points, yielding a different but related theta lift.

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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

Colleague,

The paper makes a natural move: define ω_{k+1,D} from Zagier's f_{k,D}, bundle them into a generating function Λ_k, and show it has the same Γ0(4) plus-space modularity as the Kohnen–Zagier kernel. Theorem 1.1 is solid: the splitting in (ii) and the divisor formula in (iii) are clean. Theorem 1.2 is also in decent shape, though the Vignéras eigenvalue check is only a SAGE computation and the C^2 regularity across the discriminant-zero surface is asserted rather than proved. Those are minor referee requests, not real flaws.

The serious problem is Theorem 1.3. The Petersson inner product in (2.7) is printed without a complex conjugate; I think that is a typo, because the unfolding argument only works with a sesquilinear pairing. Restore the conjugate. Then the coefficient selected by the u-integral is not ω_{k+1,D}(z) but \overline{ω_{k+1,D}(−z)}, which by the paper's own b↦−b symmetry equals \overline{ω_{k+1,D}(z)} = ω_{k+1,D}(\bar z). These are different from ω_{k+1,D}(z) in general: from Theorem 1.1(ii), ω = i f'/k + f/y, so switching z to \bar z flips the sign of the derivative term. So the theorem as stated is false for generic z. The fix is to change the kernel to Λ_k(·,−\bar z), and the same correction applies to identity (1.3). This is load-bearing — the theta lift is the advertised new result.

The construction is well-motivated and the paper is otherwise competent. The error is precise and repairable, so this deserves a serious referee rather than a desk rejection. My own verdict: major revision; I would not cite it in its current form.

Referee Report

2 major / 3 minor

Summary. The paper studies the divisor modular form attached to Zagier's hyperbolic Poincaré series f_{k,D}. It constructs a weak Maaß form ω_{k+1,D} and proves three theorems: (1.1) ω_{k+1,D} is a weak Maaß form of weight 2k+2 whose quotient by f_{k,D} is related to the divisor modular form; (1.2) the generating function Λ_k(τ,z) of the ω_{k+1,D} is modular of weight k+1/2 for Γ0(4) in Kohnen's plus space, via Vignéras' indefinite theta theorem; (1.3) a Petersson-inner-product theta lift of Λ_k(·,−z) against the plus-space Poincaré series P^+_{k+1/2,D} reproduces ω_{k+1,D}(z).

Significance. If correct, the connection between Zagier's f_{k,D} and the new weak Maaß form ω_{k+1,D}, together with the modularity of its generating function, would be a valuable addition to the literature on indefinite theta series and theta lifts. The paper is well organized, gives clear preliminary material, and includes a reproducible Sage computation in an appendix. The main theorem, however, contains a load-bearing conjugation error that makes the statement false as written; the surrounding framework is otherwise sound and appears fixable.

major comments (2)

[Section 4, proof of Theorem 1.3; Eq. (2.7)] The Petersson inner product (2.7) is sesquilinear: ⟨f,g⟩ = (1/6)∫ f(z) \overline{g(z)} y^κ dμ. The displayed line after 'Hence, we obtain' drops the conjugation on Λ_k(τ,−z). Restoring it, the unfolded integral selects \overline{ω_{k+1,D}(−z)}. Using the stated b↦−b symmetry, ω_{k+1,D}(−z)=ω_{k+1,D}(z), and since \overline{ω(z)}=ω(\bar z), the selected coefficient is ω_{k+1,D}(\bar z), not ω_{k+1,D}(z). These are not equal in general (for example D=1, k=4 gives ω(z)=2x/(y z^5)). Thus Theorem 1.3 is false as stated; the proof establishes instead the value for ⟨Λ_k(·,−z), f⟩. The lift should be defined with Λ_k(·,−\bar z), or the conclusion changed to ω_{k+1,D}(\bar z).
[Section 4, proof of Theorem 1.2] Vignéras' theorem requires p ∈ C²(R³) and the eigenfunction identity (E − Δ/(4π))p = (k−1)p on all of R³. The proof asserts C² at D=0 without a detailed argument, and the eigenfunction identity is delegated to the Sage script in Appendix A. Since Theorem 1.3 depends on this modularity, please provide a short written verification or at least justify why the piecewise-defined p is C² across the discriminant-zero surface for k>2 and why the formal symbolic identity in Sage covers the actual p (which vanishes for D≤0).

minor comments (3)

[Section 4, proof of Theorem 1.3] Once the conjugation is restored, the factor written as e^{2πiDτ}e^{-2πidτ} should be e^{2πiDτ}e^{-2πid\barτ}; the displayed v-integral is convergent only with the conjugated exponential.
[Section 4, proof of Theorem 1.2] The phrase 'p is twice continuously differentiable at D=0' deserves a one-line expansion: for k>2 the exponent k−1/2 exceeds 2, so D_+^{k−1/2} is C² across the cone. Also, the Sage code uses symbolic complex powers for D<0; clarify that the verification applies to the D>0 branch and is extended by continuity.
[General] There are a few typographical issues: 'iff satsifies' in Definition 2.4 and inconsistent use of τ vs. z in some integrals. These do not affect the mathematics.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the theta lift is a direct coefficient computation, not a fit or self-citation chain.

full rationale

The paper's derivation is self-contained in the relevant sense. It defines ω_{k+1,D} independently as a sum over binary quadratic forms, proves its Maaß-form properties termwise in Theorem 1.1, and derives the divisor-formula relation using the external Bruinier–Kohnen–Ono theorem. Theorem 1.2 constructs p explicitly so that Vignéras' theta theorem produces Λ_k, with the eigenfunction condition verified independently by a SAGE computation rather than assumed from prior work. Theorem 1.3 is a direct Petersson coefficient formula computation: the inner product of P^+_{k+1/2,D} against Λ_k(·,−z) unfolds and the u-integral selects the D-th Fourier coefficient of Λ_k, which by definition is D^{k-1/2}ω_{k+1,D}(z). No parameter is fitted to the claimed identity, no uniqueness theorem is imported from the authors' own work, and the cited self-references appear only as motivation or related context, not as load-bearing justification. The only non-circular weaknesses are proof-completeness concerns — the asserted C^2 regularity of p on the discriminant-zero surface and the reliance on SAGE for the eigenfunction identity — neither of which makes the argument circular.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The central claims rest on standard external theorems rather than fitted constants or ad hoc physical entities. The only notable internal assumption is the C^2 regularity of p and the SAGE-checked eigenfunction identity, both made explicit in Section 4 and Appendix A.

assumptions (6)

standard math Vigneras' theorem on modularity of indefinite theta series (Theorem 2.5, [25])
Used in the proof of Theorem 1.2 to establish modularity of Lambda_k in tau.
standard math Bruinier-Kohnen-Ono divisor modular form identity (Eq. 1.2, [10])
Used in the proof of Theorem 1.1(iii) to express the divisor modular form of f_{k,D}.
standard math Petersson coefficient formula and Kohnen projection pr+ properties
Basis of the theta lift computation in Theorem 1.3, citing [5, (2.5)].
standard math Absolute convergence of f_{k,D} for k > 2 (Zagier [26])
Used to justify termwise operations and the convergence argument in Proposition 3.1.
domain assumption The function p defined in Section 4 is C^2 on R^3, including at D = 0
Vigneras' theorem requires p in C^2; the paper asserts this without a full derivation, relying on the high vanishing order of q^(k-1/2).
standard math Direct computations in Lemmas 2.1 and 2.2
Elementary algebraic identities used throughout the proofs.

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Cite this review

Pith. "Pith review of On a Divisor Modular Form and a Theta Lift." pith.science (2026). https://pith.science/paper/CUZ35VEC

@misc{pith2026250901378,
  author       = {Pith},
  title        = {Pith review of: On a Divisor Modular Form and a Theta Lift},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CUZ35VEC}},
  note         = {Machine review of arXiv:2509.01378}
}

abstract

In 1975, Zagier introduced the highly influential hyperbolic Poincar\'e series $f_{k,D}$. We connect the divisor modular form of $f_{k,D}$ to a new weak Maass form $\omega_{k+1,D}$. Furthermore, we show that the generating function of $\omega_{k+1,D}$ has the same modularity properties as Kohnen and Zagier's fruitful theta kernel generating the $f_{k,D}$'s. This yields a new theta lift.

Figures

Figures reproduced from arXiv: 2509.01378 by the authors.

**Figure 1.** Relations between ωk+1,D, fk,D, and their generating functions Another connection between ωk+1,D and fk,D can be found in (3.2). This paper is organized as follows. In Section 2, we summarize the framework related to the topics of this paper. Section 3 is devoted to the proof of Theorem 1.1. Section 4 gives the proofs of Theorems 1.2 and 1.3. Acknowledgements The authors would like to thank Winfried Kohnen, Sander Z… view at source ↗

**Figure 2.** Verification of Vignéras’ differential equation References [1] T. Asai, M. Kaneko, and H. Ninomiya, Zeros of certain modular functions and an application, Comment. Math. Univ. St. Paul. 46 (1997), no. 1, 93–101. [2] P. Bengoechea, Corps quadratiques et formes modulaires, Ph.D. Thesis, Université Pierre et Marie Curie, 2013. [3] K. Bringmann, A. Folsom, K. Ono, and L. Rolen, Harmonic Maass forms and mock modular form… view at source ↗

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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.

A direct proof of Mono-Rolen-Stumpenhusen and new constructions via the Maass raising operators
math.NT 2026-06 unverdicted novelty 6.0 of 10

Direct proof via Maass raising operators that ω_{k+1,D} are images of f_{k,D}, with extensions to local Maass forms.

Reference graph

Works this paper leans on

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Pith tools

Reviewed August 5, 2026 · model on record in the stance chip above.

[1] [1]

T. Asai, M. Kaneko, and H. Ninomiya,Zeros of certain modular functions and an application, Comment. Math. Univ. St. Paul.46 (1997), no. 1, 93–101

work page 1997

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work page 2013

[3] [3]

Bringmann, A

K. Bringmann, A. Folsom, K. Ono, and L. Rolen,Harmonic Maass forms and mock modular forms: theory and applications, American Mathematical Society Colloquium Publications, vol. 64, American Mathematical Society, Providence, RI, 2017

work page 2017

[4] [4]

Bringmann, B

K. Bringmann, B. Kane, S. Löbrich, K. Ono, and L. Rolen,On divisors of modular forms, Adv. Math. 329 (2018), 541–554

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[5] [5]

Bringmann, B

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