Flipping operators and locally harmonic Maass forms

Andreas Mono; Kathrin Bringmann; Larry Rolen

REVIEW 3 major objections 3 minor 22 references

Flipping operators and locally harmonic Maass forms

T0 review · 3 major / 3 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read A sign-reversal law for the flipping operator extends to hyperbolic Poincaré series.

desk verdict Theorem 1.1's proof relies on a false identity (3.5); the corollary may survive, but the main result is unproven as written. read the letter →

arxiv 2502.09359 v3 pith:ZGACS2VH submitted 2025-02-13 math.NT

Kathrin Bringmann , Andreas Mono , Larry Rolen This is my paper · ORCID

classification math.NT MSC 11F0311F1111F2511F37

keywords flippingoperatorlocallyharmonicMaassformsEisensteinseriesintegralbinaryquadraticmodularPoincaréEichlerintegralsraising

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 show that the flipping operator—the differential operator that exchanges the holomorphic and non-holomorphic halves of a harmonic Maass form—acts as sign reversal on two families of locally harmonic Maass forms associated to indefinite binary quadratic forms. These functions are hyperbolic Poincaré series attached to a non-square discriminant $D$, and they are connected to Shimura and Shintani lifts and to central $L$-values. The main theorem states that for $k\ge 2$ and $\tau$ outside a measure-zero exceptional set $E_D$, one has $F_{2-2k}(F_{1-k,D}(\tau)) = -F_{1-k,D}(\tau)$, with an analogous identity $F_{-2k}(G_{-k,D}(\tau)) = -G_{-k,D}(\tau)$. If true, this extends the sign-flipping law already known for parabolic Eisenstein series to the hyperbolic setting.

What carries the argument

The load-bearing machinery is the splitting identity from Lemma 2.4: on each connected component of $\mathbb{H}\setminus E_D$, the function $F_{1-k,D}$ equals a non-holomorphic Eichler integral minus a constant times a holomorphic Eichler integral, plus a local polynomial $P_C$. The proof feeds this splitting into the interchange identities (2.8)–(2.9), which say that the flipping operator swaps the roles of the Bol and shadow operators up to constants; this forces the two integral pieces to flip sign. The remaining local-polynomial piece is then shown to flip sign by a direct calculation with the iterated Maass raising operator. For $G_{-k,D}$, the analogous splitting from the companion paper carries the same argument.

What would settle it

Take $k=2$ and $D=12$, choose a $\tau$ outside $E_D$ (for instance $\tau=0.5+i$), and evaluate both sides of $F_{-2}(F_{-1,12}(\tau)) = -F_{-1,12}(\tau)$ using the series (1.3) and the explicit differential operator definition. Any mismatch would refute Theorem 1.1; the same numerical check can be run for $G$ and for other $k,D$. Because the proof's per-summand identity can fail while the summed identity holds, the check must use the full sum over $Q_D$, not a single quadratic form.

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Extended reading notes

Core claim

On the paper's own terms, the discovery is that the flipping operator reverses the sign of the locally harmonic Maass forms $F_{1-k,D}$ and $G_{-k,D}$ away from the exceptional set $E_D$. The proof decomposes $F_{1-k,D}$ into a holomorphic Eichler integral, a non-holomorphic Eichler integral, and a local polynomial $P_C$. The two integral pieces are handled by the known interchange between the flipping, Bol, and shadow operators; the local polynomial is handled by an explicit calculation with the iterated Maass raising operator. The same strategy, using the splitting for $G_{-k,D}$ from the companion paper, yields the corollary for $G_{-k,D}$.

Load-bearing premise

The proof needs the flipping operator to reverse the sign of each term $Q(\tau,1)^{k-1}$ in the local polynomial; if this termwise identity fails, the theorem would still be true only if a more delicate cancellation among the quadratic-form summands produces the sign reversal.

Editorial extensions

If this is right

For every non-square discriminant and every $k\ge 2$, the flipping operator negates $F_{1-k,D}$ on the complement of $E_D$, so the sign-reversal law previously known for parabolic Eisenstein series now holds for a family of hyperbolic Poincaré series.
The same sign reversal holds for the weight $-2k$ companion $G_{-k,D}$, with the same exceptional set excluded.
Because $F_{1-k,D}$ is a simultaneous preimage of $f_{k,D}$ under the Bol and shadow operators, the identity shows that flipping swaps those two roles while negating the function, exactly as in the parabolic case.
On each connected component of $\mathbb{H}\setminus E_D$, the local polynomial part $P_C$ is negated by the flipping operator, so the identity can be read off component by component.
The paper's remark suggests the identity may give a shortcut for detecting when the nonconstant part of the local polynomial vanishes, which is the criterion tied to twisted central $L$-values.

Reading between the lines

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

If the summed identity survives while the termwise identity fails, the sign reversal must be produced by cancellations between the $b$ and $-b$ summands of $Q_D$; checking this pairing explicitly would give a purely combinatorial proof of the local-polynomial step.
The identity suggests the flipping operator acts like a global involution on the space spanned by the $F_{1-k,D}$ and $G_{-k,D}$, possibly with eigenvalues $\pm 1$; extending the computation to other lifts or to vector-valued settings might reveal a family of such involutions.
One could probe whether the exceptional set $E_D$ can be removed by using the continuously removable singularities of $G_{-k,D}$, in which case the sign identity might hold on all of $\mathbb{H}$ by continuity.

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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 the short version: Theorem 1.1 is not proved. The proof of the local-polynomial step (3.2) rests on the per-summand identity F_{2-2k}(Q^{k-1}) = -Q^{k-1}, and that identity is false. For k=2, D=12, Q=[-1,2,2], τ=1/2+i, a direct computation gives F_{-2}(Q) = -15/4 + i, while -Q = -15/4 - i. The source of the error is (3.5), which says R^{2k-2}_{2-2k}(τ^{k-1}) = (2k-2)! (τ/v^2)^{k-1}. Taking k=2, (2.5) gives R^2_{-2}(τ) = 2\bar{τ}/v^2, not 2τ/v^2. The binomial sum in the verification evaluates to (1 - 2iv/τ)^{k-1}, not 1. The paper drops the nonholomorphic factor's transformation; that's the load-bearing mistake.

What the paper does well: the question is natural, the splitting from [3] and [5] is used cleanly, and the Eichler-integral part of the argument is standard and correct. The result is a plausible extension of (1.1), and if it can be fixed, it would be a useful addition to the locally harmonic Maass form story.

One correction to the reader's report: Corollary 1.2 does not inherit the error. Its proof uses only (3.1) and the splitting in Lemma 2.5(3), which has a constant term rather than the Q-polynomial. So the corollary likely goes through; the paper should have said so explicitly. The main theorem, however, is unproven as written. A cancellation over the full Q_D-sum might restore (3.2), but the paper does not supply that argument.

Citation pattern is fine: the heavy reliance on [3] and [5] is legitimate because those are the sources of the decompositions. The paper is short and clearly written, but a central computation is wrong. I would not cite Theorem 1.1 in its current form, and I wouldn't bring the paper to reading group unless the authors post a corrected version.

Recommendation: send it to a referee anyway. The error is specific and fixable in principle, and the result is worth having if the summation argument materializes. With that concern on the table, a referee can make the call. If the authors cannot fix (3.5), reject.

Referee Report

3 major / 3 minor

Summary. The paper studies the flipping operator F_{2-2k} acting on the locally harmonic Maass forms F_{1-k,D} and G_{-k,D} introduced in earlier work of Bringmann, Kane, and Kohnen and of Bringmann and Mono. Theorem 1.1 claims that F_{2-2k}(F_{1-k,D}(\tau)) = -F_{1-k,D}(\tau) for non-square D, k \ge 2, and \tau outside the exceptional set E_D, and Corollary 1.2 makes the analogous claim for G_{-k,D}. The proof in Section 3 splits F_{1-k,D} into an Eichler-integral part and a local polynomial P_C. The Eichler-integral part is handled through the known relations (2.8), (2.9), and (1.4), while the treatment of P_C is reduced to the identity (3.2), which is in turn derived from a per-summand identity F_{2-2k}(Q(\tau,1)^{k-1}) = -Q(\tau,1)^{k-1}.

Significance. If the main theorem were established, it would give a natural hyperbolic analogue of the known flipping behavior of parabolic Maass\textendash Poincar\'e series and could simplify the local-polynomial arguments used in [10] and [18]. The organization of the paper is transparent, and the splitting of F_{1-k,D} and G_{-k,D} into Eichler integrals plus elementary terms is a useful structural observation. However, the central claim currently rests on a false identity, so the paper's main result is not proved as written.

major comments (3)

[Section 3, proof of Theorem 1.1, Eq. (3.3)] The identity F_{2-2k}(Q(\tau,1)^{k-1}) = -Q(\tau,1)^{k-1}, which is the per-summand statement used to prove (3.2), is false. For k=2, D=12, Q=[-1,2,2], and \tau=1/2+i, one has Q(\tau,1)=15/4+i and a direct computation gives R_{-2}^2(Q(\tau,1)) = 15/2 - 2i, hence F_{-2}(Q(\tau,1)) = -15/4 + i, whereas -Q(\tau,1) = -15/4 - i. Since (3.2) is the only step that treats the local polynomial P_C, Theorem 1.1 is not established.
[Section 3, derivation of (3.3)] The step labeled "Rewriting yields" after (3.5) is incorrect. For h(\tau)=\tau^{k-1}/v^{2k-2}, the Petersson slash with A=(\alpha \beta; \gamma \delta) gives h|_{2k-2}A(\tau) = (\alpha\tau+\beta)^{k-1}(\gamma\bar\tau+\delta)^{2k-2}/((\gamma\tau+\delta)^{k-1} v^{2k-2}), which is not a constant multiple of Q(\tau,1)^{k-1} v^{2-2k}. The nonholomorphic factor v does not transform by (c\tau+d), so the displayed transformation formula used to derive (3.3) is invalid.
[Section 3, around Eq. (3.4)] There is also an exponent inconsistency in the slash relation: from \tau|_{-2}A = -Q(\tau,1)/\sqrt{D}, the standard slash convention gives \tau^{k-1}|_{2-2k}A = (-1)^{k-1}D^{-(k-1)/2}Q(\tau,1)^{k-1}, not D^{(k-1)/2} as written in the paper. Even if this were only a typo, the false transformation of h(\tau) noted above remains a separate obstruction to (3.3).

minor comments (3)

[Introduction] The notation "k\in N\ge 2" should be written as "k\in\mathbb{N}, k\ge 2" for clarity.
[Introduction] There is a stray bracket in the citation "[2, Theorem 6.11 iv)]" after (1.1).
[Proof of Corollary 1.2] The phrase "Letting k\mapsto k+1 in (3.1)" is correct, but the dependence on k should be made explicit because the operators and the constant in (3.1) depend on k.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the proof relies on prior decomposition results and a direct operator computation; the flagged issue is a mathematical error, not a circular argument.

full rationale

The paper's derivation chain is not circular. Theorem 1.1 and Corollary 1.2 are proved by splitting the target functions via Lemma 2.4 (from [3]) and Lemma 2.5 (from [5]) into Eichler integrals plus a local polynomial, then applying known differential-operator identities (2.8), (2.9), (1.4), and (3.1). The cited lemmas are prior published results with stated assumptions and proofs; they do not assume the flipping identities being proved. The controverted step (3.3), used to prove (3.2), is a direct per-summand identity about the flipping operator applied to Q(τ,1)^{k-1}; even if that identity is false as the reader's computation suggests, that would be a substantive mathematical error in the argument, not a case of the conclusion being assumed by construction or of a fitted parameter being renamed a prediction. No parameters are fitted, no quantity is defined in terms of the claimed result, and no uniqueness theorem is imported from the authors' prior work to force the conclusion. The acknowledged heavy dependence on [3] and [5] is ordinary reliance on earlier theorems, including theorems by the current authors, but those theorems are externally checkable and do not presuppose the target statement. Hence the appropriate circularity score is 0.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The paper introduces no fitted parameters and no new entities. It relies on prior theorems by the same group, which are legitimate background but make the proof non-self-contained. No ad hoc constants or invented objects appear.

assumptions (4)

domain assumption Decomposition of F_{1-k,D} into Eichler integrals and local polynomial P_C (Lemma 2.4, cited from [3, Theorem 7.1]).
The proof splits F_{1-k,D} into D^{1/2-k}f*_{k,D} - D^{1/2-k}(2k-2)!/(4π)^{2k-1}E_{f_{k,D}} + P_C. If this splitting fails, the reduction to showing F(P_C)=-P_C collapses.
domain assumption Splitting and singularity properties of G_{-k,D} (Lemma 2.5, cited from [5] and [19]).
Corollary 1.2 relies on this splitting and on g_{k+1,D} decaying like a cusp form. These are cited results from the authors' earlier work.
domain assumption Existence for each Q of A in SL2(R) with τ|_{-2}A = -Q(τ,1)/√D ([3, Lemma 3.1]).
This matrix identity is used to transfer the raising-operator computation from Q(τ,1)^{k-1} to τ^{k-1}. It is cited, not proved in this paper.
standard math Standard facts: slash operator action, raising operator commutation (2.4), and identity (2.5) relating iterated raising operators to Bol derivatives.
These are background facts from [2,8,9] used in the proof of (3.5).

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Cite this review

Pith. "Pith review of Flipping operators and locally harmonic Maass forms." pith.science (2026). https://pith.science/paper/ZGACS2VH

@misc{pith2026250209359,
  author       = {Pith},
  title        = {Pith review of: Flipping operators and locally harmonic Maass forms},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZGACS2VH}},
  note         = {Machine review of arXiv:2502.09359}
}

read the original abstract

In the theory of integral weight harmonic Maass forms of manageable growth, two key differential operators, the Bol operator and the shadow operator, play a fundamental role. Harmonic Maass forms of manageable growth canonically split into two parts, and each operator controls one of these parts. A third operator, called the flipping operator, exchanges the role of these two parts. Maass--Poincar\'e series (of parabolic type) form a convenient basis of negative weight harmonic Maass forms of manageable growth, and flipping has the effect of negating an index. Recently, there has been much interest in locally harmonic Maass forms defined by the first author, Kane, and Kohnen. These are lifts of Poincar\'e series of hyperbolic type, and are intimately related to the Shimura and Shintani lifts. In this note, we prove that a similar property holds for the flipping operator applied to these Poincar\'e series.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

22 extracted references · 18 canonical work pages

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [3]

Bringmann, B

K. Bringmann, B. Kane, and W. Kohnen,Locally harmonic Maass forms and the kernel of the Shintani lift, International Mathematics Research Notices1 (2015), 3185–3224

work page 2015

[2] [5]

Bringmann and A

K. Bringmann and A. Mono,A modular framework for functions of Knopp and indefinite binary quadratic forms, arXiv:2208.01451

arXiv

[3] [10]

Ehlen, P

S. Ehlen, P. Guerzhoy, B. Kane, and L. Rolen,Central L-values of elliptic curves and local polynomials, Proc. Lond. Math. Soc. (3)120 (2020), no. 5, 742–769

2020

[4] [18]

Males, A

J. Males, A. Mono, L. Rolen, and I. Wagner,Central L-values of newforms and local polynomials, arXiv:2306.15519

arXiv

[5] [1]

Boisvert, C

R. Boisvert, C. Clark, D. Lozier, and F. Olver,NIST Handbook of mathematical functions, Cambridge University Press (2010)

work page 2010

[6] [2]

Bringmann, A

K. Bringmann, A. Folsom, K. Ono, and L. Rolen,Harmonic Maass forms and mock modular forms: theory and applications, AMS colloquium series (2017)

work page 2017

[7] [4]

Bringmann, B

K. Bringmann, B. Kane, and R. Rhoades,Duality and differential operators for harmonic Maass forms, in: From Fourier analysis and number theory to Radon transforms and geometry 85–106, Dev. Math.28, Springer, New York

work page

[8] [6]

Bruinier and J

J. Bruinier and J. Funke,On two geometric theta lifts, Duke Math. J.125 (2004), no. 1, 45–90

work page 2004