REVIEW 1 major objections 2 minor 10 references
A direct proof of Mono-Rolen-Stumpenhusen and new constructions via the Maass raising operators
T0 review · 1 major / 2 minor · reviewed 2026-06-26 · grok-4.3
Pith's one-line read The Maass raising operator maps quadratic form Poincaré series directly to the functions ω_{k+1,D}, yielding their modularity and eigenvalues.
desk verdict Bringmann and Kane give a direct proof of the MRS result by realizing the ω functions as Maass raising images of the quadratic Poincaré series, plus some extensions to local forms. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The Maass raising operator, which sends the quadratic form Poincaré series f_{k,D} to ω_{k+1,D} while transferring modularity and eigenvalue data via its commutation relations with the group action.
What would settle it
Compute the explicit action of the Maass raising operator on f_{k,D} for small values of k and D and verify whether the result matches the closed-form expression for ω_{k+1,D}.
Extended reading notes
Core claim
We realize their functions ω_{k+1,D} as images of the quadratic form Poincaré series f_{k,D} under the Maass raising operator. This perspective gives a natural explanation for the modularity and Laplace eigenvalue properties of ω_{k+1,D}. We further extend these results by investigating the images of more general local Maass forms under the Maass raising and lowering operators.
Load-bearing premise
Applying the Maass raising operator to f_{k,D} produces exactly the functions ω_{k+1,D} defined in the earlier paper.
Editorial extensions
If this is right
- Modularity of ω_{k+1,D} follows at once from that of f_{k,D} together with the known transformation law of the raising operator.
- The Laplace eigenvalue of ω_{k+1,D} is obtained by shifting the eigenvalue of f_{k,D} by the explicit amount contributed by the operator.
- The construction applies verbatim to images of arbitrary local Maass forms, producing new families with controlled transformation properties under the modular group.
- Raising and lowering operators can be iterated, generating sequences of such functions in adjacent weights.
Reading between the lines
- Fourier expansions of the ω_{k+1,D} could be read off from the known expansions of the Poincaré series after applying the raising operator term by term.
- The same operator viewpoint may relate the ω functions to other families of Maass forms constructed by different differential operators.
- Applying the lowering operator to the resulting objects could produce descent relations between forms of different weights.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper claims to give a direct conceptual proof of the main result of Mono, Rolen, and Stumpenhusen by realizing the functions ω_{k+1,D} as the images of the quadratic-form Poincaré series f_{k,D} under the Maass raising operator R_k. This is said to explain the modularity and Laplace eigenvalue properties of ω_{k+1,D} via known commutation relations. The work further extends the approach by studying the images of more general local Maass forms under the raising and lowering operators.
Significance. If the central identification is exact, the approach supplies a conceptual explanation for the properties of ω_{k+1,D} using standard differential operators on Maass forms, which is a useful contribution to the literature on quadratic-form series and their modular properties. The extension to general local Maass forms is a natural outgrowth that may prove useful for further constructions.
major comments (1)
- [proof of the main identification (around the statement that ω_{k+1,D} is realized as the image under the raising operato] The central claim that R_k(f_{k,D}) equals ω_{k+1,D} exactly (rather than up to scalar or additive kernel term) is load-bearing for the 'direct proof.' Commutation with the group action and the shift in the Laplace eigenvalue are established by standard properties of the raising operator, but these do not by themselves determine the normalization constants or confirm that the linear combination over the relevant quadratic forms matches the definition of ω_{k+1,D} in the prior work. An explicit coefficient comparison or normalization check is required.
minor comments (2)
- Recall or cite the precise normalization conventions for both the Poincaré series f_{k,D} and the functions ω_{k+1,D} at the point where the identification is stated, to make the comparison self-contained.
- Clarify the precise definition of 'local Maass forms' used in the extension section, including any growth or support conditions that are preserved or altered by the operators.
Simulated Author's Rebuttal
We thank the referee for the careful reading and constructive comments. The referee correctly notes that establishing the exact identification R_k(f_{k,D}) = ω_{k+1,D} requires more than the general commutation relations. We address this below and will revise the manuscript accordingly.
read point-by-point responses
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Referee: The central claim that R_k(f_{k,D}) equals ω_{k+1,D} exactly (rather than up to scalar or additive kernel term) is load-bearing for the 'direct proof.' Commutation with the group action and the shift in the Laplace eigenvalue are established by standard properties of the raising operator, but these do not by themselves determine the normalization constants or confirm that the linear combination over the relevant quadratic forms matches the definition of ω_{k+1,D} in the prior work. An explicit coefficient comparison or normalization check is required.
Authors: We agree that the exact equality, including normalization constants and the precise linear combination over quadratic forms, must be verified explicitly rather than inferred solely from commutation. In the revised manuscript we will insert a direct computation of R_k applied to the Fourier expansion of f_{k,D}, followed by a term-by-term coefficient comparison with the definition of ω_{k+1,D} from Mono-Rolen-Stumpenhusen. This will confirm the identification holds without scalar factors or kernel terms and will be placed immediately after the statement of the main theorem. revision: yes
Circularity Check
No circularity: independent realization via standard operators
full rationale
The paper supplies a direct proof of a result from Mono-Rolen-Stumpenhusen by exhibiting the target functions ω_{k+1,D} as the image of the independently defined quadratic-form Poincaré series f_{k,D} under the Maass raising operator. This step relies on the known commutation relations of the operator with the relevant group actions and the Laplace operator; those relations are external facts, not derived inside the paper or from self-citation. No equation equates a fitted quantity to a prediction, no uniqueness theorem is imported from the authors' own prior work, and the central identification is not obtained by renaming or by construction from the output itself. The derivation therefore remains self-contained against external benchmarks.
Assumptions & free parameters
assumptions (1)
- standard math Standard properties of Maass raising and lowering operators on modular forms
Cite this review
Pith. "Pith review of A direct proof of Mono-Rolen-Stumpenhusen and new constructions via the Maass raising operators." pith.science (2026). https://pith.science/paper/X66EE7SJ
@misc{pith2026260627212,
author = {Pith},
title = {Pith review of: A direct proof of Mono-Rolen-Stumpenhusen and new constructions via the Maass raising operators},
year = {2026},
howpublished = {\url{https://pith.science/paper/X66EE7SJ}},
note = {Machine review of arXiv:2606.27212}
}
abstract
In this paper, we give a direct conceptual proof of the main result of Mono, Rolen, and Stumpenhusen, using differential operators. More precisely, we realize their functions $\omega_{k+1,D}$ as images of the quadratic form Poincar\'e series $f_{k,D}$ under the Maass raising operator. This perspective gives a natural explanation for the modularity and Laplace eigenvalue prop erties of $\omega_{k+1,D}$. We further extend these results by investigating the images of more general local Maass forms under the Maass raising and lowering operators.
Reference graph
Works this paper leans on
-
[1]
Bringmann, A
K. Bringmann, A. Folsom, K. Ono, and L. Rolen, it Harmonic Maass forms and mock modular forms: theory and applications it , Amer. Math. Soc. Colloq. Publ. 64, 2017
2017
-
[2]
Bringmann, B
K. Bringmann, B. Kane, and W. Kohnen, it Locally harmonic Maass forms and the kernel of the Shintani lift it , Int. Math. Res. Not. 2015 (2015), 3185--3224
2015
-
[3]
Bringmann, B
K. Bringmann, B. Kane, and M. Viazovska, Theta lifts and local Maass forms , Math. Res. Lett. 20 (2013), 213--234
2013
-
[4]
Bruinier and J
J. Bruinier and J. Funke, it On two geometric theta lifts it , Duke Math J. 125 (2004), 45--90
2004
-
[5]
Bruinier, W
J. Bruinier, W. Kohnen, and K. Ono, it The arithmetic of the values of modular functions and the divisors of modular forms it , Compos. Math. 140 (2004), 552--566
2004
-
[6]
Olver, A
Digital Library of Mathematical Functions, National Institute of Standards and Technology, Editors: F. Olver, A. Olde Daalhuis, D. Lozier, B. Schneider, R. Boisvert, C. Clark, B. Miller, B. Saunders, H. Cohl, and M. McClain, http://dlmf.nist.gov/
-
[7]
Kohnen, it Newforms of half-integral weight it , J
W. Kohnen, it Newforms of half-integral weight it , J. reine Angew. Math. 333 (1982), 32--72
1982
-
[8]
Kohnen and D
W. Kohnen and D. Zagier, it Values of L -series of modular forms at the center of the critical strip it , Invent. Math. 64 (1981), no. 2, 175--198
1981
Show all 10 references
-
[9]
A. Mono, L. Rolen, and J. Stumpenhusen, On a divisor modular form and a theta lift, arXiv:2509.01378
-
[10]
Zagier, it Modular forms associated to real quadratic fields it , Invent
D. Zagier, it Modular forms associated to real quadratic fields it , Invent. Math. 30 (1975), 1--46
1975
Reviewed June 26, 2026 · model on record in the stance chip above.
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