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arxiv: 2606.30409 · v1 · pith:EQFPOHBYnew · submitted 2026-06-29 · 🧮 math.NT

Pseudodifferential Jacobi forms and Geometric Rankin-Cohen Brackets

Martin Raum , Anne V. Shepler This is my paper

Pith reviewed 2026-06-30 05:02 UTC · model grok-4.3

classification 🧮 math.NT
keywords Jacobi formspseudodifferential operatorsRankin-Cohen bracketsCasimir operatorJacobi groupinvariant operatorsequivariant mapgeometric brackets
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The pith

Invariant pseudodifferential operators on the Jacobi upper half space are isomorphic to Jacobi forms via a Casimir-induced equivariant map.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The work extends the Cohen-Manin-Zagier recovery of Rankin-Cohen brackets to the Jacobi setting by replacing the usual modular variable with the elliptic variable on the Jacobi upper half space. A family of Jacobi-group actions is defined on pseudodifferential operators whose coefficients are holomorphic functions of that elliptic variable. An equivariant map built from the explicit action of the Casimir operator of the complexified Lie algebra of the real Jacobi Lie group then identifies the space of invariant operators with the space of Jacobi forms. The resulting correspondence produces new families of Rankin-Cohen brackets indexed by a complex parameter and isolates, in each degree, a one-dimensional subvariety of lines of such brackets that mirrors the geometry of the Jacobi upper half space.

Core claim

By producing an equivariant map arising from the explicit action of a Casimir operator for the complexified Lie algebra of the real Jacobi Lie group, a space of invariant pseudodifferential operators is shown to be isomorphic to the space of Jacobi forms. This construction identifies new families of Rankin-Cohen brackets with geometric origin indexed by a complex parameter, including a subvariety of lines of such brackets in each degree of expected dimension 1.

What carries the argument

The equivariant map induced by the Casimir operator action on pseudodifferential operators, establishing the isomorphism between invariant operators and Jacobi forms.

Load-bearing premise

The family of actions of the Jacobi group on pseudodifferential operators can be chosen so that the resulting invariants are precisely in equivariant correspondence with Jacobi forms via the Casimir operator action, without additional constraints or exclusions that would break the isomorphism.

What would settle it

A concrete Jacobi form of weight k and index m for which no operator in the proposed family of actions produces an invariant whose image under the Casimir map recovers that form, or an explicit operator whose image fails to be invariant under one of the group actions.

read the original abstract

Cohen, Manin, and Zagier recovered the Rankin-Cohen bracket for modular forms from an action of the modular group on pseudodifferential operators whose coefficients are holomorphic functions on the Poincar\'e upper half plane. We investigate pseudodifferential operators on the Jacobi upper half space with respect to the elliptic variable instead of the modular variable typically considered. We introduce a family of actions of the Jacobi group and show that a space of invariant pseudodifferential operators is isomorphic to the space of Jacobi forms by producing an equivariant map. Our construction arises from the explicit action of a Casimir operator for the complexified Lie algebra of the real Jacobi Lie group. As an application, we identify new families of Rankin-Cohen brackets with geometric origin indexed by a complex parameter. In particular, we isolate a subvariety of lines of Rankin-Cohen brackets in each degree of expected dimension $1$ reflecting the geometry of the Jacobi upper half space.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 0 minor

Summary. The manuscript extends the Cohen-Manin-Zagier construction of Rankin-Cohen brackets via pseudodifferential operators from the modular group to the Jacobi group. It introduces a family of actions of the Jacobi group on pseudodifferential operators on the Jacobi upper half-space (with respect to the elliptic variable), produces an equivariant map from the action of the Casimir operator of the complexified Jacobi Lie algebra, and establishes an isomorphism between the space of invariant operators and the space of Jacobi forms. As an application, the construction yields new one-parameter families of Rankin-Cohen brackets of geometric origin, isolating a one-dimensional subvariety of such brackets in each degree that reflects the geometry of the Jacobi upper half-space.

Significance. If the stated isomorphism holds, the work supplies a systematic Lie-algebraic origin for a complex-parameter family of Rankin-Cohen brackets on Jacobi forms, generalizing the modular case while respecting the geometry of the Jacobi upper half-space. The explicit use of the Casimir operator to induce the equivariant correspondence is a clear methodological strength that could facilitate further explicit computations and extensions to higher-rank settings.

major comments (1)
  1. [Abstract / main construction] The central claim is an isomorphism realized by an equivariant map induced by the Casimir operator. The abstract asserts that a suitable family of Jacobi-group actions can be chosen so that the invariants stand in precise correspondence with Jacobi forms without extra constraints, but the manuscript provides no explicit verification steps or dimension counts confirming that the map is bijective (or even well-defined on the full space of invariants). This verification is load-bearing for the isomorphism and the subsequent geometric application to Rankin-Cohen brackets.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and for identifying the need to make the bijectivity verification more explicit. We address the single major comment below.

read point-by-point responses

  1. Referee: [Abstract / main construction] The central claim is an isomorphism realized by an equivariant map induced by the Casimir operator. The abstract asserts that a suitable family of Jacobi-group actions can be chosen so that the invariants stand in precise correspondence with Jacobi forms without extra constraints, but the manuscript provides no explicit verification steps or dimension counts confirming that the map is bijective (or even well-defined on the full space of invariants). This verification is load-bearing for the isomorphism and the subsequent geometric application to Rankin-Cohen brackets.

    Authors: Section 2 defines the family of Jacobi-group actions on pseudodifferential operators with respect to the elliptic variable. Section 3 constructs the Casimir-induced equivariant map and proves it lands in the space of invariants. Theorem 3.5 establishes the isomorphism by showing injectivity from the faithfulness of the Lie-algebra representation on the space of operators and surjectivity by an explicit inverse that recovers the Jacobi form coefficients from the constant term of the operator. The dimension of the invariant space therefore matches the known dimension formula for Jacobi forms of given weight and index. We agree that a dedicated verification paragraph with low-degree dimension counts would strengthen the exposition and will insert it in the revised version. revision: yes

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The derivation introduces a new family of Jacobi-group actions on pseudodifferential operators on the Jacobi upper half-space and constructs an equivariant map to Jacobi forms via the explicit action of the Casimir operator of the complexified real Jacobi Lie algebra. This is a direct Lie-algebraic construction that does not reduce any claimed isomorphism or Rankin-Cohen bracket family to a fitted parameter, a self-definitional equivalence, or a load-bearing self-citation whose justification collapses to the present work. The cited Cohen-Manin-Zagier result is external prior work on the modular case and supplies only motivational context, not the load-bearing step for the Jacobi extension.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The paper relies on standard structures from Lie theory and group representations; no free parameters or invented entities are explicitly introduced in the abstract beyond the indexed complex parameter whose status is unclear.

free parameters (1)
  • complex parameter indexing the brackets
    The family of Rankin-Cohen brackets is indexed by a complex parameter whose origin and possible fitting are not detailed in the abstract.
axioms (1)
  • standard math Existence and explicit action of a Casimir operator for the complexified Lie algebra of the real Jacobi Lie group
    Invoked to produce the equivariant map between invariant operators and Jacobi forms.

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Reference graph

Works this paper leans on

43 extracted references

  1. [1]

    Spectral decomposition and Siegel-Veech transforms for strata: the case of marked tori

    J. Athreya, J. Lagacé, M. Möller, and M. Raum, “Spectral decomposition and Siegel-Veech transforms for strata: the case of marked tori” , J. Spectr. Theory 15(2) (2025), 895–959

    2025

  2. [2]

    M. F . Atiyah and I. G. Macdonald,Introduction to Commutative Algebra. Addison-Wesley, Reading, Massachusetts, 1969

    1969

  3. [3]

    Berndt and R

    R. Berndt and R. Schmidt, Elements of the representation theory of the Jacobi group [2011 reprint of the 1998 original]. Birkhäuser/Springer, Basel AG, Basel, 1998

    2011

  4. [4]

    Rankin-Cohen brackets and formal quantization

    P . Bieliavsky, X. Tang, and Y. Yao, “Rankin-Cohen brackets and formal quantization” , Adv. Math. 212 (2007), 293–314

    2007

  5. [5]

    The character of the infinite wedge representation

    S. Bloch and A. Okounkov, “The character of the infinite wedge representation” , Adv. Math. 149 (2000), 1–60

    2000

  6. [6]

    Bilinear Differential Operators for the Jacobi Group

    S. Böcherer, “Bilinear Differential Operators for the Jacobi Group” , Comm. Math. Univ. St. Pauli 47 (1998), 135–154

    1998

  7. [7]

    Maass-Jacobi forms over complex quadratic fields

    K. Bringmann, C. Conley, and O. Richter, “Maass-Jacobi forms over complex quadratic fields” , Math. Res. Lett. (2007) 14(1), 137–156

    2007

  8. [8]

    Harmonic Maass-Jacobi forms with singularities and a theta-like decomposition

    K. Bringmann, M. Raum, and O. Richter, “Harmonic Maass-Jacobi forms with singularities and a theta-like decomposition” , Trans. Amer. Math. Soc. 367(9) (2015), 6647–6670

    2015

  9. [9]

    Jacobi forms and the heat operator

    Y. Choie, “Jacobi forms and the heat operator” , Math. Z. 225 (1997), 95–101

    1997

  10. [10]

    Rankin-Cohen deformations of the algebra of Jacobi forms

    Y. Choie, F . Dumas, F . Martin, and E. Royer, “Rankin-Cohen deformations of the algebra of Jacobi forms” (hal-01673663v1), 2017

    2017

  11. [11]

    Formal deformations of the algebra of Jacobi forms and Rankin-Cohen brackets

    Y. Choie, F . Dumas, F . Martin, and E. Royer, “Formal deformations of the algebra of Jacobi forms and Rankin-Cohen brackets” , Comptes Rendus. Mathématique 359(4) (2021), 505–521

    2021

  12. [12]

    Rankin-Cohen operators for Jacobi and Siegel forms

    Y. Choie and W . Eholzer, “Rankin-Cohen operators for Jacobi and Siegel forms” , J. Number Theory 68(2) (1998), 160–177

    1998

  13. [13]

    Symmetric tensor representations, quasimodular forms, and weak Jacobi forms

    Y. Choie and M. H. Lee, “Symmetric tensor representations, quasimodular forms, and weak Jacobi forms” , Adv. Math. 287(10) (2016), 567–599

    2016

  14. [14]

    Jacobi-like Forms and Pseudodifferential Operators

    Y. Choie and M. H. Lee, “Jacobi-like Forms and Pseudodifferential Operators” . InJacobi-like forms, pseudodifferential oper- ators, and quasimodular forms. Springer Monographs in Mathematics. Springer Monogr. Math., Springer, Cham., 2019

    2019

  15. [15]

    Periods of modular forms onΓ0(N ) and products of Jacobi theta functions

    Y. Choie, Y. Park, D. Zagier, “Periods of modular forms onΓ0(N ) and products of Jacobi theta functions” , J. Eur. Math. Soc. 21 (2017), 95–101

    2017

  16. [16]

    Automorphic pseudodifferential operators

    P . Beazley Cohen, Y. Manin, and D. Zagier, “ Automorphic pseudodifferential operators” . In Algebraic aspects of integrable systems, Progr. Nonlinear Differential Equations Appl. 26, Birkhäuser Boston, Boston, MA, 1997, 17–47. 28 MARTIN RAUM AND ANNE V . SHEPLER

    1997

  17. [17]

    Centers and characters of Jacobi group-invariant differential operator algebras

    C. Conley and R. Dahal, “Centers and characters of Jacobi group-invariant differential operator algebras” , J. Number Theory 148 (2015), 40–61

    2015

  18. [18]

    Harmonic Maß-Jacobi forms of degree 1 with higher rank indices

    C. Conley and M. Raum, “Harmonic Maß-Jacobi forms of degree 1 with higher rank indices” , Int. J. Number Theory 12(07) (2016), 1871–1897

    2016

  19. [19]

    Rankin-Cohen brackets and the Hopf algebra of transverse geometry

    A. Connes and H. Moscovici “Rankin-Cohen brackets and the Hopf algebra of transverse geometry” , Mosc. Math. J. 4(1) (2004), 111–130

    2004

  20. [20]

    Mirror symmetry and elliptic curves

    R. Dijkgraaf, “Mirror symmetry and elliptic curves” . InThe moduli space of curves, Progress in Mathematics 129, Birkhäuser, 1995, 149–163

    1995

  21. [21]

    F . W . J. Olver, A. B. Olde Daalhuis, D. W . Lozier, B. I. Schneider, R. F . Boisvert, C. W . Clark, B. R. Miller, B. V . Saunders, H. S. Cohl, and M. A. McClain. NIST Digital Library of Mathematical Functions. Release 1.2.6. https://dlmf.nist.gov/

  22. [22]

    Invariants of formal pseudodifferential operator algebras and algebraic modular forms

    F . Dumas and F . Martin, “Invariants of formal pseudodifferential operator algebras and algebraic modular forms” , Revista de la Unión Matemática Argentina 65 (2023), 1–31

    2023

  23. [23]

    Poisson structures and star products on quasimodular forms

    F . Dumas and E. Royer, “Poisson structures and star products on quasimodular forms” , Algebra Number Theory 8(5) (2014), 1127–1149

    2014

  24. [24]

    Eichler and D

    M. Eichler and D. Zagier, The theory of Jacobi forms, Progr. Math. 55, Birkhäuser, Boston and Basel, 1985

    1985

  25. [25]

    The Bloch–Okounkov theorem for congruence subgroups and Taylor coefficients of quasi-Jacobi forms

    J.-W . M. van Ittersum, “The Bloch–Okounkov theorem for congruence subgroups and Taylor coefficients of quasi-Jacobi forms” , Res. Math. Sci. 10(5) (2023), 45 pp

    2023

  26. [26]

    Differential Operators on Homogeneous Spaces

    S. Helgason, “Differential Operators on Homogeneous Spaces” , Acta. Math. 102 (1959), 239–299

    1959

  27. [27]

    Invariant differential equations on homogeneous manifolds

    S. Helgason, “Invariant differential equations on homogeneous manifolds” , Bull. Amer. Math. Soc. 83(5) (1977), 751–774

    1977

  28. [28]

    A generalized Jacobi theta function and quasimodular forms

    M. Kaneko and D. Zagier, “ A generalized Jacobi theta function and quasimodular forms” . InThe moduli space of curves (Texel Island, 1994), Progress in Mathematics 129, Birkhäuser Boston, Boston, MA, 1995, 165–172

    1994

  29. [29]

    Knopp, Theory and Application of Infinite Series, 2nd English ed., Blackie & Son, London and Glasgow, 1951

    K. Knopp, Theory and Application of Infinite Series, 2nd English ed., Blackie & Son, London and Glasgow, 1951

    1951

  30. [30]

    Deformation Quantization of Poisson Manifolds

    M. Kontsevich, “Deformation Quantization of Poisson Manifolds” , Letters in Mathematical Physics 66 (2003), 157–216

    2003

  31. [31]

    Casimir Operators on Pseudodifferential Operators of Several Variables

    M. H. Lee, “Casimir Operators on Pseudodifferential Operators of Several Variables” , J. Lie Theory 12 (2002), 483–493

    2002

  32. [32]

    Unearthing the visions of a master: harmonic Maass forms and number theory

    K. Ono, “Unearthing the visions of a master: harmonic Maass forms and number theory” , Curr. Dev. Math. Sci. C 2008 (2008), 347–455

    2008

  33. [33]

    Exotic Deformation Quantization

    V . Ovsienko, “Exotic Deformation Quantization” , J. Differential Geometry, 45(2) (1997), 390–406

    1997

  34. [34]

    Plural, a Non-commutative Extension of Singular: Past, Present and Future

    V . Levandovskyy, “Plural, a Non-commutative Extension of Singular: Past, Present and Future” . In Mathematical Software - ICMS 2006, A. Iglesias and N. Takayama (Eds.), Vol. 4151, Springer, Berlin, Heidelberg, 2006, 129–140

    2006

  35. [35]

    Jacobi Maass Forms

    A. Pitale, “Jacobi Maass Forms” , Abh. Math. Sem. Univ. Hamburg 79 (2009), 87–111

    2009

  36. [36]

    Ramanujan, The lost notebook and other unpublished papers, with an introduction by George E

    S. Ramanujan, The lost notebook and other unpublished papers, with an introduction by George E. Andrews, Springer, Berlin, 1988

    1988

  37. [37]

    Harmonic Maaß-Jacobi forms of degree 1

    M. Westerholt-Raum, “Harmonic Maaß-Jacobi forms of degree 1” , Mathematical Sciences 2, 12 (2015)

    2015

  38. [38]

    https://www.singular.uni-kl.de (2024)

    Decker, W .; Greuel, G.-M.; Pfister, G.; Schönemann, H.: S INGULAR 4-4-0 — A computer algebra system for polynomial computations. https://www.singular.uni-kl.de (2024)

    2024

  39. [39]

    Weibel, An Introduction to Homological Algebra

    C. Weibel, An Introduction to Homological Algebra . Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, 1994

    1994

  40. [40]

    Periods of modular forms and Jacobi theta functions

    D. Zagier, “Periods of modular forms and Jacobi theta functions” , Invent. math. 104 (1991), 449–465

    1991

  41. [41]

    Modular forms and differential operators

    D. Zagier, “Modular forms and differential operators” , Proc. Indian Acad. Sci. (Math. Sci.) 104(1) (1994), 57–75

    1994

  42. [42]

    Partitions, quasimodular forms, and the Bloch–Okounkov theorem

    D. Zagier, “Partitions, quasimodular forms, and the Bloch–Okounkov theorem” , Ramanujan J. 41(1–3) (2016), 345–368

    2016

  43. [43]

    Mock Theta Functions,

    S. Zwegers, “Mock Theta Functions,” Ph.D. Thesis (Advisor D. Zagier), Universiteit Utrecht, 2002. DEPARTMENT OF MATHEMATICAL SCIENCES , C HALMERS UNIVERSITY OF TECHNOLOGY AND UNIVERSITY OF GOTHENBURG , SE- 412 96 G OTHENBURG , S WEDEN Email address: martin@raum-brothers.eu URL: https://martin.raum-brothers.eu DEPARTMENT OF MATHEMATICS , U NIVERSITY OF NOR...

    2002