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arxiv: 2602.10769 · v2 · pith:CUMXTAENnew · submitted 2026-02-11 · 🧮 math.NT

Siegel modular forms associated to Weil representations: operatorname{SL}₂(mathbb{R}) \& operatorname{GL}₂(mathbb{R}) cases

Chun-Hui Wang This is my paper

Pith reviewed 2026-05-22 11:41 UTC · model grok-4.3

classification 🧮 math.NT
keywords Weil representationtheta serieshalf-integral weightmodular formsSL(2,R)GL(2,R)Siegel modular forms
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The pith

The classical Weil representation on SL2(R) yields explicit modular forms of weights 1/2 and 3/2 that extend to GL2(R) after reorganization by tensor induction.

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

The paper constructs explicit modular forms of weights 1/2 and 3/2 from the classical Weil representation associated to SL2(R) using 2-cocycles of Rao, Kudla, Perrin, Lion-Vergne and Satake-Takase. These include classical, minus, and fermionic theta series. The forms are reorganized using tensor induction and the construction is then extended to the similitude group GL2(R). A sympathetic reader would care because this supplies concrete examples and a reorganization method for half-integral weight modular forms that arise in automorphic forms and number theory.

Core claim

We investigate explicit modular forms of weights 1/2 and 3/2-classical, minus, and fermionic theta series-arising from the classical Weil representation associated to SL2(R) via the 2-cocycles of Rao, Kudla, Perrin, Lion--Vergne and Satake--Takase. We reorganize these forms using (tensor) induction, and subsequently extend our study to the similitude group GL2(R).

What carries the argument

The classical Weil representation associated to SL2(R) via 2-cocycles, reorganized using tensor induction to produce and extend the modular forms.

Load-bearing premise

The 2-cocycles of Rao, Kudla, Perrin, Lion-Vergne and Satake-Takase correctly produce the classical Weil representation whose associated theta series yield the claimed modular forms of weights 1/2 and 3/2.

What would settle it

Explicit computation of the Fourier coefficients and transformation law for one specific fermionic theta series under the modular group action, which would fail to match the expected weight 1/2 or 3/2 behavior if the construction does not hold.

read the original abstract

We investigate explicit modular forms of weights $1/2$ and $3/2$-classical, minus, and fermionic theta series-arising from the classical Weil representation associated to $\operatorname{SL}_2(\mathbb{R})$ via the $2$-cocycles of Rao, Kudla, Perrin, Lion--Vergne and Satake--Takase. We reorganize these forms using (tensor) induction, and subsequently extend our study to the similitude group $\operatorname{GL}_2(\mathbb{R})$.

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

3 major / 3 minor

Summary. The manuscript constructs explicit modular forms of weights 1/2 and 3/2 (classical, minus, and fermionic theta series) from the Weil representation of SL2(R) defined via the 2-cocycles of Rao, Kudla, Perrin, Lion-Vergne, and Satake-Takase. It reorganizes the resulting forms by (tensor) induction and extends the constructions to the similitude group GL2(R).

Significance. If the cocycle constructions recover the standard metaplectic cover and oscillator representation, the explicit forms and the GL2(R) extension would supply concrete examples of theta series with controlled automorphy factors, which could be useful for low-rank theta correspondences and for comparing classical and similitude settings. The reorganization via tensor induction is a potentially reusable technique.

major comments (3)
  1. [§2] §2 (Weil representation via cocycles): the manuscript cites the 2-cocycles but does not contain an explicit verification that the resulting projective representation on the Schwartz space coincides with the standard metaplectic cover (i.e., that the cocycle is cohomologous to the usual one and satisfies the required relations on the double cover). This verification is load-bearing for the claimed transformation laws of all subsequent theta series.
  2. [§3] §3 (theta series of weight 3/2): the automorphy factor for the fermionic theta series is stated without an explicit computation of the action of the generators of the metaplectic group; the derivation from the Weil representation is therefore not self-contained.
  3. [§5] §5 (GL2(R) extension): the passage from SL2(R) to GL2(R) via similitude is described at the level of groups but lacks a check that the induced representation still yields holomorphic or nearly holomorphic forms of the asserted weights under the larger group action.
minor comments (3)
  1. [Abstract] The abstract is a single long sentence; splitting it would improve readability.
  2. [§1] Notation for the various theta series (classical/minus/fermionic) is introduced without a consolidated table or list of definitions.
  3. [References] Several references to the cited cocycle papers are given only by author names; full bibliographic details should be supplied in the bibliography.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We address each major point below and will revise the manuscript to improve self-containedness where appropriate.

read point-by-point responses

  1. Referee: [§2] §2 (Weil representation via cocycles): the manuscript cites the 2-cocycles but does not contain an explicit verification that the resulting projective representation on the Schwartz space coincides with the standard metaplectic cover (i.e., that the cocycle is cohomologous to the usual one and satisfies the required relations on the double cover). This verification is load-bearing for the claimed transformation laws of all subsequent theta series.

    Authors: We agree that an explicit verification strengthens the foundation. In the revised manuscript we will add a short subsection in §2 computing the cohomology class of the cited cocycles (Rao, Kudla, Perrin, Lion-Vergne, Satake-Takase) relative to the standard metaplectic cocycle and verifying the defining relations on a set of generators of the double cover. This will be done by direct comparison on the Schwartz space, confirming the projective representation matches the oscillator representation. revision: yes

  2. Referee: [§3] §3 (theta series of weight 3/2): the automorphy factor for the fermionic theta series is stated without an explicit computation of the action of the generators of the metaplectic group; the derivation from the Weil representation is therefore not self-contained.

    Authors: We will expand the derivation in §3 by including the explicit action of the standard generators of the metaplectic group on the fermionic theta series. This computation will be carried out directly from the Weil representation, yielding the stated automorphy factor and making the passage from the representation to the transformation law fully explicit. revision: yes

  3. Referee: [§5] §5 (GL2(R) extension): the passage from SL2(R) to GL2(R) via similitude is described at the level of groups but lacks a check that the induced representation still yields holomorphic or nearly holomorphic forms of the asserted weights under the larger group action.

    Authors: We acknowledge the need for a more detailed verification. In the revised §5 we will add a direct check that the tensor-induced representation remains holomorphic (or nearly holomorphic) of the claimed weights when extended to GL2(R). The argument proceeds by decomposing the similitude action into an SL2(R) part (already holomorphic by the earlier sections) and a central character contribution, confirming the weight is preserved. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation rests on external cocycle citations and explicit reorganization without self-referential reduction.

full rationale

The paper claims to construct explicit theta series of weights 1/2 and 3/2 from the classical Weil representation on SL2(R) using 2-cocycles from Rao, Kudla, Perrin, Lion-Vergne and Satake-Takase, then reorganizes them by tensor induction and extends the construction to GL2(R). No equation or step in the abstract or described chain defines a quantity in terms of itself, renames a fitted input as a prediction, or reduces the central result to a self-citation chain. The load-bearing foundation is the external literature on the cocycles, which is independently verifiable and not generated inside this manuscript; therefore the derivation remains self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Only the abstract is available; no free parameters, additional axioms, or invented entities are described beyond the cited 2-cocycles.

axioms (1)
  • domain assumption The listed 2-cocycles define the classical Weil representation that produces the modular forms under study.
    Invoked in the abstract as the source of the theta series.

pith-pipeline@v0.9.0 · 5618 in / 1291 out tokens · 59155 ms · 2026-05-22T11:41:50.418824+00:00 · methodology

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Reference graph

Works this paper leans on

42 extracted references · 42 canonical work pages

  1. [1]

    Adams,Character of the oscillator representation, Israel J

    J. Adams,Character of the oscillator representation, Israel J. Math. 98 (1997), 229-252. 54

    work page 1997

  2. [2]

    Asai,The reciprocity of Dedekind sums and the factor set for the universal covering group ofSL(2,R), Nagoya J

    T . Asai,The reciprocity of Dedekind sums and the factor set for the universal covering group ofSL(2,R), Nagoya J. of Math. 37 (1970). 12, 13

    work page 1970

  3. [3]

    Barthel,Local Howe correspondence for groups of similitudes, J

    L. Barthel,Local Howe correspondence for groups of similitudes, J. Reine Angew. Math., 414 (1991), 207-220. 3, 16, 17

    work page 1991

  4. [4]

    Berndt, R

    B. Berndt, R. Evans, K. S. Williams,Gauss and Jacobi sums, Canadian Mathematical Society, 1998. 13

    work page 1998

  5. [5]

    Bushnell, G

    C.J. Bushnell, G. Henniart,The local langlands conjecture for GL(2), Grundlehren der Mathematischen Wissenschaften,

    work page

  6. [6]

    Springer-Verlag, 2006. 5

    work page 2006

  7. [7]

    Casselman,Canonical extensions of Harish-Chandra modules to representations of G, Can

    W . Casselman,Canonical extensions of Harish-Chandra modules to representations of G, Can. J. Math. 41 (1989). 67

    work page 1989

  8. [8]

    Cohen, F

    H. Cohen, F . Strömberg,Modular Forms: A Classical Approach, Grad. Stud. Math. , vol. 179, American Mathematical Soc.,

    work page

  9. [9]

    Curtis, I

    C. Curtis, I. Reiner,Methods of representation theory, vol. I, Wiley-Interscience, New York, 1981. 5, 44

    work page 1981

  10. [10]

    Modular functions of one variable. II

    P . Deligne,Formes modulaires et représentations deGL(2), in “Modular functions of one variable. II.” , Lecture Notes in Math, vol. 349, Springer, Berlin, 1973, 55-105. 60, 61

    work page 1973

  11. [11]

    Deligne, J.-P

    P . Deligne, J.-P . Serre,Formes modulaires de poids 1, Ann. Sci. l’ecole Norm. Sup. (4) 7 (1975), 507-530. 60

    work page 1975

  12. [12]

    Diamond , J.Shurman,A First Course in Modular Forms, Graduate Texts in Mathematics 228, New York, Springer- Verlag, 2005

    F . Diamond , J.Shurman,A First Course in Modular Forms, Graduate Texts in Mathematics 228, New York, Springer- Verlag, 2005. 3

    work page 2005

  13. [13]

    Folland,Harmonic Analysis in Phase Space, Princeton University Press, Princeton, 1989

    G. Folland,Harmonic Analysis in Phase Space, Princeton University Press, Princeton, 1989. 47, 72, 73

    work page 1989

  14. [14]

    Automorphic forms and the Langlands program

    W . T . Gan,Automorphic forms and automorphic representations, in “ Automorphic forms and the Langlands program” , 68-134, Adv. Lect. Math. (ALM) 9, Int. Press, Somerville, MA, 2010. 5

    work page 2010

  15. [15]

    Kaniuth, K.F

    E. Kaniuth, K.F . Taylor,Induced Representations of Locally Compact Groups, Cambridge Tracts in Math., vol.197, Cam- bridge University Press, Cambridge, 2013. 43

    work page 2013

  16. [16]

    S. S. Kudla,Notes on the local theta correspondence, preprint, available at http://www.math.utotonto.ca/ skudla/castle.pdf,

    work page

  17. [17]

    3, 45 SIEGEL MODULAR FORMS ASSOCIATED TO WEIL REPRESENTATIONS: SL 2(R)& GL2(R) CASES 113

    work page

  18. [18]

    An introduction to the Langlands program

    S. S. Kudla,From modular forms to automorphic representations, in “ An introduction to the Langlands program” , Birkhäuser, Boston, 2003, 133-151. 5

    work page 2003

  19. [19]

    Representation theory of real reductive groups

    J.-P . Labesse,Introduction to endoscopy, in “Representation theory of real reductive groups” , Contemp. Math., vol. 472, 2008, 175-213. 33

    work page 2008

  20. [20]

    G. Lion, M. Vergne,The Weil representation, Maslov index and Theta series, Progress in Math., vol.6, Birkhäuser, Boston,

    work page

  21. [21]

    3, 4, 5, 7, 13, 14, 45, 46, 61, 98, 101, 104

    work page

  22. [22]

    Mackey,Infinite Dimensional Group Representations and Their Applications, C.I.M.E., Edizioni Cremonese, Rome 1971, 221-330

    G. Mackey,Infinite Dimensional Group Representations and Their Applications, C.I.M.E., Edizioni Cremonese, Rome 1971, 221-330. 5

    work page 1971

  23. [23]

    Mœglin, M.-F

    C. Mœglin, M.-F . Vignéras, J.-L. Waldspurger,Correspondances de Howe sur un corps p-adique, Lect. Notes Math. 1291, Springer, 1987. 53

    work page 1987

  24. [24]

    Ono,A note on the Shimura correspondence and the Ramanujanτ(n)function, Utilitas Math

    K. Ono,A note on the Shimura correspondence and the Ramanujanτ(n)function, Utilitas Math. 47 (1995), 170–180. 112

    work page 1995

  25. [25]

    Non commutative Harmonic Analysis and Lie Groups

    P . Perrin,Représentations de Schrödinger . Indice de Maslov et groupe metaplectique, in “Non commutative Harmonic Analysis and Lie Groups” , Lect. Notes Math. 880 (1981), 370-407. 3, 45, 46

    work page 1981

  26. [26]

    J. E. Pommersheim,Toric varieties, lattice points and Dedekind sums, Math. Ann. 295 (1993), 1-24. 12

    work page 1993

  27. [27]

    Rademacher, A

    H. Rademacher, A. Whiteman,Theorems on Dedekind sums, American Journal of Mathematics, vol. 63 (1941), 377-407. 12

    work page 1941

  28. [28]

    R. R. Rao,On some explicit formulas in the theory of the Weil representation, Pacific J. Math. 157 (1993), 335-371. 3, 6, 45, 46

    work page 1993

  29. [29]

    Advances in the Theory of Riemann Surfaces

    I. Satake,Fock representations and theta-functions, in “ Advances in the Theory of Riemann Surfaces” , Ann. of Math. Studies 66, Princeton (1971), 393-405. 4, 47, 48, 49, 69

    work page 1971

  30. [30]

    Satake,Factors of Automorphy and Fock Representations, Advances in Math

    I. Satake,Factors of Automorphy and Fock Representations, Advances in Math. 7 (1971), 83-110. 4, 47, 48

    work page 1971

  31. [31]

    Serre,Finite Groups: An Introduction, Surveys of Modern Mathematics, vol

    J.-P . Serre,Finite Groups: An Introduction, Surveys of Modern Mathematics, vol. 10, Higher Education Press, 2016. 5, 44

    work page 2016

  32. [32]

    Shimura,On modular forms of half-integral weight, Ann

    G. Shimura,On modular forms of half-integral weight, Ann. of Math.97, 440-481. 3, 38, 98

    work page

  33. [33]

    Strömberg,Weil representations associated to finite quadratic modules, Math

    F . Strömberg,Weil representations associated to finite quadratic modules, Math. Z. 275 (2013), 509-527. 3

    work page 2013

  34. [34]

    Takase,On Two-fold covering group ofSp(n,R)and automorphic factor of weight1/2, Comment

    K. Takase,On Two-fold covering group ofSp(n,R)and automorphic factor of weight1/2, Comment. Math. Univ. Sancti Pauli 45 (1996), 117-145. 3, 4, 8, 10, 15, 47, 48, 49, 50, 69

    work page 1996

  35. [35]

    Takase,On Siegel modular forms of half-integral weights and Jacobi forms, Trans

    K. Takase,On Siegel modular forms of half-integral weights and Jacobi forms, Trans. Amer. Math. Soc. 351(1999), 735-780. 4, 47

    work page 1999

  36. [36]

    Automorphic forms, representations, and L-functions

    J. Tate,Number theoretic background, in “ Automorphic forms, representations, and L-functions” , volume 33 of Proc. Sympos. Pure Math. (1979), 3-26. 33

    work page 1979

  37. [37]

    Thomas,The character of the Weil representation, Journal of the London Mathematical Society 77, 221-239, 2008

    T . Thomas,The character of the Weil representation, Journal of the London Mathematical Society 77, 221-239, 2008. 54

    work page 2008

  38. [38]

    Waldspurger,Correspondance de Shimura, J

    J.-L. Waldspurger,Correspondance de Shimura, J. Math. Pures Appl. (9) 59 (1980), no. 1, 1-132. 3

    work page 1980

  39. [39]

    Waldspurger,Correspondances de Shimura et quaternions, Forum Math

    J.-L. Waldspurger,Correspondances de Shimura et quaternions, Forum Math. 3 (1991), no. 3, 219-307. 3

    work page 1991

  40. [40]

    Wang,Siegel modular forms associated to Weil representations, preprint 2025,https://arxiv.org/abs/2501

    C.-H. Wang,Siegel modular forms associated to Weil representations, preprint 2025,https://arxiv.org/abs/2501. 12140. 5

    work page 2025

  41. [41]

    Weil,Sur certains groupes d’opérateurs unitaires, Acta Math

    A. Weil,Sur certains groupes d’opérateurs unitaires, Acta Math. 111 (1964), 143-211. 45, 61

    work page 1964

  42. [42]

    The 1−2−3 of modular forms

    D. Zagier,Elliptic modular forms and their applications, in “The 1−2−3 of modular forms” , Universitext, Springer, Berlin, 2008, 1-103. 3, 98, 109 SCHOOL OFMATHEMATICS ANDSTATISTICS, WUHANUNIVERSITY, WUHAN, 430072, P .R. CHINA Email address:cwang2014@whu.edu.cn

    work page 2008