scholarly journals RANKIN–COHEN TYPE DIFFERENTIAL OPERATORS FOR SIEGEL MODULAR FORMS

1998 ◽  
Vol 09 (04) ◽  
pp. 443-463 ◽  
Author(s):  
WOLFGANG EHOLZER ◽  
TOMOYOSHI IBUKIYAMA
Keyword(s):  
Half Space ◽  
Modular Forms ◽  
Automorphic Form ◽  
Differential Operators ◽  
Automorphic Forms ◽  
Siegel Modular Forms ◽  
Elliptic Case ◽  
Vector Valued

Let ℍn be the Siegel upper half space and let F and G be automorphic forms on ℍn of weights k and l, respectively. We give explicit examples of differential operators D acting on functions on ℍn × ℍn such that the restriction of [Formula: see text] to Z = Z1 = Z2 is again an automorphic form of weight k + l + v on ℍn. Since the elliptic case, i.e. n = 1, has already been studied some time ago by R. Rankin and H. Cohen we call such differential operators Rankin–Cohen type operators. We also discuss a generalisation of Rankin–Cohen type operators to vector valued differential operators.

scholarly journals Generic differential operators on Siegel modular forms and special polynomials

2020 ◽  
Vol 26 (5) ◽  
Author(s):  
Tomoyoshi Ibukiyama
Keyword(s):  
Modular Forms ◽  
Differential Operators ◽  
Siegel Modular Forms ◽  
Diagonal Block ◽  
Jacobi Forms ◽  
Taylor Coefficients ◽  
Special Polynomials ◽  
Generating Series ◽  
Block Variables ◽  
Vector Valued

AbstractHolomorphic vector valued differential operators acting on Siegel modular forms and preserving automorphy under the restriction to diagonal blocks are important in many respects, including application to critical values of L functions. Such differential operators are associated with vectors of new special polynomials of several variables defined by certain harmonic conditions. They include the classical Gegenbauer polynomial as a prototype, and are interesting as themselves independently of Siegel modular forms. We will give formulas for all such polynomials in two different ways. One is to describe them using polynomials characterized by monomials in off-diagonal block variables. We will give an explicit and practical algorithm to give the vectors of polynomials through these. The other one is rather theoretical but seems much deeper. We construct an explicit generating series of polynomials mutually related under certain mixed Laplacians. Here substituting the variables of the polynomials to partial derivatives, we obtain the generic differential operator from which any other differential operators of this sort are obtained by certain projections. This process exhausts all the differential operators in question. This is also generic in the sense that for any number of variables and block partitions, it is given by a recursive unified expression. As an application, we prove that the Taylor coefficients of Siegel modular forms with respect to off-diagonal block variables, or of corresponding expansion of Jacobi forms, are essentially vector valued Siegel modular forms of lower degrees, which are obtained as images of the differential operators given above. We also show that the original forms are recovered by the images of our operators. This is an ultimate generalization of Eichler–Zagier’s results on Jacobi forms of degree one. Several more explicit results and practical construction are also given.

scholarly journals Higher pullbacks of modular forms on orthogonal groups

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Brandon Williams
Keyword(s):  
Modular Forms ◽  
Differential Operators ◽  
Jacobi Form ◽  
Siegel Modular Forms ◽  
Orthogonal Groups

Abstract We apply differential operators to modular forms on orthogonal groups O ⁢ ( 2 , ℓ ) {\mathrm{O}(2,\ell)} to construct infinite families of modular forms on special cycles. These operators generalize the quasi-pullback. The subspaces of theta lifts are preserved; in particular, the higher pullbacks of the lift of a (lattice-index) Jacobi form ϕ are theta lifts of partial development coefficients of ϕ. For certain lattices of signature ( 2 , 2 ) {(2,2)} and ( 2 , 3 ) {(2,3)} , for which there are interpretations as Hilbert–Siegel modular forms, we observe that the higher pullbacks coincide with differential operators introduced by Cohen and Ibukiyama.

scholarly journals Boundary behaviour of special cohomology classes arising from the Weil representation

2012 ◽  
Vol 12 (3) ◽  
pp. 571-634 ◽  
Author(s):  
Jens Funke ◽  
John Millson
Keyword(s):  
Modular Forms ◽  
Symmetric Spaces ◽  
Theta Functions ◽  
Theta Series ◽  
Siegel Modular Forms ◽  
Boundary Behaviour ◽  
Orthogonal Groups ◽  
Local Coefficients ◽  
Generating Series ◽  
Vector Valued

AbstractIn our previous paper [J. Funke and J. Millson, Cycles with local coefficients for orthogonal groups and vector-valued Siegel modular forms, American J. Math. 128 (2006), 899–948], we established a correspondence between vector-valued holomorphic Siegel modular forms and cohomology with local coefficients for local symmetric spaces $X$ attached to real orthogonal groups of type $(p, q)$. This correspondence is realized using theta functions associated with explicitly constructed ‘special’ Schwartz forms. Furthermore, the theta functions give rise to generating series of certain ‘special cycles’ in $X$ with coefficients.In this paper, we study the boundary behaviour of these theta functions in the non-compact case and show that the theta functions extend to the Borel–Sere compactification $ \overline{X} $ of $X$. However, for the $ \mathbb{Q} $-split case for signature $(p, p)$, we have to construct and consider a slightly larger compactification, the ‘big’ Borel–Serre compactification. The restriction to each face of $ \overline{X} $ is again a theta series as in [J. Funke and J. Millson, loc. cit.], now for a smaller orthogonal group and a larger coefficient system.As an application we establish in certain cases the cohomological non-vanishing of the special (co)cycles when passing to an appropriate finite cover of $X$. In particular, the (co)homology groups in question do not vanish. We deduce as a consequence a sharp non-vanishing theorem for ${L}^{2} $-cohomology.

scholarly journals Siegel modular forms and theta series attached to quaternion algebras

1991 ◽  
Vol 121 ◽  
pp. 35-96 ◽  
Author(s):  
Siegfried Böcherer ◽  
Rainer Schulze-Pillot
Keyword(s):  
Modular Forms ◽  
Orthogonal Group ◽  
Automorphic Forms ◽  
Theta Series ◽  
Siegel Modular Forms ◽  
Theta Correspondence ◽  
Quaternion Algebras ◽  
Metaplectic Group

The two main problems in the theory of the theta correspondence or lifting (between automorphic forms on some adelic orthogonal group and on some adelic symplectic or metaplectic group) are the characterization of kernel and image of this correspondence. Both problems tend to be particularly difficult if the two groups are approximately the same size.

scholarly journals Siegel modular forms and theta series attached to quaternion algebras II

1997 ◽  
Vol 147 ◽  
pp. 71-106 ◽  
Author(s):  
S. Böcherer ◽  
R. Schulze-Pillot
Keyword(s):  
Modular Form ◽  
Modular Forms ◽  
Quaternion Algebra ◽  
Automorphic Forms ◽  
Multiplicative Group ◽  
Theta Series ◽  
Siegel Modular Forms ◽  
Siegel Modular Form ◽  
Quaternion Algebras

AbstractWe continue our study of Yoshida’s lifting, which associates to a pair of automorphic forms on the adelic multiplicative group of a quaternion algebra a Siegel modular form of degree 2. We consider here the case that the automorphic forms on the quaternion algebra correspond to modular forms of arbitrary even weights and square free levels; in particular we obtain a construction of Siegel modular forms of weight 3 attached to a pair of elliptic modular forms of weights 2 and 4.

scholarly journals Structure theorems for vector valued Siegel modular forms of degree 2 and weight detk ⊗Sym(10)

2016 ◽  
Vol 27 (12) ◽  
pp. 1650101
Author(s):  
Sho Takemori
Keyword(s):  
Modular Forms ◽  
Symmetric Tensor ◽  
Siegel Modular Forms ◽  
Structure Theorems ◽  
Vector Valued

We prove the explicit structure theorems of modules [Formula: see text] of vector valued Siegel modular forms of degree [Formula: see text], where [Formula: see text] runs over the set of even integers or odd integers. We also check the conjecture given by Ibukiyama [Vector valued Siegel modular forms of symmetric tensor weight of small degrees, Comment. Math. Univ. St. Pauli 61 (2012) 51–75.] for modules of vector valued Siegel modular forms of degree [Formula: see text] of weights [Formula: see text] and [Formula: see text].

scholarly journals Siegel modular forms of genus 2 and level 2

2015 ◽  
Vol 26 (05) ◽  
pp. 1550034 ◽  
Author(s):  
Fabien Cléry ◽  
Gerard van der Geer ◽  
Samuel Grushevsky
Keyword(s):  
Modular Forms ◽  
Generating Functions ◽  
Theta Functions ◽  
Siegel Modular Forms ◽  
Genus 2 ◽  
Vector Valued ◽  
Level 2

We study vector-valued Siegel modular forms of genus 2 on the three level 2 groups Γ[2] ◁ Γ1[2] ◁ Γ0[2] ⊂ Sp(4, ℤ). We give generating functions for the dimension of spaces of vector-valued modular forms, construct various vector-valued modular forms by using theta functions and describe the structure of certain modules of vector-valued modular forms over rings of scalar-valued Siegel modular forms.

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