Arithmetic Properties of Vector-Valued Siegel Modular Forms

2020 ◽  
pp. 47-58
Author(s):  
Siegfried Böcherer
Keyword(s):  
Modular Forms ◽  
Siegel Modular Forms ◽  
Arithmetic Properties ◽  
Vector Valued

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 EFFICIENTLY GENERATED SPACES OF CLASSICAL SIEGEL MODULAR FORMS AND THE BÖCHERER CONJECTURE

2010 ◽  
Vol 89 (3) ◽  
pp. 393-405 ◽  
Author(s):  
MARTIN RAUM
Keyword(s):  
Modular Forms ◽  
Class Number ◽  
Siegel Modular Forms ◽  
Generating Set ◽  
Arithmetic Properties ◽  
Fourier Expansions

AbstractWe state and verify up to weight 172 a conjecture on the existence of a certain generating set for spaces of classical Siegel modular forms. This conjecture is particularly useful for calculations involving Fourier expansions. Using this generating set, we verify the Böcherer conjecture for nonrational eigenforms and discriminants with class number greater than one. As a further application we verify another conjecture for weights up to 150 and investigate an analog of the Victor–Miller basis. Additionally, we describe some arithmetic properties of the basis we found.

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.

scholarly journals On the computation of the determinant of vector-valued Siegel modular forms

2014 ◽  
Vol 17 (A) ◽  
pp. 247-256 ◽  
Author(s):  
Sho Takemori
Keyword(s):  
Cusp Form ◽  
Modular Forms ◽  
Siegel Modular Forms ◽  
Vector Valued

AbstractLet $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}A^{0}(\Gamma _{2})$ denote the ring of scalar-valued Siegel modular forms of degree two, level $1$ and even weights. In this paper, we prove the determinant of a basis of the module of vector-valued Siegel modular forms $\bigoplus _{k \equiv \epsilon \ {\rm mod}\ {2}}A_{\det ^{k}\otimes \mathrm{Sym}(j)}(\Gamma _{2})$ over $A^{0}(\Gamma _{2})$ is equal to a power of the cusp form of degree two and weight $35$ up to a constant. Here $j = 4, 6$ and $\epsilon = 0, 1$. The main result in this paper was conjectured by Ibukiyama (Comment. Math. Univ. St. Pauli 61 (2012) 51–75).

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 Covariants of binary sextics and vector-valued Siegel modular forms of genus two

2017 ◽  
Vol 369 (3-4) ◽  
pp. 1649-1669 ◽  
Author(s):  
Fabien Cléry ◽  
Carel Faber ◽  
Gerard van der Geer
Keyword(s):  
Modular Forms ◽  
Siegel Modular Forms ◽  
Genus Two ◽  
Vector Valued
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