Are there cusp forms for the full modular group Sp(2,Z) and representations det^3 \otimes Sym^2j(\rho_standard)

Question

What are modular forms or cusps forms, resp. ?

We start with defining their common domains $\mathbb{H}_g$ as the set of symmetric $g \times g$ matrices with positive definite imaginary parts. The symplectic group $Sp(g,\mathbb{Z})$ is the subgroup of $SL(2g,\mathbb{Z})$ such that all elements satisfy $M=J_g^t M J_g$ with $J_g$ being the canonical almost complex structure.

$Sp(g,\mathbb{Z})$ acts on $\mathbb{H}_g$ by $M(Z)=(AZ+B)(CZ+D)^{-1}$ where A,B,C and D are the block matrix entries of M.

Let $\rho : GL(g,\mathbb{C}) \to GL(V)$ be a rational representation on a finite dimensional $\mathbb{C}$-vector space then the associated modular forms are the holomorphic functions $f : \mathbb{H}_g \to V$ satisfying $f(M(Z))=\rho(CZ+D)f(Z)$ for all $M \in Sp(g,\mathbb{Z})$.

Cusps forms can be easily characterized as the elements of Siegel's $\Phi$ operator's kernel.

Modular forms in genus 2

If g equals 2 then the observed representations are the ones of $GL(2,\mathbb{C})$. We know from representation theory that all irreducible representations are isomorphic to a rep of the type $det^k \otimes Sym^{2j}(\rho_{standard})$.

We denote by $\rho_{standard}$ the standard representation $X \mapsto X$. $Sym^{2j}(\rho_{standard})$ is the associated symmetric product $GL(2,\mathbb{C}) \to Sym^{2j}(\mathbb{C}^2)$. $det$ is just the 1 dimensional determinant representation $GL(2,\mathbb{C}) \to \mathbb{C}$.

For $k\geq 4$ Tsushima has given a dimension formula for the vector space of cusps forms in

Ryuji Tsushima. An explicit dimension formula for the spaces of generalized automorphic forms with respect to Sp(2, Z). Proceedings of the Japan Academy, Ser. A, Mathematical Sciences, 59:139–142, 1983.

Satoh and Ibukiyama gave (but partly didn't publish AFAIK) generators for the modules of vector valued modular forms to the representations $det^k \otimes Sym^{2j}(\rho_{standard})$ with running k and fixed j in ${1,2,3}$.

EDIT: Ibukiyama sent me the following article

Tomoyoshi Ibukiyama. Vector Valued Siegel Modular Forms of Symmetric Tensor Weight of Small Degrees. Commentarii Mathematici Universitatis Sancti Pauli, 61, No. 1:51–75, 2012.

And odd weighgt for $Sym^{6}$, i.e. $det^{2k+1} \otimes Sym^{6}(\rho_{standard})$, was treated here

Christiaan van Dorp. Generators for a module of vector-valued Siegel modular forms of degree 2, http://arxiv.org/abs/1301.2910 .

While editing let me just add Satoh's article to get a (rather) conclusive list

Takakazu Satoh. On certain vector valued Siegel modular forms of degree two. Mathematische Annalen, 274:365–387, 1986.

end of edit.

The actual question

So the next question for me was are there cusps forms to $det^3 \otimes Sym^{2j}(\rho_{standard})$ and can they ( at least a single one) be given explicitly, in particular for j=4 ?

cheers Tom

p.s. please excuse all mistakes I made but it was the first time for me publishing on such a plattform.

Answer 1

Though I haven't seen it, I've heard that Tomoya Kiyuna, a student at Kyushu University, has done this for the case $j=4$ as part of his master's thesis. In particular, he finds eighteen explicit generators of the module of vector-valued Siegel modular forms of the symmetric tensor 8.

Answer 2

In my preprint http://arxiv.org/abs/1310.2508 , Subsection 2.1, I prove that the vector bundle of Siegel cusp forms of type $\mathrm{Sym}^j \otimes \det^k$ for the full modular group in genus two has no higher cohomology for any $j \geq 0, k \geq 3$ except $(j,k)=(0,3)$. A generating function for the Euler characteristics of these bundles is known since Tsushima. This gives the following dimension formula:

$$ \sum_{j\geq 0} s_{j,3} x^j \frac{x^{36}}{(1-x^6)(1-x^8)(1-x^{10})(1-x^{12})},$$ where $s_{j,k}$ denotes the dimension of the space of cusp forms for $\mathrm{Sp}(2,\mathbf Z)$ of type $\mathrm{Sym}^j \otimes \det^k$. In particular, the first example of a vector-valued Siegel cusp form of level 1 and weight 3 comes with the representation $\mathrm{Sym}^{36} \otimes \det^3$. I don't think that an explicit construction of any of these weight three cusp forms is known.