# 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.

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Tsushima's dimension is conjectured to hold for any $k \geq 3$, $j \geq 0$. There is strong evidence for this conjecture coming from point counts of moduli spaces over finite fields, see Section 6 of Bergström, Faber, Van der Geer: Siegel modular forms of degree three adn the cohomology of local systems. If you are interested I think I might be able to give an ad hoc proof of the case $k=3$, $j \geq 0$. –  Dan Petersen Jul 27 at 10:55
I'm puzzled now if you can give an ad hoc proof of the case $k=3, j\geq 0$ then you have proven the whole conjecture in a glimpse, haven't you ? –  Tom Jul 27 at 11:15
Oh -- you're right! I was thinking of the fact that Conjecture 6.3 in B-F-VdG is open both for local systems of the form $\mathbb V_{a,0}$ and $\mathbb V_{a,a}$; the former correspond to SMFs with $k=3$ and the latter to ones with $j=0$. It is this conjecture I think I can prove for the ones of the form $\mathbb V_{a,0}$. But of course it's already known that Tsushima's formula holds for scalar valued SMFs. Anyway this is work in progress and there are still things to check, so if you are interested you can send me an e-mail. –  Dan Petersen Jul 27 at 13:32
Just sent you a mail ! It's great that I get to know people by this question. –  Tom Jul 27 at 14:16
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.
It is already a whole year since we talked about this topic ? Never mind ! I accepted your answer because Ibukiyama and Tomoya Kiyuna told me they don't expect progress for $Sym^{10}$. –  Tom Jul 27 at 10:25