# 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 '13 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 '13 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 '13 at 13:32
Just sent you a mail ! It's great that I get to know people by this question. – Tom Jul 27 '13 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.

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This is already a quite nice answer ! Do you know what kind of strategy he used ? Or do you know how to contact him or his supervisor ? To speak quite honestly I couldn't find anything on the Kyushu University homepage. I mean Satoh, Ibukiyama and their students collect Eisenstein series and sorts of Rankin Cohen brackets until they reach the dimension Tshushima has calculated 20 years ago. But this collection seems to me to be intricate especially if you raise j. – Tom Jul 19 '12 at 5:52
I wrote to Kiyuna and he tells me that there are 18 generators: 6 of them are theta series (presumably products of theta constants) and the remaining 12 are a kind of Rankin-Cohen construction. I don't know much more but he tells me there will be a preprint in the next couple of weeks. If you'd like to send me an email (see my website for my email address), I can give you his email address. – ncr Jul 23 '12 at 13:12
I just sent you an email. – Tom Jul 24 '12 at 20:17
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 '13 at 10:25

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.

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