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What are Siegel modular forms?

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. $J_g$ is also known as involution.

$Sp(g,\mathbb{Z})$ 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 $M \in \Gamma < Sp(g,\mathbb{Z})$.

Since the determinant is also a representation scalar valued modular forms are included in this definition. If $V$'s dimension is 2 or higher then we speak of a vector valued Siegel modular form.

We can spice this definition up by allowing square roots of the determinant $\sqrt{det(CZ+D)}$. Here, we have to solve the ambiguity by a multiplier system $v$. Then we have $$f(M(Z))=v(M) \cdot \left(\sqrt{det(CZ+D)}\right)^r \cdot \rho(CZ+D)f(Z).$$

Structure theorems

If we fix a representation $\rho_0$ and a subgroup $\Gamma$ but allow arbitrary powers of $\left(\sqrt{det(CZ+D)}\right)^r$ or $det(CZ+D)^k $ , resp., then all functions satisfying $$ f(M(Z))=v(M) \cdot \left(\sqrt{det(CZ+D)}\right)^r \cdot \rho_0(CZ+D)f(Z)\quad \forall M \in \Gamma $$ form a module over (a subring of) the ring of scalar modular forms of $\Gamma$ (to the multiplier system $v$).

This module is always finitely generated. But it seems hard to find such a finite set of generators and even more all the relations between them. Without these relations it is possible to miss out a nicer description of elements in the module.

For me a structure theorems is a theorem that returns for a group and a representation such a finite set and the relations.

The actual question

Which structure theorems of vector valued Siegel modular forms are known? I'm personally most interested in genus 2.

I would make the question community because I don't expect a uniform theorem but couldn't find the button.

I already got some answers but I'm not sure how to post : all in one answer, one group $\Gamma$ (and one representation $\rho_0$) per answer, one paper per answer ???

Comments on this issue are very welcome. Afterwards I'm very happy to type this half of a dozen papers.

p.s. I hope the post is not too chaotic.

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The standard rule for community wiki big lists is one answer per post. –  S. Carnahan Jul 27 '13 at 22:10
    
Ah thanks Scott ! But for example Aoki proved structure theorems for 3 different groups in one single article in 2012. And conversely, the results for the full modular group are spread over 4 papers (and over 25 years) from Satoh, Ibukiyama, van Dorp and Kiyuna. –  Tom Jul 27 '13 at 22:29
    
Okay, the convention is one result per answer, but it is broken a lot. Try not to worry about it too much, and contribute what seems reasonable. –  S. Carnahan Jul 27 '13 at 22:35
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6 Answers

Ibukiyama proved the following results around the year 2000. They are also on the full modular group $Sp(2,\mathbb{Z})$. He covers odd weight of $Sym^2$, even weight of $Sym^6$, and all of $Sym^4$

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.

In this edit, I want to comment on odd weight of $Sym^2$.

$A$ denotes again the ring $\mathbb{C}[\phi_4,\phi_6,\chi_{10},\chi_{12}]$ and abbreviate these 4 forms by $X_1, \dots , X_4$. Let's call the module of vector valued modular forms $M$. We have $$M=\sum_{1\leq i < j < k \leq 4} A \cdot [X_i,X_j,X_k]$$ where $[X_i,X_j,X_k]$ is an other Rankin Cohen bracket. I feel that the best reference for these RC brackets is van Dorp's thesis. Let $f \in M_k(\Gamma)$, $g \in M_l(\Gamma)$ and $h \in M_m(\Gamma)$ then $[f,g,h]$ equals (up to a constant) $$k \cdot f \cdot \nabla g \wedge \nabla h \quad - l \cdot g \cdot \nabla f \wedge \nabla h \quad + m \cdot h\cdot \nabla f \wedge \nabla g . $$ It is a vector valued Seigel modular form of weight $k+l+1$ w.r.t. the representation $Sym^2$. Then the subsequent properties are obvious

  • $[f,g,h]$ is alternating;
  • $-m\cdot h[f,g,j]+l \cdot g[f,h,j]-k \cdot f[g,h,j]+n \cdot j [f,g,h]=0$

    where $f \in M_k(\Gamma)$, $g \in M_l(\Gamma)$,$h \in M_m(\Gamma)$, and $j \in M_n(\Gamma)$.

Proof :

Ibukiyama proves that the above relations are generating ones with the help of Fourier Jacobi expansions. But, he states that there are many ways to prove this result. This gives him the Hilbert function of the RHS which coincides with the LHS's one. This one can be calculated by using Tshushima's famous dimension formula.

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Aoki was the first one to tackle successfully a different group. In particular he solved the problem for $Sym^2$ and 3 groups ( and reproved it for $Sp(2,\mathbb{Z})$ ):

  • $ \Gamma_{2,0}[2]:=\left\{ M \in \Gamma_2=Sp(2,\mathbb{Z}) : C \equiv 0 \mod 2 \right\} $
  • $ \Gamma_{2,0,\psi}[3]:=\left\{ M \in \Gamma_2 : C \equiv 0 \mod 3 \ ,\ \psi(M)=1 \right\} $ where $\psi(M)$ is defined by the Legendre symbol $\left(\frac{-3}{det(D)}\right)$

  • $ \Gamma_{2,0,\psi}[4]:=\left\{ M \in \Gamma_2 : C \equiv 0 \mod 4 \ ,\ \psi(M)=1 \right\} $ where $\psi(M)$ is defined by the Legendre symbol $\left(\frac{-1}{det(D)}\right)$

Hiroki Aoki. On vector valued Siegel modular forms of degree 2 with small levels. Osaka Journal of Mathematics, 49:625–651, 2012.

Each group's ring of scalar valued modular forms is generated by 5 modular forms where the first 4 are algebraically independent, let's call them again $X_1, \dots X_4$. We shall again denote the ring generated by them by $A$. We have $$M=\sum_{1\leq i < j \leq 4} A \cdot [X_i,X_j] \quad \oplus \sum_{1\leq i < j < k \leq 4} A \cdot [X_i,X_j,X_k]$$ where we use the 2 types of RC brackets known from Satoh and Ibukiyama.

Proof:

He calculates an upper bound for $M$'s Hilbert function via exact sequences and differential operators because he can't apply Tsushima's formula. But this upper bound coincide with the one derived from the RC bracket. They are just satisfying the obvious relations.

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Ibukiyama's student Tomoya Kiyuna proved $Sym^8$ for $Sp(n,\mathbb{Z})$ in his Master's thesis. Here I haven't got any link or reference.

AFAIK he uses similar techniques as Ibukiyama for $Sym^4$ and $Sym^6$.

ncr commented on the proof in this mathoverflow thread :

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.

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Ok let's start with Satoh's result which is the oldest one. It is on the full modular group $Sp(2,\mathbb{Z})$ and $Sym^2$ with even weight, i.e. $det^{2k}\otimes Sym^{2}$

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

edit : As Scott pointed out I should elaborate on the result :

Let $A$ denote the ring $\mathbb{C}[\phi_4,\phi_6,\chi_{10},\chi_{12}]$ where the first 2 are the Eisenstein series of weight 4 and 6, resp., and the latter 2 are the cusp forms of weight 10 and 12, resp. .I think it is convenient to denote these 4 forms by $X_1, \dots , X_4$. Then $A$ is a free algebra and is the correct ring to observe the module of vector valued modular forms. Let's call it $M$. We have $$M=\sum_{1\leq i < j \leq 4} A \cdot [X_i,X_j]$$ where $[X_i,X_j]$ is the Rankin Cohen bracket. Let's recall its definition for $f \in M_k(\Gamma)$ and $g \in M_l(\Gamma)$ the RC bracket $[f,g]$ equals $k\cdot f \cdot \nabla g - l \cdot g \cdot \nabla f$ (up to a constant). The RC bracket is a vector valued Seigel modular form of weight $k+l$ w.r.t. the representation $Sym^2$. Then subsequent relations are obvious

  • $[f,g]=-[g,f]$
  • $m\cdot h[f,g]=l \cdot g[f,h]+k \cdot f[h,g]$ where $f \in M_k(\Gamma)$, $g \in M_l(\Gamma)$, and $h \in M_m(\Gamma)$.

Proof :

He proves that the above relations are generating ones. This gives him the Hilbert function of the RHS which coincides with the LHS's one. This one can be calculated by using Tshushima's famous dimension formula.

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Could you say what the theorem says, instead of just remarking on its existence? Are the relations too cumbersome to write in full? –  S. Carnahan Jul 27 '13 at 23:15
    
well I should have answered in a different order then because my 'expertise' would lie in the articles that are still to come. But I won't be online for the next hours! –  Tom Jul 28 '13 at 0:14
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Van der Geer's student Christiaan van Dorp could settle odd weight of $Sym^6$ for $Sp(2,\mathbb{Z})$ in his 2011 M.Sc. thesis

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

The whole thesis can be found here. He gives a nice overview over the field and presents a shortened version of the original $Sym^2$ cases.

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In the preprint "Explicit computations of Siegel modular forms of degree two", which you can find here on arXiv, Raum, Ryan, Skoruppa and Tornaria recall quite a few structural results about Siegel modular forms in general, in degree two in particular, and with quite a few references ; it should cover this question quite nicely.

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Dear Julien, thanks for referencing this article. But unfortunately, my focus lies on the 'vector valued' part. Of these I could only find the results of Satoh and Ibukiyama in the article. –  Tom Oct 13 '13 at 16:53
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