# 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.