Siegel modular form (awaiting review)

A (scalar valued) Siegel modular form $F$ of degree $g$ (or genus $g$) is a holomorphic function on the Siegel upper half space $\Bbb{H}_g$ that satisfies a transformation property under arithmetic subgroups of $\GSp(2g)$ such as the following.

• symplectic group: $\Sp(2g,\Z)$
• Hecke subgroup: $\Gamma_0(N)\subset \Sp(2g,\Z)$
• paramodular group: $K(N)\subset \Sp(4,\Q)$

A Siegel modular form of weight $k\in\Z$ and degree $g$ with respect to a subgroup $\Gamma\subset \GSp(2g,\Q)$ is a function $F:\Bbb{H}_g\to\C$ such that for any $\gamma=\begin{pmatrix} A&B\\C&D\end{pmatrix}$ we have $F((AZ+B)(CZ+D)^{-1})=\left(\det(C Z+D)\right)^{k}F(Z)$. If $g=1$, one must also impose the following condition: there exists $\epsilon>0$ such that $$|F(x+iy)|\ll y^\epsilon\quad\text{for }y\geq 1.$$

More generally Siegel modular forms can have half-integral weights, and vector valued Siegel modular forms have representations for their weights.