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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.

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.

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• Review status: beta
• Last edited by Alex J. Best on 2018-12-13 14:24:42
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