Siegel modular form

Remarks

For $g = 1$ this is the definition of a modular form of weight $k$ for $SL_2(\Z)$. For $g \ge 2$ the Koecher principle shows that conditions 1 and 2 imply boundedness on every set $\set{Z : \operatorname{Im} Z \ge c I_g}$ with $c > 0$, so no condition at the cusp is needed. Every Siegel modular form has a Fourier expansion $f(Z) = \sum_T a(T) e^{2\pi i \operatorname{tr}(TZ)}$, over half-integral symmetric matrices $T \ge 0$. Forms of weight $k$ for subgroups of finite index of $\Gamma_g$, and vector-valued forms, are defined in the same way.

Example: a constant function $f = c$ is a Siegel modular form of weight $0$, since condition 2 reads $c = \det(CZ + D)^0 c$. Example: let $S$ be a positive definite even unimodular integer matrix of size $m$. The theta series $\theta_S(Z) = \sum_{X} e^{\pi i \operatorname{tr}(X^T S X Z)}$, over all integer $m \times g$ matrices $X$, is a Siegel modular form of weight $m/2$.

Non-example: the $j$-invariant $j(z)$ is holomorphic on $\mathbb{H}_1$ and satisfies condition 2 with $k = 0$. But $j(z) = e^{-2\pi i z} + 744 + \cdots$ is unbounded on $\operatorname{Im} z \ge 1$, so condition 3 fails.