Siegel modular forms are one generalization of modular forms to functions in more than one complex variable (another such generalization are Hilbert modular forms).

Where ordinary modular forms are locally functions on the moduli stack of elliptic curves over the complex numbers, so Siegel modular forms are locally functions on something like the moduli stack of higher-dimensional complex abelian varieties.

Definition

Let $\mathcal{H}_{g}$ be the Siegel upper half-space of degree $g$ and let $\Sp_{2g}(\mathbb{Z})$ the subgroup of elements in the symplectic group$\Sp_{2g}(\mathbb{R})$ with integral entries. Let $V$ be a finite-dimensional vector space over $\mathbb{C}$ and fix a representation

$\rho:\GL_{g}(\mathbb{C})\to \GL(V).$

A Siegel modular form of weight $\rho$ is a holomorphic function$f:\mathcal{H}_{g}\to V$ such that

$f(\gamma(\tau))=\rho(C\tau+D)f(\tau)$

for all $\tau=\begin{pmatrix}A & B\\C & D\end{pmatrix}\in\Sp_{2g}(\mathbb{Z})$ and all $\tau\in\mathcal{H}_{g}$. If $g=1$, we additionally require that $f$ is holomorphic at $\infty$. A classical Siegel modular form of weight $k$ is the special case when the representation $\rho$ is given by taking $k$-th powers of the determinant. Further specializing to $g=1$, this gives classical (also known as elliptic) modular forms.