Siegel Modular Forms

General Idea

For the usual modular forms on $SL(2,\mathbb{Z})$, replace the group $SL(2,\mathbb{Z})$ by the automorphism group $Sp(2g,\mathbb{Z})$ of a unimodular symplectic form on $\mathbb{Z}^{2n}$, the upper half plane by the Siegel upper half plane $\mathcal{H}_g$, one gets the notion of Siegel modular forms. The integer $g\geq 1$ is called the degree or genus. For $g=1$ Siegel modular forms are the usual elliptic modular forms.

Moduli space of principally polarized abelian varieties

The quotient space $Sp(2g,\mathbb{Z})\setminus \mathcal{H}_g$ is the moduli space of principally polarized abelian varieties.

The Siegel modular group

Recall that the group $SL(2,\mathbb{Z})$ is the automorphism group of the lattice $\mathbb{Z}^2$ with the standard alternating form $\langle\cdot,\cdot\rangle$ with $$ \langle (a,b),(c,d)\rangle = ad-bc. $$ Take for $g\in \mathbb{Z}_{\geq 1}$ the lattice $\mathbb{Z}^{2g}$ of rank $2g$ with basis $e_1,…,e_g,f_1,…,f_g$, s.t. for the symplectic form $\langle\cdot,\cdot\rangle$ $$ \langle e_i,e_j \rangle = 0, \langle f_i ,f_j \rangle = 0, \langle e_i,f_j\rangle = \delta_{ij}. $$

Definition(Symplectic group) . The symplectic group, or the Siegel modular group $Sp(2g,\mathbb{Z})$ is the automorphism group of the symplectic lattice $$ Sp(2g,\mathbb{Z}) := \operatorname{Aut}(\mathbb{Z}^{2g},\langle\cdot,\cdot,\rangle). $$

It is instructive to look at the matrices of the group using the basis. The matrix group elements are of the form $$ \begin{pmatrix} A& B \\ C&D \end{pmatrix} $$ with $AB^\mathsf{t} = BA^\mathsf{t}$, $CD^\mathsf{t} = DC^\mathsf{t}$ and $AD^\mathsf{t} - BC^\mathsf{t}=1_g$. Equivalently, $$ C^\mathsf{t}A - A^\mathsf{t}C=0, D^\mathsf{t}B-B^\mathsf{t}D=0, D^\mathsf{t}A - B^\mathsf{t}C=1_g. $$

Remark. This is suspiciously similar to the split orthogonal group.

Definition(Siegel upper half plane) . The Siegel upper half plane $\mathcal{H}_g$ is defined as $$ \mathcal{H}_g = \{\tau \in \operatorname{Mat}(g\times g,\mathbb{C})|\tau^\mathsf{t} = \tau, \operatorname{Im}(\tau) \gt 0\} $$ namely the space of $g\times g$ complex symmetric matrices that have positive definite imaginary part.

The positive definiteness of the imaginary part guarentees that for $\gamma \in \Gamma_g$ and $\tau \in \mathcal{H}_g$, the matrix $(C\tau + D)$ is always invertible. $\gamma$ acts on the Siegel upper half plane by a ‘fractional linear transformation’ $$ \tau \mapsto \gamma \tau = (A\tau + B)(C\tau + D)^{-1}. $$

The group $Sp(2g,\mathbb{R})/{\pm 1}$ acts on $\mathcal{H}_g$ effectively and is the biholomorphic automorphism group of $\mathcal{H}_g$. The action is transitive, with the stabilizer of $i 1_g$ being the unitary group, thus $\mathcal{H}_g$ can be regarded as the coset space $Sp(2g,\mathbb{R})/U(g)$ of a simple Lie group by a maximal compact subgroup.

The elliptic upper half plane $\mathcal{H}_1$ is analytically equivalent to the unit disc ${z\in\mathbb{C}:|z| < 1}$, while $\mathcal{H}_g$ is analytically equivalent to a bounded symmetric domain $$ D_g:= { Z\in \operatorname{Mat}(g\times g,\mathbb{C}): Z^\mathsf{t} = Z, Z^\mathsf{t}\cdot Z < 1_g} $$ with the generalized Cayley transform $$ \begin{gather} \tau \mapsto z=(\tau - i1_g)(\tau + i1_g)^{-1}\\ z\mapsto \tau = i\cdot(1_g + z)(1_g -z)^{-1}. \end{gather} $$ The ‘symmetric’ in the name refers to the existence of an involution on $D_g$: $$ \tau \mapsto -\tau^{-1}\quad (z\mapsto -z) $$ having exactly one isolated fixed point.