Siegel modular forms

In Modular Forms we introduced modular forms as certain holomorphic functions on the upper half-plane following certain transformation properties with respect to the action of the group $\mathrm{SL}_{2}(\mathbb{Z})$ (or more generally its congruence subgroups). We also saw that they are sections of certain sheaves on the compactified moduli space of elliptic curves, possibly together with extra structure, such as a basis of $N$-torsion, a point of order $N$, or a cyclic subgroup of order $N$ (see also The Moduli Space of Elliptic Curves).

In this post we shall introduce a higher-dimensional generalization of this idea. Namely, we shall introduce Siegel modular forms, which are to principally polarized abelian varieties as the usual (also called elliptic) modular forms are to elliptic curves.

Let us follow the same approach that we used to introduce modular forms, as certain functions on the upper half-space with certain transformation properties. Therefore the first thing we will need is a higher-dimensional analogue of the upper half-space.

The Siegel upper half-space of degree $g$ (or genus $g$), denoted $\mathcal{H}_{g}$ is the set of all $g\times g$ symmetric matrices whose entries are complex numbers with a positive imaginary part. If $g=1$, then this is the same as the usual upper half-space.

Now we need the analogue of the transformation properties of an elliptic modular form under the modular group $\mathrm{SL}_{2}(\mathbb{Z})$. We recall that the action of $\mathrm{SL}_{2}(\mathbb{Z})$ on the upper half-plane was inherited from the action of $\mathrm{SL}_{2}(\mathbb{R})$ via Mobius transformations. If $\begin{pmatrix}a&b\\c&d\end{pmatrix}$ is an element of $\gamma=\mathrm{SL}_{2}(\mathbb{R})$, then it maps a point $\tau$ on the upper half-plane to $\displaystyle \gamma(z)=\frac{a\tau+b}{c\tau+d}$. Then we define a modular form of weight $k$ to be a holomorphic function $f:\mathcal{H}\to\mathbb{C}$ such that $f(\gamma(\tau))=(c\tau+d)^{k}f(\tau)$ and such that $f$ is holomorphic at infinity (it is bounded as the imaginary part of $\tau$ approaches infinity).

For Siegel modular form, our group will be the Siegel modular group $\mathrm{Sp}_{2g}(\mathbb{Z})$, which is a subgroup of the symplectic group $\mathrm{Sp}_{2g}(\mathbb{R})$. The elements of the symplectic group are $2g\times 2g$ real matrices which can be written in the form $\begin{pmatrix}A&B\\C&D\end{pmatrix}$ where $A$, $B$, $C$, and $D$ are $g\times g$ real matrices satisfying $AB^{T}=BA^{T}$, $CD^{T}=DC^{T}$, and $AD^{T}-DC^{T}=I_{g}$, where the superscript ${}^{T}$ means taking the transpose and $I_{g}$ is the $g\times g$ identity matrix. Note that if $g=1$, then the first two conditions are automatically satisfied while the third condition says that the determinant of the matrix must be $1$. Therefore $\mathrm{Sp}_{2}(\mathbb{R})=\mathrm{SL}_{2}(\mathbb{R})$.

Now let $\tau$ be an element of the Siegel upper half-plane $\mathcal{H}_{g}$. Note that $\tau$ is now a $g\times g$ matrix. An element $\gamma$ of $\mathrm{Sp}_{2g}(\mathbb{R})$ sends $\tau$ to the element

$\gamma(\tau)=(A\tau+B)(C\tau+D)^{-1}$.

We are almost ready to define Siegel modular forms. Although we may define Siegel modular forms as being complex-valued just like elliptic modular forms, and they are in themselves worthwhile objects of study, it is sometimes more natural to consider Siegel modular forms as being vector-valued. This arises for example when we want to obtain Siegel modular forms as sections of the Hodge bundle, which is the pushforward of the sheaf of relative differentials of the universal principally polarized abelian variety over $\mathbb{C}$ on the moduli space of principally polarized abelian varieties over $\mathbb{C}$ (which is obtained as the quotient of $\mathcal{H}_{2g}$ by $\mathrm{Sp}_{2g}(\mathbb{Z})$).

Let $V$ be a finite-dimensional vector space over $\mathbb{C}$, and let $\rho:\mathrm{GL}_{g}(\mathbb{C})\to \mathrm{GL}(V)$ be a representation of $\mathrm{GL}_{g}(\mathbb{C})$ on $V$. A Siegel modular form of weight $\rho$ is a holomorphic function $f:\mathcal{H}_{g}\to V$ such that

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

for any $g\in\mathrm{SL}_{2}(\mathbb{Z})$, and which is holomorphic at infinity if $g=1$. If $g>1$, the holomorphicity at infinity is automatically taken care of by what is known as Kocher’s principle.

In the special case that $V=\mathbb{C}$, and $\rho$ is given by taking powers of the determinant, i.e. our Siegel modular form is a holomorphic function $f:\mathcal{H}\to\mathbb{C}$ such that

$f(\gamma(\tau))=\mathrm{det}(C\tau+D)^{k}f(\tau)$

then we say that our Siegel modular form is a classical Siegel modular form. Note that a classical Siegel modular form of degree $1$ is an elliptic modular form.

We may also consider Siegel modular forms for congruence subgroups $\Gamma(N)$ of $\mathrm{Sp}_{2g}(\mathbb{Z})$, where $\Gamma(N)$ is the subgroup of $\mathrm{Sp}_{2g}(\mathbb{Z})$ consisting of elements that become the identity matrix after reduction mod $N$.

The theory of Siegel modular forms is more complicated than the theory of elliptic modular forms, but we may use the latter to guide our study of the former. For instance, we may want to consider the Fourier expansion of Siegel modular forms. We may also want to consider its Hecke algebra (see also Hecke Operators). There are also analogues of important examples of elliptic modular forms, such as the Eisenstein series or the discriminant, for Siegel modular forms. We may also use elliptic modular forms to construct explicit examples of Siegel modular forms (a process known as lifting). All these and more will hopefully be discussed in future posts on this blog.

References:

Siegel modular form on Wikipedia

Siegel upper half-space on Wikipedia

Siegel modular variety on Wikipedia

Symplectic group on Wikipedia

Siegel modular forms by Gerard van der Greer