Siegel Half-Space and Modular Forms

Siegel Modular Forms:

Consider the Siegel upper-half space defined by

$\displaystyle \mathcal H_g = \left\{Z= X+ iY | X, Y\in Sym_{g \times g}(\mathbb R), Y>0\right \}$

where ${X, Y}$ are symmetric ${g \times g}$ real matrices, ${Y}$ is a positive definite matrix.

Convex combinations of psd matrices are positive define, hence this is a convex subset of the subspace of symmetric matrices.

For example ${n=1}$ gives the usual upper-halfplane ${\mathbb H = \{x+iy | x, y \in \mathbb R, y>0\}}$ .

We have the action of ${SL_2(\mathbb R)}$ on ${\mathbb H}$ by Mobius transformations.

We have the Cayley map that relates ${\mathbb H}$ to the unit disk. Similarly ${\mathcal H_g}$ can be mapped to the domain

$\displaystyle D_g =\{ Z| Z\in \in Sym_{g \times g}(\mathbb C), Z\bar Z -1 <0 \}$

Similarly we have the action of the Symplectic group ${\text{Sp}_{2g}(\mathbb R)}$ on the Siegel upper half-space.

The symplectic group is given by

$\displaystyle \text{Sp}_{2g}(\mathbb R)=\left\{\gamma \in G L_{2 g}(\mathbb{R}) \mid \gamma^{\mathrm{T}}\left(\begin{array}{cc} 0 & I_{g} \\ -I_{g} & 0 \end{array}\right) \gamma=\left(\begin{array}{cc} 0 & I_{g} \\ -I_{g} & 0 \end{array}\right) \right\}$

and it has discrete subgroups

$\displaystyle \Gamma_{g}(N)=\left\{\gamma \in G L_{2 g}(\mathbb{Z}) \mid \gamma^{\mathrm{T}}\left(\begin{array}{cc} 0 & I_{g} \\ -I_{g} & 0 \end{array}\right) \gamma=\left(\begin{array}{cc} 0 & I_{g} \\ -I_{g} & 0 \end{array}\right), \gamma \equiv I_{2 g} \quad \bmod N\right\}$

The action is given by

$\displaystyle \gamma Z=\left(\begin{array}{ll} A & B \\ C & D \end{array}\right) Z := (AZ+B) (CZ+D)^{-1}.$

Verify that ${CZ+D}$ is non-singular and the resulting matrix has positive imaginary part. Upto ${\pm I}$ , these symmetries ${\gamma Z}$ are the only biholomorphic automorphism of the space.

The imaginary part of ${\gamma Z}$ is given by

$\displaystyle Im(\gamma Z) = Im (Z) (CZ+D)^{-1}$

The differential form $\displaystyle dZ Y^{-1} d\bar Z Y^{-1}$ satisfies

$\displaystyle d(\gamma Z) Y(\gamma Z)^{-1} d(\bar \gamma Z) Y(\gamma Z)^{-1} ={(CZ+D)^{T}}^{-1} dZ Y^{-1} d \bar{Z} Y^{-1}(C Z+D)^{T}$

Therefore the trace is an invariant differential form- gives the invariant metric on the space.

$\displaystyle \text{tr}(dZ Y^{-1} d\bar Z Y^{-1})$

The invariant volume form is given by

$\displaystyle \frac{dXdY}{\det Y^{n}}$

$\displaystyle A=\left\{ \left(\begin{array}{cc} a & 0 \\ 0 & ^t a^{-1} \end{array}\right), a \in S p(n, \mathbb{R})\right\}$

acts on the imaginary “line” ${\{ iY, Y>0\}}$ , and acts is the conjugation action ${ Y \rightarrow aYa^{T}}$ .

The discrete subgroup

$\displaystyle \text{Sp}_{2g}(\mathbb Z)=\left\{\gamma \in G L_{2 g}(\mathbb{Z}) \mid \gamma^{\mathrm{T}}\left(\begin{array}{cc} 0 & I_{g} \\ -I_{g} & 0 \end{array}\right) \gamma=\left(\begin{array}{cc} 0 & I_{g} \\ -I_{g} & 0 \end{array}\right) \right\}$

contains the following subgroups

$\displaystyle \left\{\left(\begin{array}{cc} u & 0 \\ 0 & ^t u^{-1} \end{array}\right) \mid u \text { unimodular }\right\}$

and

$\displaystyle \left\{\left(\begin{array}{ll} 1 & n \\ 0 & 1 \end{array}\right) \mid n \text { symmetric, integral entries }\right\}$

A Siegel modular form is a holomorphic function of weight ${k}$ (holomorphic at infinity) that satisfies

$\displaystyle f (\gamma Z) = \det(CZ+D)^{k} f(Z).$

Instead of ${\det^k}$ we can consider any representation ${\rho}$ of ${GL_g}$ and have ${f}$ to take values in that representation.
That is if ${\rho: GL_g \rightarrow GL(V)}$ , we ask for ${f (\gamma Z) = \rho(CZ+D) f(Z),}$ so that ${f: \mathcal H_g \rightarrow V.}$

We can also define modular forms with a character by

$\displaystyle f (\gamma Z) =\chi(\det D) \det(CZ+D)^{k} f(Z).$

For ${g>1}$ , we don’t need to impose the condition at infinity. (Koecher Principle).

For level ${1}$ , the invariance under ${ \left\{\left(\begin{array}{cc} u & 0 \\ 0 & ^t u^{-1} \end{array}\right) \mid u \text { unimodular }\right\}}$ and ${ \left\{\left(\begin{array}{ll} 1 & n \\ 0 & 1 \end{array}\right) \mid n \text { symmetric, integral entries }\right\}}$ implies that

$\displaystyle f(uZu^{t} + n) = \det u^k f(Z)$

and we have the Fourier expansion

$\displaystyle f(Z) = \sum_{T \ge 0} a(T) e(\text{tr}(TZ))$

where ${T}$ runs over half-integral positive semi-definite symmetric matrices (diagonal entries are integers, off-diagonal are half-integers).

$\displaystyle a(T) =\int_{X \bmod 1} f(Z) e(-\text{tr}(TZ)) dX$

We have ${a(T) = a(MTM^{T})}$ for integer matrices ${M}$ , so we can collect the terms and write the sum as indexed by classes under the congujate action.

For ${g>1}$ , the contribution from ${T \not > 0}$ can be seen to be zero because of existence of elements

$\displaystyle \left(\begin{array}{cc} u & 0 \\ 0 & ^t u^{-1} \end{array}\right)$

with

$\displaystyle u =\left(\begin{array}{cc} u_1 & 0 \\ 0 & 1 \end{array}\right), u_1=\left(\begin{array}{ll} 1 & m \\ 0 & 1 \end{array}\right)$

That’s why we don’t need the conditions for boundedness at infinity, this is called Koecher principle.

Now if the sum runs over ${T>0}$ strictly positive half-integral matrices, the form ${f}$ is called a cusp-form. This is equivalent to ${f}$ not having any limits under the Siegel operators.

That is if we take the limits

$\displaystyle \lim_{j \rightarrow \infty}f(Z^ j), \quad Z^j= \left(\begin{array}{cc} Z_1 & Z_{2}^{(j)} \\ t_{Z}(j) & Z_{3}^{(j)} \end{array}\right)$

where ${Z_2}$ are bounded and the eigenvalues of ${Z_3}$ goes to infinity, we get modular forms of smaller dimension.

$\displaystyle \lim_{j \rightarrow \infty}f(Z^j) (Z_1)=\sum_{T_1 \geq 0} a\left(\begin{array}{ll} T_1 & 0 \\ 0 & 0 \end{array}\right) e(\text{tr}(T_1Z_1)$

For cusp forms ${f}$ , all these limits applied to the functions ${F(Z)= \det(CZ+D)^{-k} f(\gamma Z)}$ are zero .

The space of modular forms will be finite dimensional as in the classical case.

Examples:

For degree ${g=1}$ , we have the usual modular forms, hence the level ${1}$ forms are generated by the Eisenstein series ${E_4}$ and ${E_6}$ .

For ${g\ge 2}$ , we have the following examples.

Eisenstein Series:

$\displaystyle E_k =\sum_{C, D} \frac{1}{\det(C Z+D)^{k}}$

where ${C, D}$ run over the bottom half of the Siegel modular group ${Sp_{2g}(\mathbb Z)}$ .

Klingen Eisenstein series:

One can use a cusp form of lower degree to construct

$\displaystyle E_P=\sum_{P_r/ \Gamma} f(\gamma Z) \frac{1}{\det(C Z+D)^{k}}$

The ${f}$ will be a limit under the Siegel operator of this Eisenstein series.

Hecke Operators:

For the groups ${\gamma_0(N)}$ defined by ${C\equiv \mathbf 0 \bmod N}$ , we define the double cosets for each ${p}$

$\displaystyle T^{(n)}(p):=\Gamma_{0}^{(n)}(N)\left(\begin{array}{ll} 1_{n} & \\ & p 1_{n} \end{array}\right) \Gamma_{0}^{(n)}(N)$

$\displaystyle T_{j}^{(n)}\left(p^{2}\right):=\Gamma_{0}^{(n)}(N)\left(\begin{array}{llll} 1_{j} & & & \\ & p 1_{n-j} & & \\ & & p^{2} 1_{j} & \\ & & & p 1_{n-j} \end{array}\right) \Gamma_{0}^{(n)}(N)$

These generate the Hecke Algebra.

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