Siegel modular form

Let ${\displaystyle g,N\in \mathbb {N} }$ and define

${\displaystyle {\mathcal {H}}_{g}=\left\{\tau \in M_{g\times g}(\mathbb {C} )\ {\big |}\ \tau ^{\mathrm {T} }=\tau ,{\textrm {Im}}(\tau ){\text{ positive definite}}\right\},}$

the Siegel upper half-space. Define the symplectic group of level ${\displaystyle N}$, denoted by ${\displaystyle \Gamma _{g}(N),}$ as

${\displaystyle \Gamma _{g}(N)=\left\{\gamma \in GL_{2g}(\mathbb {Z} )\ {\big |}\ \gamma ^{\mathrm {T} }{\begin{pmatrix}0&I_{g}\\-I_{g}&0\end{pmatrix}}\gamma ={\begin{pmatrix}0&I_{g}\\-I_{g}&0\end{pmatrix}},\ \gamma \equiv I_{2g}\mod N\right\},}$

where ${\displaystyle I_{g}}$ is the ${\displaystyle g\times g}$ identity matrix. Finally, let

${\displaystyle \rho$ :{\textrm {GL}}_{g}(\mathbb {C} )\rightarrow {\textrm {GL}}(V)}

be a rational representation, where ${\displaystyle V}$ is a finite-dimensional complex vector space.

Given

${\displaystyle \gamma ={\begin{pmatrix}A&B\\C&D\end{pmatrix}}}$

and

${\displaystyle \gamma \in \Gamma _{g}(N),}$

define the notation

${\displaystyle (f{\big |}\gamma )(\tau )=(\rho (C\tau +D))^{-1}f(\gamma \tau ).}$

Then a holomorphic function

${\displaystyle f:{\mathcal {H}}_{g}\rightarrow V}$

is a Siegel modular form of degree ${\displaystyle g}$ (sometimes called the genus), weight ${\displaystyle \rho }$, and level ${\displaystyle N}$ if

${\displaystyle (f{\big |}\gamma )=f}$

for all ${\displaystyle \gamma \in \Gamma _{g}(N)}$. In the case that ${\displaystyle g=1}$, we further require that ${\displaystyle f}$ be holomorphic 'at infinity'. This assumption is not necessary for ${\displaystyle g>1}$ due to the Koecher principle, explained below. Denote the space of weight ${\displaystyle \rho }$, degree ${\displaystyle g}$, and level ${\displaystyle N}$ Siegel modular forms by

${\displaystyle M_{\rho }(\Gamma _{g}(N)).}$

For degree 1, the level 1 Siegel modular forms are the same as level 1 modular forms. The ring of such forms is a polynomial ring C[E₄,E₆] in the (degree 1) Eisenstein series E₄ and E₆.

For degree 2, (Igusa 1962, 1967) showed that the ring of level 1 Siegel modular forms is generated by the (degree 2) Eisenstein series E₄ and E₆ and 3 more forms of weights 10, 12, and 35. The ideal of relations between them is generated by the square of the weight 35 form minus a certain polynomial in the others.

For degree 3, Tsuyumine (1986) described the ring of level 1 Siegel modular forms, giving a set of 34 generators.

For degree 4, the level 1 Siegel modular forms of small weights have been found. There are no cusp forms of weights 2, 4, or 6. The space of cusp forms of weight 8 is 1-dimensional, spanned by the Schottky form. The space of cusp forms of weight 10 has dimension 1, the space of cusp forms of weight 12 has dimension 2, the space of cusp forms of weight 14 has dimension 3, and the space of cusp forms of weight 16 has dimension 7 (Poor & Yuen 2007) harv error: no target: CITEREFPoorYuen2007 (help).

For degree 5, the space of cusp forms has dimension 0 for weight 10, dimension 2 for weight 12. The space of forms of weight 12 has dimension 5.

For degree 6, there are no cusp forms of weights 0, 2, 4, 6, 8. The space of Siegel modular forms of weight 2 has dimension 0, and those of weights 4 or 6 both have dimension 1.

For small weights and level 1, Duke & Imamoḡlu (1998) give the following results (for any positive degree):

• Weight 0: The space of forms is 1-dimensional, spanned by 1.
• Weight 1: The only Siegel modular form is 0.
• Weight 2: The only Siegel modular form is 0.
• Weight 3: The only Siegel modular form is 0.
• Weight 4: For any degree, the space of forms of weight 4 is 1-dimensional, spanned by the theta function of the E₈ lattice (of appropriate degree). The only cusp form is 0.
• Weight 5: The only Siegel modular form is 0.
• Weight 6: The space of forms of weight 6 has dimension 1 if the degree is at most 8, and dimension 0 if the degree is at least 9. The only cusp form is 0.
• Weight 7: The space of cusp forms vanishes if the degree is 4 or 7.
• Weight 8:In genus 4, the space of cusp forms is 1-dimensional, spanned by the Schottky form and the space of forms is 2-dimensional. There are no cusp forms if the genus is 8.
• There are no cusp forms if the genus is greater than twice the weight.

The following table combines the results above with information from Poor & Yuen (2006) harvtxt error: no target: CITEREFPoorYuen2006 (help) and Chenevier & Lannes (2014) and Taïbi (2014).