Siegel modular form

Definition

Preliminaries

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

:\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 N, denoted by \Gamma_g(N), as

:\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 I_g is the g \times g identity matrix. Finally, let

:\rho:\textrm{GL}_g(\mathbb{C}) \rightarrow \textrm{GL}(V)

be a rational representation, where V is a finite-dimensional complex vector space.

Siegel modular form

Given

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

and

:\gamma \in \Gamma_g(N),

define the notation

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

Then a holomorphic function

:f:\mathcal{H}_g \rightarrow V

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

:(f\big|\gamma)=f

for all \gamma \in \Gamma_g(N). In the case that g=1, we further require that f be holomorphic 'at infinity'. This assumption is not necessary for g1 due to the Koecher principle, explained below. Denote the space of weight \rho, degree g, and level N Siegel modular forms by

:M_{\rho}(\Gamma_g(N)).

Examples

Some methods for constructing Siegel modular forms include:

• Eisenstein series
• Theta functions of lattices and Siegel theta series
• Saito–Kurokawa lift for degree 2
• Ikeda lift
• Miyawaki lift
• Products of Siegel modular forms.

Level 1, small degree

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, 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, 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 .

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.

Level 1, small weight

For small weights and level 1, 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.

Table of dimensions of spaces of level 1 Siegel modular forms

The following table combines the results above with information from and and .

Koecher principle

The theorem known as the Koecher principle states that if f is a Siegel modular form of weight \rho, level 1, and degree g1, then f is bounded on subsets of \mathcal{H}_g of the form

:\left{\tau \in \mathcal{H}_g \ | \textrm{Im}(\tau) \epsilon I_g \right},

where \epsilon0. Corollary to this theorem is the fact that Siegel modular forms of degree g1 have Fourier expansions and are thus holomorphic at infinity.

Applications to physics

In the D1D5P system of supersymmetric black holes in string theory, the function that naturally captures the microstates of black hole entropy is a Siegel modular form. In general, Siegel modular forms have been described as having the potential to describe black holes or other gravitational systems.

Siegel modular forms also have uses as generating functions for families of CFT2 with increasing central charge in conformal field theory, particularly the hypothetical AdS/CFT correspondence.

References

• {{citation|title=Formes automorphes et voisins de Kneser des réseaux de Niemeier
• {{citation|mr=1600030
• {{citation|mr=0871067
• {{citation|mr=2409679
• {{citation|mr=0141643
• {{citation|mr=0001251
• {{citation|mr=0853217

References

1. This was proved by [[Max Koecher]], ''Zur Theorie der Modulformen n-ten Grades I'', Mathematische. Zeitschrift 59 (1954), 455–466. A corresponding principle for [[Hilbert modular form]]s was apparently known earlier, after Fritz Gotzky, ''Uber eine zahlentheoretische Anwendung von Modulfunktionen zweier Veranderlicher'', Math. Ann. 100 (1928), pp. 411-37
2. (11 April 2017). "Siegel modular forms and black hole entropy". Journal of High Energy Physics.
3. (7 November 2018). "Siegel paramodular forms and sparseness in AdS3/CFT2". Journal of High Energy Physics.