In mathematics, Siegel modular forms are a major type of automorphic form. These generalize conventional elliptic modular forms which are closely related to elliptic curves. The complex manifolds constructed in the theory of Siegel modular forms are Siegel modular varieties, which are basic models for what a moduli space for abelian varieties (with some extra level structure) should be and are constructed as quotients of the Siegel upper half-space rather than the upper half-plane by discrete groups.
Siegel modular forms are holomorphic functions on the set of symmetric n × n matrices with positive definite imaginary part; the forms must satisfy an automorphy condition. Siegel modular forms can be thought of as multivariable modular forms, i.e. as special functions of several complex variables.
Siegel modular forms were first investigated by for the purpose of studying quadratic forms analytically. These primarily arise in various branches of number theory, such as arithmetic geometry and elliptic cohomology. Siegel modular forms have also been used in some areas of physics, such as conformal field theory and black hole thermodynamics in string theory.
Let
g,N\inN
l{H}g=\left\{\tau\inMg(C) | \tauT=\tau,rm{Im}(\tau)positivedefinite\right\},
the Siegel upper half-space. Define the symplectic group of level
N
\Gammag(N),
\Gammag(N)=\left\{\gamma\inGL2g(Z) | \gammaT\begin{pmatrix}0&Ig\ -Ig&0\end{pmatrix}\gamma=\begin{pmatrix}0&Ig\ -Ig&0\end{pmatrix}, \gamma\equivI2g\modN\right\},
where
Ig
g x g
\rho:rm{GL}g(C) → rm{GL}(V)
be a rational representation, where
V
Given
\gamma=\begin{pmatrix}A&B\ C&D\end{pmatrix}
and
\gamma\in\Gammag(N),
define the notation
(f|\gamma)(\tau)=(\rho(C\tau+D))-1f(\gamma\tau).
Then a holomorphic function
f:l{H}g → V
is a Siegel modular form of degree
g
\rho
N
(f|\gamma)=f
for all
\gamma\in\Gammag(N)
g=1
f
g>1
\rho
g
N
M\rho(\Gammag(N)).
Some methods for constructing Siegel modular forms include:
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[''E''<sub>4</sub>,''E''<sub>6</sub>] 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.
For small weights and level 1, give the following results (for any positive degree):
The following table combines the results above with information from and and .
| 0 | 1: 1 | 0: 1 | 0: 1 | 0: 1 | 0: 1 | 0: 1 | 0: 1 | 0: 1 | 0: 1 | 0: 1 | 0: 1 | 0: 1 | 0: 1 |
| 2 | 1: 1 | 0: 0 | 0: 0 | 0: 0 | 0: 0 | 0: 0 | 0: 0 | 0: 0 | 0: 0 | 0: 0 | 0: 0 | 0: 0 | 0: 0 |
| 4 | 1: 1 | 0: 1 | 0: 1 | 0: 1 | 0: 1 | 0: 1 | 0: 1 | 0: 1 | 0: 1 | 0: 1 | 0: 1 | 0: 1 | 0: 1 |
| 6 | 1: 1 | 0: 1 | 0: 1 | 0: 1 | 0: 1 | 0: 1 | 0: 1 | 0: 1 | 0: 1 | 0: 0 | 0: 0 | 0: 0 | 0: 0 |
| 8 | 1: 1 | 0: 1 | 0: 1 | 0:1 | 1: 2 | 0: 2 | 0: 2 | 0: 2 | 0: 2 | 0: | 0: | 0: | 0: |
| 10 | 1: 1 | 0: 1 | 1: 2 | 0: 2 | 1: 3 | 0: 3 | 1: 4 | 0: 4 | 1: | 0: | 0: | 0: | 0: |
| 12 | 1: 1 | 1: 2 | 1: 3 | 1: 4 | 2: 6 | 2: 8 | 3: 11 | 3: 14 | 4: 18 | 2:20 | 2: 22 | 1: 23 | 1: 24 |
| 14 | 1: 1 | 0: 1 | 1: 2 | 1: 3 | 3:6 | 3: 9 | 9: 18 | 9: 27 | |||||
| 16 | 1: 1 | 1: 2 | 2: 4 | 3: 7 | 7: 14 | 13:27 | 33:60 | 83:143 | |||||
| 18 | 1: 1 | 1: 2 | 2: 4 | 4:8 | 12:20 | 28: 48 | 117: 163 | ||||||
| 20 | 1: 1 | 1: 2 | 3: 5 | 6: 11 | 22: 33 | 76: 109 | 486:595 | ||||||
| 22 | 1: 1 | 1: 2 | 4: 6 | 9:15 | 38:53 | 186:239 | |||||||
| 24 | 1: 1 | 2: 3 | 5: 8 | 14: 22 | |||||||||
| 26 | 1: 1 | 1: 2 | 5: 7 | 17: 24 | |||||||||
| 28 | 1: 1 | 2: 3 | 7: 10 | 27: 37 | |||||||||
| 30 | 1: 1 | 2: 3 | 8: 11 | 34: 45 |
The theorem known as the Koecher principle states that if
f
\rho
g>1
f
l{H}g
\left\{\tau\inl{H}g |rm{Im}(\tau)>\epsilonIg\right\},
where
\epsilon>0
g>1
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.[2] In general, Siegel modular forms have been described as having the potential to describe black holes or other gravitational systems.[2]
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.[3]