Given and define the notation Then a holomorphic function is a Siegel modular form of degree , weight, and level if for all. In the case that, we further require that be holomorphic 'at infinity'. This assumption is not necessary for due to the Koecher principle, explained below. Denote the space of weight, degree, and level Siegel modular forms by
Examples
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 ringC in the Eisenstein seriesE4 and E6. For degree 2, showed that the ring of level 1 Siegel modular forms is generated by the 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 :
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. 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.
Weight
degree 0
degree 1
degree 2
degree 3
degree 4
degree 5
degree 6
degree 7
degree 8
degree 9
degree 10
degree 11
degree 12
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
10
1: 1
0: 1
1: 2
0 : 2
1: 3
0: 3
1: 4
0: 4
1:
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
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-
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Koecher principle
The theorem known as the Koecher principle states that if is a Siegel modular form of weight, level 1, and degree, then is bounded on subsets of of the form where. Corollary to this theorem is the fact that Siegel modular forms of degree 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.
