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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 E 8 lattice (of appropriate degree). The only cusp form is 0. Weight 5: The only Siegel modular ...
A 2D slice of a Calabi–Yau quintic. One such quintic is birationally equivalent to the compactification of the Siegel modular variety A 1,3 (2). [1]In mathematics, a Siegel modular variety or Siegel moduli space is an algebraic variety that parametrizes certain types of abelian varieties of a fixed dimension.
where T is an element of the Siegel upper half plane of degree g. This is a Siegel modular form of degree d and weight dim(L)/2 for some subgroup of the Siegel modular group. If the lattice L is even and unimodular then this is a Siegel modular form for the full Siegel modular group. When the degree is 1 this is just the usual theta function of ...
The variable τ may be a complex number in the upper half-plane in which case the theta constants are modular forms, or more generally may be an element of a Siegel upper half plane in which case the theta constants are Siegel modular forms. The theta function of a lattice is essentially a special case of a theta constant.
In mathematics, the Siegel upper half-space of degree g (or genus g) (also called the Siegel upper half-plane) is the set of g × g symmetric matrices over the complex numbers whose imaginary part is positive definite. It was introduced by Siegel . It is the symmetric space associated to the symplectic group Sp(2g, R).
It is a generalization of the Siegel modular group, and has the same relation to polarized abelian varieties that the Siegel modular group has to principally polarized abelian varieties. It is the group of automorphisms of Z 2n preserving a non-degenerate skew symmetric form. The name "paramodular group" is often used to mean one of several ...
In higher dimensions, moduli of algebraic varieties are more difficult to construct and study. For instance, the higher-dimensional analogue of the moduli space of elliptic curves discussed above is the moduli space of abelian varieties, such as the Siegel modular variety. This is the problem underlying Siegel modular form theory.
Furthermore, the theta function of an even unimodular lattice of rank n is actually a modular form of weight n/2. The theta function of an integral lattice is often written as a power series in q = e 2 i π τ {\displaystyle q=e^{2i\pi \tau }} so that the coefficient of q n gives the number of lattice vectors of norm 2 n .