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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 ...
Siegel modular varieties have been used in conformal field theory via the theory of Siegel modular forms. [11] In string theory , the function that naturally captures the microstates of black hole entropy in the D1D5P system of supersymmetric black holes is a Siegel modular form.
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 ...
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).
The Siegel Eisenstein series of degree g and weight an even integer k > 2 is given by the sum ∑ C , D 1 det ( C Z + D ) k {\displaystyle \sum _{C,D}{\frac {1}{\det(CZ+D)^{k}}}} Sometimes the series is multiplied by a constant so that the constant term of the Fourier expansion is 1.
The Selmer group is finite. This implies that the part of the Tate–Shafarevich group killed by f is finite due to the following exact sequence. 0 → B(K)/f(A(K)) → Sel (f) (A/K) → ะจ(A/K)[f] → 0. The Selmer group in the middle of this exact sequence is finite and effectively computable.
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 .
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.