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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 closed upper half-plane is the union of the upper half-plane and the real ... Yet another space interesting to number theorists is the Siegel upper half-space ...
An example is the Siegel upper half plane, where V⊂R k(k + 1)/2 is the cone of positive definite quadratic forms in R k and m = k(k + 1)/2. A Siegel domain of the second kind (or second type, or genus 2), also called a Piatetski-Shapiro domain, is the open subset of C m ×C n of elements (z,w) such that
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 ...
A half-space can be either open or closed. An open half-space is either of the two open sets produced by the subtraction of a hyperplane from the affine space. A closed half-space is the union of an open half-space and the hyperplane that defines it. The open (closed) upper half-space is the half-space of all (x 1, x 2, ..., x n) such that x n > 0
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.
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 ...
(For example, the universal cover of a real projective plane is a sphere.) ... Siegel upper half space. C 2 is the same as B 2, and C 1 is the same as B 1 and A 1.