Siegel upper half-space

In the mathematical branch of function theory of Siegel half-space or the Siegel half-plane refers to a generalization of the half-plane. This room is named after the mathematician Carl Ludwig Siegel, who systematically examined this object.

Definition

Siegel of (upper) half space of degrees is defined as the amount of complex symmetric matrices whose imaginary part is positive definite.

Comments

• In the case of the Siegel half-space is the known upper half-plane.
• The Siegel upper half-space carries a symplectic operation of the group. The quotient is the moduli space of principal- polarized abelian varieties. In the case of quotient space parametrized elliptic curves. The j- function in this case indicates the j- invariant of the curve.
• Ichirō Satake was 1957 a compactification of the seal between the half-space.

Multidimensional theta series

Siegel of the half-space plays an important role in defining the theta series in several complex variable as a generalization of the upper half-plane. The multi-dimensional theta series is a function

Which by

Is defined. This series converges normally and is therefore holomorphic.