Siegel upper half-space

Content sourced from Wikipedia, licensed under CC BY-SA 3.0.

The Siegel upper half-space Hg of degree g is the set of g×g complex symmetric matrices τ whose imaginary part is positive definite. It was introduced by Siegel in 1939. When g = 1 it becomes the usual Poincaré upper half-plane. Hg is the symmetric space associated with the real symplectic group Sp(2g, R).

Hg is an open subset of the space of g×g complex symmetric matrices, so it is a complex manifold of dimension g(g+1)/2. It is a special type of Siegel domain.

The group Sp(2g, R) acts on Hg by a natural action. This action is continuous, faithful, and transitive. The stabilizer of the point iI_g is the unitary group U(g), which is a maximal compact subgroup of Sp(2g, R). Therefore Hg is diffeomorphic to the symmetric space Sp(2g, R)/U(g). There is a natural invariant Riemannian metric on Hg.

The Siegel modular group Γg is Sp(2g, Z), the integer points of Sp(2g, R). The quotient Γg \ Hg can be understood as a moduli space: it classifies g-dimensional principally polarized complex Abelian varieties.

If τ = X + iY is in Hg (with X and Y real symmetric and Y positive definite), one can define a positive definite Hermitian form H on C^g. This form takes integral values on the lattice Z^g + Z^g τ, so the complex torus C^g / (Z^g + Z^g τ) becomes an Abelian variety together with the polarization given by H. The polarization is unimodular, i.e., principal. Conversely, every principally polarized Abelian variety arises in this way. Thus the quotient Γg \ Hg parametrizes principally polarized Abelian varieties.