Siegel Moduli Variety

The Siegel moduli variety is the coarse moduli variety of principally polarized abelian varieties of dimension . An abelian variety polarization comes from an ample line bundle and gives a homomorphism from the abelian variety to its dual abelian variety, which parametrizes algebraically trivial line bundles. The polarization is principal when this homomorphism is an isomorphism. Over , the analytic space is the quotient of the Siegel upper half-space by . The upper half-space consists of complex symmetric matrices whose imaginary parts are positive definite. Its complex dimension is .

In particular, parametrizes elliptic curves, while is a threefold parametrizing principally polarized abelian surfaces. An algebraic curve in is Hodge generic when it is contained in no proper special subvariety. The Zilber-Pink prediction for such a curve concerns finiteness of its intersections with the union of all special curves and special points.