SPECIAL SUBVARIETIES IN MUMFORD-TATE VARIETIES

Let X = Gamma\D be a Mumford-Tate variety, i.e., a quotient of a Milmford-Tate domain D = G(R)/V by a discrete subgroup Gamma. umford-Tate varieties are generalizations of Shimura varieties. We define the notion of a special subvariety Y subset of X (of Shimura type), and formulate necessary criteria for Y to be special. Our method consists in looking at finitely many compactified special curves C-i in Y, and testing whether the inclusion boolean OR(i) C-i subset of Y satisfies certain properties. One of them is the so-called relative proportionality condition. In this paper, we give a new formulation of this numerical criterion in the case of /vlumford-Tate varieties X. In this way, we give necessary and sufficient criteria for a subvariety Y of X to be a special subvariety of Shimura type in the sense of the Andre-Oort conjecture. We discuss in detail the important case where X = A(g), the moduli space of principally polarized abelian varieties.