Special Subvarieties in Mumford-Tate Varieties

Abstract

Let $X=\Gamma \backslash D$ be a Mumford-Tate variety, i.e., a quotient of a Mumford-Tate domain $D=G(\mathcal{R})/V$ by a discrete subgroup $\Gamma$. Mumford-Tate varieties are generalizations of Shimura varieties. We define the notion of a special subvariety $Y \subset X$ (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 $\bigcup_i C_i \subset Y$ 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 Mumford-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 André-Oort conjecture. We discuss in detail the important case where $X=A_g$, the moduli space of principally polarized abelian varieties.