Let $(G, X)$ be a Shimura datum, let $\Omega$ be a connected component of $X$, let $\Gamma$ be a congruence subgroup of $G(\mathbb{ Q})^{ +}$, and consider the quotient map $q: \Omega \to S:=\Gamma \backslash \Omega$. Consider the Harish-Chandra embedding $\Omega\subset\mathbb{ C}^{ N}$, where $N=\dim X$. We prove two results that give geometric conditions which if satisfied by an algebraic variety $V \subset \mathbb{ C}^{ N} \times S$, ensure that there is a Zariski dense subset of $V$ of points of the form $(x,q(x))$.
Categories: math.AG math.NT