Algebraic Varieties and Automorphic Functions

Abstract 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 that, 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 $(x,q(x))$.