Algebraic Varieties and Automorphic Functions

Abstract

Let $(G,X)$ be a Shimura datum, let $\hat{I}\text{c◯}$ be a connected component of $X$, let $\hat{I}“$ be a congruence subgroup of $G(\mathbb{Q}{)}^{+}$, and consider the quotient map $q:\hat{I}\text{c◯}\to S:=\hat{I}“\backslash \hat{I}\text{c◯}$. Consider the Harish-Chandra embedding $\hat{I}\text{c◯}\subset {\mathbb{C}}^N$, 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 $(x,q(x))$.

16 pages. Minor corrections