Algebraic Varieties and Automorphic Functions

Abstract

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