Algebraic Varieties and Automorphic Functions

TL;DR

This paper establishes geometric conditions under which algebraic varieties related to Shimura data contain Zariski dense sets of special points, advancing understanding of their structure and distribution.

Contribution

It provides new geometric criteria ensuring the density of special points in algebraic varieties associated with Shimura data.

Findings

Identifies conditions guaranteeing Zariski density of special points.

Connects geometric properties with distribution of automorphic functions.

Enhances understanding of algebraic varieties in the context of Shimura varieties.

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 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 $(x,q(x))$.