Algebraic Varieties and Automorphic Functions
Sebastian Eterović, Roy Zhao
TL;DR
The paper tackles the existential closedness problem for the Shimura quotient map $q:\Omega\to S$ by embedding $\Omega$ into $\mathbb{C}^N$ via the Harish-Chandra realization and exploiting boundary geometry. It proves two geometric criteria ensuring a Zariski-dense subset of an algebraic variety $V\subset \mathbb{C}^N\times S$ lies in the graph $E_q$: (i) if the projection $\pi(V)$ is Zariski-dense in $\mathbb{C}^N$, then $V\cap E_q$ is Zariski-dense in $V$, and (ii) for broad and free varieties of the form $L\times W$ with $L$ totally geodesic, the image $q(L)$ is dense in $W$ (and conjecturally $V\cap E_q$ is Zariski-dense). The work combines Harish-Chandra/Borel embeddings, Shilov boundary analysis, and limit-set results with Ax–Schanuel for Shimura varieties and products of subvarieties, yielding corollaries for mixed Shimura settings and abelian-family cases. These results advance understanding of ECP in arithmetic geometry, connecting geometric criteria to model-theoretic density phenomena and unlikely intersection questions. The method integrates complex-analytic, algebraic, and ergodic tools to establish density phenomena in Shimura-analytic settings, with potential applications to questions about special subvarieties and modular-exponential maps.
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 $(x,q(x))$.