Hecke Orbits on Globally Drinfeld-Uniformized Shimura Varieties Are $p$-adically Nowhere Dense

Abstract

Fix a projective unitary Shimura datum satisfying Varshavsky's global Drinfeld uniformization hypotheses at a place $v\mid p$ of its reflex field $E$. Let $\mathcal{S}h$ be the base change of the Shimura variety to $\mathbb C_p$ at a sufficiently small neat finite level. We prove that for every closed reduced algebraic subvariety $Z\subset\mathcal{S}h$ with $\operatorname{dim} Z<\operatorname{dim} \mathcal{S}h$, the $p$-adic closure of its full finite adelic Hecke orbit has empty interior in $\mathcal{S}h(\mathbb C_p)$. At certain compatible levels, the nowhere density theorem also holds for Shimura varieties associated with a related PEL datum having the same adjoint datum as Varshavsky's. Moreover, for every rational prime $p$ and every integer $d\geq2$, there is a projective unitary Shimura variety of dimension $d-1$ satisfying these uniformization hypotheses at a place above $p$ for which the nowhere density theorem holds.