6 problems

Let $\mathcal{S}h$ be a Shimura variety and let $Z\subseteq\mathcal{S}h$ be a positive-dimensional subvariety. Denote by the Hecke orbit of $Z$ the union of its images under the He…

Let $p>0$, let $g$ be a positive integer, and let $x=\left[\left(A_x,\lambda_x\right)\right]$ be a $\overline{\mathbb{F}}_p$-point of the moduli space $\mathcal{A}_g$ of $g$-dimens…

Let $S_K(G,X)$ be the Shimura variety under consideration, and let $Y\subset S_K(G,X)$ be a positive-dimensional subvariety intersecting the ordinary locus. The Hecke orbit of $Y$…

Let $\shS^i$ be the height and Artin-invariant strata in the special fiber of the orthogonal Shimura variety parametrizing degree-$2d$ polarized K3 surfaces, and let the prime-to-…

Let $\mathcal{S}$ be the canonical integral model of a Shimura variety of Hodge type with hyperspecial level, and let $\mathcal{S}_{\overline{\mathbb{F}}_p}$ be its special fiber.…

Let $k$ be an algebraically closed field of characteristic $p$, and let $S$ be a PEL type Shimura variety over $k$. For a point $x\in S$, let its prime-to-$p$ Hecke orbit be the or…