10 problems

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 $p$ be a prime of good reduction for a Shimura variety of Hodge type, and let $x$ be a point of its special fibre. Let the central leaf containing $x$ be the locus of points wh…

Chai's density conjecture. If $K_p$ is hyperspecial, then the orbit of any point of ${\cal S}_{[b_\mu]}$ under $\bold G(\mathbb A_f^p)$ is dense in $\overline{Sh(G,h)}_{\bold K}$.

Let $\mathscr{S}$ be an integral model of a Shimura variety of Hodge type over $\mathbb{F}_p$, and let $X=\mathscr{S}_{\mathbb{F}_p}$ and $Y$ be a point. The full Hecke orbit conje…

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 $S_K(G,X)$ 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 $K$ be a characteristic zero local field with ring of integers $\mathcal{O}_K$, residue field $k$, and algebraic closure $\bar{k}$. Let $\mathscr{G}$ be a $p$-divisible group o…

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 $g$ be sufficiently large, let $M$ be a weakly special subvariety of the moduli space $44\mathcal{A}_g44$, and let $H(M)$ denote its total Hecke orbit. Finite Hecke-translate c…