Towards a p-adic nowhere density conjecture of Hecke Orbits

We propose a conjecture on the $p$-adic nowhere density of the Hecke orbit of subvarieties of Hodge type Shimura varieties. By investigating the monodromy of $p$-adic Galois representations associated with points on such Shimura varieties, we prove that the locus in a formal ${\mathcal{O}}_K$-neighborhood of a mod $p$ point that has large monodromy is open dense, where $K$ is a totally ramified finite extension of ${\breve{\mathbb{Q}}}_p\mathrm{.}$