The Hecke orbit conjecture predicts that Hecke symmetries characterize the central foliation on Shimura varieties over an algebraically closed field $k$ of characteristic $p$. The conjecture predicts that on the mod $p$ reduction of a Shimura variety, any prime-to-p Hecke orbit is dense in the central leaf containing it, and was recently proved by a series of nice papers. However, the behavior of Hecke correspondences induced by isogenies between abelian varieties in characteristic $p$ and $p$-adically is significantly different from the behavior in characteristic zero and under the topology induced by Archimedean valuations. In this talk,  we will formulate a $p$-adic analog of the Hecke orbit conjecture and investigate the $p$-adic monodromy of $p$-adic Galois representations attached to points of Shimura varieties of Hodge type. We prove a density theorem for the locus of formal neighborhood associated to the mod $p$ points of the Shimura variety whose monodromy is large and use it to deduce the non-where density of Hecke orbits under certain circumstances.

[pre-talk at 3:00pm]