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]