The p-adic analog of the Hecke orbit conjecture and density theorems toward the p-adic monodromy

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 original 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.