A Hecke correspondence in characteristic zero is finite-to-finite. However in positive characteristic such a correspondence may blow-up-and-down. It is the aim of this talk to understand this phenomenon. The key to this is a foliation. We discuss examples. We see a generalization of Serre-Tate coordinates, as given by Ching-Li Chai, and we give a new proof of the fact that any polarized abelian variety can be lifted to characteristic zero.