Motivation. Isogenies with a polarized abelian variety give moduli points in a Hecke orbit. In characteristic zero any Hecke orbit is dense in this moduli space. What can be said about the Zariski closure of a Hecke orbit in positive characteristic?

This was the starting point for Ching-Li Chai who proved that for an ordinary abelian variety its Hecke orbit is dense (Inventiones Math. 1995) and for a conjecture that predicted the Zariski closure of any Hecke orbit (Questions in algebraic geometry, 1995).

Result. In joint work with Ching-Li Chai we prove this conjecture. One of basic tools is the notion of foliations. The main focus of the talk will be on describing properties of these foliations, rather well-understood by now. Many other notions and previous results will be recalled in the proofs. We give examples and open questions.