One striking feature of the moduli space $\mathcal{A}_g$ is that it has a lot of symmetries, usually attributed to Hecke. These symmetries are algebraic correspondences on $\mathcal{A}_g$ coming from the symplectic group $GSp(2g)$. Over the complex numbers these Hecke correspondences are usually studied using the complex uniformization of $\mathcal{A}_g$ described above. Over a field of positive characteristic, for instance the algebraic closure of $\mathbb{Z}/p\mathbb{Z}$, the moduli space $\mathcal{A}_g$ has fine structures not coming from characteristic $0$, because the local structure of abelian varieties in characteristic $p$ may not all ``look alike''. Typical examples of such fine structures include the Newton polygon stratification and the Ekedahl-Oort stratification. These two fine structures are both preserved by the Hecke symmetries, but in general they are not enough to characterize the orbit of the Hecke symmetries. \smallbreak

Recently Oort defined a ``foliation'' structure on the moduli space $\mathcal{A}_g$ in characteristic $p$. Conjecturally the Zariski closure of the orbits of the Hecke symmetries are exactly the closures of ``leaves'' in Oort's foliation. We will explain some recent progress on the Hecke orbit problem, including the local structure of the leaves. The latter generalizes a theorem of Serre and Tate announced in the 1964 Summer Institute of Algebraic Geometry at Woods Hole.