Hecke orbits.

Abstract: Let $p$ be a prime number and $g$ be a positive integer. Let $A_g$ be the moduli space of $g$-dimensional principally polarized abelian varieties over the algebraic closure of the finite field with $p$ elements. There is a family of finite-to-one correspondences on this moduli space generated by prime-to-$p$ isogenies between abelian varieties; these symmetries are known as Hecke correspondences. Recently F. Oort defined a family of locally closed smooth algebraic subvarieties of $A_g$, called "leaves"; a leaf is the locus in $A_g$ corresponding to a fixed isomorphism class of polarized $p$-divisible group. Clearly every leaf is stable under all prime-to-$p$ Hecke correspondences. Oort conjectured that every Hecke orbit is dense in the leaf containing it. We will explain methods motivated by this conjecture, and how these methods can be used to prove the conjecture.