| Résume | If $A$ is a $g$-dimensional principally polarized abelian variety over $\mathbb{C}$, it is not hard to show that the Hecke orbit of $A$ (the set of principally polarized abelian varieties which are isogenous to $A$) is dense in $\mathcal{A}_g(\mathbb{C})$. This fails in positive characteristic because Hecke orbits are confined to Newton strata, the loci of $\mathcal{A}_g$ where the Newton polygon of the abelian variety is constant. The Hecke orbit conjecture, now a theorem of Chai-Oort, thus asks whether Hecke orbits are dense in their respective Newton strata over an algebraically closed field of characteristic $p$. In the first part of the talk, we will explain Chai's strategy for proving the Hecke orbit conjecture, and we will present the recent results in $p$-adic monodromy that led to a proof, by D'Addezio and van Hoften, of the Hecke orbit conjecture for all Shimura varieties of Hodge type. In the second part we present a refinement of these $p$-adic monodromy techniques, with an application to the rigidity of Tate-linear varieties, which are central objects in Chai's work. |
