Uniformity in unlikely intersections and the dynamical André--Oort conjecture
Myrto Mavraki (University of Toronto)
Abstract: A rational map is postcritically finite (PCF) if its critical orbits are finite. Postcritically finite maps play an important role in dynamics. Further, it was suggested by Silverman that they play a role analogous to CM elliptic elliptic curves. Inspired in part by the Pink-Zilber conjectures in unlikely intersections, Baker and DeMarco formulated a conjecture aiming to describe the subvarieties of $M_d$ that contain a Zariski dense set of PCF points. Their conjecture, now known as dynamical André--Oort conjecture (or DAO), was recently resolved in the case of curves by Ji--Xie, but remains open in higher dimensions. In this talk we will describe recent work with DeMarco and Ye, providing uniform bounds on the configurations of PCF points in families of subvarieties in $M_d$. We also provide a gap principle in the spirit of Dimitrov--Gao--Habegger's, Kühne's, and Gao--Ge--Kühne's work on the uniform Mordell--Lang conjecture.
number theory
Audience: researchers in the topic
| Organizers: | Sol Friedberg*, Benjamin Howard, Dubi Kelmer, Spencer Leslie, Keerthi Madapusi Pera, Bjorn Poonen*, Andrew Sutherland*, Wei Zhang* |
| *contact for this listing |