Abstract

A little over a year ago Kühne, building on work of Dimitrov, Gao and Habegger, proved a uniform version of the Manin-Mumford conjecture (a theorem of Raynaud). His proof uses various ingredients, among them equi-distribution on families of abelian varieties. The latter ingredient was generalized by Yuan and Zhang to families of dynamical systems (among other things), with related work by Gauthier. Another ingredient in his proof, that is also central to the work of Dimitrov, Gao and Habegger, is the so-called Ax-Schanuel theorem. This is a powerful functional transcendence statement, motivated by the André-Oort conjecture, that was obtained over several works in the past 20 years and characterizes the intersection of algebraic varieties with the graph of the uniformization map of mixed Shimura varieties. It is however particular to those and not applicable to dynamical systems in general.

After giving a brief historical overview, I will explain how Myrto Mavraki and I overcame the restrictions imposed by a reliance on Ax-Schanuel in recent work and proved uniform versions of the dynamical Manin-Mumford conjecture for one-parameter families of endomorphisms of the projective line. I will explain aspects of our new strategy that also applies in the abelian context and generalizes results of Habegger as well as results of Baker and Call-Silverman. Time permitting, I will also report on very recent results for higher dimensional bases.