Towards uniformity in the dynamical Bogomolov conjecture.

SEMINARS, NUMBER THEORY

Speaker:

Niki Myrto Mavraki - Harvard

Inspired by an analogy between torsion and preperiodic points, Zhang has proposed a dynamical generalization of the classical Manin-Mumford and Bogomolov conjectures. A special case of these conjectures, for `split' maps, has recently been established by Nguyen, Ghioca and Ye. In particular, they show that two rational maps have at most finitely many common preperiodic points, unless they are `related'. Recent breakthroughs by Dimitrov, Gao, Habegger and Kühne have established that the classical Bogomolov conjecture holds uniformly across curves of given genus.

In this talk we discuss uniform versions of the dynamical Bogomolov conjecture across 1-parameter families of split maps and curves. To this end, we establish instances of a 'relative dynamical Bogomolov conjecture'. This is joint work with Harry Schmidt (University of Basel).