A dynamical variant of the Pink-Zilber conjecture

This talk will present results from the paper of Ghioca and Nguyen "A dynamical variant of the Pink–Zilber conjecture". Pink-Zilber conjecture relates to the theory of unlikely intersections in Diophantine Geometry and is a generalisation of Manin-Mumford conjecture. The latter states there can't be too many special points in the variety unless this variety is special as well. For example, for abelian varieties special = torsion, so Manin-Mumford will state that if the set of torsion points is dense in a subvariety X of an abelian variety, then X is a torsion subvariety (translation of an abelian subvariety by a torsion point). Instead of searching torsion points inside X, we can look at the intersections of X with torsion subvarieties - Pink-Zilber states that if X intersects the set of torsion subvarieties of some chosen dimensions in a dense subset, then X is contained in a torsion subvariety.

The analogue of Zilber-Pink conjecture described in the article considers a map of affine spaces A^n \to A^n of a particular, quite simple form and states that an irreducible variety X is contained in a periodic subvariety of A^n if X intersects the union of periodic subvarieties of some chosen dimension in a dense subset. The proof is by induction and involves some basic algebraic geometry/number theory as properties of finite morphisms and Weil heights.

The talk will begin with a presentation of some different variants of Manin-Mumford and Pink-Zilber conjecture and then outline the sketch of the proof of the main theorem from the article of Ghioca and Nguyen.

 

This is an informal seminar on arithmetic algebraic geometry and related fields.

The purpose is to present work in progress or to learn about some existing piece of mathematics. Typically the talks are parts of a longer series.