From Manin–Mumford to Unlikely Intersections: The Relative Case and Beyond

In the first part of the talk, I will give a brief survey of the Manin–Mumford conjecture, which predicts the distribution of torsion points in commutative algebraic groups. I will then motivate a relative version proposed by Pink and Zannier, emphasizing the guiding philosophy of unlikely intersections. From this perspective, I will also discuss several concrete examples. In the second part of the talk, I will explain how the relative Manin–Mumford conjecture can fail for families of semi-abelian varieties, as shown by a counterexample of Bertrand and Edixhoven. Nevertheless, the Zilber–Pink conjecture suggests that their construction should, in a precise sense, account for all possible counterexamples. This part is based on joint work with Kaiyuan Gu.

Go to http://people.math.harvard.edu/~demarco/AlgebraicDynamics/ for more information