General (Main) Seminar MSRI program on DDC-Unlikely intersections in families of abelian varieties

Organizer: V. Harizanov and A. Shlapentokh

For link contact [email protected]

Thursday, October 29, 12:00-1:00 ET on Zoom

Laura Capuano, Politecnico di Torino, Italy

Title: Unlikely intersections in families of abelian varieties

Abstract: Manin Mumford conjecture about the distribution of torsion points on subvarieties of semiabelian varieties has a natural analogue in families, and one can formulate more general conjectures in this setting. I will treat the relative Manin-Mumford results proved by Masser and Zannier for curves in a one-parameter family of abelian varieties and more general results obtained in this setting in collaboration with F. Barroero. The proofs of these results uses a method introduced for the first time by Pila and Zannier who gave an alternative proof of Manin-Mumford conjecture; this is based on the combination of tools coming from the theory of o-minimality, in particular a theorem of Pila and Wilkie about counting rational points of bounded height on certain transcendental varieties with other Diophantine ingredients.