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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.