Unlikely Intersections in Elliptic Surfaces and Problems of Masser

Abstract

This chapter turns from the multiplicative-group context to the context of abelian varieties. There are here entirely similar results and conjectures: we have already recalled the Manin–Mumford conjecture, and pointed out that the Zilber conjecture also admits an abelian exact analogue. Actually, abelian varieties have moduli, which introduce new issues with respect to the toric case. The chapter focuses mainly on some new problems, raised by Masser, which represent a relative case of Manin–Mumford–Raynaud, where the relevant abelian variety is no longer fixed but moves in a family. The unlikely intersections of Masser's questions occur in the special case of elliptic surfaces (i.e., families of elliptic curves), and can be dealt with by a method that has been recently introduced.