Joint PU/IAS Number Theory

I will talk about a unification of two bounded height results around abelian varieties. One is due to Silverman from 1983, which states, for an abelian scheme A/C with no fixed part over a curve C, that the set of points on C where the generic Mordell Weil group fails to specialize injectively has bounded height. The other is by Habegger in 2008 inside one abelian variety: a “geometrically nondegenerate” subvariety intersected with all torsion cosets up to complementary dimension gives a set of bounded height. I shall take the point of view from unlikely intersections (more precisely, just likely intersections) and discuss the key idea of the arithmetic part of the proof by homomorphism approximations.