Bjorn Poonen

Let $X$ be a subvariety of an abelian variety $A$ over a number field $k$. The Mordell-Lang conjecture is concerned with the intersection of $X(\overline{k})$ with the division group of a finitely generated subgroup of $A(\overline{k})$. The (generalized) Bogomolov conjecture is concerned with the set of points in $X(\overline{k})$ of small canonical height. These have both been proven, although only the former has been completely generalized to semiabelian varieties. We will prove a result that contains both conjectures.