The Bogomolov conjecture for totally degenerate abelian varieties
  We prove the Bogomolov conjecture for an abelian variety A over a function field which is totally degenerate at a place v. We adapt Zhang's proof of the number field case replacing the complex analytic tools by tropical analytic geometry. A key step is the tropical equidistribution theorem for A at the totally degenerate place. As an application, we obtain finiteness of torsion points with coordinates in the maximal unramified algebraic extension over v. 
Walter Gubler