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.Comment: 21 pages; submitted. Minor errors corrected, applications in Section
6 adde
