We prove the Bogomolov conjecture for a totally degenerate abelian variety A over a function field. 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 v. As a corollary, we obtain finiteness of torsion points with coordinates in the maximal unramified algebraic extension over v.
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