The Bogomolov conjecture for totally degenerate abelian varieties
In two subsequent publications [Invent. Math. 169, No. 2, 321-376 (2007; Zbl 1153.14036); Invent. Math. 169, No. 2, 377-400 (2007; Zbl 1153.14029)], Walter Gubler proves the Bogomolov conjecture in the case where the abelian variety is totally degenerate at some place of the ground field.
The first paper “Tropical varieties for non-archimedean analytic spaces” provides three main theorems, the third of which is a key step in the proof of the Bogomolov Conjecture which is carried out in the second paper. Read the detailed reviews of Paul Vojta [ Zbl 1153.14036; Zbl 1153.14029] to get a first understanding of interplay of the two papers and the broad variety of techniques.