The Bogomolov conjecture for totally degenerate abelian varieties

Abstract

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.

arXiv comment21 pages; submitted. Minor errors corrected, applications in Section 6 added

References in corpus (2)

• The Bogomolov conjecture for totally degenerate abelian varieties
• Tropical varieties for non-archimedean analytic spaces

Cited by in corpus (27)

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• Filtered geometric lattices and Lefschetz Section Theorems over the tropical semiring
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• Arakelov geometry, heights, equidistribution, and the Bogomolov conjecture
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• Heights and arithmetic dynamics over finitely generated fields
• The geometric Bogomolov conjecture
• Faithful tropicalization of the Grassmannian of planes
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• The arithmetic Hodge Index Theorem and rigidity of dynamical systems over function fields
• Equidistribution of torsion points in abelian varieties
• A Dynamical Bogomolov Property
• Zhang's Conjecture and the Effective Bogomolov Conjecture over function fields
• Monge-Ampère measures for toric metrics on abelian varieties