On the group of purely inseparable points of an abelian variety defined over a function field of positive characteristic II

Abstract

Let $A$ be an abelian variety over the function field $K$ of a curve over a finite field. We describe several mild geometric conditions ensuring that the group $A(K^{\mathrm{perf}})$ is finitely generated and that the $p$-primary torsion subgroup of $A(K^{\mathrm{sep}})$ is finite. This gives partial answers to questions of Scanlon, Ghioca and Moosa, and Poonen and Voloch. We also describe a simple theory (used to prove our results) relating the Harder-Narasimhan filtration of vector bundles to the structure of finite flat group schemes of height one over projective curves over perfect fields. Finally, we use our results to give a complete proof of a conjecture of Esnault and Langer on Verschiebung divisibility of points in abelian varieties over function fields.

This third version contains a complete proof of the conjecture of Esnault and Langer Verschiebung divisibility (not only over the algebraic closure of a finite field, as in the previous version)

References in corpus (2)

• Overconvergent unit-root F-isocrystals and isotriviality
• Le groupe de Selmer des isogénies de hauteur un

Cited by in corpus (5)

• Parabolicity conjecture of $F$-isocrystals
• Maximal tori of monodromy groups of $F$-isocrystals and an application to abelian varieties
• Positivity of Hodge bundles of abelian varieties over some function fields
• Perfect points of abelian varieties
• Finiteness and cofiniteness of fine Selmer groups over function fields