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 perf ) is finitely generated and that the p-primary torsion subgroup of A(K 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 in the situation where the base field is the algebraic closure of a finite field.
We prove that in positive characteristic, the Manin-Mumford conjecture implies the Mordell-Lang conjecture, in the situation where the ambient variety is an abelian variety defined over the function field of a smooth curve over a finite field and the relevant group is a finitely generated group. In particular, in the setting of the last sentence, we provide a proof of the Mordell-Lang conjecture, which does not depend on tools coming from model theory.
We give a new proof of the Adams-Riemann-Roch theorem for a smooth projective morphism X → Y , in the situation where Y is a regular scheme, which is quasi-projective over F p . We also partially answer a question of B. Köck.
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