The Selmer group of isogenies of rank one

I will describe the proof of the fact that the Selmer group of an isogeny of height one between abelian varieties over a global function field of positive characteristic can be embedded in the group of homomorphisms between two natural vector bundles on the smooth projective model of the function field. This can be seen as a refinement of a special case of a theorem of Artin-Milne (see Prop. 1.1 in M. Artin and J.S. Milne, Duality in the flat cohomology of curves, Invent. Math. 35 (1976), 111–129). This result can for instance be applied to the relative Frobenius morphism. I will then describe some arithmetic applications of this result.