Local flat duality of abelian varieties

Let K be a field complete for a discrete valuation and with algebraically closed residue field of positive characteristic p. We prove the existence of a non-degenerate pairing between the first (flat) cohomology group of an abelian variety A over K and the fundamental group of the Néron model of the dual abelian variety. This pairing extends to the p-primary components a pairing introduced by Shafarevich. We relate this pairing with Grothendieck’s pairing.