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 _K 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 in 16]. We relate this pairing with Grothendieck's pairing. Received: 7 January 2002 / Revised version: 6 December 2002 Published online: 24 April 2003 Mathematics Subject Classification (2000): 14k05, 14F20, 14G22