On the Tate-Shafarevich pairing in abelian varieties over

We study the Tate-Shafarevich pairing in abelian varieties over complete discretely valued fields with pseudofinite residue field. It turns out that in this case the Tate-Shafarevich pairing induces dualities between the fixed n-torsion subgroups of the Weil-Chatelet group of abelian variety and the quotient group of the group of rational point by its image under multiplication by n. The similar result holds for higher dimensional complete discretely valued fields with pseudofinite residue field. Some applications of this result will be discussed.