On the Tate pairing associated to an isogeny between abelian varieties over pseudofinite field

Abstract

In this note, we consider the Tate pairing associated to an isogeny between abelian varieties over pseudofinite field. P. Bruin [1] defined this pairing over finite field \(k\): \(\mathrm{ker}\,\hat{\phi}(k) \; \times \; \mathrm{coker}\,(\phi(k)) \longrightarrow k^*\), and proved its perfectness over finite field. We prove perfectness of the Tate pairing associated to an isogeny between abelian varieties over pseudofinite field, with help of the method, used by P. Bruin in the case of finite ground field [1].