Geodesic
On the Tate pairing associated to an isogeny between abelian varieties over pseudofinite field
Algebra and discrete mathematics, Tome 16 (2013) no. 1, pp. 103-106

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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].
Keywords: pseudofinite field, isogeny, Tate pairing associated to an isogeny. 
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     author = {V. Nesteruk},
     title = {On the {Tate} pairing associated to an isogeny between abelian varieties over pseudofinite field},
     journal = {Algebra and discrete mathematics},
     pages = {103--106},
     publisher = {mathdoc},
     volume = {16},
     number = {1},
     year = {2013},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ADM_2013_16_1_a10/}
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V. Nesteruk. On the Tate pairing associated to an isogeny between abelian varieties over pseudofinite field. Algebra and discrete mathematics, Tome 16 (2013) no. 1, pp. 103-106. http://geodesic.mathdoc.fr/item/ADM_2013_16_1_a10/