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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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/

[1] P. Bruin, “The Tate pairing for abelian varieties over finite fields”, Journal de theorie des nombres de Bordeaux, 23:2 (2011), 323–328 | DOI | MR | Zbl

[2] M. Fried, M. Jarden, Field arithmetic, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics, 11, Second edition, Springer-Verlag, Berlin, 2005 | MR

[3] F. Hess, “A Note on the Tate Pairing of Curves over Finite Fields”, Archiv der Mathematik, 82:1 (2004), 28–32 | DOI | MR | Zbl

[4] J. S. Milne, Abelian Varieties, 2008 http://www.jmilne.org/math/

[5] V. Nesteruk, “On nondegeneracy of Tate product for curves over pseudofinite fields”, Visnyk of the Lviv Univ. Series Mechanics and Mathematics, 2010, no. 72, 195–200 (in Ukrainian) | Zbl

[6] V. Platonov, A. Rapinchuk, Algebraic groups and number theory, 1991 (in Russian)

[7] E. F. Schaefer, “A new proof for the non-degeneracy of the Frey-Rück pairing and a connection to isogenies over the base field”, Computational Aspects of Algebraic Curves (University of Idaho), Lecture Notes Series in Computing, 13, ed. T. Shaska, World Scientific Publishing, Hackensack, NJ, 2005, 1–12 | MR | Zbl

[8] J. H. Silverman, The arithmetic of elliptic curves, Graduate Texts in Mathematics, 106, Springer-Verlag, New York, 1986 | DOI | MR | Zbl