The Tate pairing for Abelian varieties over finite fields
Peter Bruin1
1 Université Paris-Sud 11 Département de Mathématiques Bâtiment 425 91405 Orsay cedex France
Journal de théorie des nombres de Bordeaux, Volume 23 (2011) no. 2, pp. 323-328.

In this expository note, we describe an arithmetic pairing associated to an isogeny between Abelian varieties over a finite field. We show that it generalises the Frey–Rück pairing, thereby giving a short proof of the perfectness of the latter.

Nous décrivons un accouplement arithmétique associé à une isogenie entre variétés abéliennes sur un corps fini. Nous montrons qu’il généralise l’accouplement de Frey et Rück, donnant ainsi une démonstration brève de la perfection de ce dernier.

DOI: 10.5802/jtnb.764

Peter Bruin 1

1 Université Paris-Sud 11 Département de Mathématiques Bâtiment 425 91405 Orsay cedex France 
@article{JTNB_2011__23_2_323_0,
     author = {Peter Bruin},
     title = {The {Tate} pairing for {Abelian} varieties over finite fields},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {323--328},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {23},
     number = {2},
     year = {2011},
     doi = {10.5802/jtnb.764},
     mrnumber = {2817932},
     zbl = {1246.11123},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.764/}
}
TY  - JOUR
AU  - Peter Bruin
TI  - The Tate pairing for Abelian varieties over finite fields
JO  - Journal de théorie des nombres de Bordeaux
PY  - 2011
SP  - 323
EP  - 328
VL  - 23
IS  - 2
PB  - Société Arithmétique de Bordeaux
UR  - https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.764/
DO  - 10.5802/jtnb.764
LA  - en
ID  - JTNB_2011__23_2_323_0
ER  - 
%0 Journal Article
%A Peter Bruin
%T The Tate pairing for Abelian varieties over finite fields
%J Journal de théorie des nombres de Bordeaux
%D 2011
%P 323-328
%V 23
%N 2
%I Société Arithmétique de Bordeaux
%U https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.764/
%R 10.5802/jtnb.764
%G en
%F JTNB_2011__23_2_323_0
Peter Bruin. The Tate pairing for Abelian varieties over finite fields. Journal de théorie des nombres de Bordeaux, Volume 23 (2011) no. 2, pp. 323-328. doi : 10.5802/jtnb.764. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.764/

[1] G. Frey and H.-G. Rück, A remark concerning m-divisibility and the discrete logarithm in class groups of curves. Mathematics of Computation 62 (1994), 865–874. | MR | Zbl

[2] F. Heß, A note on the Tate pairing of curves over finite fields. Archiv der Mathematik 82 (2004), no. 1, 28-32. | MR | Zbl

[3] E. W. Howe, The Weil pairing and the Hilbert symbol. Mathematische Annalen 305 (1996), 387–392. | EuDML | MR | Zbl

[4] S. Lang, Abelian varieties over finite fields. Proceedings of the National Academy of Sciences of the U.S.A. 41 (1955), no. 3, 174–176. | MR | Zbl

[5] S. Lang, Abelian Varieties. Interscience, New York, 1959. | MR | Zbl

[6] S. Lichtenbaum, Duality theorems for curves over p-adic fields. Inventiones Mathematicae 7 (1969), 120–136. | EuDML | MR | Zbl

[7] D. Mumford, Abelian Varieties. Tata Institute of Fundamental Research, Bombay, 1970. | MR | Zbl

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

[9] J-P. Serre, Corps locaux. Hermann, Paris, 1962. | MR | Zbl

[10] J. Tate, WC-groups over 𝔭-adic fields. Séminaire Bourbaki, exposé 156. Secretariat mathématique, Paris, 1957. | Numdam | MR | Zbl

Cited by Sources: