The Tate pairing for Abelian varieties over finite fields

Peter Bruin[1]

  • [1] Université Paris-Sud 11 Département de Mathématiques Bâtiment 425 91405 Orsay cedex France

Journal de Théorie des Nombres de Bordeaux (2011)

  • Volume: 23, Issue: 2, page 323-328
  • ISSN: 1246-7405

Abstract

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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.

How to cite

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Bruin, Peter. "The Tate pairing for Abelian varieties over finite fields." Journal de Théorie des Nombres de Bordeaux 23.2 (2011): 323-328. <http://eudml.org/doc/219774>.

@article{Bruin2011,
abstract = {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.},
affiliation = {Université Paris-Sud 11 Département de Mathématiques Bâtiment 425 91405 Orsay cedex France},
author = {Bruin, Peter},
journal = {Journal de Théorie des Nombres de Bordeaux},
keywords = {Tate pairing; abelian varieties},
language = {eng},
month = {6},
number = {2},
pages = {323-328},
publisher = {Société Arithmétique de Bordeaux},
title = {The Tate pairing for Abelian varieties over finite fields},
url = {http://eudml.org/doc/219774},
volume = {23},
year = {2011},
}

TY - JOUR
AU - Bruin, Peter
TI - The Tate pairing for Abelian varieties over finite fields
JO - Journal de Théorie des Nombres de Bordeaux
DA - 2011/6//
PB - Société Arithmétique de Bordeaux
VL - 23
IS - 2
SP - 323
EP - 328
AB - 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.
LA - eng
KW - Tate pairing; abelian varieties
UR - http://eudml.org/doc/219774
ER -

References

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  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. Zbl0813.14045MR1218343
  2. F. Heß, A note on the Tate pairing of curves over finite fields. Archiv der Mathematik 82 (2004), no. 1, 28-32. Zbl1051.11030MR2034467
  3. E. W. Howe, The Weil pairing and the Hilbert symbol. Mathematische Annalen 305 (1996), 387–392. Zbl0854.11031MR1391223
  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. Zbl0064.03902MR71115
  5. S. Lang, Abelian Varieties. Interscience, New York, 1959. Zbl0516.14031MR106225
  6. S. Lichtenbaum, Duality theorems for curves over p -adic fields. Inventiones Mathematicae 7 (1969), 120–136. Zbl0186.26402MR242831
  7. D. Mumford, Abelian Varieties. Tata Institute of Fundamental Research, Bombay, 1970. Zbl0583.14015MR282985
  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. Zbl1154.14320MR2181869
  9. J-P. Serre, Corps locaux. Hermann, Paris, 1962. Zbl0137.02601MR354618
  10. J. Tate, WC-groups over 𝔭 -adic fields. Séminaire Bourbaki, exposé 156. Secretariat mathématique, Paris, 1957. Zbl0091.33701MR105420

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