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Isogenies of abelian varieties over finite fields

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Abstract

In this paper we give conditions under which two abelian varieties that are defined over a finite field \(F\), and are isogenous over some larger field, are \(F\)-isogenous. Further, we give conditions under which a given isogeny is defined over \(F\).

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Notes

  1. We have just learned about similar (but different) counterexamples with \(F={\mathbb {R}}\) and \({\mathbb {Q}}\) in a recent book of Chai, Conrad and Oort [3], Example 1.2.6 on pp. 22–23 and Example 1.6.4 on pp. 74–75].

  2. After this paper appeared on the arXiv and was submitted, we learned that this proposition was proved in [3], Prop. 1.2.6.1 on p. 23]. The proof there is completely different from ours.

References

  1. Chow W.L.: Abelian varieties over function fields. Trans. Am. Math. Soc. 78, 253–275 (1955).

  2. Cohen H., Frey G. et al. (eds.): Handbook of Elliptic and Hyperelliptic Curve Cryptography. Chapman & Hall/CRC, Boca Raton (2006).

  3. Chai C.-L., Conrad B., Oort F.: Complex multiplication and lifting problems. Mathematical Surveys and Monographs 195. American Mathematical Society, Providence (2014).

  4. Curtis C.W., Reiner I.: Methods of Representation Theory—with Applications to Finite Groups and Orders, vol. I. Wiley, New York (1981).

  5. Galbraith S.D., Pujolas J., Ritzenthaler C., Smith B.A.: Distortion maps for genus two curves. J. Math. Cryptol. 3, 1–18 (2009).

  6. Hoffstein J., Pipher J., Silverman J.: An Introduction to Mathematical Cryptography. Springer, New York (2008).

  7. Lang S.: Abelian Varieties, 2nd edn. Springer, New York (1983).

  8. Mumford D.: Abelian Varieties, Tata Institute of Fundamental Research Studies in Mathematics No. 5. Oxford University Press, London (1970).

  9. Oort F.: Abelian varieties over finite fields. In: Higher-dimensional geometry over finite fields, pp. 123–188, NATO Sci. Peace Secur. Ser. D Inf. Commun. Secur., 16, IOS, Amsterdam (2008). http://www.staff.science.uu.nl/~oort0109/A-AVffGoe07-2-oort.ps.

  10. Serre J-P.: Rigidité du foncteur de Jacobi d’echelon \(n \ge 3\), Appendix to A. Grothendieck, Techniques de construction en géométrie analytique, X. Construction de l’espace de Teichmüller, Séminaire Henri Cartan, 1960/61, no. 17. http://numdam.org/numdam-bin/fitem?id=SHC_1960-1961__13_2_A4_0.

  11. Shimura G., Taniyama Y.: Complex multiplication of abelian varieties and its application to number theory. Publ. Math. Soc. Jpn, vol. 6, Tokyo (1961).

  12. Silverberg A.: Fields of definition for homomorphisms of abelian varieties. J. Pure Appl. Algebra 77, 253–262 (1992).

  13. Silverberg A., Zarhin Y.G.: Isogenies of abelian varieties. J. Pure Appl. Algebra 90, 23–37 (1993).

  14. Silverberg A., Zarhin Y.G.: Variations on a theme of Minkowski and Serre. J. Pure Appl. Algebra 111, 285–302 (1996).

  15. Silverberg A., Zarhin Y.G: Semistable reduction and torsion subgroups of abelian varieties. Ann. Inst. Fourier 45, 403–420 (1995).

  16. Silverman J.H.: The arithmetic of elliptic curves. Grad. Texts Math, vol. 106, Springer, New York (1986).

  17. Tate J.: Endomorphisms of abelian varieties over finite fields. Invent. math. 2, 134–144 (1966).

  18. Verheul E. R.: Evidence that XTR is more secure than supersingular elliptic curve cryptosystems. In: Pfitzmann B. (ed.) Advances in Cryptology—EUROCRYPT 2001. Lecture Notes in Computer Science, vol. 2045, pp. 195–210. Springer, Berlin (2001).

  19. Weil A.: Variétés Abéliennes et Courbes Algébriques. Hermann, Paris (1948).

  20. Weil A.: Foundations of Algebraic Geometry, 2nd edn. American Mathematical Society, Providence (1962).

  21. Weil A.: Adeles and Algebraic Groups, Progress in Mathematics, vol. 23. Birkhäuser, Boston, Basel, Stuttgart (1982).

  22. Zarhin Y.G.: Homomorphisms of abelian varieties. In: Aubry Y., Lachaud G. (eds.) Arithmetic, Geometry and Coding Theory (AGCT 2003), Sémin. Congr. vol. 11, pp. 189–215. Soc. Math. France (2005).

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Acknowledgments

Silverberg was partially supported by the National Science Foundation. Zarhin was partially supported by the Simons Foundation (Grant #246625 to Yuri G. Zarhin). The authors thank the referees for helpful comments.

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Authors and Affiliations

  1. Department of Mathematics, University of California, Irvine, CA, 92697-3875, USA

    Alice Silverberg

  2. Department of Mathematics, Pennsylvania State University, University Park, PA, 16802, USA

    Yuri G. Zarhin

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  1. Alice Silverberg
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  2. Yuri G. Zarhin
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Correspondence to Alice Silverberg.

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This paper is dedicated to the memory of Scott Vanstone.

This is one of several papers published in Designs, Codes and Cryptography comprising the “Special Issue on Cryptography, Codes, Designs and Finite Fields: In Memory of Scott A. Vanstone”.

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Silverberg, A., Zarhin, Y.G. Isogenies of abelian varieties over finite fields. Des. Codes Cryptogr. 77, 427–439 (2015). https://doi.org/10.1007/s10623-015-0078-2

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