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Almost Prime Orders of CM Elliptic Curves Modulo p

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Algorithmic Number Theory (ANTS 2008)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 5011))

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Abstract

Given an elliptic curve over \(\mathbb Q\) with complex multiplication by \({\mathcal O}_K\), the ring of integers of the quadratic imaginary field K, we analyze the integer d E  =gcd\(\{|E(\mathbb F_p)|:p\) splits in \({\mathcal O}_K\}\), where \(|E(\mathbb F_p)|\) is the size of the group of rational \(\mathbb F_p\) points, and prove that it can be bigger than the common factor that comes from the torsion of the curve. Then, we prove that #{p ≤ x, p splits in \({{\mathcal O}_K}:{\frac {1}{d_E}}|{\it E}({\mathbb F}_{p})|=P_2\}\gg {\it x}/(\log x)^2\) hence extending the results in [16]. This is the best known result in the direction of the Koblitz conjecture about the primality of \(|E(\mathbb F_p)|\).

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References

  1. Atkin, A.O.L., Morain, F.: Elliptic curves and primality proving. Math. Comp. 61(203), 29–68 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  2. Balog, A., Cojocaru, A., David, C.: Average twin prime conjecture for elliptic curves, http://arXiv.org/abs/0709.1461

  3. Borosh, I., Moreno, C.J., Porta, H.: Elliptic curves over finite fields. II, Math. Comput. 29, 951–964 (1975)

    Article  MATH  MathSciNet  Google Scholar 

  4. Cai, Y.C.: On Chen’s theorem (II). J. Number Theory (2007) (Available online)

    Google Scholar 

  5. Chen, J.R.: On the representation of a larger even integer as the sum of a prime and the product of at most two primes. Sci. Sinica 16, 157–176 (1973)

    MATH  MathSciNet  Google Scholar 

  6. Cojocaru, A.C.: Questions about the reductions modulo primes of an elliptic curve. In: Number Theory, CRM Proc. Lecture Notes, vol. 36, pp. 61–79. Amer. Math. Soc., Providence, RI (2004)

    Google Scholar 

  7. Cojocaru, A.C.: Reductions of an elliptic curve with almost prime orders. Acta Arith. 119, 265–289 (2005)

    MATH  MathSciNet  Google Scholar 

  8. Cojocaru, A.C.: Cyclicity of Elliptic Curves Modulo p. PhD thesis, Queen’s University (2002)

    Google Scholar 

  9. Greaves, G.: Sieves in Number Theory. Springer, Berlin (2001)

    MATH  Google Scholar 

  10. Friedlander, J., Iwaniec, H.: The Sieve (preprint)

    Google Scholar 

  11. Gupta, R., Murty, M.R.: Primitive points on elliptic curves. Compositio Math. 58, 13–44 (1986)

    MATH  MathSciNet  Google Scholar 

  12. Gupta, R., Murty, M.R.: Cyclicity and generation of points modulo p on elliptic curves. Invent. Math. 101, 225–235 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  13. Ireland, K., Rosen, M.: A Classical Introduction to Modern Number Theory. In: GTM, vol. 84, Springer, Heidelberg (1982)

    Google Scholar 

  14. Iwaniec, H.: Sieve Methods (notes for a graduate course in Rutgers University) (1996)

    Google Scholar 

  15. Iwaniec, H., Kowalski, E.: Analytic Number Theory, vol. 53. Colloquim Publications, AMS (2004)

    MATH  Google Scholar 

  16. Iwaniec, H., Jiménez Urroz, J.: Orders of CM elliptic curves modulo p with at most two primes, https://upcommons.upc.edu/e-prints/bitstream/2117/1169/1/p2incmellip906.pdf

  17. Johnson, D.: Mean values of Hecke L-functions. J. reine angew. Math. 305, 195–205 (1979)

    MATH  MathSciNet  Google Scholar 

  18. Katz, N.: Galois properties of torsion points on abelian varieties. Invent. Math. 62, 481–502 (1981)

    Article  MathSciNet  Google Scholar 

  19. Koblitz, N.: Primality of the number of points on an elliptic curve over a finite field. Pacifc J. Math. 131, 157–165 (1988)

    MATH  MathSciNet  Google Scholar 

  20. Lang, S., Trotter, H.: Frobenius distributions in GL2-extensions. Lecture Notes in Math., vol. 504. Springer, Berlin, New York (1976)

    Google Scholar 

  21. Miri, S.A., Murty, V.K.: An application of sieve methods to elliptic curves. In: Pandu Rangan, C., Ding, C. (eds.) INDOCRYPT 2001. LNCS, vol. 2247, pp. 91–98. Springer, Heidelberg (2001)

    Chapter  Google Scholar 

  22. Murty, M.R.: Artin’s Conjecture and Non-Abelian Sieves. PhD Thesis, MIT (1980)

    Google Scholar 

  23. Rubin, K., Silverberg, A.: Point counting on reductions of CM elliptic curves, http://arxiv.org/abs/0706.3711v1

  24. Serre, J.-P.: Résumé des cours de 1977–1978, Ann. Collège France. Collège de France, Paris, p. 6770ff (1978)

    Google Scholar 

  25. Stark, H.M.: Counting points on CM elliptic curves. Rocky Mountain J. Math. 26(3), 1115–1138 (1996)

    MATH  MathSciNet  Google Scholar 

  26. Steuding, J., Weng, A.: On the number of prime divisors of the order of elliptic curves modulo p. Acta Arith. 117(4), 341–352 (2005)

    MATH  MathSciNet  Google Scholar 

  27. Steuding, J., Weng, A.: Erratum: On the number of prime divisors of the order of elliptic curves modulo p. Acta Arith. 119(4), 407–408 (2005)

    MathSciNet  Google Scholar 

  28. Wu, J.: Chen’s double sieve, Goldbach’s conjecture and the twin prime problem. Acta Arith. 114, 215–273 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  29. Wu, J.: Chen’s double sieve. Goldbach’s conjecture and the twin prime problem 2, http://arXiv.org/abs/0709.3764

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Authors and Affiliations

  1. CRM, Universite de Montreal,  

    Jorge Jiménez Urroz

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  1. Jorge Jiménez Urroz
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Alfred J. van der Poorten Andreas Stein

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Jiménez Urroz, J. (2008). Almost Prime Orders of CM Elliptic Curves Modulo p . In: van der Poorten, A.J., Stein, A. (eds) Algorithmic Number Theory. ANTS 2008. Lecture Notes in Computer Science, vol 5011. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-79456-1_4

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  • DOI: https://doi.org/10.1007/978-3-540-79456-1_4

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-79455-4

  • Online ISBN: 978-3-540-79456-1

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