The Ramanujan Journal

, Volume 30, Issue 2, pp 223–242 | Cite as

The distribution of the number of points modulo an integer on elliptic curves over finite fields

Article

Abstract

Let \(\mathbb{F}_{q}\) be a finite field, and let b and N be integers. We prove explicit estimates for the probability that the number of rational points on a randomly chosen elliptic curve E over \(\mathbb{F}_{q}\) equals b modulo N. The underlying tool is an equidistribution result on the action of Frobenius on the N-torsion subgroup of E. Our results subsume and extend previous work by Achter and Gekeler.

Keywords

Elliptic curves Finite fields Frobenius statistics Modular curves 

Mathematics Subject Classification

14H52 14K10 

Notes

Acknowledgements

The authors are very grateful to the anonymous referee of a prior submission of this document, to the anonymous referee of the current submission, to Hendrik W. Lenstra for suggesting the use of Chebotarev’s density theorem, and to Barry Mazur and Bjorn Poonen for their helpful comments on modular curves. Both authors thank F.W.O.-Vlaanderen for its financial support. The first author thanks the Massachusetts Institute of Technology for its hospitality.

References

  1. 1.
    Achter, J.: The distribution of class groups of function fields. J. Pure Appl. Algebra 204(2), 316–333 (2006) MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Achter, J.: Results of Cohen–Lenstra type for quadratic function fields. In: Lauter, K., Ribet, K. (eds.) Computational Arithmetic Geometry. Contemporary Mathematics, vol. 463, pp. 1–8. American Mathematical Society, Providence (2008) CrossRefGoogle Scholar
  3. 3.
    Achter, J., Sadornil, D.: On the probability of having rational -isogenies. Arch. Math. 90, 511–519 (2008) MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Birch, B.: How the number of points of an elliptic curve over a fixed prime field varies. J. Lond. Math. Soc. 43, 57–60 (1968) MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Carayol, H.: La conjecture de Sato–Tate. Séminaire Bourbaki 977, 59ème année (2006–2007) Google Scholar
  6. 6.
    Castryck, W., Folsom, A., Hubrechts, H., Sutherland, A.V.: The probability that the number of points on the Jacobian of a genus 2 curve is prime. Proc. Lond. Math. Soc. 104(6), 1235–1270 (2012) MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Chavdarov, N.: The generic irreducibility of the numerator of the zeta function in a family of curves with large monodromy. Duke Math. J. 87(1), 151–180 (1997) MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Crandall, R., Pomerance, C.: Prime Numbers: A Computational Perspective, 2nd edn. Springer, Berlin (2005) MATHGoogle Scholar
  9. 9.
    Deligne, P.: La conjecture de Weil: II. Publ. Math. IHES 52, 137–252 (1980) MathSciNetMATHGoogle Scholar
  10. 10.
    Deligne, P., Rapoport, M.: Les schémas de modules de courbes elliptiques. In: Modular Functions of One Variable, II (Proc. Int. Summer School Antwerp). Lecture Notes in Math., vol. 349, pp. 143–174. Springer, Berlin (1973) CrossRefGoogle Scholar
  11. 11.
    Fried, M., Jarden, M.: Field Arithmetic, 3rd edn. Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge, vol. 11. Springer, Berlin (1986) MATHGoogle Scholar
  12. 12.
    Galbraith, S., McKee, J.: The probability that the number of points on an elliptic curve over a finite field is prime. J. Lond. Math. Soc. 62(3), 671–684 (2000) MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Gekeler, E.-U.: Frobenius distributions of elliptic curves over finite prime fields. Int. Math. Res. Not. 37, 1999–2018 (2003) MathSciNetCrossRefGoogle Scholar
  14. 14.
    Gekeler, E.-U.: The distribution of group structures on elliptic curves over finite prime fields. Doc. Math. 11, 119–142 (2006) MathSciNetMATHGoogle Scholar
  15. 15.
    Gekeler, E.-U.: Statistics about elliptic curves over finite prime fields. Manuscr. Math. 127, 55–67 (2008) MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Howe, E.: On the group orders of elliptic curves over finite fields. Compos. Math. 85, 229–247 (1993) MathSciNetMATHGoogle Scholar
  17. 17.
    Katz, N., Mazur, B.: Arithmetic Moduli of Elliptic Curves. Princeton University Press, Princeton (1985) MATHGoogle Scholar
  18. 18.
    Katz, N., Sarnak, P.: Random Matrices, Frobenius Eigenvalues, and Monodromy. Colloquium Publications, vol. 45. Am. Math. Soc., Providence (1998) Google Scholar
  19. 19.
    Kedlaya, K., Sutherland, A.V.: Hyperelliptic curves, L-polynomials, and random matrices. In: Lachaud, G., Ritzenthaler, C., Tsfasman, M. (eds.) Proceedings of AGCT-11. Contemporary Mathematics, vol. 487, pp. 119–162. American Mathematical Society, Providence (2009) Google Scholar
  20. 20.
    Lang, S., Trotter, H.: Frobenius Distributions in GL2-Extensions. Springer, Berlin (1976) Google Scholar
  21. 21.
    Lenstra, H.W.: Factoring integers with elliptic curves. Ann. Math. 126(2), 649–673 (1987) MathSciNetMATHCrossRefGoogle Scholar
  22. 22.
    Rosser, J.B., Schoenfeld, L.: Approximate formulas for some functions of prime numbers. Ill. J. Math. 6(1), 64–94 (1962) MathSciNetMATHGoogle Scholar
  23. 23.
    Silverman, J.: The Arithmetic of Elliptic Curves. Graduate Texts in Mathematics, vol. 106. Springer, Berlin (1985) Google Scholar
  24. 24.
    Vlǎduţ, S.: Cyclicity statistics for elliptic curves over finite fields. Finite Fields Appl. 5, 13–25 (1999) MathSciNetCrossRefGoogle Scholar
  25. 25.
    Yoshida, H.: On an analogue of the Sato conjecture. Invent. Math. 19, 261–277 (1973) MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2012

Authors and Affiliations

  1. 1.Departement WiskundeKatholieke Universiteit LeuvenLeuven (Heverlee)Belgium

Personalised recommendations