Skip to main content

On the representation of the number of integral points of an elliptic curve modulo a prime number

Abstract

In this paper we shall investigate the problem of the representation of the number of integral points of an elliptic curve modulo a prime number p. We present a way of expressing an exponential sum which involves polynomials of third degree, in explicit non-exponential terms. In the process, we prove explicit formulas for the calculation of certain series involving the Riemann zeta function.

This is a preview of subscription content, access via your institution.

Notes

  1. By f′(x) we denote the derivative df(x)/dx.

  2. By r′ we denote the derivative dr(x,p)/dx.

References

  1. Apostol, T.: Introduction to Analytic Number Theory. Springer, New York (1984)

    Google Scholar 

  2. Atkin, A.O.L., Moralin, F.: Elliptic curves and primality proving. Math. Comput. 61, 29–68 (1993)

    Article  MATH  Google Scholar 

  3. Elkies, N.D.: Elliptic and modular curves over finite fields and related computational issues. In: Buell, D.A., Teitelbaum, J.T. (eds.) Computational Perspectives on Number Theory. Proc. Conf. in honor of A.O.L. Atkin, pp. 21–76. AMS, Providence (1998)

    Google Scholar 

  4. Goldwasser, S., Kilian, J.: Almost all primes can be quickly certified. In: Proc. 18th STOC, Berkeley, May 28–30, 1986, pp. 316–329. ACM, New York (1986)

    Google Scholar 

  5. Hasse, H.: Math. Z. 31, 565–582 (1930)

    Article  MATH  MathSciNet  Google Scholar 

  6. Havil, J.: Gamma, Exploring Euler’s Constant. Princeton University Press, Princeton (2003)

    MATH  Google Scholar 

  7. Ireland, K., Rosen, M.: A Classical Introduction to Modern Number Theory, 2nd edn. GTM, vol. 84. Springer, New York (1990)

    Book  MATH  Google Scholar 

  8. Ivić, A.: The Riemann Zeta-Function: The Theory of the Riemann Zeta-Function with Applications. Wiley, New York (1985)

    MATH  Google Scholar 

  9. Iwaniec, H., Kowalski, E.: Analytic Number Theory. AMS Colloq. Publ., vol. 53. AMS, Providence (2004)

    MATH  Google Scholar 

  10. Koblitz, N.: Elliptic curve cryptosystems. Math. Comput. 48, 203–209 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  11. Koblitz, N.: A Course in Number Theory and Cryptography. Springer, New York (1994)

    Book  MATH  Google Scholar 

  12. Korobov, N.: Exponential Sums and Their Applications. Kluwer Academic, Dordrecht (1992)

    Book  MATH  Google Scholar 

  13. Kowalski, E.: Exponential sums over finite fields. I. Elementary methods. http://www.math.ethz.ch/~kowalski/exp-sums.pdf

  14. Lenstra, H.W.: Factoring integers with elliptic curves. Ann. Math. 126(3), 649–673 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  15. Miller, V.: Uses of elliptic curves in cryptography. In: Advances in Cryptology, Proc. of Crypto’85. Lecture Notes in Computer Science, vol. 218, pp. 417–426. Springer, New York (1986)

    Chapter  Google Scholar 

  16. Miller, S.J., Takloo-Bighash, R.: An Invitation to Modern Number Theory. Princeton University Press, Princeton (2006)

    MATH  Google Scholar 

  17. Montgomery, P.L.: Speeding the Pollard and elliptic curve methods for factorizations. Math. Comput. 48, 243–264 (1987)

    Article  MATH  Google Scholar 

  18. Rassias, M.Th.: Problem-Solving and Selected Topics in Number Theory: In the Spirit of the Mathematical Olympiads. Springer, New York (2011)

    Book  Google Scholar 

  19. Schoof, R.: Elliptic curves over finite fields and the computation of square roots mod p. Math. Comput. 44(170), 483–494 (1985)

    MATH  MathSciNet  Google Scholar 

  20. Silverman, J.: The Arithmetic of Elliptic Curves. Graduate Texts in Mathematics, vol. 106. Springer, New York (1986)

    MATH  Google Scholar 

  21. Srivastava, H.M., Choi, J.: Series Associated with the Zeta and Related Functions. Kluwer Academic, Dordrecht (2001)

    Book  MATH  Google Scholar 

  22. Washington, L.C.: Elliptic Curves—Number Theory and Cryptography. CRC Press, Boca Raton (2008)

    MATH  Google Scholar 

Download references

Acknowledgements

I would like to thank my Ph.D. advisor Professor E. Kowalski who suggested to me this area of research and for his very useful advice.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Michael T. Rassias.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Rassias, M.T. On the representation of the number of integral points of an elliptic curve modulo a prime number. Ramanujan J 36, 483–499 (2015). https://doi.org/10.1007/s11139-013-9524-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11139-013-9524-9

Keywords

  • Elliptic curves
  • Integral points
  • Exponential sums
  • Riemann zeta function

Mathematics Subject Classification

  • 11L03
  • 11L07
  • 14H52
  • 11M06
  • 11Yxx
  • 11T71