Skip to main content

Bounds for exponential sums modulo p 2

Abstract

In this paper we consider exponential sums over subgroups G ⊂ ℤ *q . Using Stepanov’s method, we obtain nontrivial bounds for exponential sums in the case where q is a square of a prime number.

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

References

  1. 1.

    J. Bourgain, Exponential Sum Estimates over Subgroups of *q , q Arbitrary, preprint.

  2. 2.

    J. Bourgain and S. Konyagin, “Estimates for the number of sums and products and for exponential sums over subgroups in fields of prime order,” C. R. Math. Acad. Sci. Paris, 337, 75–80 (2003).

    MATH  MathSciNet  Google Scholar 

  3. 3.

    D. R. Heath-Brown, “An estimate for Heilbronn’s exponential sum,” in: Analytic Number Theory: Proc. Conf. in Honor of Heini Halberstam, Birkhäuser, Boston (1996), pp. 451–463.

    Google Scholar 

  4. 4.

    D. R. Heath-Brown and S. V. Konyagin, “New bounds for Gauss sums derived from k th powers, and for Heilbronn’s exponential sums,” Quart J. Math., 51, 221–235 (2000).

    MATH  Article  MathSciNet  Google Scholar 

  5. 5.

    A. A. Karatsuba, “Fractional parts of functions of a special form,” Izv. Ross. Akad. Nauk, Ser. Mat., 59, No. 4, 93–102 (1995).

    Google Scholar 

  6. 6.

    A. A. Karatsuba, “Double Kloosterman sums,” Mat. Zametki, 66, No. 5, 682–687 (1999).

    MathSciNet  Google Scholar 

  7. 7.

    S. V. Konyagin, “Estimates for trigonometric sums over subgroups and for Gaussian sums,” in: IV Int. Conf. “Modern Problems in Number Theory and Its Applications,” Topical Problems, Part III [in Russian], Moscow (2002), pp. 86–114.

  8. 8.

    S. Konyagin and I. Shparlinski, Character Sums with Exponential Functions, Cambridge University Press, Cambridge (1999).

    MATH  Google Scholar 

  9. 9.

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

    MATH  Google Scholar 

  10. 10.

    I. E. Shparlinski, “Estimates for Gauss sums,” Math. Notes, 50, 140–146 (1991).

    MathSciNet  Google Scholar 

  11. 11.

    S. A. Stepanov, “The number of points of a hyperelliptic curve over a prime field,” Izv. Akad. Nauk SSSR, Ser. Mat., 33, 1171–1181 (1969).

    MATH  MathSciNet  Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Yu. V. Malykhin.

Additional information

__________

Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 11, No. 6, pp. 81–94, 2005.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Malykhin, Y.V. Bounds for exponential sums modulo p 2 . J Math Sci 146, 5686–5696 (2007). https://doi.org/10.1007/s10958-007-0385-x

Download citation

Keywords

  • Prime Number
  • Hyperelliptic Curve
  • Singular Case
  • Regular Case
  • Distinct Pair