Israel Journal of Mathematics

, Volume 218, Issue 1, pp 273–297 | Cite as

Frobenius nonclassicality of Fermat curves with respect to cubics

Article
First Online:
Received:
Revised:

Abstract

For Fermat curves F: aX n + bY n = Z n defined over F q , we establish necessary and sufficient conditions for F to be F q -Frobenius nonclassical with respect to the linear system of plane cubics. In the new F q -Frobenius nonclassical cases, we determine explicit formulas for the number N q (F) of F q -rational points on F. For the remaining Fermat curves, nice upper bounds for N q (F) are immediately given by the Stöhr–Voloch Theory.

This is a preview of subscription content, log in to check access.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    N. Arakelian and H. Borges, Frobenius nonclassicality with respect to linear systems of curves of arbitrary degree, Acta Arith. 167, (2015) 43–66.MathSciNetCrossRefMATHGoogle Scholar
  2. [2]
    H. Borges, On complete (N,d)-arcs derived from plane curves, Finite Fields Appl. 15, (2009) 82–96.MathSciNetCrossRefMATHGoogle Scholar
  3. [3]
    T. Cochrane and C. Pinner, Explicit bounds on monomial and binomial exponential sums, Q. J. Math. 62, (2011) 323–349.MathSciNetCrossRefMATHGoogle Scholar
  4. [4]
    A. García and J. F. Voloch, Wronskians and linear independence in fields of prime characteristic, Manuscripta Math. 59, (1987) 457–469.MathSciNetCrossRefMATHGoogle Scholar
  5. [5]
    A. García and J. F. Voloch, Fermat curves over finite fields, J. Number Theory 30, (1988) 345–356.MathSciNetCrossRefMATHGoogle Scholar
  6. [6]
    M. Giulietti, F. Pambianco, F. Torres and E. Ughi, On complete arcs arising from plane curves, Des. Codes Cryptogr. 25, (2002) 237–246.MathSciNetCrossRefMATHGoogle Scholar
  7. [7]
    J. W. P. Hirschfeld, G. Korchmáros and F. Torres, Algebraic curves over a finite field, Princeton Series in Applied Mathematics, Princeton University Press, Princeton, NJ, 2008.Google Scholar
  8. [8]
    G. Korchmáros and T. Szőnyi, Fermat curves over finite fields and cyclic subsets in high-dimensional projective spaces, Finite Fields Appl. 5, (1999) 206–217.MathSciNetCrossRefMATHGoogle Scholar
  9. [9]
    H. W. Lenstra, On a problem of Garcia, Stichtenoth, and Thomas, Finite Fields Appl. 8, (2002) 166–170.MathSciNetCrossRefMATHGoogle Scholar
  10. [10]
    S. Mattarei, On a bound of Garcia and Voloch for the number of points of a Fermat curve over a prime field, Finite Fields Appl. 13, (2007) 773–777.MathSciNetCrossRefMATHGoogle Scholar
  11. [11]
    M. Namba, Geometry of projective algebraic curves, Monographs and Textbooks in Pure and Applied Mathematics, Vol. 88, Marcel Dekker, Inc., New York, 1984.Google Scholar
  12. [12]
    R. Pardini, Some remarks on plane curves over fields of finite characteristic, Compositio Math. 60, (1986) 3–17.MathSciNetMATHGoogle Scholar
  13. [13]
    I. E. Shparlinskiĭ, Estimates for Gauss sums, Mat. Zametki 50, (1991) 122–130.MathSciNetMATHGoogle Scholar
  14. [14]
    K.-O. Stöhr and J. F. Voloch, Weierstrass points and curves over finite fields, Proc. London Math. Soc. 52 (3), (1986) 1–19.MathSciNetCrossRefMATHGoogle Scholar
  15. [15]
    J. F. Voloch, A note on (k,n)-arcs, Discrete Math. 294, (2005) 223–224.MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Hebrew University of Jerusalem 2017

Authors and Affiliations

  1. 1.Centro de Matemàtica, Computaçao e CogniçaoUniversidade Federal do ABCS˜ao Carlos SPBrazil
  2. 2.Instituto de Ciências Matemàticas e de ComputaçaoUniversidade de S˜ao PauloS˜ao Carlos SPBrazil

Personalised recommendations