Advertisement

L-Polynomials of the Curve \(\displaystyle y^{q^n}-y=\gamma x^{q^h+1} - \alpha \) over \({\mathbb F}_{q^m}\)

  • Ferruh ÖzbudakEmail author
  • Zülfükar Saygı
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9061)

Abstract

Let \(\chi \) be a smooth, geometrically irreducible and projective curve over a finite field \({\mathbb F}_q\) of odd characteristic. The L-polynomial \(L_\chi (t)\) of \(\chi \) determines the number of rational points of \(\chi \) not only over \({\mathbb F}_q\) but also over \({\mathbb F}_{q^s}\) for any integer \(s \ge 1\). In this paper we determine L-polynomials of a class of such curves over \({\mathbb F}_q\).

Keywords

Algebraic curves L-polynomials Rational points 

Notes

Acknowledgment

The first author was partially supported by TÜBİTAK under Grant No. TBAG-112T011.

References

  1. 1.
    Coşgun, A., Özbudak, F., Saygı, Z.: Rational points of some algebraic curves over finite fields, in preparationGoogle Scholar
  2. 2.
    Huppert, B., Blackburn, N.: Finite Groups II. Springer-Verlag, Berlin, Heidelberg, New York (1982)CrossRefzbMATHGoogle Scholar
  3. 3.
    Kedlaya, K.S.: Counting points on hyperelliptic curves using Monsky-Washnitzer cohomology. J. Ramanujan Math. Soc. 16, 323–338 (2001)zbMATHMathSciNetGoogle Scholar
  4. 4.
    Lauder, A.G.B., Wan, D.: Computing zeta functions of Artin-Schreier curves over finite fields. Lond. Math. Soc. JCM 5, 34–55 (2002)zbMATHMathSciNetGoogle Scholar
  5. 5.
    Manin, Y.I.: The theory of commutative formal groups over fields of finite characteristic. Uspekhi Mat. Nauk 18:6(114), 3–90 (1963)MathSciNetGoogle Scholar
  6. 6.
    Niederreiter, H., Xing, C.: Rational Points on Curves over Finite Fields: Theory and Applications. Cambridge University Press, Cambridge (2001)CrossRefGoogle Scholar
  7. 7.
    Niederreiter, H., Xing, C.: Algebraic Geometry in Coding Theory and Cryptography. Princeton University Press, Princeton (2009)zbMATHGoogle Scholar
  8. 8.
    Özbudak, F., Saygı, Z.: Rational points of the curve \(y^{q^n}-y=\gamma x^{q^h+1} - \alpha \) over \(_{q^m}\). In: Larcher, G., Pillichshammer, F., Winterhof, A., Xing, C.P. (eds.) Applied Algebra and Number Theory. Cambridge Univesity Press, Cambridge (2014)Google Scholar
  9. 9.
    Rück, H.-G.: A note on elliptic curves over finite fields. Math. Comp. 49, 301–304 (1987)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Rück, H.-G., Stichtenoth, H.: A characterization of Hermitian function fields over finite fields. J. Reine Angew. Math. 457, 185–188 (1994)zbMATHMathSciNetGoogle Scholar
  11. 11.
    Stichtenoth, H.: Die Hasse-Witt-Invariante eines Kongruenzfunktionenkorpers. Arch. Math. 33, 357–360 (1979)CrossRefMathSciNetGoogle Scholar
  12. 12.
    Stichtenoth, H.: Algebraic Function Fields and Codes. Springer-Verlag, Berlin (2009)zbMATHGoogle Scholar
  13. 13.
    Tate, J.: Endomorphisms of abelian varieties over finite fields. Invent. Math. 2(2), 134–144 (1966)CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Tsfasman, M.A., Vladut, S.G., Nogin, D.: Algebraic Geometric Codes: Basic Notions. American Mathematical Society, Providence (2007)CrossRefGoogle Scholar
  15. 15.
    Voloch, J.F.: A note on elliptic curves over finite fields. Bull. Soc. Math. France 116, 455–458 (1989)MathSciNetGoogle Scholar
  16. 16.
    Waterhouse, W.: Abelian varieties over finite fields. Ann. Sci. Ecole Norm. 2(4), 521–560 (1969)zbMATHMathSciNetGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Department of MathematicsMiddle East Technical UniversityAnkaraTurkey
  2. 2.Institute of Applied MathematicsMiddle East Technical UniversityAnkaraTurkey
  3. 3.Department of MathematicsTOBB University of Economics and TechnologyAnkaraTurkey

Personalised recommendations