Designs, Codes and Cryptography

, Volume 79, Issue 3, pp 423–441 | Cite as

Further results on rational points of the curve \(\displaystyle y^{q^n}-y=\gamma x^{q^h+1} - \alpha \) over \({\mathbb {F}}_{q^m}\)

  • Ayhan Coşgun
  • Ferruh Özbudak
  • Zülfükar Saygı


Let q be a positive power of a prime number. For arbitrary positive integers hnm with n dividing m and arbitrary \(\gamma ,\alpha \in {\mathbb {F}}_{q^m}\) with \(\gamma \ne 0\) the number of \({\mathbb {F}}_{q^m}\)-rational points of the curve \(y^{q^n}-y=\gamma x^{q^h+1} - \alpha \) is determined in many cases (Özbudak and Saygı, in: Larcher et al. (eds.) Applied algebra and number theory, 2014) with odd q. In this paper we complete some of the remaining cases for odd q and we also present analogous results for even q.


Artin–Schreier type curve Rational points Algebraic curves Finite fields 

Mathematics Subject Classification

11G20 14G05 14G50 



We would like to thank the anonymous referees for their insightful and helpful comments that improved the presentation of this paper.


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Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  • Ayhan Coşgun
    • 1
  • Ferruh Özbudak
    • 2
  • Zülfükar Saygı
    • 3
  1. 1.Department of MathematicsMiddle East Technical UniversityAnkaraTurkey
  2. 2.Department of Mathematics and Institute of Applied MathematicsMiddle East Technical UniversityAnkaraTurkey
  3. 3.Department of MathematicsTOBB University of Economics and TechnologyAnkaraTurkey

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