Superspecial Hyperelliptic Curves of Genus 4 over Small Finite Fields

  • Momonari KudoEmail author
  • Shushi Harashita
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11321)


In this paper, we enumerate superspecial hyperelliptic curves of genus 4 over finite fields \(\mathbb {F}_q\) for small q. This complements our preceding results in the non-hyperelliptic case. We give a feasible algorithm to enumerate superspecial hyperelliptic curves of genus g over \(\mathbb {F}_q\) in the case that q and \(2g+2\) are coprime and \(q>2g+1\). We executed the algorithm for \((g,q)= (4,11^2)\), \((4,13^2)\), \((4,17^2)\) and (4, 19) with our implementation on a computer algebra system Magma. Moreover, we found many maximal hyperelliptic curves and some minimal hyperelliptic curves over \(\mathbb {F}_{q}\) from among enumerated superspecial curves.


Hyperelliptic curves Superspecial curves Maximal curves 


  1. 1.
    Bettale, L., Faugère, J.-C., Perret, L.: Hybrid approach for solving multivariate systems over finite fields. J. Math. Crypt. 3, 177–197 (2009)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Bosma, W., Cannon, J., Playoust, C.: The Magma algebra system. I. The user language. J. Symb. Comput. 24, 235–265 (1997)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Deuring, M.: Die Typen der Multiplikatorenringe elliptischer Funktionenkörper. Abh. Math. Sem. Univ. Hamburg 14(1), 197–272 (1941)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Ekedahl, T.: On supersingular curves and abelian varieties. Math. Scand. 60, 151–178 (1987)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Fuhrmann, R., Garcia, A., Torres, F.: On maximal curves. J. Number Theory 67, 29–51 (1997)MathSciNetCrossRefGoogle Scholar
  6. 6.
    van der Geer, G., et al.: Tables of Curves with Many Points (2009). Accessed 5 Apr 2018
  7. 7.
    González, J.: Hasse-Witt matrices for the Fermat curves of prime degree. Tohoku Math. J. 49(2), 149–163 (1997). MR 1447179 (98b:11064)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Hartshorne, R.: Algebraic Geometry, GTM 52. Springer, Heidelberg (1977). Scholar
  9. 9.
    Hashimoto, K.: Class numbers of positive definite ternary quaternion Hermitian forms. Proc. Japan Acad. Ser. A Math. Sci. 59(10), 490–493 (1983)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Hashimoto, K., Ibukiyama, T.: On class numbers of positive definite binary quaternion Hermitian forms II. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 28(3), 695–699 (1982)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Hurt, N.E.: Many Rational Points: Coding Theory and Algebraic Geometry. Kluwer Academic Publishers, Dordrecht (2003)CrossRefGoogle Scholar
  12. 12.
    Ibukiyama, T.: On rational points of curves of genus \(3\) over finite fields. Tohoku Math. J. 45, 311–329 (1993)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Ibukiyama, T., Katsura, T.: On the field of definition of superspecial polarized abelian varieties and type numbers. Compositio Math. 91(1), 37–46 (1994)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Kudo, M., Harashita, S.: Superspecial curves of genus \(4\) in small characteristic. Finite Fields Their Appl. 45, 131–169 (2017)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Kudo, M., Harashita, S.: Enumerating superspecial hyperelliptic curves of genus \(4\) over small finite fields, in preparationGoogle Scholar
  16. 16.
    Kudo, M. and Harashita, S.: Enumerating superspecial curves of genus \(4\) over prime fields, arXiv: 1702.05313 [math.AG] (2017)
  17. 17.
    Kudo, M., Harashita, S.: Enumerating superspecial curves of genus \(4\) over prime fields (abstract version of [16]). In: Proceedings of The Tenth International Workshop on Coding and Cryptography 2017 (WCC 2017), 18–22 September 2017, Saint-Petersburg, Russia (2017).
  18. 18.
    Li, K.-Z., Oort, F.: Moduli of Supersingular Abelian Varieties. Lecture Notes in Mathematics, vol. 1680. Springer, Berlin (1998). Scholar
  19. 19.
    Manin, Y. I.: On the theory of Abelian varieties over a field of finite characteristic, AMS Transl. Ser. 2 50, 127–140 (1966). Translated by G. Wagner (originally published in Izv. Akad. Nauk SSSR Ser. Mat. 26, 281–292 (1962))Google Scholar
  20. 20.
    Nygaard, N.O.: Slopes of powers of Frobenius on crystalline cohomology. Ann. Sci. École Norm. Sup. 14(4), 369–401 (1982, 1981)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Özbudak, F., Saygı, Z.: Explicit maximal and minimal curves over finite fields of odd characteristics. Finite Fields Their Appl. 42, 81–92 (2016)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Serre, J.-P.: Nombre des points des courbes algebrique sur \(\mathbb{F}_{q}\). Théor. Nombres Bordeaux 83(2), 22 (1983, 1982)Google Scholar
  23. 23.
    Tafazolian, S.: A note on certain maximal hyperelliptic curves. Finite Fields Their Appl. 18, 1013–1016 (2012)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Tafazolian, S., Torres, F.: On the curve \(y^n=x^m +x\) over finite fields. J. Number Theory 145, 51–66 (2014)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Xue, J., Yang, T.-C., Yu, C.-F.: On superspecial abelian surfaces over finite fields. Doc. Math. 21, 1607–1643 (2016)MathSciNetzbMATHGoogle Scholar
  26. 26.
    Yui, N.: On the Jacobian varieties of hyperelliptic curves over fields of characterisctic \(p>2\). J. Algebr. 52, 378–410 (1978)CrossRefGoogle Scholar
  27. 27.
    Data base of superspecial curves of genus \(4\) over finite fields and their algebraic closures.

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Authors and Affiliations

  1. 1.Kobe City College of TechnologyNishi-ku, KobeJapan
  2. 2.Graduate School of Environment and Information SciencesYokohama National UniversityHodogaya-ku, YokohamaJapan

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