Abstract
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.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Bettale, L., Faugère, J.-C., Perret, L.: Hybrid approach for solving multivariate systems over finite fields. J. Math. Crypt. 3, 177–197 (2009)
Bosma, W., Cannon, J., Playoust, C.: The Magma algebra system. I. The user language. J. Symb. Comput. 24, 235–265 (1997)
Deuring, M.: Die Typen der Multiplikatorenringe elliptischer Funktionenkörper. Abh. Math. Sem. Univ. Hamburg 14(1), 197–272 (1941)
Ekedahl, T.: On supersingular curves and abelian varieties. Math. Scand. 60, 151–178 (1987)
Fuhrmann, R., Garcia, A., Torres, F.: On maximal curves. J. Number Theory 67, 29–51 (1997)
van der Geer, G., et al.: Tables of Curves with Many Points (2009). http://www.manypoints.org. Accessed 5 Apr 2018
González, J.: Hasse-Witt matrices for the Fermat curves of prime degree. Tohoku Math. J. 49(2), 149–163 (1997). MR 1447179 (98b:11064)
Hartshorne, R.: Algebraic Geometry, GTM 52. Springer, Heidelberg (1977). https://doi.org/10.1007/978-1-4757-3849-0
Hashimoto, K.: Class numbers of positive definite ternary quaternion Hermitian forms. Proc. Japan Acad. Ser. A Math. Sci. 59(10), 490–493 (1983)
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)
Hurt, N.E.: Many Rational Points: Coding Theory and Algebraic Geometry. Kluwer Academic Publishers, Dordrecht (2003)
Ibukiyama, T.: On rational points of curves of genus \(3\) over finite fields. Tohoku Math. J. 45, 311–329 (1993)
Ibukiyama, T., Katsura, T.: On the field of definition of superspecial polarized abelian varieties and type numbers. Compositio Math. 91(1), 37–46 (1994)
Kudo, M., Harashita, S.: Superspecial curves of genus \(4\) in small characteristic. Finite Fields Their Appl. 45, 131–169 (2017)
Kudo, M., Harashita, S.: Enumerating superspecial hyperelliptic curves of genus \(4\) over small finite fields, in preparation
Kudo, M. and Harashita, S.: Enumerating superspecial curves of genus \(4\) over prime fields, arXiv: 1702.05313 [math.AG] (2017)
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). http://wcc2017.suai.ru/proceedings.html
Li, K.-Z., Oort, F.: Moduli of Supersingular Abelian Varieties. Lecture Notes in Mathematics, vol. 1680. Springer, Berlin (1998). https://doi.org/10.1007/BFb0095931
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))
Nygaard, N.O.: Slopes of powers of Frobenius on crystalline cohomology. Ann. Sci. École Norm. Sup. 14(4), 369–401 (1982, 1981)
Özbudak, F., Saygı, Z.: Explicit maximal and minimal curves over finite fields of odd characteristics. Finite Fields Their Appl. 42, 81–92 (2016)
Serre, J.-P.: Nombre des points des courbes algebrique sur \(\mathbb{F}_{q}\). Théor. Nombres Bordeaux 83(2), 22 (1983, 1982)
Tafazolian, S.: A note on certain maximal hyperelliptic curves. Finite Fields Their Appl. 18, 1013–1016 (2012)
Tafazolian, S., Torres, F.: On the curve \(y^n=x^m +x\) over finite fields. J. Number Theory 145, 51–66 (2014)
Xue, J., Yang, T.-C., Yu, C.-F.: On superspecial abelian surfaces over finite fields. Doc. Math. 21, 1607–1643 (2016)
Yui, N.: On the Jacobian varieties of hyperelliptic curves over fields of characterisctic \(p>2\). J. Algebr. 52, 378–410 (1978)
Data base of superspecial curves of genus \(4\) over finite fields and their algebraic closures. http://www2.math.kyushu-u.ac.jp/~m-kudo/Ssp-curves-genus-4.html
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2018 Springer Nature Switzerland AG
About this paper
Cite this paper
Kudo, M., Harashita, S. (2018). Superspecial Hyperelliptic Curves of Genus 4 over Small Finite Fields. In: Budaghyan, L., Rodríguez-Henríquez, F. (eds) Arithmetic of Finite Fields. WAIFI 2018. Lecture Notes in Computer Science(), vol 11321. Springer, Cham. https://doi.org/10.1007/978-3-030-05153-2_3
Download citation
DOI: https://doi.org/10.1007/978-3-030-05153-2_3
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-05152-5
Online ISBN: 978-3-030-05153-2
eBook Packages: Computer ScienceComputer Science (R0)