Ternary and Quaternary Curves of Small Fixed Genus and Gonality With Many Rational Points

References

• Bosma, W., Cannon, J., Playoust, C. (1997). The magma algebra system. I. The user language. J. Symbol. Comput. 24(3/4): 235–265. doi:10.1006/jsco.1996.0125
   Web of Science ®Google Scholar
• Castryck, W., Tuitman, J. (2016). Point counting on curves using a gonality preserving lift. arXiv: 1605.02162v2 [math.NT].
   Google Scholar
• Castryck. W., Tuitman, J. (2018). Point counting on curves using a gonality preserving lift. Q. J. Math. 69(1): 33–74. doi:10.1093/qmath/hax031
   Web of Science ®Google Scholar
• Dickson, L.E. (1958). Linear Groups: With An Exposition of the Galois Field Theory. New York: Dover Publications. With an introduction by W. Magnus
   Google Scholar
• Faber, X., Grantham, J. (2021). Binary curves of small fixed genus and gonality with many rational points. arXiv: 2005.07054v2 [math.NT]. To appear in J. Algebra.
   Google Scholar
• Faber, X., Grantham, J. (2021). Extremely pointless curves. In preparation.
   Google Scholar
• Hart,W., Johansson, F., Pancratz, S. (2020). FLINT: Fast Library for Number Theory. Version 2.6.0. Available at: http://flintlib.org.
   Google Scholar
• Howe, E. W., Lauter, K. E. (2003). Improved upper bounds for the number of points on curves over finite fields. Ann. Inst. Fourier (Grenoble). 53(6):1677–1737. doi:10.5802/aif.1990
   Web of Science ®Google Scholar
• Howe, E. W., Lauter, K. E. (2012). New methods for bounding the number of points on curves over finite fields. In C. Faber, G. Farkas, and R. de Jong (Editors), Geometry and Arithmetic, EMS Ser. Congr. Rep. Zürich: Eur. Math. Soc., pp. 173–212.
   Google Scholar
• Howe, E.W., Lauter, K.E., Top, J. (2005). Pointless curves of genus three and four. In Y. Aubry and G. Lachaud (Editors), Arithmetic, Geometry and Coding Theory (AGCT 2003). Vol. 11 of Sémin. Congr. Paris: Soc. Math. France, pp. 125–141.
   Google Scholar
• Ihara, Y. (1981/1982). Some remarks on the number of rational points of algebraic curves over finite fields. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 28(3): 721–724.
   Google Scholar
• Lauter, K. (2000). Non-existence of a curve over F3 of genus 5 with 14 rational points. Proc. Amer. Math. Soc. 128(2): 369–374.
   Web of Science ®Google Scholar
• Lauter, K. (2000). Zeta functions of curves over finite fields with many rational points. In Y. Aubry and G. Lachaud (Editors), Coding Theory, Cryptography and Related Areas (Guanajuato, 1998). Berlin: Springer, pp. 167–174.
   Google Scholar
• Lauter, K. (2001). Geometric methods for improving the upper bounds on the number of rational points on algebraic curves over finite fields. J. Algebraic Geom. 10(1): 19–36. With an appendix in French by J.-P. Serre.
   Web of Science ®Google Scholar
• Liu, Q. (2002). Algebraic Geometry and Arithmetic Curves. Vol. 6 of Oxford Graduate Texts in Mathematics. Oxford: Oxford University Press. Translated from the French by Reinie Erné, Oxford Science Publications.
   Google Scholar
• Niederreiter, H., Xing, C. (1997). Cyclotomic function fields, Hilbert class fields, and global function fields with many rational places. Acta Arith. 79(1): 59–76. doi:10.4064/aa-79-1-59-76
   Web of Science ®Google Scholar
• Niederreiter, H., Xing, C. (2001). Rational Points on Curves Over Finite Fields: Theory and Applications. Vol. 285 of London Mathematical Society Lecture Note Series, Cambridge: Cambridge University Press.
   Google Scholar
• Rigato, A. (2010). Uniqueness of low genus optimal curves over F2. In Arithmetic, Geometry, Cryptography and Coding Theory 2009. Vol. 521 of Contemp. Math. Providence, RI: Amer. Math. Soc., pp. 87–105.
   Google Scholar
• Ritzenthaler, C. (2003). Existence d’une courbe de genre 5 sur F3 avec 13 points rationels. arXiv: math/0302147 [math.NT].
   Google Scholar
• Savitt, D. (2003). The maximum number of points on a curve of genus 4 over F8 is 25. Canad. J. Math. 55(2): 331–352. With an appendix by Kristin Lauter.
   Web of Science ®Google Scholar
• Serre, J. P. (1983). Sur le nombre des points rationnels d’une courbe algébrique sur un corps fini. C. R. Acad. Sci. Paris Sér. I Math. 296(9): 397–402.
   Google Scholar
• Serre, J. P. (1985). Rational points on curves over finite fields. Course notes by F. Q. Gouvêa.
   Google Scholar
• Serre, J. P. (2020). Rational Points on Curves Over Finite Fields. Vol. 18 of Documents Mathématiques. Paris: Société Mathématique de France. With contributions by Everett Howe, Joseph Oesterlé, and Christophe Ritzenthaler.
   Google Scholar
• The Sage Developers. (2019). SageMath, the Sage Mathematics Software System (Version 8.7). Available at: http://www.sagemath.org.
   Google Scholar
• van der Geer, G., van der Vlugt, M. (2000). Tables of curves with many points. Math. Comp. 69(230): 797–810.
   Web of Science ®Google Scholar
• Voight, J. (2005). Curves over finite fields with many points: an introduction. In Y. Aubry and G. Lachaud (Editors), Computational Aspects of Algebraic Curves. Vol.13 of Lecture Notes Ser. Comput., Hackensack, NJ: World Sci. Publishers, pp. 124–144.
   Google Scholar
• Xarles, X. (2020). A census of all genus 4 curves over the field with 2 elements. arXiv: 2007.07822v1 [math.AG].
   Google Scholar