Howe, Everett W. The maximum number of points on a curve of genus eight over the field of four elements. (English) Zbl 1464.11061 J. Number Theory 220, 320-329 (2021). Let \(q\) be a power of a prime number, \(g\) be a non-negative integer, and denote by \(N_{q}(g)\) the maximum number of rational points on a curve of genus \(g\) over the finite field \(\mathbb{F}_{q}\). Studying \(N_{q}(g)\) from both its assimptotic behavior and computation of its actual value is a noteworthy, and often challenging, problem.For \(q=4\) and \(g=8\), the Oesterlé bound shows that \[ N_{4}(8)\leqslant 24, \] while H. Niederreiter and C. Xing proves in [Acta Arith. 79, No. 1, 59–76 (1997; Zbl 0891.11057)] that there exists a curve of genus \(8\) over the finite field \(\mathbb{F}_4\) with \(21\) rational points.In this paper, the author improves the previous upper and lower bounds by showing that \[ 22\leqslant N_{4}(8)\leqslant 23, \] where to prove the lower bound, a genus-\(8\) curve over \(\mathbb{F}_{2}\) with \(22\) points over \(\mathbb{F}_{4}\) is constructed as a degree-\(3\) Kummer extension of a genus-\(2\) curve. Reviewer: Mariana Coutinho (São Carlos) MSC: 11G20 Curves over finite and local fields 14G05 Rational points 14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture) 14G15 Finite ground fields in algebraic geometry Keywords:curve; Jacobian; Weil polynomial; points Citations:Zbl 0891.11057 Software:manYPoints PDFBibTeX XMLCite \textit{E. W. Howe}, J. Number Theory 220, 320--329 (2021; Zbl 1464.11061) Full Text: DOI arXiv References: [1] van der Geer, G.; Howe, E. W.; Lauter, K. E.; Ritzenthaler, C., — Table of curves with many points, retrieved 1 April 2020 (2009) [2] Howe, E. W.; Lauter, K. E., Improved upper bounds for the number of points on curves over finite fields, Ann. Inst. Fourier (Grenoble), 53, 6, 1677-1737 (2003), correction in [3] · Zbl 1065.11043 [3] Howe, E. W.; Lauter, K. E., Corrigendum to: Improved upper bounds for the number of points on curves over finite fields, Ann. Inst. Fourier (Grenoble), 57, 3, 1019-1021 (2007) · Zbl 1248.11043 [4] Howe, E. W.; Lauter, K. E., New methods for bounding the number of points on curves over finite fields, (Faber, C.; Farkas, G.; de Jong, R., Geometry and Arithmetic. Geometry and Arithmetic, EMS Ser. Congr. Rep. (2012), Eur. Math. Soc.: Eur. Math. Soc. Zürich), 173-212 · Zbl 1317.11065 [5] Ihara, Y., Some remarks on the number of rational points of algebraic curves over finite fields, J. Fac. Sci., Univ. Tokyo, Sect. 1A, Math., 28, 3, 721-724 (1981), (1982) · Zbl 0509.14019 [6] Lauter, K. E., Zeta functions of curves over finite fields with many rational points, (Buchmann, J.; Høholdt, T.; Stichtenoth, H.; Tapia-Recillas, H., Coding Theory, Cryptography and Related Areas. Coding Theory, Cryptography and Related Areas, Guanajuato, 1998 (2000), Springer: Springer Berlin), 167-174 · Zbl 1009.11049 [7] Manin, Y. I., What is the maximum number of points on a curve over \(\mathbb{F}_2\)?, J. Fac. Sci., Univ. Tokyo, Sect. 1A, Math., 28, 3, 715-720 (1981), (1982) · Zbl 0527.14021 [8] Niederreiter, H.; Xing, C., Cyclotomic function fields, Hilbert class fields, and global function fields with many rational places, Acta Arith., 79, 1, 59-76 (1997) · Zbl 0891.11057 [9] Rigato, A., Uniqueness of low genus optimal curves over \(\mathbb{F}_2\), (Kohel, D.; Rolland, R., Arithmetic, Geometry, Cryptography and Coding Theory 2009. Arithmetic, Geometry, Cryptography and Coding Theory 2009, Contemp. Math., vol. 521 (2010), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI), 87-105 · Zbl 1219.11097 [10] Serre, J.-P., Rational points on curves over finite fields (1985), Harvard University, unpublished notes by F.Q. Gouvêa of lectures at [11] Serre, J.-P., Rational Points on Curves over Finite Fields (2020), Soc. Math. France: Soc. Math. France Paris, in press, edited by A. Bassa, E. Lorenzo García, C. Ritzenthaler, and R. Schoof, with contributions by Everett Howe, Joseph Oesterlé, and Christophe Ritzenthaler · Zbl 1475.11002 [12] Tate, J., Endomorphisms of Abelian varieties over finite fields, Invent. Math., 2, 134-144 (1966) · Zbl 0147.20303 [13] Vlâdut, S. G.; Drinfel’d, V. G., The number of points of an algebraic curve, Funktsional. Anal. i Prilozhen., 17, 1, 68-69 (1983), English translation in [14] · Zbl 0522.14011 [14] Vlâdut, S. G.; Drinfel’d, V. G., The number of points of an algebraic curve, Funct. Anal. Appl., 17, 53-54 (1983), English translation of [13] · Zbl 0522.14011 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.