Abstract
The number N of rational points on an algebraic curve of genus g over a finite field \({\mathbb{F}}_q \) satisfies the Hasse–Weil bound \(N \leqslant q + 1 + 1g\sqrt q \). A curve that attains this bound is called maximal. With \(g_0 = \frac{1}{2}(q - \sqrt q )\) and \(g_1 = \frac{1}{4}(\sqrt q - 1)^2 \), it is known that maximalcurves have \(g = g_0 or g \leqslant {\text{ }}g_1 \). Maximal curves with \(g = g_0 or g_1 \) have been characterized up to isomorphism. A natural genus to be studied is \(g_2 = \frac{1}{8}(\sqrt q - 1)(\sqrt q - 3),\) and for this genus there are two non-isomorphic maximal curves known when \(\sqrt q \equiv 3 (\bmod 4)\). Here, a maximal curve with genus g 2 and a non-singular plane model is characterized as a Fermat curve of degree \(\frac{1}{2}(\sqrt q + 1)\).
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References
Abdon, M. and Torres, F.: On maximal curves in characteristic two, Manuscripta Math. 99, (1998), 39-53.
Fuhrmann, R., Garcia, A. and Torres, F.: On maximal curves, J. Number Theory 67 (1997), 29-51.
Fuhrmann, R. and Torres, F.: The genus of curves over finite fields with many rational points, Manuscripta Math. 89 (1996), 103-106.
Fuhrmann, R. and Torres, F.: On Weierstrass points and optimal curves, Rend. Circ. Mat. Palermo Suppl. 51 (1998), 25-46.
Garcia, A. and Homma, M.: Frobenius order-sequences of curves, In: G. Frey and J. Ritter (eds), Algebra and Number Theory, de Gruyter, Berlin, 1994, pp. 27-41.
Garcia, A. and Viana, P.: Weierstrass points on certain non-classical curves, Arch. Math. 46 (1986), 315-322.
Garcia, A. and Voloch, J. F.: Wronskians and linear independence in fields of prime characteristic, Manuscripta Math. 59 (1987), 457-469.
Hirschfeld, J. W. P.: ProjectiveGeometriesOver Finite Fields, 2nd edn, Oxford Univ. Press, Oxford, 1998.
Hirschfeld, J. W. P. and Korchmáros, G.: Embedding an arc into a conic in a finite plane, Finite Fields Appl. 2 (1996), 274-292.
Hirschfeld, J. W. P. and Korchmáros, G.: On the number of points on an algebraic curve over a finite field, Bull. Belg. Math. Soc. Simon Stevin 5 (1998), 313-340.
Homma, M.: A souped-up version of Pardini's theorem and its applications to funny curves, Compositio Math. 71 (1989), 295-302.
Ihara, Y.: Some remarks on the number of rational points of algebraic curves over finite fields, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 28 (1981), 721-724.
Lachaud, G.: Sommes d'Eisenstein et nombre de points de certaines courbes algébriques sur les corps finis, C.R. Acad. Sci. Paris Sér. I Math. 305 (1987), 729-732.
Orzech, G. and Orzech, O.: Plane Algebraic Curves, M. Dekker, New York, 1981.
Pardini, R.: Some remarks on plane curves over fields of finite characteristic, Compositio Math. 60 (1986), 3-17.
Rück, H. G. and Stichtenoth, H.: A characterization of Hermitian function fields over finite fields, J. Reine Angew. Math. 457 (1994), 185-188.
Seidenberg, A.: Elements of the Theory of Algebraic Curves, Addison-Wesley, Reading, Mass, 1969.
Stichtenoth, H.: Algebraic Function Fields and Codes, Springer-Verlag, Berlin, 1993.
Stichtenoth, H. and Xing, C. P.: The genus of maximal function fields, Manuscripta Math. 86 (1995), 217-224.
Stöhr, K. O. and Voloch, J. F.: Weierstrass points and curves over finite fields, Proc. London Math. Soc. 52 (1986), 1-19.
Walker, R. J.: Algebraic Curves, Princeton University Press, Princeton, 1950 (Dover, New York, 1962).
Weil, A., Courbes Algébriques et Variétés Abeliennes, Hermann, Paris, 1948.
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Cossidente, A., Hirschfeld, J.W.P., Korchmáros, G. et al. On Plane Maximal Curves. Compositio Mathematica 121, 163–181 (2000). https://doi.org/10.1023/A:1001826520682
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DOI: https://doi.org/10.1023/A:1001826520682