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Maximal number of rational points that a (smooth, geometrically irreducible) curve of genus 3 over the finite field GF(q) can have, where q is the n-th prime power >= 2.
(Formerly M4338)
1

%I M4338 #19 May 02 2024 04:31:48

%S 7,10,14,16,20,24,28,28,32,38,40,44,48,56,56,60,62,64,72,78,80,87,92,

%T 96,102,107,113,116,120,122,131,136,136,144,155

%N Maximal number of rational points that a (smooth, geometrically irreducible) curve of genus 3 over the finite field GF(q) can have, where q is the n-th prime power >= 2.

%D R. Auer and J. Top, Some genus 3 curves with many points, pp. 163-171 of ANTS 2002, Lect. Notes Computer Sci. 2369 (2002).

%D J. W. P. Hirschfeld, Linear codes and algebraic codes, pp. 35-53 of F. C. Holroyd and R. J. Wilson, editors, Geometrical Combinatorics. Pitman, Boston, 1984. See N_q(3) on page 51.

%D J.-P. Serre, Sur le nombre des points rationnels d'une courbe algébrique sur un corps fini, Compt. Rend. Acad. Sci. Paris, 296 (1983), 397-402; Oeuvres, vol. 3, pp. 658-663.

%D J.-P. Serre, Nombres de points des courbes algébriques sur F_q, Semin. Theorie Nombres Bordeaux, 1982/83, No. 22; Oeuvres, vol. 3, pp. 664-669.

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%D W. C. Waterhouse, Abelian varieties over finite fields. Ann. Sci. Ecole Norm. Sup. (4) 2 1969, 521-560.

%H K. Lauter and J.-P. Serre, <a href="https://arxiv.org/abs/math/0104086">The maximum or minimum number of rational points on curves of genus three over finite fields</a>, arXiv:math/0104086 [math.AG], 2001; Compos. Math.134 (p. 87-111) 2002.

%H Jean-Pierre Serre, <a href="https://gallica.bnf.fr/ark:/12148/bpt6k55351747/f35.item">Sur le nombre des points rationnels d'une courbe algébrique sur un corps fini</a>, C. R. Acad. Sci. Paris Ser. I Math. 296 (1983), no. 9, 397-402.

%H Jaap Top, <a href="https://arxiv.org/abs/math/0301264">Curves of genus 3 over small finite fields</a>, arXiv:math/0301264 [math.NT], 2003.

%e For q=23 the value is 48: this maximum is attained by the following curve (due to Serre): x^4+y^4+z^4-5(x^2y^2 +y^2z^2 + z^2x^2)=0, over the field with 23 elements.

%Y Cf. A080205.

%K nonn,more

%O 1,1

%A _N. J. A. Sloane_.

%E More terms from A. E. Brouwer, Sep 15 1997.

%E Edited by _Dean Hickerson_, Feb 05 2003 and Feb 23 2003, adding more terms from the paper by Jaap Top.