The Fricke-Macbeath curve is a smooth projective algebraic curve of genus with automorphism group . We recall two models of it (introduced, respectively, by Maxim Hendriks and by Bradley Brock) defined over , and we establish an explicit isomorphism defined over between these models. Moreover, we decompose up to isogeny over the jacobian of one of these models. As a consequence we obtain a simple formula for the number of points over on (the reduction of) this model, in terms of the elliptic curve with equation . Moreover, twists by elements of of the curve over finite fields are described. The curve leads to a number of new records as maintained on manYPoints of curves of genus with many rational points over finite fields.
La courbe de Fricke-Macbeath est une courbe projective lisse de genre avec groupe d’automorphismes . Nous rappelons deux modèles de cette courbe (introduits respectivement par Maxim Hendriks et Bradley Brock) définis sur , et nous établissons un isomorphisme explicite, défini sur , entre ces deux modèles. De plus, nous décomposons à isogénie sur près la jacobienne de l’un des modèles. Comme une conséquence nous obtenons une formule simple pour le nombre de points sur de (la réduction de) ce modèle, en termes de la courbe elliptique d’équation . Enfin, des tordus de cette courbe par des éléments de sur des corps finis sont décrits. La courbe donne un certain nombre de nouveaux records maintenus par manYPoints de courbes de genre avec beaucoup de points rationnels sur des corps finis.
Revised:
Accepted:
Published online:
DOI: 10.5802/jtnb.1019
Mots-clés : Hurwitz curve, automorphism group, jacobian, point counting
Jaap Top 1; Carlo Verschoor 1
CC-BY-ND 4.0
@article{JTNB_2018__30_1_117_0,
author = {Jaap Top and Carlo Verschoor},
title = {Counting points on the {Fricke{\textendash}Macbeath} curve over finite fields},
journal = {Journal de th\'eorie des nombres de Bordeaux},
pages = {117--129},
publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
volume = {30},
number = {1},
year = {2018},
doi = {10.5802/jtnb.1019},
zbl = {1408.11093},
mrnumber = {3809712},
language = {en},
url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1019/}
}
TY - JOUR AU - Jaap Top AU - Carlo Verschoor TI - Counting points on the Fricke–Macbeath curve over finite fields JO - Journal de théorie des nombres de Bordeaux PY - 2018 SP - 117 EP - 129 VL - 30 IS - 1 PB - Société Arithmétique de Bordeaux UR - https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1019/ DO - 10.5802/jtnb.1019 LA - en ID - JTNB_2018__30_1_117_0 ER -
%0 Journal Article %A Jaap Top %A Carlo Verschoor %T Counting points on the Fricke–Macbeath curve over finite fields %J Journal de théorie des nombres de Bordeaux %D 2018 %P 117-129 %V 30 %N 1 %I Société Arithmétique de Bordeaux %U https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1019/ %R 10.5802/jtnb.1019 %G en %F JTNB_2018__30_1_117_0
Jaap Top; Carlo Verschoor. Counting points on the Fricke–Macbeath curve over finite fields. Journal de théorie des nombres de Bordeaux, Volume 30 (2018) no. 1, pp. 117-129. doi : 10.5802/jtnb.1019. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1019/
[1] The Magma algebra system. I. The user language, J. Symb. Comput., Volume 24 (1997) no. 3-4, pp. 235-265 | DOI | MR | Zbl
[2] A canonical curve of genus 7, Proc. Lond. Math. Soc., Volume 17 (1967), pp. 207-225 | DOI | MR | Zbl
[3] Shimura curve computations, Algorithmic number theory. 3rd international symposium (Lect. Notes Comput. Sci.), Volume 1423 (1998), pp. 1-47 | DOI | MR | Zbl
[4] Ueber eine einfache Gruppe von Operationen, Math. Ann., Volume 52 (1899), pp. 321-339 | DOI | MR | Zbl
[5] Tables of Curves with Many Points, 2009 (http://manypoints.org)
[6] Fields of definition of uniform dessins on quasiplatonic surfaces, Riemann and Klein surfaces, automorphisms, symmetries and moduli spaces (Contemporary Mathematics), Volume 629, American Mathematical Society, 2015, pp. 155-170 | Zbl
[7] Platonic Maps of Low Genus, Eindhoven University of Technology, The Netherlands (2013) (Ph. D. Thesis)
[8] Edmonds maps on Fricke-Macbeath curve, Ars Math. Contemp., Volume 8 (2015) no. 2, pp. 275-289 | DOI | MR | Zbl
[9] Curves of medium genus with many points (2016) (https://arxiv.org/abs/1609.01838v2)
[10] Ueber die Transformation siebenter Ordnung der elliptischen Functionen, Clebsch Ann., Volume XIV (1879), pp. 428-471 | Zbl
[11] On a curve of genus 7, Proc. Lond. Math. Soc., Volume 15 (1965), pp. 527-542 | DOI | MR | Zbl
[12] Twists of genus three curves over finite fields, Finite Fields Appl., Volume 16 (2010) no. 5, pp. 347-368 | DOI | MR | Zbl
[13] 2-extensions with many points (2000) (http://arxiv.org/abs/math/0011067v1)
[14] Appendix to “Square-root parametrization of plane curves”, Algebraic geometry and its applications, Springer, 1994, pp. 85-88
[15] Construction of class fields and zeta functions of algebraic curves, Ann. Math., Volume 85 (1967), pp. 58-159 | DOI | MR | Zbl
[16] Algebraic Curves over Finite Fields, University of Groningen, The Netherlands (2013) (Ph. D. Thesis)
[17] Twists of the Klein Curve and of the Fricke Macbeath Curve, University of Groningen, The Netherlands (2015) (Masters thesis)
[18] Macbeath’s Curve and the Modular Group, Glasg. Math. J., Volume 27 (1985), pp. 239-247 corrigendum in ibid., 28 (1986), p. 241 | DOI | MR | Zbl
Cited by Sources: