Counting points on the Fricke–Macbeath curve over finite fields

Top, Jaap; Verschoor, Carlo

Journal de théorie des nombres de Bordeaux, Tome 30 (2018) no. 1, pp. 117-129.

La courbe de Fricke-Macbeath est une courbe projective lisse de genre $7$ avec groupe d’automorphismes ${\mathrm{PSL}}_2\left({𝔽}_8\right)$. Nous rappelons deux modèles de cette courbe (introduits respectivement par Maxim Hendriks et Bradley Brock) définis sur $\mathbb{Q}$, et nous établissons un isomorphisme explicite, défini sur $\mathbb{Q}\left(\sqrt{-7}\right)$, entre ces deux modèles. De plus, nous décomposons à isogénie sur $\mathbb{Q}$ près la jacobienne de l’un des modèles. Comme une conséquence nous obtenons une formule simple pour le nombre de points sur ${𝔽}_q$ de (la réduction de) ce modèle, en termes de la courbe elliptique d’équation $y^2=x^3+x^2-114x-127$. Enfin, des tordus de cette courbe par des éléments de ${\mathrm{PSL}}_2\left({𝔽}_8\right)$ sur des corps finis sont décrits. La courbe donne un certain nombre de nouveaux records maintenus par manYPoints de courbes de genre $7$ avec beaucoup de points rationnels sur des corps finis.

The Fricke-Macbeath curve is a smooth projective algebraic curve of genus $7$ with automorphism group ${\mathrm{PSL}}_2\left({𝔽}_8\right)$. We recall two models of it (introduced, respectively, by Maxim Hendriks and by Bradley Brock) defined over $\mathbb{Q}$, and we establish an explicit isomorphism defined over $\mathbb{Q}\left(\sqrt{-7}\right)$ between these models. Moreover, we decompose up to isogeny over $\mathbb{Q}$ the jacobian of one of these models. As a consequence we obtain a simple formula for the number of points over ${𝔽}_q$ on (the reduction of) this model, in terms of the elliptic curve with equation $y^2=x^3+x^2-114x-127$. Moreover, twists by elements of ${\mathrm{PSL}}_2\left({𝔽}_8\right)$ 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 $7$ with many rational points over finite fields.

Classification : 11N56, 14G42
Mots clés : Hurwitz curve, automorphism group, jacobian, point counting

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