Geometry of the del Pezzo surface $y^2=x^3+Am^6+Bn^6$

Desjardins, Julie; NaskrĘcki, Bartosz

doi:10.5802/aif.3635

Geometry of the del Pezzo surface

y^{2} = x^{3} + A m^{6} + B n^{6}

[Géométrie de la surface de Del Pezzo

y^{2} = x^{3} + A m^{6} + B n^{6}

]

Desjardins, Julie ¹ ; NaskrĘcki, Bartosz ^2,³

¹ Mathematical and Computational Sciences University of Toronto Mississauga Deerfield Hall Mississauga, ON L5L 3E2 (Canada)
² Faculty of Mathematics and Computer Science Adam Mickiewicz University in Poznań ul. Uniwersytetu Poznańskiego 4 61-614, Poznań (Poland)
³ Mathematical Institute Polish Academy of Sciences ul. Śniadeckich 8 00-656, Warszawa (Poland)

Annales de l'Institut Fourier, Tome 74 (2024) no. 5, pp. 2231-2274.

Résumé
Abstract

Dans cet article, nous donnons un algorithme efficace et efficient qui en entrée prend des entiers non nuls $A$ et $B$ et en sortie produit les générateurs du groupe Mordell–Weil de la courbe elliptique sur $ℚ (t)$ donnée par une équation de la forme $y^{2} = x^{3} + A t^{6} + B$ . Notre méthode utilise la correspondance entre les $240$ courbes exceptionnelles d’une surface del Pezzo de degré $1$ et les sections de hauteur de Shioda minimale sur la surface elliptique correspondante sur $ℚ$ . Pour la plupart des surfaces elliptiques rationnelles, la densité des points rationnels est démontrée par diverses personnes, mais les résultats sont partiels dans le cas où la surface a un modèle minimal qui est une surface del Pezzo de degré $1$ . En particulier, les surfaces données par l’équation de Weierstrass $y^{2} = x^{3} + A^{6} + B$ , sont parmi les rares pour lesquelles la question n’est pas résolue, parce que le signe de l’équation fonctionnelle des fibres peut être constant. Notre résultat prouve la densité des points rationnels dans beaucoup de ces cas où elle était auparavant inconnue.

In this paper, we give an effective and efficient algorithm which on input takes non-zero integers $A$ and $B$ and on output produces the generators of the Mordell–Weil group of the elliptic curve over $ℚ (t)$ given by an equation of the form $y^{2} = x^{3} + A t^{6} + B$ . Our method uses the correspondence between the 240 lines of a del Pezzo surface of degree 1 and the sections of minimal canonical height on the corresponding elliptic surface over $\bar{ℚ}$ .

For most rational elliptic surfaces, the density of the rational points is proven by various authors, but the results are partial in case when the surface has a minimal model that is a del Pezzo surface of degree 1. In particular, the ones given by the Weierstrass equation $y^{2} = x^{3} + A t^{6} + B$ , are among the few for which the question is unsolved, because the root number of the fibres can be constant. Our result proves the density of the rational points in many of these cases where it was previously unknown.

Reçu le : 2019-11-25
Révisé le : 2021-12-15
Accepté le : 2023-05-15
Publié le : 2024-10-11

DOI : 10.5802/aif.3635

Classification : 14G05, 14J26, 14J27, 14D10, 11G05
Keywords: del Pezzo surfaces, density of rational points, elliptic surfaces, Mordell–Weil groups.
Mot clés : Surface de Del Pezzo, densité des points rationnels, surfaces elliptiques, groupes de Mordell–Weil.

Affiliations des auteurs :

Desjardins, Julie ¹ ; NaskrĘcki, Bartosz ^{2, 3}

Licence :

CC-BY-ND 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     journal = {Annales de l'Institut Fourier},
     pages = {2231--2274},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
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Desjardins, Julie; NaskrĘcki, Bartosz. Geometry of the del Pezzo surface $y^2=x^3+Am^6+Bn^6$. Annales de l'Institut Fourier, Tome 74 (2024) no. 5, pp. 2231-2274. doi : 10.5802/aif.3635. https://aif.centre-mersenne.org/articles/10.5802/aif.3635/

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