In this paper, we give an effective and efficient algorithm which on input takes non-zero integers and and on output produces the generators of the Mordell–Weil group of the elliptic curve over given by an equation of the form . 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 .
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 , 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.
Dans cet article, nous donnons un algorithme efficace et efficient qui en entrée prend des entiers non nuls et et en sortie produit les générateurs du groupe Mordell–Weil de la courbe elliptique sur donnée par une équation de la forme . Notre méthode utilise la correspondance entre les courbes exceptionnelles d’une surface del Pezzo de degré et les sections de hauteur de Shioda minimale sur la surface elliptique correspondante sur . 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 . En particulier, les surfaces données par l’équation de Weierstrass , 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.
Revised:
Accepted:
Online First:
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.
Desjardins, Julie 1; NaskrĘcki, Bartosz 2, 3
@unpublished{AIF_0__0_0_A85_0, author = {Desjardins, Julie and Naskr\k{E}cki, Bartosz}, title = {Geometry of the del {Pezzo} surface $y^2=x^3+Am^6+Bn^6$}, journal = {Annales de l'Institut Fourier}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, year = {2024}, doi = {10.5802/aif.3635}, language = {en}, note = {Online first}, }
Desjardins, Julie; NaskrĘcki, Bartosz. Geometry of the del Pezzo surface $y^2=x^3+Am^6+Bn^6$. Annales de l'Institut Fourier, Online first, 44 p.
[1] Detecting perfect powers in essentially linear time, Math. Comput., Volume 67 (1998) no. 223, pp. 1253-1283 | DOI | MR | Zbl
[2] Lie groups and Lie algebras. Chapters 4–6, Elements of Mathematics, Springer, 2002, xii+300 pages (translated from the 1968 French original by Andrew Pressley) | DOI | MR | Zbl
[3] Rational points on del Pezzo surfaces of degree one, MSc thesis, Leiden University (2018)
[4] On the density of rational points on rational elliptic surfaces, Acta Arith., Volume 189 (2019) no. 2, pp. 109-146 | DOI | MR | Zbl
[5] On the variation of the root number in families of elliptic curves, J. Lond. Math. Soc., Volume 99 (2019) no. 2, pp. 295-331 | DOI | MR | Zbl
[6] Verification code in MAGMA of the calculations in the paper Geometry of the del Pezzo surface (2019) (https://zenodo.org/records/10659434)
[7] Density of rational points on a family of del Pezzo surfaces of degree one (with an appendix by Jean-Louis Colliot-Thélène), Adv. Math., Volume 405 (2022), 108489, 20 pages | DOI | MR | Zbl
[8] Root numbers and parity of ranks of elliptic curves, J. Reine Angew. Math., Volume 658 (2011), pp. 39-64 | DOI | MR | Zbl
[9] Classical algebraic geometry. A modern view, Cambridge University Press, 2012, xii+639 pages | DOI | MR | Zbl
[10] On the behavior of the root numbers in families of elliptic curves (2003) (https://arxiv.org/abs/math/0408141)
[11] Minimal models of rational surfaces over arbitrary fields, Izv. Akad. Nauk SSSR, Ser. Mat., Volume 43 (1979) no. 1, p. 19-43, 237 | MR
[12] Rational points on some elliptic surfaces, Acta Arith., Volume 153 (2012) no. 1, pp. 93-108 | DOI | MR | Zbl
[13] Rational curves on algebraic varieties, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge, 32, Springer, 1996, viii+320 pages | DOI | MR
[14] Quadratic families of elliptic curves and unirationality of degree 1 conic bundles, Am. J. Math., Volume 139 (2017) no. 4, pp. 915-936 | DOI | MR | Zbl
[15] Root numbers of fibers of elliptic surfaces, Compos. Math., Volume 99 (1995) no. 1, pp. 33-58 | Numdam | MR | Zbl
[16] Cubic forms: algebra, geometry, arithmetic, North-Holland Mathematical Library, 4, North-Holland; Elsevier, 1974, vii+292 pages (translated from the Russian by M. Hazewinkel) | MR | Zbl
[17] The basic theory of elliptic surfaces, Dottorato di Ricerca in Matematica, ETS Editrice, 1989, vi+108 pages | MR | Zbl
[18] The Mordell–Weil lattice of a rational elliptic surface, Comment. Math. Univ. St. Pauli, Volume 40 (1991) no. 1, pp. 83-99 | MR | Zbl
[19] Density of rational points on del Pezzo surfaces of degree one, Adv. Math., Volume 261 (2014), pp. 154-199 | DOI | MR | Zbl
[20] On the unirationality of del Pezzo surfaces of degree 2, J. Lond. Math. Soc., Volume 90 (2014) no. 1, pp. 121-139 | DOI | MR | Zbl
[21] On the Mordell–Weil lattices, Comment. Math. Univ. St. Pauli, Volume 39 (1990) no. 2, pp. 211-240 | MR | Zbl
[22] Advanced topics in the arithmetic of elliptic curves, Graduate Texts in Mathematics, 151, Springer, 1994, xiv+525 pages | DOI | MR | Zbl
[23] Algorithm for determining the type of a singular fiber in an elliptic pencil, Modular functions of one variable, IV (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972) (Lecture Notes in Mathematics), Volume 476 (1975), pp. 33-52 | DOI | MR | Zbl
[24] Rational points on certain elliptic surfaces, Acta Arith., Volume 129 (2007) no. 2, pp. 167-185 | DOI | MR | Zbl
[25] Rational points on certain del Pezzo surfaces of degree one, Glasg. Math. J., Volume 50 (2008) no. 3, pp. 557-564 | DOI | MR | Zbl
[26] Weak approximation on del Pezzo surfaces of degree 1, Adv. Math., Volume 219 (2008) no. 6, pp. 2123-2145 | DOI | MR | Zbl
[27] Density of rational points on isotrivial rational elliptic surfaces, Algebra Number Theory, Volume 5 (2011) no. 5, pp. 659-690 | DOI | MR | Zbl
Cited by Sources: