Geometry of the del Pezzo surface y 2 =x 3 +Am 6 +Bn 6
[Géométrie de la surface de Del Pezzo y 2 =x 3 +Am 6 +Bn 6 ]
Desjardins, Julie1 ; NaskrĘcki, Bartosz2, 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, Online first, 44 p.

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 +At 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 +At 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 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 +At 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 :
Révisé le :
Accepté le :
Première publication :
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.
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) 
@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},
}
TY  - UNPB
AU  - Desjardins, Julie
AU  - NaskrĘcki, Bartosz
TI  - Geometry of the del Pezzo surface $y^2=x^3+Am^6+Bn^6$
JO  - Annales de l'Institut Fourier
PY  - 2024
PB  - Association des Annales de l’institut Fourier
N1  - Online first
DO  - 10.5802/aif.3635
LA  - en
ID  - AIF_0__0_0_A85_0
ER  - 
%0 Unpublished Work
%A Desjardins, Julie
%A NaskrĘcki, Bartosz
%T Geometry of the del Pezzo surface $y^2=x^3+Am^6+Bn^6$
%J Annales de l'Institut Fourier
%D 2024
%I Association des Annales de l’institut Fourier
%Z Online first
%R 10.5802/aif.3635
%G en
%F AIF_0__0_0_A85_0
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] Bernstein, Daniel J. Detecting perfect powers in essentially linear time, Math. Comput., Volume 67 (1998) no. 223, pp. 1253-1283 | DOI | MR | Zbl

[2] Bourbaki, Nicolas 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] Bulthuis, Jelle Rational points on del Pezzo surfaces of degree one, MSc thesis, Leiden University (2018)

[4] Desjardins, Julie On the density of rational points on rational elliptic surfaces, Acta Arith., Volume 189 (2019) no. 2, pp. 109-146 | DOI | MR | Zbl

[5] Desjardins, Julie 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] Desjardins, Julie; Naskręcki, B. Verification code in MAGMA of the calculations in the paper Geometry of the del Pezzo surface y 2 =x 3 +Am 6 +Bn 6 (2019) (https://zenodo.org/records/10659434)

[7] Desjardins, Julie; Winter, Rosa 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] Dokchitser, Tim; Dokchitser, Vladimir Root numbers and parity of ranks of elliptic curves, J. Reine Angew. Math., Volume 658 (2011), pp. 39-64 | DOI | MR | Zbl

[9] Dolgachev, Igor V. Classical algebraic geometry. A modern view, Cambridge University Press, 2012, xii+639 pages | DOI | MR | Zbl

[10] Helfgott, H. A. On the behavior of the root numbers in families of elliptic curves (2003) (https://arxiv.org/abs/math/0408141)

[11] Iskovskih, Vasily A. 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] Jabara, Enrico Rational points on some elliptic surfaces, Acta Arith., Volume 153 (2012) no. 1, pp. 93-108 | DOI | MR | Zbl

[13] Kollár, János Rational curves on algebraic varieties, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge, 32, Springer, 1996, viii+320 pages | DOI | MR

[14] Kollár, János; Mella, Massimiliano 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] Manduchi, Elisabetta Root numbers of fibers of elliptic surfaces, Compos. Math., Volume 99 (1995) no. 1, pp. 33-58 | Numdam | MR | Zbl

[16] Manin, Yuriĭ I. 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] Miranda, Rick The basic theory of elliptic surfaces, Dottorato di Ricerca in Matematica, ETS Editrice, 1989, vi+108 pages | MR | Zbl

[18] Oguiso, Keiji; Shioda, Tetsuji 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] Salgado, Cecília; van Luijk, Ronald Density of rational points on del Pezzo surfaces of degree one, Adv. Math., Volume 261 (2014), pp. 154-199 | DOI | MR | Zbl

[20] Salgado, Cecília; Testa, Damiano; Várilly-Alvarado, Anthony 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] Shioda, Tetsuji On the Mordell–Weil lattices, Comment. Math. Univ. St. Pauli, Volume 39 (1990) no. 2, pp. 211-240 | MR | Zbl

[22] Silverman, Joseph H. Advanced topics in the arithmetic of elliptic curves, Graduate Texts in Mathematics, 151, Springer, 1994, xiv+525 pages | DOI | MR | Zbl

[23] Tate, John 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] Ulas, Maciej Rational points on certain elliptic surfaces, Acta Arith., Volume 129 (2007) no. 2, pp. 167-185 | DOI | MR | Zbl

[25] Ulas, Maciej 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] Várilly-Alvarado, Anthony Weak approximation on del Pezzo surfaces of degree 1, Adv. Math., Volume 219 (2008) no. 6, pp. 2123-2145 | DOI | MR | Zbl

[27] Várilly-Alvarado, Anthony Density of rational points on isotrivial rational elliptic surfaces, Algebra Number Theory, Volume 5 (2011) no. 5, pp. 659-690 | DOI | MR | Zbl

Cité par Sources :