An elliptic surface of Mordell-Weil rank $8$ over the rational numbers
Journal de Théorie des Nombres de Bordeaux, Tome 6 (1994) no. 1, pp. 1-8.

Néron showed that an elliptic surface with rank $8$, and with base $B={P}_{1}ℚ$, and geometric genus $=0$, may be obtained by blowing up $9$ points in the plane. In this paper, we obtain parameterizations of the coefficients of the Weierstrass equations of such elliptic surfaces, in terms of the $9$ points. Manin also describes bases of the Mordell-Weil groups of these elliptic surfaces, in terms of the $9$ points ; we observe that, relative to the Weierstrass form of the equation, ${Y}^{2}={X}^{3}+A{X}^{2}+BX+C$ (with $deg\left(A\right)\le 2,deg\left(B\right)\le 4$, and $deg\left(C\right)\le 6\right)$ a basis $\left\{\left({X}_{1},{Y}_{1}\right),\cdots ,\left({X}_{8},{Y}_{8}\right)\right\}$ can be found with ${X}_{i}$ and ${Y}_{i}$ polynomial of degree $\le 2,\le 3$, respectively. One explicit example is computed, showing that for almost every elliptic surface given by a Weierstrass equation of the above form, a basis may be found with ${X}_{i}$ and ${Y}_{i}$ polynomial of degree $\le 2,\phantom{\rule{0.277778em}{0ex}}\le 3$, respectively.

@article{JTNB_1994__6_1_1_0,
author = {Schwartz, Charles F.},
title = {An elliptic surface of {Mordell-Weil} rank $8$ over the rational numbers},
journal = {Journal de Th\'eorie des Nombres de Bordeaux},
pages = {1--8},
publisher = {Universit\'e Bordeaux I},
volume = {6},
number = {1},
year = {1994},
doi = {10.5802/jtnb.102},
mrnumber = {1305284},
language = {en},
url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.102/}
}
Charles F. Schwartz. An elliptic surface of Mordell-Weil rank $8$ over the rational numbers. Journal de Théorie des Nombres de Bordeaux, Tome 6 (1994) no. 1, pp. 1-8. doi : 10.5802/jtnb.102. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.102/

[1] D.A. Cox and S. Zucker, Intersection numbers of sections of elliptic surfaces, Invent. Math., 53 (1979), 1-44. | MR 538682 | Zbl 0444.14004

[2] W. Fulton, Algebraic Curves, an introduction to algebroic geometry, Benjamin, W. A. (1959), (Mathematics lecture note series). | Zbl 0181.23901

[3] A. Kas, On the deformation types of regular elliptic surfaces, Preprint (1976). | MR 569686

[4] Ju. I. Manin, The Tate height of points on an Abelian variety; its variants and applications, AMS Translations (series 2) 59 (1966), 82-110. | Zbl 0192.26801

[5] L.J. Mordell, Diophantine Equations, Academic Press, London (1969). | MR 249355 | Zbl 0188.34503

[6] A. Néron, Les propriétés du rang des courbes algibriques dans les corps de degré de transcendance fini, Centre National de la Recherche Scientifique, (1950), 65-69. | MR 41477 | Zbl 0040.16001

[7) A. Néron, Propriétés arithmétiques de certaines familles de courbes algébriques, Proc. Int. Congress, Amsterdam, III (1954), 481-488. | MR 87210 | Zbl 0074.15901

[8] C.F. Schwartz, A Mordell-Weil group of rank 8, and a subgmup of finite index, Nagoya Math. J. 93 (1984), 19-26. | MR 738915 | Zbl 0504.14031

[9] T. Shioda, On elliptic modular surfaces, J. Math. Soc. Japan 24 (1972), 20-59. | MR 429918 | Zbl 0226.14013

[10] T. Shioda, An infinite family of elliptic curves over Q with large rank via Néron's method, Invent. Math. 106 (1991), 109-119. | MR 1123376 | Zbl 0766.14024

[11] K. Oguiso and T. Shioda, The Mordell-Weil lattice of a rational elliptic surface, Comment. Math. Univ. St. Pauli 40 (1991), 83-99. | MR 1104782 | Zbl 0757.14011