An elliptic surface of Mordell-Weil rank $8$ over the rational numbers

Charles F. Schwartz

doi:10.5802/jtnb.102

An elliptic surface of Mordell-Weil rank

8

over the rational numbers

Charles F. Schwartz

Journal de Théorie des Nombres de Bordeaux, Tome 6 (1994) no. 1, pp. 1-8.

Résumé

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, $Y^{2} = X^{3} + A X^{2} + B X + C$ (with $deg (A) \leq 2, deg (B) \leq 4$ , and $deg (C) \leq 6)$ a basis $\{(X_{1}, Y_{1}), \dots, (X_{8}, Y_{8})\}$ can be found with $X_{i}$ and $Y_{i}$ polynomial of degree $\leq 2, \leq 3$ , 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 $\leq 2, \leq 3$ , respectively.

MR 1305284

DOI : https://doi.org/10.5802/jtnb.102

@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/

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