On a family of elliptic surfaces with Mordell-Weil rank 4

Schwartz, Charles F.

doi:10.1090/S0002-9939-1988-0915705-8

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ISSN 1088-6826(online) ISSN 0002-9939(print)

On a family of elliptic surfaces with Mordell-Weil rank

Author: Charles F. Schwartz
Journal: Proc. Amer. Math. Soc. 102 (1988), 1-8
MSC: Primary 14J27,; Secondary 11D41,11G99,14D10,14G25
MathSciNet review: 915705
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Abstract: In this paper, we find bases for the Mordell-Weil groups of a family of elliptic surfaces. In particular, let ${E_{(a,b)}} \to B$ be the elliptic surface given by

$\displaystyle {y^2} = 4\left[ {{x^3} - \sum\limits_{i = 0}^2 {{a_i}{u^i}x + } \sum\limits_{j = 0}^3 {{b_j}{u^j}} } \right].$

If the elliptic surface has Mordell-Weil rank 4 over ${\mathbf{C}}$ , then we find a basis $\{ {\sigma _i} = ({x_i},{y_i})\vert 1 \leq i \leq 4\}$ with ${x_i}$ and ${y_i}$ , linear in . We do this by finding a parametrization of this family of elliptic surfaces; furthermore, if the parameters are rational numbers, then the Mordell-Weil group is rational over ${\bf {Q}}$

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