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

Schwartz, Charles F.

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

On a family of elliptic surfaces with Mordell-Weil rank $4$
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Proc. Amer. Math. Soc. 102 (1988), 1-8 Request permission

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 \[ {y^2} = 4\left [ {{x^3} - \sum \limits _{i = 0}^2 {{a_i}{u^i}x + } \sum \limits _{j = 0}^3 {{b_j}{u^j}} } \right ].\] If the elliptic surface has Mordell-Weil rank 4 over ${\mathbf {C}}$, then we find a basis $\{ {\sigma _i} = ({x_i},{y_i})|1 \leq i \leq 4\}$ with ${x_i}$ and ${y_i}$, linear in $u$. 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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