Infinite rank of elliptic curves over $\mathbf{Q}^{\ab}$

If $E$ is an elliptic curve defined over a quadratic field $K$, and the $j$-invariant of $E$ is not 0 or 1728, then $E(\mathbf{Q}^{\ab})$ has infinite rank. If $E$ is an elliptic curve in Legendre form, $y^2 = x(x-1)(x-λ)$, where $\mathbf{Q}(λ)$ is a cubic field, then $E(K \mathbf{Q}^{\ab})$ has infinite rank. If $λ\in K$ has a minimal polynomial $P(x)$ of degree 4 and $v^2 = P(u)$ is an elliptic curve of positive rank over $\bbq$, we prove that $y^2 = x(x-1)(x-λ)$ has infinite rank over $K\bbq^{\ab}$.

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