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

Abstract

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, $y^2 = x(x-1)(x-\lambda)$, where $\mathbf{Q}(\lambda)$ is a cubic field, then $E(K \mathbf{Q}^{\ab})$ has infinite rank. If $\lambda\in K$ 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 $y^2 = x(x-1)(x-\lambda)$ has infinite rank over $K\bbq^{\ab}$.