Infinite rank of elliptic curves over $\mathbb{Q}^{\mathrm{ab}}$
Volume 158 / 2013
Acta Arithmetica 158 (2013), 49-59
MSC: Primary 11G05; Secondary 14H52.
DOI: 10.4064/aa158-1-3
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 $E(\mathbb{Q}^{\mathrm{ab}})$ has infinite rank. If $E$ is an elliptic curve in Legendre form, $y^2 = x(x-1)(x-\lambda)$, where $\mathbb{Q}(\lambda)$ is a cubic field, then $E(K \mathbb{Q}^{\mathrm{ab}})$ 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 $\mathbb{Q}$, we prove that $y^2 = x(x-1)(x-\lambda)$ has infinite rank over $K\mathbb{Q}^{\mathrm{ab}}$.
