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

TL;DR

This paper proves that elliptic curves over certain fields have infinite rank in their rational points over the maximal abelian extension, expanding understanding of their arithmetic properties.

Contribution

It establishes new infinite rank results for elliptic curves over quadratic and cubic fields, including specific forms and conditions involving minimal polynomials.

Findings

Elliptic curves over quadratic fields with non-special j-invariants have infinite rank over .

Legendre form elliptic curves over cubic fields have infinite rank over the compositum with .

Elliptic curves defined by quartic polynomials with positive rank over  have infinite rank over the compositum with .

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 $K\bbq^{\ab}$.