If E is an elliptic curve defined over a quadratic field K, and the j-invariant of E is not 0 or 1728, then has infinite rank. If E is an elliptic curve in Legendre form, y² = x(x-1)(x-λ), where ℚ(λ) is a cubic field, then has infinite rank. If λ ∈ K has a minimal polynomial P(x) of degree 4 and v² = P(u) is an elliptic curve of positive rank over ℚ, we prove that y² = x(x-1)(x-λ) has infinite rank over .
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-aa158-1-3,
author = {Bo-Hae Im and Michael Larsen},
title = {Infinite rank of elliptic curves over $$\mathbb{Q}$^{ab}$
},
journal = {Acta Arithmetica},
volume = {161},
year = {2013},
pages = {49-59},
zbl = {1272.11077},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-aa158-1-3}
}
Bo-Hae Im; Michael Larsen. Infinite rank of elliptic curves over $ℚ^{ab}$
. Acta Arithmetica, Tome 161 (2013) pp. 49-59. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-aa158-1-3/