On products of quadratic twists and ranks of elliptic curves over large fields

In this paper, we give examples of elliptic curves E / K over a number field K satisfying the property that there exist P 1 , P 2 ∈ K [ t ] such that the twistsE P 1, E P 2andE P 1 P 2are of positive rank over K ( t ). As a consequence of this result on twists, we show that for those elliptic curves E / K , and for each σ ∈ G a l ( K ¯ / K ) , the rank of E over the fixed field ( K ab ) σ under σ is infinite, where K ab is the maximal abelian extension of K .