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

Abstract

In this paper, we give examples of elliptic curves E/K over a number field K satisfying the property that there exist P1, P2 K[t] such that the twists and are 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 , the rank of E over the fixed field (Kab) under is infinite, where Kab is the maximal abelian extension of K.