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 P(1), P(2) is an element of K[t] such that the twists E(P1), E(P2) and E(P1P2) 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 sigma is an element of Gal((K) over bar /K), the rank of E over the fixed field (K(ab))(sigma) under sigma is infinite, where K(ab) is the maximal abelian extension of K