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.