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

摘要

In this paper, we give examples of elliptic curves E/K over a number field K satisfying theproperty that there exist P~1,P~2E K[t] such that the twistsE~(P1),E~(P2)andE~(P1 are of positiverank over K(t). As a consequence of this result on twists, we show that for those elliptic curvesE/K, and for each a E Gal(K/K), the rank of E over the fixed field (K~(ab)~underais infinite,where Kab is the maximal abelian extension of K.