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Journal of the London Mathematical Society Advance Access published online on July 31, 2008
Journal of the London Mathematical Society, doi:10.1112/jlms/jdn048
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© 2008 London Mathematical Society

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

Bo-Hae Im

Department of Mathematics
Chung-Ang University
221, Heukseok-dong
Dongjak-gu
Seoul 156-756
South Korea

Álvaro Lozano-Robledo

Department of Mathematics
584 Malott Hall
Cornell University
Ithaca
NY 14853
USA
alozano@math.cornell.edu

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

Bo-Hae Im was supported by the Korea Science and Engineering Foundation(KOSEF) grant funded by the Korea government(MOST) (No. R01-2007-000-10660-0).

2000 Mathematics Subject Classification 11G05 (primary).

Received April 5, 2007; revised November 5, 2007;
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