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Vladimir Dokchitser, Root Numbers of Non-Abelian Twists of Elliptic Curves, Proceedings of the London Mathematical Society, Volume 91, Issue 2, September 2005, Pages 300–324, https://doi.org/10.1112/S0024611505015261
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Abstract
We study the global root number of the complex L-function of twists of elliptic curves over Q by real Artin representations. We obtain examples of elliptic curves over Q which, while not having any rational points of infinite order, conjecturally must have points of infinite order over the fields Q \sqrt[3] {m} )$ for every cube-free m > 1. We describe analogous phenomena for elliptic curves over the fields Q( \sqrt[r] {m} )$, and in the towers $(\mathbb{Q}( \sqrt[r^n] {m})_{n \ge 1} )$ and $(\mathbb{Q}( \sqrt[r^n] {m}, \mu_{r^n})_{n \ge 1})$, where r ≥ 3 is prime. 2000 Mathematics Subject Classification 11G40, 11G05.
