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Mordell–Weil Groups and the Rank of Elliptic Curves over Large Fields
Published online by Cambridge University Press: 20 November 2018
Abstract
Let $K$ be a number field,
$\bar{K}$ an algebraic closure of
$K$ and
$E/K$ an elliptic curve defined over
$K$. In this paper, we prove that if
$E/K$ has a
$K$-rational point
$P$ such that
$2P\ne O$ and
$3P\ne O$, then for each
$\sigma \,\in \,\text{Gal(}\overline{K}/K\text{)}$, the Mordell–Weil group
$E({{\overline{K}}^{\sigma }})$ of
$E$ over the fixed subfield of
$\bar{K}$ under
$\sigma $ has infinite rank.
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- Research Article
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- Copyright © Canadian Mathematical Society 2006
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