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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

Bo-Hae Im
Bo-Hae Im*
Affiliation:
Department of Mathematics, University of Utah, Salt Lake City, UT 84112 e-mail: [email protected]
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Abstract

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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.

Keywords

11G05
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 58 , Issue 4 , 01 August 2006 , pp. 796 - 819
Copyright
Copyright © Canadian Mathematical Society 2006

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