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Descending Rational Points on Elliptic Curves to Smaller Fields
Published online by Cambridge University Press: 20 November 2018
Abstract
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In this paper, we study the Mordell-Weil group of an elliptic curve as a Galois module. We consider an elliptic curve 
$E$ defined over a number field 
$K$ whose Mordell-Weil rank over a Galois extension 
$F$ is 1, 2 or 3. We show that $E$ acquires a point (points) of infinite order over a field whose Galois group is one of
${{C}_{n}}\times {{C}_{m}}(n=1,\,2,\,3,\,4,\,6,\,m\,=\,1,\,2),\,{{D}_{n}}\times {{C}_{m}}(n=2,\,3,\,4,\,6,\,m=\,1,\,2),\,{{A}_{4}}\times {{C}_{m}}(m=1,\,2),\,{{S}_{4}}\,\times \,{{C}_{m}}(m=1,\,2)$
. Next, we consider the case where $E$ has complex multiplication by the ring of integers 
$\mathcal{O}$ of an imaginary quadratic field 
$\Re $ contained in $K$. Suppose that the $\mathcal{O}$-rank over a Galois extension $F$ is 1 or 2. If 
$\Re \ne \mathbb{Q}(\sqrt{-1})$ and 
$\mathbb{Q}(\sqrt{-3})$ and 
${{h}_{\Re }}$ (class number of $\Re $) is odd, we show that $E$ acquires positive $\mathcal{O}$-rank over a cyclic extension of $K$ or over a field whose Galois group is one of
$\text{S}{{\text{L}}_{2}}(\mathbb{Z}/3\mathbb{Z})$
, an extension of
$\text{S}{{\text{L}}_{2}}(\mathbb{Z}/3\mathbb{Z})$
by 
$\mathbb{Z}/2\mathbb{Z}$, or a central extension by the dihedral group. Finally, we discuss the relation of the above results to the vanishing of 
$L$-functions.
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- Research Article
- Information
- Copyright
- Copyright © Canadian Mathematical Society 2001
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