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Descending Rational Points on Elliptic Curves to Smaller Fields


Published:20010601
Printed: Jun 2001
Amir Akbary
V. Kumar Murty
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Abstract
In this paper, we study the MordellWeil group of an elliptic curve
as a Galois module. We consider an elliptic curve $E$ defined over a
number field $K$ whose MordellWeil 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 $\o$
of an imaginary quadratic field $\k$ contained in $K$. Suppose that the
$\o$rank over a Galois extension $F$ is $1$ or $2$. If $\k\neq\Q(\sqrt{1})$
and $\Q(\sqrt{3})$ and $h_{\k}$ (class number of $\k$) is odd, we show that
$E$ acquires positive $\o$rank over a cyclic extension of $K$ or over a
field whose Galois group is one of $\SL_2(\Z/3\Z)$, an extension of
$\SL_2(\Z/3\Z)$ by $\Z/2\Z$, or a central extension by the dihedral group.
Finally, we discuss the relation of the above results to the vanishing of
$L$functions.