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.