  Let k be a global field, $\bar{k}$ a separable closure of k, and $G_k$ the
absolute Galois group $\Gal(\bar{k}/k)$ of $\bar{k}$ over k. For every g in
$G_k$, let $\bar{k}^g$ be the fixed subfield of $\bar{k}$ under g. Let E/k be
an elliptic curve over k. We show that for each g in $G_k$, the Mordell-Weil
group $E(\bar{k}^g)$ has infinite rank in the following two cases. Firstly when
k is a global function field of odd characteristic and E is parametrized by a
Drinfeld modular curve, and secondly when k is a totally real number field and
E/k is parametrized by a Shimura curve. In both cases our approach uses the
non-triviality of a sequence of Heegner points on E defined over ring class
fields.
