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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 $\sigma \,\in \,{{G}_{K}}$ , let ${{\bar{k}}^{\sigma }}$ be the fixed subfield of $\bar{k}$ under $\sigma$ . Let $E/k$ be an elliptic curve over $k$ . It is known that the Mordell–Weil group $E({{\bar{k}}^{\sigma }})$ has infinite rank. We present a new proof of this fact in the following two cases. First, 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.

Heegner Points and the Rank of Elliptic Curves over Large Extensions of Global Fields