Heegner points and Mordell-Weil groups of elliptic curves over large fields

Let $E/\mathbb{Q}$ be an elliptic curve defined over $\mathbb{Q}$ of conductor $N$ and let $\operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ be the absolute Galois group of an algebraic closure $\overline{\mathbb{Q}}$ of $\mathbb{Q}$. For an automorphism $\sigma\in \operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$, we let $\overline{\mathbb{Q}}^{\sigma}$ be the fixed subfield of $\overline{\mathbb{Q}}$ under $\sigma$. We prove that for every $\sigma\in \operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$, the Mordell-Weil group of $E$ over the maximal Galois extension of $\mathbb{Q}$ contained in $\overline{\mathbb{Q}}^{\sigma}$ has infinite rank, so the rank of $E(\overline{\mathbb{Q}}^{\sigma})$ is infinite. Our approach uses the modularity of $E/\mathbb{Q}$ and a collection of algebraic points on $E$ - the so-called Heegner points - arising from the theory of complex multiplication. In particular, we show that for some integer $r$ and for a prime $p$ prime to $rN$, the rank of $E$ over all the ring class fields of a conductor of the form $rp^n$ is unbounded, as $n$ goes to infinity.