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

Abstract:

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.