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

http://dx.doi.org/10.4153/CJM-2008-023-0
Canad. J. Math. 60(2008), 481-490

  Published:2008-06-01
 Printed: Jun 2008

Let $k$ be a global field, $\overline{k}$ a separable closure of $k$, and $G_k$ the absolute Galois group $\Gal(\overline{k}/k)$ of $\overline{k}$ over $k$. For every $\sigma\in G_k$, let $\ks$ be the fixed subfield of $\overline{k}$ under $\sigma$. Let $E/k$ be an elliptic curve over $k$. It is known that the Mordell--Weil group $E(\ks)$ 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.