Heegner points and the rank of elliptic curves over large extensions of global fields

Abstract

Let k be a global field, (k) over bar a separable closure of k, and G(k) the absolute Galois group Gal((k) over bar /k) of k over k. For every sigma is an element of G(k), let (k) over bar (sigma) be the fixed subfield of (k) over bar under sigma. Let E/k be ail elliptic curve over k. It is known that the Mordell-Weil group E((k) over bar (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 Elk 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