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

Let k be a global field, k a separable closure of k, and Gk the absolute Galois group Gal(k/k) of k over k. For every σ ∈ Gk, let k σ be the fixed subfield of k under σ. Let E/k be an elliptic curve over k. We show that for each σ ∈ Gk, the Mordell-Weil group E(k σ ) has infinite rank in the following two cases. Firstly 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.