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

الأصل · EN

Let k be a global field, k a separable closure of k, and Gₖ the absolute Galois group (k/k) of k over k. For every g in Gₖ, let kᵍ be the fixed subfield of k under g. Let E/k be an elliptic curve over k. We show that for each g in Gₖ, the Mordell-Weil group E(kᵍ) 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.