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

Abstract

AbstractLetkbe a global field,a separable closure ofk, andGkthe absolute Galois groupofoverk. For everyσ ∈ Gk, letbe the fixed subfield ofunderσ. LetE/kbe an elliptic curve overk. It is known that the Mordell–Weil grouphas 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 andEis parametrized by a Drinfeld modular curve, and secondly whenkis a totally real number field andE/kis parametrized by a Shimura curve. In both cases our approach uses the non-triviality of a sequence of Heegner points onEdefined over ring class fields.