Mordell-Weil groups and the rank of elliptic curves over large fields

Abstract

Let K be a number field, (K) over bar an algebraic closure of K and E/K an elliptic curve defined over K. In this paper, we prove that if E/K has a K-rational point P such that 2P not equal O and 3P not equal O, then for each sigma is an element of Gal ((K) over bar /K), the Mordell-Weil group E((K) over bar (sigma)) of E over the fixed subfield of (K) over bar under sigma has infinite rank