The rank of elliptic curves with rational 2-torsion points over large fields

Let K be a number field, K an algebraic closure of K, G(K) the absolute Galois group Gal(<(K)overbar >/K), K-ab the maximal abelian extension of K and E/K an elliptic curve defined over K. In this paper, we prove that if all 2-torsion points of E/K are K-rational, then for each sigma epsilon G(K), E((K-ab)(sigma)) has infinite rank, and hence E( K s) has infinite rank.