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

Let $K$ be a number field, $\overline{K}$ an algebraic closure of $K$, $G_K$ the absolute Galois group $\operatorname{Gal}(\overline{K}/K)$, $K_{ab}$ 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 $\sigma\in G_K$, $E((K_{ab})^{\sigma})$ has infinite rank, and hence $E(\overline{K}^{\sigma})$ has infinite rank.