# Torsion of elliptic curves and abelian extensions

Let $$L/K$$ be an abelian $$p$$-extension of number fields and $$E$$ be an elliptic curve over $$\Bbb Q$$. If $$E[p](K)=0$$, does it follow that $$E[p](L)=0$$ ? The converse is obviously true, but I don't have any reference for my problem.

• Not sure but you may try : replace $L$ by $K(E[p]) \cap L$, $G = Gal(L/K)$ is an abelian subgroup of $GL_2(\mathbb{F}_p)$. Does $g \in G$ of order $p$ mean $\langle g \rangle = u T u^{-1}$ where $T = \{\pmatrix{1 & b \\ 0 & 1 }\}$ ? Then only $DT = \{\pmatrix{ a & b \\ 0 & a }\}$ commute with $T$. So $G = u T u^{-1}$ which means $E(K)[p]$ contains a non-trivial element. – reuns Jan 21 at 15:03
• The case of $char K=p$ is easier since $rank E[p](\bar{K})$ is $0$ or $1$. The group $Aut(\mathbb F_p)$ has no order $p$ elements. – eduard Jan 21 at 15:47