Elliptic curve and restriction

Question

Let $E$ be an elliptic curve. Let $\xi$ be a class of $H^{1}(\mathbb{Q}, E[m])$ unramified at the prime $\ell$. Then $\xi$ restricted to $H^{1}(I_{\ell}, E[m])$ where $I_{\ell}\subset \operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ is the inertia group for $\ell$. Let $\operatorname{res}_{\ell}(\xi)$ be the restriction of $\xi$ to $\operatorname{Gal}(\overline{\mathbb{Q}_{\ell}}/\mathbb{Q}_{\ell})$, does it follow that $\operatorname{res}_{\ell}(\xi) = 0$?

Answer 1

The kernel of the restriction $H^1(\mathbf{Q}_\ell,E[m])\rightarrow H^1(I_\ell,E[m])$ is $H^1(\mathbf{Q}_\ell^{\mathrm{unr}}/\mathbf{Q}_\ell,E[m]^{I_\ell})$, which is the module of $\mathrm{Frob}_\ell$-coinvariants of $E[m]^{I_\ell}$, $E[m]^{I_\ell}/(\mathrm{Frob}_\ell-1)E[m]^{I_\ell}$. This has the same order as $H^0(\mathbf{Q}_\ell,E[m])$, so it will fail to be zero if $E$ has a non-trivial $\mathbf{Q}_\ell$-rational $m$-torsion point.