Let $E$ be an elliptic curve over $\mathbb{Q}_p$ with good reduction at $p$, i.e., there exists an elliptic curve $\mathcal{E}$ over $\mathbb{Z}_p$ whose generic fiber is isomorphic to $E$. I have seen on page 76, just above Theorem 5.38, of a course note by Schraen that the action of $G_{\mathbb{Q}_p}$ on $E[N](\overline{\mathbb{Q}_p})$ is unramified for $p\nmid N$. I would like to know if my understanding of his argument is correct:
Proof. Take $\overline{s}:=\mathrm{Spec}(\overline{\mathbb{Q}_p})$ be a geometric point of $\mathrm{Spec}{\mathbb{Q}_p}$ and $\mathrm{Spec}{\mathbb{Z}_p}$. Since $1/N\in \mathbb{Z}_p$, $\mathcal{E}[N]$ is finite étale over $\mathbb{Z}_p$. Taking its fiber $\mathcal{E}[N]_{\overline{s}} := \mathrm{Hom}_{\mathbb{Z}_p}(\overline{s},\mathcal{E}[N])=\mathcal{E}[N](\overline{\mathbb{Q}_p})$ above $\overline{s}$, we get a representation of $\pi_1(\mathrm{Spec}{\mathbb{Z}_p,\overline{s}})$, which is induced by its action on the étale cover $\mathcal{E}[N]$. But at the same time, $\mathcal{E}[N](\overline{\mathbb{Q}_p})$ is naturally $E[N](\overline{\mathbb{Q}_p}) = \mathrm{Hom}_{\mathbb{Q}_p}(\overline{s},E[N]) =: E[N]_{\overline{s}}$, which is a representation of $\pi_1(\mathrm{Spec} \mathbb{Q}_p,\overline{s})\simeq G_{\mathbb{Q}_p}$, such that the actions of these fundamental groups are related by the natural map $\pi_1(\mathrm{Spec} \mathbb{Q}_p,\overline{s})\to \pi_1(\mathrm{Spec} \mathbb{Z}_p,\overline{s})$ which is surjective with kernel being the inertia subgroup by [SGA 1, Proposition V.8.2]. Thus, the action of the inertia subgroup of $G_{\mathbb{Q}_p}$ on $E[N](\overline{\mathbb{Q}_p})$ is trivial. $\square$
After writing down this argument, I do have one
Question: is this action of $G_{\mathbb{Q}_p}$ on $E[N](\overline{\mathbb{Q}_p})$ really just the coordinate-wise Galois action on the torsion points? The main point of my confusion is perhaps that I don't understand the quotient group $\mathrm{Aut}(E[N]/\mathrm{Spec}\mathbb{Q}_p)$ of $\pi_1(\mathbb{Q}_p,\overline{s})\simeq G_{\mathbb{Q}_p}$... Could anyone make this explicit? Thanks in advance!