# Maximal unramified quotient of $E[p]$ for the action of $G_{\mathbb{Q}_p}$

Let $$E$$ be an elliptic curve with good and ordinary reduction at an odd prime $$p$$. Suppose $$E[p]$$ denotes the $$p$$-torsion points of $$E$$ and $$G_{\mathbb{Q}_p} := \text{Gal}(\overline{\mathbb{Q}_p}/\mathbb{Q}_p)$$.

In the article Selmer group and congruences (page 6)’, Greenberg says that one can characterize $$\widetilde{E}[p]$$ as the maximal unramified quotient of $$E[p]$$ for the action of $$G_{\mathbb{Q}_p}$$ where $$\widetilde{E}$$ denotes the reduction of $$E$$ in $$\mathbb{F}_p$$.

This is so because $$p$$ is assumed to be odd and therefore the action of the inertia subgroup of $$G_{\mathbb{Q}_p}$$ on the kernel of the reduction map $$\pi: E[p] \longrightarrow \widetilde{E}[p]$$ is nontrivial.

It will be every helpful if someone can explain how $$p$$ being odd’ is playing a role in proving the non trivial action of the inertia subgroup on the kernel of the reduction map $$\pi$$ ?