$p$ adic representation which satisfies $p^2V＝0$ p adic representation is defined as finitely generated $ \Bbb{Z}_p$module with continuous linear action of absolute galois group $G_{ \Bbb{Q}_p}$.
I know a few examples, like $ \Bbb{Z}_p(1)$, $T_p(E)$($E$ is elliptic curve).
I want to know an example of $p$ adic representation $V$ which satisfies $p^2V＝0$.
Thank you for your help.
(My thought (?) Just  $ \Bbb{Z}_p(1)/p^2\Bbb{Z}_p(1)$, $T_p(E)/p^2T_p(E)$ is ? But I don't know $G_{ \Bbb{Q}_p}$ acts continuous linearly ?
Are there other typical examples?)
 A: Let $n$ be a positive integer and $K/\mathbb Q_p$ be a finite extension of degree $n$ with valuation ring $R$. Now $R/p^2R$ is a finite $\mathbb Z_p$-module via the composition $\mathbb Z_p \hookrightarrow R \twoheadrightarrow R/p^2R$.
By the defnition of the topology of pro-finite group $G_{\mathbb Q_p}$, there is a continuous surjection $G_{\mathbb Q_p} \twoheadrightarrow \text{Gal}(K/\mathbb Q_p)$. The latter Galois group acts on $R/p^2R$ continuously too (because the kernel is still a finite-index subgroup in $G_{\mathbb Q_p}$). This makes $R/pR$ an example of a $p$-adic Galois representation $V$ such that $p^2V=0$.
More generally you could take some kind of geometric object ( like $\mathbb G_m$ or an Elliptic Curve or just simply Spec $R$ for example.) over $R$ and reduce it mod $p^2R$ and study it's etale cohomology with $p^2$-torsion coefficients. This will (by functoriality) naturally have the structure of a $p$-adic representation and also be $p^2$-torsion. Your examples are exactly of this kind.
Of course, you can play this entire game mutatis mutandis with $2$ replaced by any positive integer $k$.
