Let $A$ be an abelian variety over a $p$-adic field $K\_v$, i.e., $K\_v$ is a finite field extension of $\mathbb Q\_p$, for $p$ a prime number. Denote by $k\_v$ the residue field of $K\_v$ and let $\mathcal{A}\_v^{0}(k\_v)$ be the smooth part of the $k\_v$-rational points of the modulo $v$ reduced variety $A$, i.e., the $k\_v$-rational points of the connected component of the identity section of the special fiber at $v$ of the Néron model $\mathcal{A}/\mathcal{O}\_K$ of $A$. Denote by $A\_0(K\_v)$ the preimage of the reduction-mod-$v$-map on $\mathcal{A}\_v^{0}(k\_v)$ and by $A\_1(K\_v)$ the kernel, hence we have the short exact sequence
$$0 \rightarrow A\_1(K\_v) \rightarrow A\_0(K\_v) \rightarrow \mathcal{A}\_v^{0}(k\_v) \rightarrow 0.$$
Why is $A\_1(K\_v)$ a pro-$p$ group?


(Is there a standard reference for this fact?)



Why is the kernel of reduction a pro-p group?
The kernel of reduction $A\_1(K\_v)$ is isomorphic to the group $\hat A(\mathfrak{m}\_v)$ associated to the formal group $\hat A$ of $A$ defined over the valuation ring $\mathcal O\_v$ of $K\_v$ with maximal ideal $\mathfrak m\_v$. By standard properties of formal groups, the multiplication-by-$m$-endomorphism on $\hat A(\mathfrak m\_v)$ is an isomorphism, if $m$ is coprime to the characteristic of the residue field, i.e., to $p$. (See for example Silverman, AEC, IV. Prop. 2.3) It is an easy excercise to check that any profinite group, such that for all primes $\ell \neq p$ the multiplication-by-$\ell$-map is an isomorphism, is a pro-$p$ group. Hence $A\_1(K\_v)$ is.


