Why is the kernel of reduction a pro-p group?

Question

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?)

Answer 1

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.