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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?)

Stefan Keil
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  • Sorry, I have assumed in my answer that $A_0(K_v)$ is pro-p itself, don't ask me why, perhaps because it's compact? It was mentioned that this is not true. I have deleted my answer, since it was useless and because then your question gets more attention again. Best, Marc – Marc Palm Apr 16 '12 at 20:57
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    Because it is the evaluation of the formal group of $A$ at the maximal ideal of $K_v$. (You should take the Neron model of $A$ over $O_v$ to be able to talk about "reduction" properly.) – Chris Wuthrich Apr 16 '12 at 22:25
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    A. Mattuck, Abelian varieties over p-adic ground fields. Annals of Mathematics 62 (1955) 92–119. – Felipe Voloch Apr 17 '12 at 01:36
  • @Marc: Thanks for the try. @Chris: Thank you for your comment. I meant the Néron model and added a sentence of explanation. Also I will state the answer using this property. @Felipe: Thanks for the reference, but I am very sorry, I couldn't find the answer of my question in it. – Stefan Keil Apr 19 '12 at 12:39
  • Really? It's right there in italics on the first page. It reduces to the additive group of the integers of the field, where the result is clear. – Felipe Voloch Apr 19 '12 at 14:15
  • Well, you are perfectly right, thanks again. To be honest, I was just skimming through the article if it contains literally the statement I was asking for. (And where in the article is it mentioned that this existing subgroup isomorphic to some copies of the valuation ring is the same as the kernel of reduction?) – Stefan Keil Apr 23 '12 at 11:00

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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.

Stefan Keil
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