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By a theorem of Mattuck [Abelian Varieties over $p$-Adic Ground Fields, Annals of Mathematics, Second Series, Vol. 62, No. 1 (Jul., 1955), pp. 92-119], for an Abelian variety $A$ of dimension $g$ over a $p$-adic local field $K$ with ring of integers $O_K$, there is an exact sequence $0 \to O_K^g \to A(K) \to \{\mathrm{finite}\} \to 0$.

Is there an analogue of this theorem for Abelian varieties over $\mathbf{F}_q(\!(t)\!)$ (e.g. with $O_K^g$ replaced by a pro-$p$-group)? The only thing I found is a remark in [Milne, Arithmetic Duality Theorems, Remark I.3.6, bottom of p. 45]. For elliptic curves, there is Proposition 10.2.26 (Filtration of $E(K)$) in [Liu, Algebraic Geometry and Arithmetic Curves, 2nd ed.].

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    $\begingroup$ The kernel is given, in some appropriate sense, by the formal group of $A$. This must be covered in many sources. $\endgroup$
    – Lubin
    Sep 5, 2018 at 13:28
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    $\begingroup$ Following up on Jon Lubin's comment, Serre, Lie Algebras and Lie Groups, Part II, Ch IV covers formal groups. $\endgroup$ Sep 5, 2018 at 17:36
  • $\begingroup$ @Lubin: Thanks (also Felipe Voloch). I proved it for $1$-dimensional formal groups and for $n$-dimensional commutative formal groups $\mathcal{F}$ of finite height over $\mathbf{F}_q[[t]]$, I need theorems on the power series $[p]_\mathcal{F}(X_1,\ldots,X_n)$. Can you recommend me some references, please? $\endgroup$
    – user19475
    Sep 15, 2018 at 4:26
  • $\begingroup$ Sorry, I’m no longer well-informed. In general, you handle higher-dimensional formal groups by looking at their Dieudonné module. Perhaps @FelipeVoloch can point you in the right direction. $\endgroup$
    – Lubin
    Sep 15, 2018 at 4:48
  • $\begingroup$ I found [Lazard, Commutative formal groups], but it introduces a lot of notation and is therefore difficult to read. $\endgroup$
    – user19475
    Sep 15, 2018 at 4:53

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By [Serre, Lie Algebras and Lie Groups], p. 116, Theorem and p. 118, Corollary 2, there is an open subgroup of $A(K)$ which is a pro-$p$-group. Therefore, it suffices to show that the torsion subgroup of $A(K)$ is finite, because then there is an exact sequence (algebraically and topologically) $$0 \to P \to A(K) \to F \to 0$$ with $P$ a torsion-free pro-$p$-group and $F$ finite (shrink the pro-$p$-group such that it contains no torsion). By potential semistable reduction, one can assume $A/K$ semi-stable, and since it is clear for tori, $A/K$ of good reduction. Then use [Clark, Pete L. and Xarles, Xavier: Local bounds for torsion points on abelian varieties. In: Can. J. Math., 60(3) (2008), 532–555], Proposition 10 to show that the torsion subgroup of $A(K)$ is finite.

Note that in contrast to $p$-adic local fields, $A(K)/p$ is not finite, but uncountably infinite. (Use flat cohomology and the Kummer sequence for $A/K$ giving $$0 \to A(K)/p \to \mathrm{H}^1_\mathrm{fppf}(K,A[p]) \to \mathrm{H}^1(K,A)[p] \to 0$$ and [Milne, Arithmetic Duality Theorems], Theorem III.7.8.)

By [Serre, Galois cohomology], §1.1 exercise 1), $P$ is isomorphic to an infinite product of copies of $\mathbf{Z}_p$.

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