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Theorem : Let $A$ be an abelian varieties of dimension $d$ over a field $k$, non-archimedian valued complete, i.e. $\mathbb{Q}_p$, then $A(k)$ contains a subgroup of finite index analytically isomorphic and homeomorphic to $I \oplus I \oplus \ldots \oplus I$ ($d$ summands) where $I$ is the ring of integers of $k$.

This is a theorem written in a paper by Arthur Mattock in 1955. I was wondering if someone knows where one can find any other proof of his theorem.

P.S. the name of the paper is Abelian varieties over $p$-adic ground fields.

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  • $\begingroup$ What do you mean by a "new proof?" $\endgroup$ Jan 18, 2015 at 0:15
  • $\begingroup$ Fixed!!. Noah S $\endgroup$ Jan 18, 2015 at 0:17
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    $\begingroup$ You omitted the necessary hypothesis that $k$ has characteristic 0. Over any such field, Serre's book "Lie groups and Lie algebras" proves something much more general: every analytic group manifold over such $k$ has a canonically associated Lie algebra (vanishing bracket in the commutative case) and has an open subgroup isomorphic to to the "Campbell-Baker-Hausdorff" group law. For trivial Lie bracket this is what you seek (up to scaling the coordinates), but only finite index for $A(k)$ when $k$ is finite degree over $\mathbf{Q}_p$. $\endgroup$
    – user74230
    Jan 18, 2015 at 1:27

1 Answer 1

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Answered in the comments by user74230:

You omitted the necessary hypothesis that $k$ has characteristic $0$. Over any such field, Serre's book "Lie groups and Lie algebras" proves something much more general: every analytic group manifold over such $k$ has a canonically associated Lie algebra (vanishing bracket in the commutative case) and has an open subgroup isomorphic to to the "Campbell-Baker-Hausdorff" group law. For trivial Lie bracket this is what you seek (up to scaling the coordinates), but only finite index for $A(k)$ when $k$ is finite degree over $\mathbb{Q}_{p}$.

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  • $\begingroup$ Thank you very much for you answer. I really cant understand one point. How exactly one defines the Campbell-Baker-Hausdorff formal group chunk!!! $\endgroup$ Jan 21, 2015 at 13:27

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