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*.



Abelian varieties over $p$-adic fields

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> *Answered in the comments by user74230:*
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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 $\mathbb{Q}\_{p}$.


