Classification of abelian $p$-adic Lie groups?

Question

Every real connected abelian Lie group is products of $\Bbb R/ \Bbb Z$ and $\Bbb R$. Are there any classification theorem of $p$-adic Lie groups (i.e those topological groups containing open uniform groups, or equivalently those p-analytic manifolds with compatible group structures) like the real case? As there is no global exp, things may be more difficult.

Answer 1

As I have said in response to other questions, others will be able to answer much better, and more fully, than I do below. But:

The fact that $p$-adic spaces are totally disconnected makes it very difficult to answer your question satisfactorily. If $k$ is a $p$-adic field, with ring of integers $\mathfrak o$, and if your Lie group is of dimension $n$ as a $k$-space, then there will be an open subgroup isomorphic to $\mathfrak o^n$, $n$-fold set-theoretic product. But such a subgroup is far from being unique.

The above relates to $p$-adic Lie groups, but if your Lie group has additional structure, for instance if it’s an algebraic group, or even better, an Abelian variety, much more can be said. I’ll leave it at that.