Let $F$ be a finite extension of $\mathbb{Q}_p$ and $G$ the group of rational points of an algebraic group defined over $F$, endowed with the natural topology. Any Zariski closed subgroup $H \subset G$ is $\text{exp } \mathfrak{h}$ for some subalgebra $\mathfrak{h}$ of the Lie algebra $\mathfrak{g}$ of $G$, but what about subgroups which are closed in the $p$adic topology? For instance, the compact open subgroups of $G$ are very important, and I wonder if such a subgroup arises by exponentiating a compact open subgroup of $\mathfrak{g}$ (i.e. an $\mathcal{O}_F$lattice) closed under the Lie bracket. Is the situation better when $G$ is unipotent?

3$\begingroup$ Your statement about exp's is false: padic exp has severe convergence problems even for GL$_n$. Read Serre's book "Lie groups and Lie algebras", in which he develops a good Lie correspondence over any nonarchimedean field of characteristic 0 from scratch (and carries along the archimedean case, clarifying the special role of $\mathbf{Q}_p$ much as $\mathbf{R}$ has "better" features than $\mathbf{C}$ for a Lie correspondence, due to the density of $\mathbf{Q}$ in $\mathbf{R}$). Also see Bourbaki Lie Ch. III. You cannot expect to nail down exactly subgroups that are exp of their Lie algebra. $\endgroup$– grpSep 27, 2012 at 21:27
1 Answer
Let $G$ be a $p$adic Lie group. There is a 1:1 correspondence between $p$adically closed subgroups up to finite index of $G$ and Lie subalgebras of its Lie algebra (which is a Lie algebra over $\mathbb Q_p$). The correspondence works just as in the classical setting. This is shown in a paper by Mattuck from the fifties, and in much greater generality in Lazard's thesis (beware, beware).
Attention: The image of a closed subgroup $H \subseteq G$ under the logarithm map is not always a $\mathbb Z_p$ submodule of the Lie algebra. Example: take $p=2$ and $H \subseteq \mathrm{GL}_3$ the unipotent radical of the Borel. Then $\mathrm{log}(H(\mathbb Z_p))$ is not stable under $+$.
You cannot hope for such a correspondence for strictly all closed subgroups. Although the exponential map is a homeomorphism locally around zero, it can in general not be extended to a surjective map. There is already a problem with $\mathbb Z_p^\ast$.
Also, this does not make much sense over extensions of $\mathbb Q_p$. If $F$ is some nontrivial finite extension of $\mathbb Q_p$, then $F$ viewed as a Lie group under addition has many closed subgroups (all the $\mathbb Q_p$linear subspaces), but the Lie algebra (which is also $F$) has no proper $F$linear subalgebra.

2$\begingroup$ Concerning your final paragraph: things do make good sense over extensions $F$ of $\mathbf{Q}_p$ provided one replaces the purely topological viewpoint of "closed subgroups" with the more analytic viewpoint of "closed $F$analytic subgroups" taken up to clopen subgroups (and Lie $F$subalgebras of the ambient Lie algebra). This is discussed nicely in both Serre's book and Bourbaki. It is analogous to the fact that one has a good Lie correspondence over $\mathbf{C}$ but it requires going beyond the purely topological formulation that works well over $\mathbf{R}$. $\endgroup$– grpSep 28, 2012 at 2:26