For certain subcategories of LCA groups, we have nice descriptions of the dual category under Pontryagin duality (all groups are implicitly assumed to be abelian):

finite groups $\leftrightarrow$ finite groups

discrete groups $\leftrightarrow$ compact groups

discrete torsion groups $\leftrightarrow$ profinite groups

discrete groups where each element is annhilated by some power of $p$ $\leftrightarrow$ pro $p$-groups

etc.

So I was wondering if we have a similar description of the Pontryagin dual of the category of abelian locally profinite groups, i.e. locally compact totally disconnected groups. Since locally profinite groups include discrete groups and profinite groups, the dual category will need to include discrete torsion groups and compact groups. Is there more we can say?

locally ellipticif each of its compact subsets is contained in a compact subgroup (which can be chosen to be open). A table mentioning this correspondence (totally disconnected vs locally elliptic) can be found p6 of arxiv.org/abs/1011.5333 – YCor Jun 11 at 19:08