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?



What are the LCA groups that are the Pontryagin dual of a locally profinite abelian group?
Totally disconnected LCA groups are (profinite)-by-(discrete) LCA groups. Hence their Pontryagin dual are (compact)-by-(discrete torsion) groups. These are precisely **locally elliptic** LCA groups.


(I'm using the kernel-by-quotient convention.)


A locally compact group is called locally elliptic if each of its compact subsets is contained in a compact (open) subgroup. That is, a directed union of compact open subgroups.


