Extension of the invertible-adele action to the completed level-structured lattice space

Determine whether the action of the group of invertible finite adeles on the space of rank-r lattices with level structure extends to the completion LLNRi in a sufficiently nice way, and, if so, construct such an extension; in particular, establish whether this extension is possible at all.

Background

The paper constructs an action of the group of invertible finite adeles on the space of rank-r lattices with level structure, and separately shows that the induced action of fractional ideals on the completed space of lattices of rank at most r is by homeomorphisms. The completion LLNRi additionally records inverse level structures, so extending the adele action requires controlling how these level structures behave under passage to boundary strata.

The authors explicitly state that they have not obtained an extension to LLNRi that is sufficiently well behaved and even suspect that no such extension exists, while leaving that suspicion unproved. Resolving this issue would determine whether the adelic or fractional-ideal action can be carried to the completed level-structured space and consequently to the associated modular forms.