I am curious to know if we can somehow relate to the study of local class field theory through Lubin-Tate formal group with the study of class number of a field (global field in general) in class field theory.

As far as I have understood Lubin-Tate formal group, it gives a parallel interpretation to local class field theory by construction of maximal abelian extension of a local field and then defining the local artin reciprocity map.

While class numbers are generally discussed in the context of global fields (number fields), is there an analogous local notion in the study of local fields?

May be in that case, one can study the class number pattern of the tower of extensions of a local field generated through the $p^n$-torsion points can be connected to the structure of Lubin-Tate formal group. For example, Theorem 1 is a result where Lubin-Tate tower finds its place. What is your opinion?

Thanks