Is there any relationship between the study of class number of a number field with the study of class field theory through Lubin-Tate formal group?

Question

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

Answer 1

The ring of integers in a local field is a PID: every nonzero ideal is a power of the maximal ideal and the maximal ideal is principal.

The Hilbert class field of a number field $K$ is its maximal abelian extension unramified at all places of $K$. Its Galois group over $K$ is isomorphic to the ideal class group, so that Galois group is abelian and need not be cyclic. Local fields, on the other hand, have unramified extensions of every degree and they are all cyclic extensions, making the maximal unramified extension of a local field a pro-cyclic extension.

In a positive direction, global objects of a number field $K$ can be seen in some structure related to completions. The class number of a totally real abelian number field $K$ appears as a factor in the formula for the residue of the $p$-adic zeta-function of $K$ for each prime $p$. See Theorem 5.24 in Washington's book on cyclotomic fields. This can be used to prove theorems about the $p$-divisibility of some class numbers,such as $p \mid h(\mathbf Q(\zeta_p)) \Leftrightarrow p \mid h^+(\mathbf Q(\zeta_p))$. And Iwasawa theory provides formulas for the $p$-power in the class numbers of every sufficiently high layer in a $\mathbf Z_p$-extension of a number field.