Studying a connection between Iwasawa theory and the $K(n)$-local Picard group by some geometry on the Lubin-Tate stack

Question

Let $Pic_n^0$ denote the even part of the $K(n)$-local Picard group, and let $Pic_n^*$ denote $Hom(Pic_n^0, W(\mathbb{F}_{p^n})^x)$. Denote by $L$ the profinite group ring $\mathbb{Z}_p[[Pic_n^*]] $. Let $\lambda$ be an element of $Pic_n$; this gives a map $Pic_n^* \to W(\mathbb{F}_{p^n})^x$, given by evaluation at $\lambda$. This induces a ring map $L \to W(\mathbb{F}_{p^n})$. We can consider the $Pic_n$-graded homotopy group of a ($K(n)$-local) spectrum $X$. What we want is a "universal" $L$-module $P(X)$ such that $P (x)_{L}W(\mathbb{F}_{p^n})$, where we regard $W(\mathbb{F}_{p^n})$ as a $L$-module via $\lambda$.

At height $1$, this problem is not too hard: we know that $Pic_1^0 = \mathbb{Z}_p^x$, so that $Pic_1^* = \mathbb{Z}_p^x$. If $\psi$ is any topological generator of $\mathbb{Z}_p$, then $L$ is isomorphic to $p-2$ copies of $\mathbb{Z}_p[[T]]$, where $T$ corresponds to $\psi-1$. Let $P$ denote $\mathbb{Z}_p$, with the trivial $Pic_1^*$-action. The other action on $\mathbb{Z}_p$ coming from an element $\lambda$ is given by $g*x = g(\lambda) x$; here, we are thinking of $End(\mathbb{Z}_p^x)$ as $\mathbb{Z}_p^x$. In this case, for a fixed lambda with $\lambda(k) = k^m, P (x)_L \mathbb{Z}_p = \mathbb{Z}_p[[T]] = \mathbb{Z}_p/(k^m-1)$. But $k^m-1$ is a unit unless $(p-1)|m$, and in that case, $k^{(p-1)*q}-1 = u*p*q$ where $u$ denotes the unit. It follows that $\pi_{(p-1)*q} L_K(1) S = \mathbb{Z}/(p*q)$, as standard computations tell us.

Can this connection with Iwasawa theory be studied via some geometry on the Lubin-Tate stack?

Answer 1

A long time ago there were various attempts to make this sort of approach work, but they were not very successful. If you want to try again then you should make sure that you are familiar with the phenomena observed by computation in the height two case (which are quite complex). The computations are due to Shimomura, Yabe and their collaborators. One perspective on the answer is explained in Section 15.2 of the memoir Morava $K$-theories and localisation (by Mark Hovey and myself), and another in the paper The homotopy groups of the $E(2)$ local sphere at $p > 3$ revisited (by Mark Behrens).