Honda formal group as Galois module in unramified extensions of local fields

Abstract

For given rational prime number $p$ consider the tower of finite extensions of fields $K_0/{\mathbb{Q}}_p,$ $K/K_0,L/K,M/L$, where $K/K_0$ is unramified and $M/L$ is a Galois extension with Galois group $G$. Suppose one dimensional Honda formal group over the ring ${\mathcal{O}}_K$, relative to the extension $K/K_0$ and uniformizer $\pi \in K_0$ is given. The operation $x\underset{F}{+}y=F(x,y)$ sets a new structure of abelian group on the maximal ideal ${\mathfrak{p}}_M$ of the ring ${\mathcal{O}}_M$ which we will denote by $F({\mathfrak{p}}_M)$. In this paper the structure of $F({\mathfrak{p}}_M)$ as ${\mathcal{O}}_{K_0}[G]$-module is studied for specific unramified $p$-extensions $M/L$.