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

TL;DR

This paper investigates the module structure of Honda formal groups over unramified extensions of local fields, revealing how these structures behave as Galois modules in specific unramified p-extensions.

Contribution

It provides a detailed analysis of Honda formal groups as Galois modules in unramified extensions, a novel perspective in local field theory.

Findings

Describes the module structure of Honda formal groups over unramified extensions

Establishes the Galois module properties of formal groups in specific extensions

Enhances understanding of formal groups in local field Galois theory

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 $M/L$.