Honda Formal Module in an Unramified p-Extension of a Local Field as a Galois Module

Abstract

For a fixed rational prime number p, consider a chain of finite extensions of fields K₀/ℚ_p, K/K₀, L/K, and M/L, where K/K₀ is an unramified extension and M/L is Galois extension with Galois group G. Suppose that a one-dimensional Honda formal group F over the ring \(\mathcal{O}_K\) relative to the extension K/K₀ and a uniformizing element π ∈ K₀ is given. This paper studies the structure of \(F(\mathfrak{m}_M)\) as an \(\mathcal{O}_{K_0}\)[G]-module for an unramified p-extension M/L provided that \(W_F\cap{F({\frak{m}}_L)}=W_F\cap{F({\frak{m}}_M)}=W_F^s\) for some s ≥ 1, where W_F^s is the π^s-torsion and W_F = ∪_n=1^∞W_Fⁿ is the complete π-torsion of a fixed algebraic closure K^alg of the field K.