Honda formal group in unramified p-extension of local field as Galois module

Abstract

Let p be a rational prime. Let K/Qp, L/K, and M/L be finite extensions of local fields, let M/L be a Galois extension with Galois group G, and let F be a one-dimensional formal group law over the ring OK. We denote by F(mM) the abelian group whose underlying set is the maximal ideal mM of OM and the group operation is defined by the formula x+F y = F(x, y). The group F(mM) can be regarded as an EndO K(F)[G]-module with the natural action of G. The case of the multiplicative formal group law F = Fm is considered in detail in the paper of Borevich (1965), where the author describes the above-mentioned module structure by generators and defining relations when K = Qp and M/L is a cyclic p-extension. It is well known that if F is a Lubin—Tate formal group law, then there is an embedding of rings OK → EndO K(F), which makes it possible to consider F(mM) as an OK[G]-module. One natural generalization to the case of Lubin—Tate formal group laws is the paper of Vostokov (2014), where the structure of the OK[G]-module F(mM) is described in the case of an unramified cyclic p-extension of irregular fields M and L with equal indices of irregularity. Let K0/Qp be a finite extension, K/K0 be an unramified extension, π ∈ K0 be a uniformizing element in K0, F be a Honda formal group of finite height over the ring OK for the extension K/K0 and the prime π. Let Kalg be a fixed algebraic closure of K, mKalg be the valuation ideal, Wn F = ker[πn]F ⊂ F(mKalg ) be the πn-torsion submodule, and WF = ∞ n=1Wn F . In the present paper, we generalize the results of Vostokov to the case of a Honda formal group F. In particular, we describe the structure of the OK0 [G]-module F(mM) in the case of an unramified p-extension M/L for which the condition WF ∩ F(mL) = WF ∩ F(mM) = Ws F is valid for some s ≥ 1.