Honda formal group in unrami?ed p-extension of local ?eld as Galois module
Let p be a rational prime. Let K/Qp, L/K, and M/L be ﬁnite extensions of local ﬁelds, letM/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 formulax + y = F (x, y). The group F (mM ) can be regarded as an EndOK (F )[G]-module with theFnatural 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 isan embedding of rings OK '→ EndOK (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 Lwith 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 ﬁxedalgebraic closure of K, mKalg be the valuation ideal, W n = ker[πn]F ⊂ F (m) be theFπn-torsion submodule, and WF = l∞ nKalgn=1 WF. In the present paper, we generalize the resultsF is valid for some s ≥ 1.F is valid for some s ≥ 1.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 )= W s