Groups of automorphisms of local fields of period p and nilpotent class < p, I.

Abstract

Suppose KK is a finite field extension of Qpℚp containing a primitive ppth root of unity. Let K<pK<p be a maximal pp-extension of KK with the Galois group of period pp and nilpotent class <p<p. In this paper, we develop formalism which allows us to study the structure of Γ<p=Gal(K<p/K)Γ<p=Gal(K<p/K) via methods of Lie theory. In particular, we introduce an explicit construction of a Lie Fp????p-algebra LL and an identification Γ<p=G(L)Γ<p=G(L), where G(L)G(L) is a pp-group obtained from the elements of LL via the Campbell–Hausdorff composition law. In the next paper, we apply this formalism to describe the ramification filtration {Γ(v)<p}v≥0{Γ<p(v)}v≥0 and an explicit form of the Demushkin relation for Γ<pΓ<p.