Nilpotent local class field theory

Abstract

n=1 Ln(A) be the universal graded Lie algebra associated to A (see §2 for exact definitions). Any homomorphism φ of A into G/G(2) gives rise to a homomorphism φ∗ of L(A) into L(G). In this paper we study the special situation that A is the profinite completion K× of the multiplicative group K× of a local field K, i.e. a field which is complete with respect to a discrete valuation with finite residue class field. The group G is the absolute Galois group GK of K and φ is the Artin isomorphism of K× onto GK/G (2) K . The surjectivity of φ implies the same for φ∗. The goal of this paper is the determination of the kernel of φ∗. This is equivalent to the determination of the kernel of component homomorphisms