Une théorie du corps de classes local non abélien

Abstract

A non-Abelian generalization of local class field theory, which can describe non-abelian Galois groups in terms of the ground field, is given. The absolute Galois group of a local field is canonically defined up to an inner automorphism and the natural approach of describing class functions on it leads to Langland's philosophy. Another way to describe the absolute Galois group is to rigidify its structure by means of an arbitrary lifting of the Frobenius automorphism. This point of view is used in the metabelian local class field theory of Koch and de Shalit and its generalization by Gurevich. Iterating a slightly modified version of metabelian local class field theory on some fields of norms leads to a ‘non-abelian local class field theory’.