On the outer automorphism groups of the absolute Galois groups of mixed-characteristic local fields

Abstract

In the present paper, we study the outer automorphism groups of the absolute Galois groups of mixed-characteristic local fields from the point of view of anabelian geometry. Let us recall that it is well-known that the natural homomorphism from the automorphism group of a mixed-characteristic local field to the outer automorphism group of the absolute Galois group of the given mixed-characteristic local field is injective. One main result of the present paper is that if the mixed-characteristic local field satisfies certain conditions, then the set of conjugates of the image of this injective homomorphism in the outer automorphism group is infinite, which thus implies that the image of this injective homomorphism is not normal in the outer automorphism group. In particular, one may conclude that it is impossible to establish a functorial group-theoretic reconstruction, from the absolute Galois group, of the “field-theoretic” subgroup, i.e., the image of this injective homomorphism, of the outer automorphism group.