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

TL;DR

This paper investigates the outer automorphism groups of absolute Galois groups of mixed-characteristic local fields, revealing new structural properties and connections to mapping class groups within anabelian geometry.

Contribution

It introduces novel results on the non-normality of certain automorphism images and constructs specific representations, extending previous work with a new analogy to mapping class groups.

Findings

The image of the automorphism group is not a normal subgroup.

Existence of a continuous automorphism leading to a non-Hodge-Tate representation.

Generalization of prior results by Hoshi and Nishio.

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. In particular, we show that, under certain mild assumptions, the image of the natural homomorphism from the automorphism group of a mixed-characteristic local field to the outer automorphism group of the associated absolute Galois group is not a normal subgroup. Furthermore, we show that, for the absolute Galois group of a mixed-characteristic local field satisfying certain assumptions, there exist a continuous representation and a continuous automorphism of the group such that the former is irreducible, abelian, and crystalline, but the continuous representation obtained as the composite of the former with the latter is not even Hodge-Tate. These results significantly generalize previous works by Hoshi and Nishio. A key observation in obtaining these results is to focus on the analogy between the mapping class groups of topological surfaces and the outer automorphism groups of the absolute Galois groups of mixed-characteristic local fields. To the best of the author's knowledge, this is the first work applying results from the theory of mapping class groups to the anabelian geometry of mixed-characteristic local fields, going beyond a mere analogy between the two.