On the Outer Automorphism Groups of the Absolute Galois Groups of 2-adic local Fields
Yu Nishio

TL;DR
This paper investigates the structure of outer automorphism groups of absolute Galois groups of 2-adic local fields, revealing non-normal subgroups and constructing specific Galois representations with particular properties.
Contribution
It proves that certain subgroups arising from field automorphisms are not normal in the outer automorphism group of 2-adic local fields and constructs specific Galois representations related to automorphisms.
Findings
Subgroups from field automorphisms are not normal in the outer automorphism group.
The set of conjugates of these subgroups is infinite under certain conditions.
Existence of special Galois representations and automorphisms for p=2.
Abstract
In the present paper, we study the outer automorphism groups of the absolute Galois groups of 2-adic 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. Moreover, Hoshi and the author of the present paper proved that, for absolutely abelian mixed-characteristic local fields with odd residue characteristic and even extension degree over , this subgroup arising from field automorphisms is not normal in the outer automorphism group and has infinitely many distinct conjugates. As a result in this direction, one of the main results of the present paper is the assertion that if a 2-adic local field satisfies certain…
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Taxonomy
TopicsAlgebraic Geometry and Number Theory · Advanced Algebra and Geometry · advanced mathematical theories
