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

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 $p$ and even extension degree over ${\mathbb Q}_p$, 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 conditions, then the set of conjugates of the subgroup arising from field automorhisms in the outer automorphism group is infinite, which thus implies that this subgroup is not normal in the outer automorphism group. On the other hand, for an odd prime number $p$, Hoshi proved the existence of an irreducible Hodge-Tate $p$-adic representation of dimension two of the absolute Galois group of a $p$-adic local field and an automorphism of the absolute Galois group such that the $p$-adic Galois representation obtained by pulling back the given $p$-adic Galois representation by the given automorphism is not Hodge-Tate. As a result in a direction that is different from the above direction, another main result of the present paper is the existence of a pair of a representation and an automorphism that is similar to the above pair in the case where $p=2$.