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

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.

💡 Research Summary

The paper investigates the outer automorphism groups Out(Gₖ) of absolute Galois groups Gₖ attached to mixed‑characteristic local fields k (a finite extension of ℚₚ). The central theme is to understand how the field‑theoretic automorphism group Aut(k) sits inside Out(Gₖ) and to explore the impact of outer automorphisms on p‑adic Hodge‑theoretic properties of continuous representations of Gₖ.

Background and Motivation
Classical anabelian geometry asks whether a field can be recovered from its absolute Galois group. While the functor k ↦ Gₖ is faithful, it is not full in general; there exist non‑isomorphic fields with isomorphic absolute Galois groups. The subgroup Aut(k) embeds naturally into Out(Gₖ) via the canonical map Aut(k) → Aut(Gₖ) → Out(Gₖ). Earlier work (Neukirch–Uchida, Hoshi–Nishio) gave ring‑theoretic characterizations of Aut(k) inside Out(Gₖ) but did not describe the difference between the two groups.

Main Results

1. Theorem I – Non‑normality and infinite conjugacy classes
   Assumptions: p is an odd prime, k/ℚₚ is a finite Galois extension of degree > 1.
   Conclusion: The subgroup Aut(k) is not normal in Out(Gₖ). Moreover, the set of Out(Gₖ)‑conjugates of Aut(k) is infinite; consequently there are infinitely many distinct subgroups of Out(Gₖ) isomorphic to Aut(k).
   Proof strategy:
   – When