The m-step solvable anabelian geometry of mixed-characteristic local fields

Let K_∘ and K_• be mixed-characteristic local fields, and let m be an integer ≥ 0. For each ∈ \ ∘, • \, let K_ᵐ denote the maximal m-step solvable extension of K_, and write GK_ᵐ for the maximal m-step solvable quotient of the absolute Galois group GK_ of K_; we equip GK_ and its quotients with filtration defined by the ramification groups in upper numbering. It is known from the previous bi-anabelian result due to Mochizuki that the natural map from the set of field isomorphisms K_∘→ K_• into the set of outer isomorphisms GK_∘ → GK_• of filtered profinite groups is bijective; in particular, the isomorphism class of K_∘ is determined by the isomorphism class of the filtered profinite group GK_∘. In this paper, we prove (with some mono-anabelian results) that the isomorphism class of K_∘ᵐ (hence of K_∘) is determined by the isomorphism class of GK_∘ᵐ⁺² as a filtered profinite group, and that the field isomorphism K_∘ᵐ→ K_•ᵐ is determined functorially by the isomorphism GK_∘ᵐ⁺³→ GK_•ᵐ⁺³ of filtered profinite groups.