Anabelian geometry has been developed over a much wider class of fields than Grothendieck, who is the originator of anabelian geometry, conjectured. So, it is natural to ask the following question: What kinds of fields are suitable for the base fields of anabelian geometry?In the present paper, we consider this problem for higher local fields. First, to consider "anabelianness" of higher local fields themselves, we give mono-anabelian reconstruction algorithms of various invariants of higher local fields from their absolute Galois groups. As a result, the isomorphism classes of certain types of higher local fields are completely determined by their absolute Galois groups. Next, we prove that mixed-characteristic higher local fields are Kummer-faithful. This result affirms the above question for these higher local fields to a certain extent.

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Let K be a eld, G K the absolute Galois group of K, X a hyperbolic curve over K, and π 1 (X) the étale fundamental group of X. The absolute Grothendieck conjecture in anabelian geometry asks: Is it possible to recover X group-theoretically, solely from π 1 (X) (not π 1 (X) ↠ G K)? When K is a p-adic eld (i.e. a nite extension of Q p), this conjecture (called the p-adic absolute Grothendieck conjecture) is unsolved. To approach this problem, we introduce a certain p-adic analytic invariant dened by Serre (which we call i-invariant). Then, the absolute p-adic Grothendieck conjecture can be reduced to the following problems: (A) determining whether a proper hyperbolic curve admits a rational point from the data of i-invariants of the sets of rational points of the curve and its coverings; (B) recovering the i-invariant of the set of rational points of a proper hyperbolic curve group-theoretically. The main results of the present paper give a complete armative answer to (A) and a partial armative answer to (B).