Topics in the anabelian geometry of mixed-characteristic local fields

Abstract

In the present paper, we study the anabelian geometry of mixed-characteristic local fields by an algorithmic approach. We begin by discussing some generalities on log-shells of mixed-characteristic local fields. One main topic of this discussion is the difference between the log-shell and the ring of integers. This discussion concerning log-shells allows one to establish mono-anabelian reconstruction algorithms for constructing some objects related to the p-adic valuations. Next, we consider open homomorphisms between profinite groups of MLF-type. This consideration leads us to a bi-anabelian result for absolutely unramified mixed-characteristic local fields. Next, we establish some mono-anabelian reconstruction algorithms related to each of absolutely abelian mixed-characteristic local fields, mixed-characteristic local fields of degree one, and Galois-specifiable mixed-characteristic local fields. For instance, we give a mono-anabelian reconstruction algorithm for constructing the Norm map with respect to the finite extension determined by the uniquely determined minimal mixed-characteristic local subfield. Finally, we apply various results of the present paper to prove some facts concerning outer automorphisms of the absolute Galois groups of mixed-characteristic local fields that arise from field automorphisms of the mixed-characteristic local fields.