Topics in the anabelian geometry of mixed-characteristic local fields

Abstract

In the present paper, we study the anabelian geometry of mixedcharacteristic 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 mixedcharacteristic local fields that arise from field automorphisms of the mixed-characteristic local fields.