The m-step solvable anabelian geometry of number fields

Abstract

Given a number field K and an integer m≥0, let Km denote the maximal m-step solvable Galois extension of K and write GKm for the maximal m-step solvable Galois group Gal(Km/K) of K. In this paper, we prove that the isomorphy type of K is determined by the isomorphy type of GK3. Further, we prove that Km/K is determined functorially by GKm+3 (resp. GKm+4) for m≥2 (resp. m≤1. This is a substantial sharpening of a famous theorem of Neukirch and Uchida. A key step in our proof is the establishment of the so-called local theory, which in our context characterises group-theoretically the set of decomposition groups (at nonarchimedean primes) in GKm, starting from GKm+2.