The m-step solvable anabelian geometry of number fields

Given a number field K and an integer m >= 0, let K-m denote the maximal m-step solvable Galois extension of K and write G(K)(m) for the maximal m-step solvable Galois group Gal (K-m/K) of K. In this paper, we prove that the isomorphy type of K is determined by the isomorphy type of G(K)(3). Further, we prove that K-m/K is determined functorially by G(K)(m+3) (resp. G(K)(m+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 G(K)(m), starting from G(K)(m+)(2).