The m-step solvable anabelian geometry of number fields

Abstract

Abstract Given a number field K and an integer m ≥ 0 {m\geq 0} , let K m {K_{m}} denote the maximal m-step solvable Galois extension of K and write G K m {G_{K}^{m}} for the maximal m-step solvable Galois group Gal ⁡ ( K m / K ) {\operatorname{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 {G_{K}^{3}} . Further, we prove that K m / K {K_{m}/K} is determined functorially by G K m + 3 {G_{K}^{m+3}} (resp. G K m + 4 {G_{K}^{m+4}} ) for m ≥ 2 {m\geq 2} (resp. m ≤ 1 {m\leq 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 {G_{K}^{m}} , starting from G K m + 2 {G_{K}^{m+2}} .