The m-step Solvable Mono-anabelian Geometry of Number Fields

Abstract

In 1970’s, it was proven by Neukirch and Uchida that number fields are determined up to isomorphism by their absolute Galois groups which is known as the Neukirch-Uchida theorem. A pro-solvable version of the Neukirch-Uchida theorem was proven by Uchida in 1976. However, the structures of the Galois groups considered in both cases are unknown. In 2019, Saïdi and Tamagawa proved an m-step (for suitable positive integer m) solvable version of the Neukirch-Uchida theorem. In particular, they proved that number fields are determined up to isomorphism by the maximal 3-step solvable quotients of their absolute Galois groups. However, all results above do not provide a group-theoretic algorithm to reconstruct a number field from their (various quotients of) absolute Galois groups. In 2021, Hoshi established a group-theoretic algorithm, to reconstruct a number field (together with its maximal pro-solvable extension) from the maximal pro-solvable quotient of its absolute Galois group. The goal of this thesis is to develop a group-theoretic algorithm, to reconstruct a number field (together with its maximal m-step solvable extension) from the maximal m+6-step solvable quotient of its absolute Galois group. If K is an imaginary quadratic field or Q, we establish a group-theoretic reconstruction of K from the maximal 3-step solvable quotient of its absolute Galois group. Furthermore, we proved that an open continuous homomorphism between the maximal m+3-step solvable quotients of absolute Galois groups of number fields determines an open continuous homomorphism between the corresponding maximal m-step solvable quotients that arises from field embedding if and only if it is compatible with the cyclotomic characters.