Two-step nilpotent extensions are not anabelian

TL;DR

The paper demonstrates that certain two-step nilpotent quotients of absolute Galois groups do not uniquely determine the underlying number fields, showing non-anabelian phenomena in number theory.

Contribution

It constructs explicit examples of non-isomorphic number fields with isomorphic two-step nilpotent Galois quotients and describes these groups combinatorially using a generalized Rado graph.

Findings

Existence of non-isomorphic fields with isomorphic two-step nilpotent Galois quotients

Explicit combinatorial description of these Galois groups

Application of the back-and-forth method from model theory

Abstract

We prove the existence of two non-isomorphic number fields $K$ and $L$ such that the maximal two-step nilpotent quotients of their absolute Galois groups are isomorphic. In particular, one may take $K$ and $L$ to be any of the imaginary quadratic number fields of discriminant -11, -19, -43, -67, -163. Furthermore, we give an explicit combinatorial description of these Galois groups in terms of a generalization of the Rado graph. A critical ingredient in our proofs is the back-and-forth method from model theory.