Schur sigma-groups with abelian quotient invariants (9,3)

TL;DR

This paper constructs specific non-metabelian Schur sigma-groups with particular properties to provide evidence for a new class of imaginary quadratic fields having a three-stage 3-class field tower, differing from previously known examples.

Contribution

It introduces a novel construction of Schur sigma-groups with type (9,3) and demonstrates their connection to new imaginary quadratic fields with three-stage 3-class towers.

Findings

Existence of non-metabelian Schur sigma-groups with lo(S)=21 and cl(S)=9.

Identification of a new class of imaginary quadratic fields with specific 3-class group structure.

These fields have a three-stage 3-class field tower with a unique principalization type.

Abstract

By the construction of suitable non-metabelian Schur sigma-groups S of type (9,3) with log order lo(S) = 21 and nilpotency class cl(S) = 9, evidence is provided of a new class of imaginary quadratic fields K with 3-class group Cl(3,K) ~ C(9) * C(3) and punctured principalization type kappa ~ (1,4,4;4) whose 3-class field tower consists of precisely three stages. In contrast, previous examples of three-stage towers were associated with kappa in { (1,1,2;2), (1,1,2;3), (1,1,4;2), (1,2,3;1) }, lo(S) = 9 and cl(S) = 5.