# Harmonically balanced capitulation over quadratic fields of type (9,9)

## TL;DR

This paper investigates the structure of Galois groups of 3-class field towers over quadratic fields with specific class group types, using Artin patterns and computational group theory to identify new phenomena and bounds.

## Contribution

It introduces a method to determine Galois group types via Artin patterns and extends the SmallGroups database to analyze groups with harmonic transfer kernels.

## Key findings

- Identification of new group phenomena with harmonic transfer kernels
- Bounds for the relation rank depending on quadratic field signatures
- Extension of the SmallGroups database for larger p-groups

## Abstract

The isomorphism type of the Galois group G of finite 3-class field towers of quadratic number fields with 3-class group of type (9,9) is determined by means of Artin patterns which contain information on the transfer of 3-classes to unramified abelian 3-extensions. First, as an approximation of the group G, its metabelianization M=G/G", which is isomorphic to the Galois group of the second Hilbert 3-class field, is sought by sifting the SmallGroups library with the aid of pattern recognition. In cases with order |M|>3^8, the SmallGroups database must be extended by means of the p-group generation algorithm, which reveals new phenomena of groups with harmonically balanced transfer kernels and trees with periodic trifurcations. Bounds for the relation rank d2(M) of M in dependence on the signature of the quadratic base field admit the decision whether the derived length of G is dl(G)=2 or dl(G)>=3.