3-class field towers with 2 or 3 stages

TL;DR

This paper characterizes the Galois groups of unramified 3-class field towers over quadratic fields with specific 3-class groups and principalization types, identifying conditions for tower length and structure.

Contribution

It provides necessary and sufficient conditions for the Galois group to coincide with the second-level Galois group, and explores the structure of non-metabelian towers with unbounded derived length.

Findings

Conditions for Galois group coincidence established

Infinite non-metabelian Galois groups with shared metabelianization identified

Minimal discriminants for fixed tower length determined experimentally

Abstract

For quadratic fields \(k=\mathbb{Q}(\sqrt{d})\) with discriminant \(d\), \(3\)-class group \(\mathrm{Cl}_3(k)\simeq (\mathbb{Z}/3\mathbb{Z})^2\), and four \textit{simple} \(3\)-principalization types \(\varkappa(k)\in\lbrace (1122),(3122),(1231),(2231)\rbrace\), we establish necessary and sufficient conditions for the Galois group \(S=\mathrm{Gal}(\mathrm{F}_3^\infty(k)/k)\) of the unramified Hilbert \(3\)-class field tower of \(k\) to coincide with the Galois group \(M=\mathrm{Gal}(\mathrm{F}_3^2(k)/k)\) of the maximal metabelian unramified \(3\)-extension of \(k\). In the case of non-coincidence, we study the path between \(M\) and \(S\) in the descendant tree of the elementary bicyclic \(3\)-group \((\mathbb{Z}/3\mathbb{Z})^2\). For two \textit{complex} \(3\)-principalization types \(\varkappa(k)\in\lbrace (2122),(4231)\rbrace\), we show that infinitely many non-metabelian possible Galois groups \(S=\mathrm{Gal}(\mathrm{F}_3^\infty(k)/k)\) with presumably unbounded derived length \(\mathrm{dl}(S)\) share a common metabelianization \(M=S/S^{\prime\prime}\), whence only partial criteria can be stated. Minimal discriminants \(d>0\) with assigned simple \(3\)-principalization type \(\varkappa(k)\) and fixed length \(\ell_3(k)\in\lbrace 2,3\rbrace\) of the \(3\)-class field tower are determined experimentally for nilpotency class \(\mathrm{cl}(M)\in\lbrace 5,7,9,11\rbrace\) under assumption of the generalized Riemann hypothesis.