Heuristics for $2$-class Towers of Cyclic Cubic Fields

TL;DR

This paper investigates the Galois groups of maximal unramified and ramified 2-extensions of cyclic cubic fields, proposing heuristics and conjectures about their distribution based on data and group-theoretic considerations.

Contribution

It introduces heuristics for the distribution of Galois groups in 2-class towers of cyclic cubic fields, extending Cohen-Lenstra heuristics to this context.

Findings

Identification of candidate pro-2 groups for Galois groups

Data supporting conjectural distribution patterns

Observations on the structure of 2-class groups

Abstract

We consider the Galois group $G_2(K)$ of the maximal unramified $2$-extension of $K$ where $K/\mathbb{Q}$ is cyclic of degree $3$. We also consider the group $G^+_2(K)$ where ramification is allowed at infinity. In the spirit of the Cohen-Lenstra heuristics, we identify certain types of pro-$2$ group as the natural spaces where $G_2(K)$ and $G^+_2(K)$ live when the $2$-class group of $K$ is $2$-generated. While we do not have a theoretical scheme for assigning probabilities, we present data and make some observations and conjectures about the distribution of such groups.