Heuristics for 2-class Towers of Cyclic Cubic Fields

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 groups 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.

Keywords:

• Cohen–Lenstra heuristics
• class field tower
• cyclic cubic field
• ideal class group

2010 Mathematics Subject Classification:

• 11R29
• 11R16

Correction Statement

This article has been republished with minor changes. These changes do not impact the academic content of the article.

Acknowledgments

We thank Brandon Alberts, Jordan Ellenberg, Farshid Hajir, Yuan Liu, John Voight, Jiuya Wang, and Melanie Matchett Wood for helpful conversations and feedback. We also thank the anonymous referees for their careful reading of the article and for making a number of helpful suggestions that improved the exposition. The work of MRB was partially supported by summer Lenfest Grants from Washington and Lee University.

Declaration of Interest

No potential conflict of interest was reported by the author(s).

Additional information

Funding

We thank Brandon Alberts, Jordan Ellenberg, Farshid Hajir, Yuan Liu, John Voight, Jiuya Wang, and Melanie Matchett Wood for helpful conversations and feedback. We also thank the anonymous referees for their careful reading of the article and for making a number of helpful suggestions that improved the exposition. The work of MRB was partially supported by summer Lenfest Grants from Washington and Lee University.