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arXiv · 2012.14824

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

Nigel Boston·Michael R. Bush
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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.

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Nigel Boston, Michael R. Bush. 2020-12-29. Heuristics for $2$-class Towers of Cyclic Cubic Fields. https://arxiv.org/abs/2012.14824

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