A Cohen-Lenstra Heuristic for Schur $\sigma$-Groups

TL;DR

This paper proposes a probabilistic heuristic model for the entire Galois group of maximal unramified pro-p extensions of imaginary quadratic fields, extending Cohen-Lenstra heuristics from abelian to non-abelian groups with symmetries.

Contribution

It develops a new heuristic framework for Schur σ-groups, analyzing their structure via probability measures on classes of pro-p groups with symmetries, generalizing Cohen-Lenstra's ideas.

Findings

Constructed a probability space for σ-isomorphism classes of weak Schur σ-groups.

Computed measures of basic subsets inversely proportional to automorphism group sizes.

Showed that classes with finite abelianization in all open subgroups have measure 1.

Abstract

For any odd prime $p$ and any imaginary quadratic field $K$, the $p$-tower group $G_K$ associated to $K$ is the Galois group over $K$ of the maximal unramified pro-$p$-extension of $K$. This group comes with an action of a finite group $\{1,\sigma\}$ of order $2$ induced by complex conjugation and is known to possess a number of other properties, making it a so-called Schur $\sigma$-group. Its maximal abelian quotient is naturally isomorphic to the $p$-primary part of the narrow ideal class group of ${\mathcal O}_K$, and the Cohen-Lenstra heuristic gives a probabilistic explanation for how often this group is isomorphic to a given finite abelian $p$-group. The present paper develops an analogue of this heuristic for the full group $G_K$. It is based on a detailed analysis of general pro-$p$-groups with an action of $\{1,\sigma\}$, which we call $\sigma$-pro-$p$-groups. We construct a probability space whose underlying set consists of $\sigma$-isomorphism classes of weak Schur $\sigma$-groups and whose measure is constructed from the principle that the relations defining $G_K$ should be randomly distributed according to the Haar measure. We also compute the measures of certain basic subsets, the result being inversely proportional to the order of the $\sigma$-automorphism group of a certain finite $\sigma$-$p$-group, as has often been observed before. Finally, we show that the $\sigma$-isomorphism classes of weak Schur $\sigma$-groups for which each open subgroup has finite abelianization form a subset of measure $1$.