A Cohen-Lenstra Heuristic for Schur $σ$-Groups

TL;DR

This work extends the Cohen–Lenstra philosophy from abelian class groups to the full non-abelian $p$-tower groups $G_K$ attached to imaginary quadratic fields. It develops the theory of σ-pro-$p$-groups, constructs a probabilistic space $\mathrm{Sch}$ of σ-isomorphism classes via Haar-distributed random relations, and defines a limiting measure $\mu_\infty$ that assigns explicit masses to basic events. A key result is that the subset of strong Schur σ-groups has measure 1, supporting a coherent non-abelian heuristic for $G_K$ and aligning abelian and non-abelian statistics through masses like $\mu_\infty(\{[G]\})=C_\infty/|\mathrm{Aut}_σ(G)|$. The framework yields exact mass formulas for finite quotients, connects with known Golod–Shafarevich finiteness phenomena, and provides a principled probabilistic prediction for the distribution of $G_K$ among imaginary quadratic fields, including finite vs infinite behavior and special Zassenhaus types. Overall, the paper supplies a rigorous probabilistic model for non-abelian $p$-tower groups and a bridge between Cohen–Lenstra-type abelian predictions and non-abelian arithmetic statistics.

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 $\{1,σ\}$ of order $2$ induced by complex conjugation and is known to possess a number of other properties, making it a so-called Schur $σ$-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 $\{1,σ\}$, which we call $σ$-pro-$p$-groups. We construct a probability space whose underlying set consists of $σ$-isomorphism classes of weak Schur $σ$-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 $σ$-automorphism group of a certain finite $σ$-$p$-group, as has often been observed before. Finally, we show that the $σ$-isomorphism classes of weak Schur $σ$-groups for which each open subgroup has finite abelianization form a subset of measure $1$.

Theorems & Definitions (62)

• Proposition 2.1
• Proposition 2.2
• Proposition 3.3
• Proposition 3.4
• Proposition 3.5
• Proposition 3.6
• Proposition 3.7
• Proposition 4.1
• Proposition 4.2
• Proposition 4.3