Schur $\sigma$-groups of type $(3,3)$ for $p=3$

TL;DR

This paper classifies certain Galois groups of imaginary quadratic fields, showing most are finite or related to $ m PGL_2$ over $Q_3$, supporting conjectures about their structure and distribution.

Contribution

It proves that for most cases, the Galois groups are either finite or related to $ m PGL_2$, providing evidence for conjectures on their structure and distribution.

Findings

Most Galois groups are finite or isomorphic to open subgroups of $ m PGL_2(Q_3)$.

Explicit computations support the heuristic predictions for fields with discriminant up to $-10^8$.

Results lend credence to conjectures relating to Schur $\sigma$-groups and the Fontaine-Mazur conjecture.

Abstract

For any imaginary quadratic field $K$, the Galois group $G_K$ of its maximal unramified pro-$3$-extension is a Schur $\sigma$-group. If this has Zassenhaus type $(3,3)$, there are 13 possibilities for the isomorphism class of the finite quotient $G_K/D_4(G_K)$. We prove that for 10 of these 13 cases $G_K$ is either finite or isomorphic to an open subgroup of a form of $\mathop{\rm PGL}_2$ over $\mathbb{Q}_3$. Combined with the Fontaine-Mazur conjecture, or with earlier work on an analogue of the Cohen--Lenstra heuristic for Schur $\sigma$-groups, this lends credence to the "if" part of a conjecture of McLeman. Using explicit computations of triple Massey products, we also test the heuristic for all imaginary quadratic fields $K$ with $d(G_K)=2$ and discriminant $-10^8 < d_K < 0$ and find a reasonably good agreement.