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

TL;DR

This paper characterizes certain Schur σ-groups of type (3,3), showing they are isomorphic to open subgroups of PGL_2 over Q_p, and discusses implications for number theory conjectures.

Contribution

It proves that Schur σ-groups of Zassenhaus type (3,3) with finite abelianization of open subgroups are isomorphic to open subgroups of PGL_2 over Q_p.

Findings

Schur σ-groups of type (3,3) are isomorphic to open subgroups of PGL_2 over Q_p.

Supports conjectures related to Cohen-Lenstra heuristics and Fontaine-Mazur.

Provides structural classification of these groups.

Abstract

For any odd prime $p$, the Galois group of the maximal unramified pro-$p$-extension of an imaginary quadratic field is a Schur $\sigma$-group. But Schur $\sigma$-groups can also be constructed and studied abstractly. We prove that if $p>3$, any Schur $\sigma$-group of Zassenhaus type $(3,3)$, for which every open subgroup has finite abelianization, is isomorphic to an open subgroup of a form of ${\rm PGL}_2$ over ${\mathbb Q}_p$. Combined with earlier work on an analogue of the Cohen-Lenstra heuristic for Schur $\sigma$-groups, or with the Fontaine-Mazur conjecture, this lends credence to the ``if'' part of a conjecture of McLeman.