Schur $\ddot{I}ƒ$-groups of type (3,3)

Abstract

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