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

Abstract

For any imaginary quadratic field $K$, the Galois group $G_K$ of its maximal unramified pro-$3$-extension is a Schur $\ddot{I}ƒ$-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 ${\mathrm{PGL}}_2$ over ${\mathbb{Q}}_3$. Combined with the Fontaine-Mazur conjecture, or with earlier work on an analogue of the Cohen--Lenstra heuristic for Schur $\ddot{I}ƒ$-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.

31 pages