Weekly Bulletin

Fix an odd prime $p$. For any imaginary quadratic field $K$, the Galois group $G_K$ of its maximal unramified pro-$p$-extension is a Schur $\sigma$-group. If the minimal number of generators of $G_K$ is at least two, then work of Golod–Shafarevich and many others implies that $G_K$ is infinite unless it has Zassenhaus type $(3,3)$, $(3,5)$, or $(3,7)$. In these three remaining cases, there is no characterization of the pairs $(K,p)$ for which $G_K$ is finite. In most cases when the Zassenhaus type is $(3,3)$, we already know that $G_K$ is $p$-adic analytic and should therefore be finite according to the Fontaine-Mazur conjecture. Regardless of that, it is an interesting problem to classify all Schur $\sigma$-groups of type $(3,3)$. In this talk, I will present recent and ongoing joint work with Richard Pink towards such a classification.