Heuristics for $p$-class towers of imaginary quadratic fields, with an Appendix by Jonathan Blackhurst

Cohen and Lenstra have given a heuristic which, for a fixed odd prime $p$, leads to many interesting predictions about the distribution of $p$-class groups of imaginary quadratic fields. We extend the Cohen-Lenstra heuristic to a non-abelian setting by considering, for each imaginary quadratic field $K$, the Galois group of the $p$-class tower of $K$, i.e. $G_K:=\mathrm{G}\mathrm{a}\mathrm{l}(K_{\mathrm{\infty }}\mathrm{/}K)$ where $K_{\mathrm{\infty }}$ is the maximal unramified $p$-extension of $K$. By class field theory, the maximal abelian quotient of $G_K$ is isomorphic to the $p$-class group of $K$. For integers $c\ge 1$, we give a heuristic of Cohen-Lentra type for the maximal $p$-class $c$ quotient of ${\text{G}}_K$ and thereby give a conjectural formula for how frequently a given $p$-group of $p$-class $c$ occurs in this manner. In particular, we predict that every finite Schur $\sigma$-group occurs as $G_K$ for infinitely many fields $K$. We present numerical data in support of these conjectures.