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

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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. Gₖ:=Gal(K∞/K) where K∞ is the maximal unramified p-extension of K. By class field theory, the maximal abelian quotient of Gₖ is isomorphic to the p-class group of K. For integers c≥ 1, we give a heuristic of Cohen-Lentra type for the maximal p-class c quotient of 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 σ-group occurs as Gₖ for infinitely many fields K. We present numerical data in support of these conjectures.