Cohen–Lenstra Moments for Some Nonabelian Groups

Abstract

Cohen and Lenstra detailed a heuristic for the distribution of odd p-class groups for imaginary quadratic fields. One such formulation of this distribution is that the expected number of surjections from the class group of an imaginary quadratic field k to a fixed odd abelian group is 1. Class field theory tells us that the class group is also the Galois group of the Hilbert class field, the maximal unramified abelian extension of k, so we could equivalently say the expected number of unramified G-extensions of k is 1/#Aut(G) for a fixed abelian group G. We generalize this to asking for the expected number of unramified G-extensions Galois over $\mathbb{Q}$ for a fixed finite group G, with no restrictions placed on G. We review cases where the answer is known or conjectured by Boston-Wood, Boston-Bush-Hajir, and Bhargava, then answer this question in several new cases. In particular, we show when the expected number is zero and give a nontrivial family of groups realizing this. Additionally, we prove the expected number for the quaternion group $Q_8$ and dihedral group $D_4$ of order 8 is infinite. Lastly, we discuss the special case of groups generated by elements of order 2 and give an argument for an infinite expected number based on Malle's conjecture.