A Prescribed Class-Group Structure for an Imaginary Quadratic Field

The study of class groups of quadratic fields goes back to Gauss, whose conjectures about them have helped shape number theory for over two centuries. The size of the class group $G$ of the imaginary quadratic field with discriminant $-D$ is known by Brauer–Siegel to be $\asymp D^{1/2+o(1)}$. The structure of these class groups is much more mysterious. Gauss’ genus theory allows us to determine the $2$-torsion subgroup $G[2]$ of $G$, in terms of the number of prime factors of $D$, but very little is known about other torsion pieces or the group structure as a whole. Since the structure of the $2$-torsion subgroups of class groups is well understood, it is common to instead study the group $C(D):=G/G[2]$.

The Cohen–Lenstra heuristics provide a detailed framework to study the statistics of the groups $C(D)$, and they predict that - keeping the size fixed groups $H$ should occur with frequency proportional to

Normalizing for Brauer–Siegel, we therefore expect a group $H$ of size to occur an expected $\asymp |H|^2/|\operatorname{Aut}(H)|$ number of times, at least on average over $|H|$. (This passes a quick sanity check: we expect most groups to be ~cyclic, and $\mathbb{Z}/n\mathbb{Z}$ is predicted to occur $\sim n$ times, which accounts for $\sim X^2$ groups of size $\leq X$, in line with Brauer–Siegel.) Note that this heuristic (which goes beyond Cohen–Lenstra) is discussed in the work of Holmin, Jones, Kurlberg, McLeman, and Petersen; see also Soundararajan’s work on imaginary quadratic fields with a given class number.

Our question requires finding $D$ such that $C(D)$ is isomorphic to $(\mathbb{Z}/n\mathbb{Z})^2\times(\mathbb{Z}/n^3\mathbb{Z})$ for some $n$. The size of this group is $n^5$ and the automorphism group of this group has size $\sim n^{11}$. Thus the odds for each $n$ of this group occurring is $\sim \frac 1n$, and we therefore expect this group to occur, on average, a constant number of times in each dyadic range. In particular, we expect that groups of this form occur infinitely often as $C(D)$. We call such $D$’s good.

Currently, it is completely unknown how to construct infinitely many quadratic fields while constraining the entire class group. The most famous example of this is the Gauss class number $1$ conjecture for real quadratic fields: Gauss conjectured that there are infinitely many real quadratic fields with trivial class group, and this is completely open. Consequently, we (the authors of this problem) have no better way of finding good $D$’s than by doing a brute force search. This becomes impractical in large ranges, where we expect good $D$’s to become (logarithmically) sparse.

Empirical testing seems to be in rough accordance with the above: the set of good $D$’s up to a billion is quite small – there are just $65$ of them. These include $19$ with $n=2$, $35$ with $n=3$, $9$ with $n=4$, and $2$ with $n=5$. These are also quite reasonably distributed in dyadic ranges. The top nine dyadic ranges going down from a billion contain respectively $14$, $9$, $6$, $5$, $7$, $7$, $5$, $4$, and $7$ occurrences of good $D$’s, with just a single good $D$ below $5^9$.