A prescribed class-group structure for imaginary quadratic fields

Class groups of imaginary quadratic fields are well understood on average but very hard to control one field at a time. This problem asks for three huge fields, with discriminants beyond $10^{40}$, whose class groups modulo 2-torsion all have the unusual shape $(\mathbb{Z}/n)^2 \times \mathbb{Z}/n^3$, for three different values of $n$.

Problem

For a squarefree integer $D > 0$, let $G = \mathrm{Cl}(\mathbb{Q}(\sqrt{-D}))$ be the class group and $C(D) = G / G[2]$ its quotient by the 2-torsion subgroup.

Task. Find three distinct squarefree integers $D_1, D_2, D_3 > 10^{40}$ such that

$$C(D_i) \cong (\mathbb{Z}/n_i\mathbb{Z})^2 \times \mathbb{Z}/n_i^3\mathbb{Z} \qquad (i = 1, 2, 3)$$

for integers $n_1, n_2, n_3 > 1$ that are pairwise distinct.

Background

Gauss's genus theory determines $G[2]$ from the prime factors of $D$, so it is natural to study $C(D)$, whose structure is far more mysterious. The Cohen–Lenstra heuristics [1] predict that, among groups of a given size, a group $H$ occurs with frequency proportional to $1/|\mathrm{Aut}(H)|$; work of Holmin, Jones, Kurlberg, McLeman and Petersen [2] and of Soundararajan [3] studies how often individual groups occur. Combining this with the Brauer–Siegel size $|G| = D^{1/2 + o(1)}$, the problem's authors estimate that a group of this shape (order $n^5$, automorphism group of order about $n^{11}$) should appear a constant number of times on average in each dyadic range of $D$, hence infinitely often. Yet no method is known for constructing infinitely many quadratic fields with a prescribed whole class group; even Gauss's conjecture that infinitely many real quadratic fields have class number 1 is open.

What is known

• Up to $10^9$ there are 65 such $D$: 19 with $n = 2$, 35 with $n = 3$, 9 with $n = 4$ and 2 with $n = 5$, spread fairly evenly over dyadic ranges (Epoch, by brute-force search).
• Some abelian groups never occur as class groups of imaginary quadratic fields [2], so heuristics must be used with care.
• Brute force becomes impractical far beyond $10^9$, where suitable $D$ should become sparse; the problem's authors know no better method than search.

What would count as progress

• Three integers $D_1, D_2, D_3$. Each is checked by computing a class group of discriminant size about $10^{40}$, which is routine with standard computer algebra when one assumes the generalized Riemann hypothesis, and by reading off $C(D_i)$.
• Epoch scores submissions with its own verifier, given the three integers.
• A construction that explains why the fields have this class group would be far more valuable than a lucky search, and might extend to other prescribed groups.

References

1. H. Cohen and H. W. Lenstra Jr., Heuristics on class groups of number fields, in Number Theory Noordwijkerhout 1983, Lecture Notes in Math. 1068 (1984), 33–62. doi:10.1007/BFb0099440
2. S. Holmin, N. Jones, P. Kurlberg, C. McLeman and K. L. Petersen, Missing class groups and class number statistics for imaginary quadratic fields, Experiment. Math. 28 (2019), 233–254. doi:10.1080/10586458.2017.1383952
3. K. Soundararajan, The number of imaginary quadratic fields with a given class number, 2007. arXiv:0707.0237

Source

Imported from FrontierMath: Open Problems, square-class-groups, curated by Epoch AI. Open according to the source as of 2026-10-20. The statement above is written in our own words; see the source for the original wording, history and discussion.