Imaginary quadratic fields whose class groups have large ℓ-rank

The class group of a number field measures how far its ring of integers is from unique factorisation. Heuristics predict that for every prime ℓ and every r, some imaginary quadratic fields have class groups containing r independent elements of order ℓ, but explicit examples run out quickly: for ℓ = 3 the record is rank 8. The task is to beat the records.

Problem

For a positive integer $D$ such that $-D$ is a fundamental discriminant, let $\mathrm{Cl}(K)$ be the class group of $K = \mathbb{Q}(\sqrt{-D})$, identified with the group of equivalence classes of positive definite binary quadratic forms $ax^2 + bxy + cy^2$ of discriminant $b^2 - 4ac = -D$ under composition. For an odd prime $\ell$, the ℓ-rank of $\mathrm{Cl}(K)$ is the integer $r$ with $\mathrm{Cl}(K)[\ell] \cong (\mathbb{Z}/\ell\mathbb{Z})^r$.

Task. For as many pairs $(\ell, r)$ as possible from the challenge set

• $\ell = 3$ and $r = 9, \dots, 16$;
• $\ell = 5$ and $r = 6, \dots, 12$;
• $\ell = 7$ and $r = 5, \dots, 10$;
• $\ell = 11$ or $\ell = 13$, and $r = 4, \dots, 8$;

find $D$ together with $r$ forms of order exactly $\ell$ in $\mathrm{Cl}(\mathbb{Q}(\sqrt{-D}))$ that generate a subgroup isomorphic to $(\mathbb{Z}/\ell\mathbb{Z})^r$. This certifies that the ℓ-rank is at least $r$. The larger goal is a method that produces such fields for any $(\ell, r)$.

Background

Cohen and Lenstra's heuristics (1984) predict that for each odd prime $\ell$ and each $r$, a positive proportion of imaginary quadratic fields have ℓ-rank exactly $r$ [1]; this is not proved for any $\ell$. Producing fields of large rank on demand therefore tests how well class groups can be steered by construction rather than found by search. For $\ell = 3$ there is a link with elliptic curves: descent by 3-isogeny on a Mordell curve $y^2 = x^3 + k$ of large rank forces large 3-ranks in the class groups of related quadratic fields [2].

What is known

• The record 3-rank for a quadratic field is 8, for $\mathbb{Q}(\sqrt{-3D})$ with $D = 72513834653847828539450325493$. Elkies obtained it from a Mordell curve of rank 16 and computed the class group unconditionally, using the Burgess bounds (2025) [2].
• For every ℓ, families with ℓ-rank at least 3 are known, but for all but a few ℓ no example of rank 4 or more is known (Epoch's summary).
• Known constructions give $D$ growing exponentially with ℓ even for rank 1, whereas the heuristics suggest polynomial growth (Epoch's summary).
• Every pair in the challenge set is one for which no field is currently known.

What would count as progress

• For a pair $(\ell, r)$: a discriminant $-D$ and $r$ forms $(a_i, b_i, c_i)$ with $b_i^2 - 4a_ic_i = -D$, each of order $\ell$ and jointly independent. Composition and reduction of forms make this quick to check, even for very large $D$.
• Epoch scores submissions with its own verifier: one certificate per line in the form ℓ|r|D|a,b,c|..., with credit for each certified pair of the challenge set.
• A general construction taking $\ell$ and $r$ and returning $D$ would be the real prize, and might shed light on the Cohen–Lenstra conjecture itself.

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. N. D. Elkies, Rank of an elliptic curve and 3-rank of a quadratic field via the Burgess bounds, Research in Number Theory 11 (2025), article 70. doi:10.1007/s40993-024-00601-x

Source

Imported from FrontierMath: Open Problems, class-group-rank, 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.