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Imaginary quadratic fields whose class groups have large ℓ-rank
Exhibit imaginary quadratic fields whose class groups have record ℓ-rank, such as 3-rank 9 or 5-rank 6, certified by independent forms.
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| 1 | +# Imaginary quadratic fields whose class groups have large ℓ-rank | |
| 2 | + | |
| 3 | +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. | |
| 4 | + | |
| 5 | +## Problem | |
| 6 | + | |
| 7 | +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 $\mathrm{Cl}(K)[\ell] \cong (\mathbb{Z}/\ell\mathbb{Z})^r$. | |
| 8 | + | |
| 9 | +**Task.** For as many pairs $(\ell, r)$ as possible from the challenge set | |
| 10 | + | |
| 11 | +- $\ell = 3$ and $r = 9, \dots, 16$; | |
| 12 | +- $\ell = 5$ and $r = 6, \dots, 12$; | |
| 13 | +- $\ell = 7$ and $r = 5, \dots, 10$; | |
| 14 | +- $\ell = 11$ or $\ell = 13$, and $r = 4, \dots, 8$; | |
| 15 | + | |
| 16 | +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 $(\ell, r)$. | |
| 17 | + | |
| 18 | +## Background | |
| 19 | + | |
| 20 | +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]. | |
| 21 | + | |
| 22 | +## What is known | |
| 23 | + | |
| 24 | +- 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]. | |
| 25 | +- 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). | |
| 26 | +- Known constructions give $D$ growing exponentially with ℓ even for rank 1, whereas the heuristics suggest polynomial growth (Epoch's summary). | |
| 27 | +- Every pair in the challenge set is one for which no field is currently known. | |
| 28 | + | |
| 29 | +## What would count as progress | |
| 30 | + | |
| 31 | +- 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 $D$. | |
| 32 | +- 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. | |
| 33 | +- 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. | |
| 34 | + | |
| 35 | +## References | |
| 36 | + | |
| 37 | +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](https://doi.org/10.1007/BFb0099440) | |
| 38 | +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](https://doi.org/10.1007/s40993-024-00601-x) | |
| 39 | + | |
| 40 | +## Source | |
| 41 | + | |
| 42 | +Imported from [FrontierMath: Open Problems, class-group-rank](https://epoch.ai/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. | |