About the problem
This problem asks for an algorithm to do the following: given an odd prime \(\ell\) and a positive integer \(r\), return a square-free positive integer \(D\) such that the number field \(K=\mathbb{Q}(\sqrt{-D})\) has class group \(\operatorname{Cl}_K\) with \(\ell\)-rank \(\geq r,\) i.e. \(\dim_{\mathbb{F}_\ell} \operatorname{Cl}_K/\ell \operatorname{Cl}_K\geq r.\)
A paper of Cohen and Lenstra conjectures that for each \(\ell,r\) there is a \(D\) such that \(\operatorname{Cl}_K\) has \(\ell\)-rank exactly \(r\). In fact, they conjecture that there are infinitely many such \(D\), and that such \(D\) form a positive proportion of all \(D\). Nevertheless, is no \(\ell\) for which this conjecture is known.
For each \(\ell\) there are infinitely many \(D\) that give rank \(\geq 3\) [Kim 2015], though already for rank \(1\) the known constructions yield \(D\) growing exponentially with \(\ell\), while the Cohen–Lenstra heuristics suggest \(D\) should grow only polynomially. For all but a few \(\ell\), no example of rank \(\geq 4\) is known.
For \(\ell=3\), the current record rank is \(8\), due to Elkies in 2025. This was found not by direct search but via a connection with elliptic curves of high rank.
The hope is that a solution to this problem could contain ideas that help lead to a proof of the Cohen–Lenstra conjecture, i.e. that there is some \(D\) such that \(\operatorname{Cl}_K\) has \(\ell\)-rank \(r\) for each \(r\).
The verifier tests a solution on a finite set of values of \(\ell\) and \(r\) for which no \(D\) is known. Even a single such instance would likely be of interest.
Prompt
Find imaginary quadratic fields K = Q(sqrt(-D)) whose class group Cl(K) has large ℓ-rank.
For a prime ℓ, the ℓ-rank of Cl(K) is the integer r such that the ℓ-torsion subgroup Cl(K)[ℓ] is
isomorphic to (Z/ℓZ)^r. Exhibit fields achieving ℓ-rank at least r for the (ℓ, r) pairs in the
challenge set below. You demonstrate that the ℓ-rank is at least r by giving r independent
class-group elements, each of order exactly ℓ, so that together they generate a subgroup
isomorphic to (Z/ℓZ)^r.
## Challenge set
Each (ℓ, r) pair below is a separate challenge:
ℓ = 3 r = 9, 10, 11, 12, 13, 14, 15, 16
ℓ = 5 r = 6, 7, 8, 9, 10, 11, 12
ℓ = 7 r = 5, 6, 7, 8, 9, 10
ℓ = 11 r = 4, 5, 6, 7, 8
ℓ = 13 r = 4, 5, 6, 7, 8
Each challenge you certify earns credit; certifying every pair in the set earns full credit, and
any nonempty subset earns partial credit. Completing more is better.
## Submission
Write your answer to a file, one certificate per line, with fields separated by the pipe
character, and submit the file's path with the submit tool:
ℓ|r|D|gen_1|gen_2|...|gen_r
- ℓ — the prime. Every generator on the line must have order exactly ℓ in Cl(K).
- r — the number of generators that follow. They must be independent.
- D — a positive integer such that -D is a fundamental discriminant; the field is K = Q(sqrt(-D)).
- gen_i — an element of Cl(K), given as a binary quadratic form of discriminant -D: three
comma-separated integers "a,b,c" meaning a·x² + b·x·y + c·y², with b² - 4·a·c = -D.
Only lines whose (ℓ, r) is in the challenge set earn credit; you may include others (for example
the one below) but they are ignored.
### Example line
3|3|4447704|390,-96,2857|921,-786,1375|346,-68,3217
This is ℓ = 3, r = 3, discriminant -4447704, and three independent order-3 forms — a well-formed
example of the format. (3, 3) is below the challenge set, so it verifies without counting.