Groups appear with probability inversely proportional to their automorphisms
Class groups do not behave like random finite abelian groups of a given order. The Cohen–Lenstra heuristics propose instead that a group appears with weight inversely proportional to the size of its automorphism group — which strongly favours cyclic groups and makes large p-ranks rare. The predictions match computed tables closely, and they are now used in the opposite direction: as a check on whether a computation has found the whole class group.
Learning objectives
- State the weighting principle behind the heuristics.
- Quote the predicted frequency of odd class number divisibility.
- Explain why the prime 2 is excluded from the odd-part predictions.
- Use the heuristics to sanity-check computed class group tables.
- Identify what has been proved and what remains conjectural.
Section 01The weighting principle
The central proposal is that a finite abelian group G occurs among class groups with weight proportional to 1/|Aut(G)|.
| Group of order 4 | |Aut| | Relative weight |
|---|---|---|
| ℤ/4ℤ | 2 | High — cyclic groups are favoured |
| ℤ/2 ⊕ ℤ/2 | 6 | Three times less likely than the cyclic group |
The effect compounds: a group of p-rank r has an automorphism group of size roughly pr2, so high rank is heavily suppressed. The heuristics therefore predict that class groups are usually cyclic or close to it — which is exactly what tables show.
It is the natural measure on isomorphism classes of objects with symmetry: counting each group once over-counts the symmetric ones, since they arise from fewer distinct presentations. The same weighting appears throughout enumerative mathematics, which is part of why the heuristic is persuasive.
Section 02Predictions
For imaginary quadratic fields, the heuristics apply to the odd part of the class group and predict, for an odd prime p, that the proportion of discriminants with p dividing h approaches
Genus theory determines the 2-rank of the class group exactly, from the number of prime factors of the discriminant. It is not random at all, so the heuristics deliberately apply only to the odd part. Applying them to the 2-part produces predictions that are simply wrong.
Section 03Use as a validation tool
Because the predictions are sharp and the data sets are large, deviation from them is informative. A table of computed class numbers whose distribution departs materially from the heuristics is far more likely to contain a systematic computational error than to have discovered new mathematics.
Divisibility frequencies
Compare the observed proportion of h divisible by 3, 5, 7 against the predicted values over a large discriminant range.
Rank distribution
High p-ranks should be rare. An excess of them suggests relations were missed, producing spurious extra generators.
Cyclic proportion
The predicted proportion of cyclic class groups is high; a shortfall suggests the structure determination is incomplete.
Most Cohen–Lenstra predictions remain unproven. Significant partial results exist — notably the average 3-torsion in quadratic fields, established by Davenport and Heilbronn well before the heuristics were formulated, and the function field analogues, which have been proved. The general statements are conjectural, and computations that agree with them are evidence for them rather than the reverse.
ReferenceFrequently asked questions
Do the heuristics apply to higher degree fields?
Extensions have been proposed for higher degree and for relative extensions, with the weighting adjusted for the Galois module structure. They are less thoroughly tested than the quadratic case, and the correct formulation in the presence of extra automorphisms is subtle.
Why do real and imaginary fields have different predictions?
Because in the real case the unit group contributes an extra constraint, effectively imposing one additional relation. The heuristics account for this by shifting the exponent, which is why real quadratic fields have a much higher predicted proportion of trivial class group.
Can the heuristics prove anything about a specific field?
No. They describe distributions over families and say nothing whatsoever about an individual discriminant. Their computational value is entirely in checking aggregate behaviour of large tables.
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