Math Conjectures

Cohen–Lenstra Class-Group Heuristics for Quadratic Fields

OPENLandmarkConjectureProposed 1983 · Canonical special case

Canonical statement

Fix an odd prime pp and a finite abelian pp-group AA. For fundamental discriminants DD, ordered by ∣D∣|D|, put KD=Q(D)K_D=\mathbb Q(\sqrt D). Then
lim⁡X→∞#{−X<D<0:D fundamental, Cl⁡(KD)[p∞]≃A}#{−X<D<0:D fundamental}=∏i=1∞(1−p−i)∣Aut⁡(A)∣, \lim_{X\to\infty} \frac{\#\{-X<D<0:D\text{ fundamental},\ \operatorname{Cl}(K_D)[p^\infty]\simeq A\}}{\#\{-X<D<0:D\text{ fundamental}\}} =\frac{\prod_{i=1}^{\infty}(1-p^{-i})}{|\operatorname{Aut}(A)|},
and
lim⁡X→∞#{0<D<X:D fundamental, Cl⁡(KD)[p∞]≃A}#{0<D<X:D fundamental}=∏i=2∞(1−p−i)∣A∣ ∣Aut⁡(A)∣. \lim_{X\to\infty} \frac{\#\{0<D<X:D\text{ fundamental},\ \operatorname{Cl}(K_D)[p^\infty]\simeq A\}}{\#\{0<D<X:D\text{ fundamental}\}} =\frac{\prod_{i=2}^{\infty}(1-p^{-i})}{|A|\,|\operatorname{Aut}(A)|}.
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Fix an odd prime \(p\) and a finite abelian \(p\)-group \(A\). For fundamental discriminants \(D\), ordered by \(|D|\), put \(K_D=\mathbb Q(\sqrt D)\). Then
\[
 \lim_{X\to\infty}
 \frac{\#\{-X<D<0:D\text{ fundamental},\ \operatorname{Cl}(K_D)[p^\infty]\simeq A\}}{\#\{-X<D<0:D\text{ fundamental}\}}
 =\frac{\prod_{i=1}^{\infty}(1-p^{-i})}{|\operatorname{Aut}(A)|},
\]
and
\[
 \lim_{X\to\infty}
 \frac{\#\{0<D<X:D\text{ fundamental},\ \operatorname{Cl}(K_D)[p^\infty]\simeq A\}}{\#\{0<D<X:D\text{ fundamental}\}}
 =\frac{\prod_{i=2}^{\infty}(1-p^{-i})}{|A|\,|\operatorname{Aut}(A)|}.
\]
NT-ARITH-040:[CohenLenstra1984ClassGroups][CohenMartinet1990ClassGroups][Wood2022RandomGroups][Ellenberg2026CohenLenstra]
Number theoryProbability theory

Notes

Cohen and Lenstra proposed that the odd-primary parts of quadratic class groups behave like random finite abelian groups weighted by inverse automorphism count [CohenLenstra1984ClassGroups]. For imaginary quadratic fields the weight is proportional to ∣Aut⁡(A)∣−1|\operatorname{Aut}(A)|^{-1}; for real quadratic fields the unit rank contributes the additional factor ∣A∣−1|A|^{-1}. This predicts exact limiting probabilities, not merely average class numbers.

The heuristics have generated a broad theory of moments, random matrices, function-field analogues, and distributions in special families [Wood2022RandomGroups]. Cohen and Martinet extended the framework to wider families of number fields [CohenMartinet1990ClassGroups], but roots of unity and family-specific Galois structure can invalidate naive universal extensions. A current survey emphasizes that the exact quadratic distributions remain largely unproved despite substantial progress on selected moments and low torsion [Ellenberg2026CohenLenstra].

This entry therefore fixes an odd prime and the imaginary- and real-quadratic limiting laws. It excludes p=2p=2, where genus theory changes the distribution, and does not assert an unrestricted Cohen–Lenstra–Martinet law for arbitrary number-field families.

References (4)

The boxed statement is the canonical open formulation — not a stronger variant or a related research program. The status reflects the catalog's last review; do your own literature search before investing serious effort.