Cohen–Lenstra–Gerth Heuristics

• Cohen–Lenstra–Gerth heuristics are predictive models for class group distributions in number fields and function fields, based on automorphism counts and random matrix theory.
• They refine original predictions by incorporating corrections for unit groups, p=2 phenomena, composite torsion, and nonabelian field extensions.
• Supported by empirical data and rigorous proofs in special cases, these heuristics bridge analytic, algebraic, and topological methods through models like Hurwitz spaces.

The Cohen–Lenstra–Gerth heuristics constitute a suite of predictions regarding the statistical distribution of class groups and their generalizations in families of number fields and function fields. Rooted in the foundational Cohen–Lenstra philosophy for odd primes $p$, these heuristics conjecture (in essence) that the $p$-part of the ideal class group of a "random" quadratic field is distributed like a random finite abelian $p$-group, with probability proportional to the inverse order of its automorphism group. Gerth's extension addressed the $p=2$ case, and broader refinements accommodate unit corrections, roots of unity, ray class variants, and nonabelian or higher-tower extensions, with far-reaching analogues in random matrix theory, symplectic group eigenspaces, and topological (Hurwitz space) methods.

1. Origin and Statement of the Heuristics

The original Cohen–Lenstra heuristics predict that for a fixed odd prime $p$, the probability that the Sylow $p$-subgroup of the class group $\operatorname{Cl}_K$ of an imaginary quadratic field $K$ is isomorphic to a finite abelian $p$-group $G$ is

$$\mu_{\mathrm{CL}}(G) = \frac{1}{|\operatorname{Aut}(G)|} \prod_{k=1}^\infty (1 - p^{-k}) \qquad [1301.1239], [1504.04391], [1404.2447].$$

This measure extends multiplicatively across primes and is supported by both empirical data and, in special cases, rigorous theorems (e.g., for $p=3$ or for 3-torsion arising from cubic field parametrizations).

For real quadratic fields, additional corrections involving unit groups appear, leading to a "shifted" version: $$\mu_{\mathrm{CL}}^{(+)}(G) = \frac{|G|}{|\operatorname{Aut}(G)|} \prod_{k=1}^\infty (1 - p^{-k-1}) .$$ Gerth further generalized this to systematically handle the $p=2$ case, where genus theory and 2-rank phenomena force essential modifications (Babu et al., 2024).

A generalized heuristic for composite torsion predicts that, for $n = \prod p_i^{r_i}$, the average size of the $n$-torsion parts of class groups factors as the product of the averages for the $p_i^{r_i}$-parts (Koymans et al., 26 Dec 2025).

2. Refinements for $p=2$ and the Gerth Correction

For $p=2$, the naive Cohen–Lenstra predictions fail due to the systematic presence of large 2-torsion from genus theory. Gerth proposed replacing the direct study of $\operatorname{Cl}_K$ by its square subgroup $\operatorname{Cl}^2_K$ and analyzing the quotient $V_K = \operatorname{Cl}^2_K/\operatorname{Cl}^4_K$, a vector space over $\mathbb{F}_2$ of 4-rank $r_4(K)$. He predicted, and Fouvry–Klüners later proved, that higher moments of the distribution of $r_4$ stabilize to explicit constants: $$L_m = \prod_{k=1}^\infty (1 + 2^{-k})^{-1} \sum_{r=0}^\infty 2^{mr + \frac{r(r+1)}{2}} \prod_{k=1}^r (1 - 2^{-k})^{-1} ,$$ with analogous results in more general settings for quadratic extensions of Galois number fields with $2$ split and class number $1$ (Babu et al., 2024). These results underpin the general principle that suitable averaging over vector spaces of ambiguous classes restores a universal statistical model for the "refined" $2$-torsion.

3. Function Field Analogues and Homological Stability

In the setting of quadratic extensions of function fields $K = \mathbb{F}_q(t)$, the probabilistic measure on finite abelian $\ell$-groups (for odd $\ell$) takes the exact same form as the original Cohen–Lenstra law: $$\mu(A) = \frac{\prod_{i\geq 1}(1 - \ell^{-i})}{|\operatorname{Aut}(A)|} .$$ These distributions were justified unconditionally for surjective moments as $q \to \infty$ by point counting on Hurwitz schemes, with the key technical input being homological stability of the associated Hurwitz spaces (Randal-Williams, 2019, Landesman et al., 2024). The core mechanism is that, in large characteristic, the number of rational components of these moduli spaces matches the expected number of surjective morphisms from class groups to fixed target groups, mirroring the number field picture.

4. Random Matrix Models and Automorphism Count Laws

A major advance is the realization that the same limiting measures arise as the distribution of the cokernel of large random matrices over rings such as $\mathbb{Z}_p$ or complete discrete valuation rings, a universality principle extending far beyond number field arithmetic. Specifically, as $n \to \infty$, the cokernel of an $n \times n$ random matrix with independent, appropriately non-degenerate entries over $\mathbb{Z}_p$ converges in law to the Cohen–Lenstra measure on finite abelian $p$-groups (Maples, 2013, Wood, 2015, Cheong et al., 2018, Mészáros, 2023):

Model	$p$-torsion Distribution	Reference
Random i.i.d. matrices	$\mu_{\mathrm{CL}}(G)$ as above	(Maples, 2013)
Haar-random endos	$\mu_{\mathrm{CL}}(G)$, with explicit automorphism count	(Cheong et al., 2018)
Determinantal bias	Asymptotic matches Cohen–Lenstra for $p \ge 5$	(Mészáros, 2023)

Random matrix models justify these measures as arising from high-dimensional symmetries, not just from field arithmetic.

5. Correction Terms: Roots of Unity and Schur Multiplier Invariants

In arithmetic settings where the base field contains nontrivial roots of unity, the heuristics deviate from the naive automorphism-count law. Adam–Malle introduced a modification based on the distribution of $1$-eigenspaces in generalized symplectic groups; the predictions interpolate between the original Cohen–Lenstra law and explicit corrections reflecting the Galois module structure of units and roots of unity (Adam et al., 2014).

For instance, when $K_0$ contains $\mu_p$ but not $\mu_{p^2}$, the moments are shifted, reflecting the symplectic structure: $$P_{1,p}^{(u)}(G) = (p^2)_\infty (p)_\infty \cdot \frac{|G|^u}{|\operatorname{Aut}(G)|}.$$ Empirically, fields with roots of unity (e.g., cyclic cubic fields) have more frequent occurrences of certain class group structures, such as $C_2 \times C_2$ when $p=2$, than predicted by the naive model (Rubinstein-Salzedo, 2012). These corrections can often be described using invariants in quotients of the Schur multiplier group, decomposing families of extensions into components indexed by $H_2(G,c)$, with standard moments holding on each component and overall frequency matching observed data upon averaging.

6. Nonabelian, Higher-Tower, and Composite Heuristics

Modern generalizations treat nonabelian Galois module structures, towers of fields (e.g., $p$-class towers), and composite torsion. Boston–Bush–Hajir formulated nonabelian Cohen–Lenstra heuristics predicting the distribution of $p$-class tower groups for imaginary quadratic fields via group-theoretic weights involving automorphism centralizers and Schur $\sigma$-groups (Boston et al., 2011). For $2$-class towers in cyclic cubic fields, analogous data-driven conjectures identify explicit small families of valid groups, assign rational masses to their occurrences, and observe consistency with abelian-level predictions (Boston et al., 2020).

Gerth's heuristics for the average $n$-torsion in quadratic class groups with $n$ composite predict factorizations, now confirmed for $n=6$ (i.e., for $6$-torsion) in recent work (Koymans et al., 26 Dec 2025). The constant governing the average is a product of the odd prime local factors, manifesting the conjectural independence of the $p$-parts for distinct $p$.

7. Higher Moments, Rank Laws, and Universal Implications

The Cohen–Lenstra–Gerth heuristics conjecture not only distributional laws for class groups, but also exact formulas for all joint moments and the joint distributions of the successive $p^j$-ranks. The moments take the form

$$\mathbb{E}\bigl[|\operatorname{Cl}(K)[p]|^{m_1} \cdots |\operatorname{Cl}(K)[p^\ell]|^{m_\ell}\bigr] = \sum_{\mu \subseteq \lambda} C_{\lambda, \mu}(p)$$

with $C_{\lambda, \mu}(p)$ polynomials defined combinatorially (Delaunay et al., 2013). Knowledge of all such moments determines the limiting rank distribution, i.e., the vector $(\mathrm{rk}_{p^1}, ..., \mathrm{rk}_{p^\ell})$, via the solution of infinite linear systems with Gaussian-type decay.

These moment laws are compatible with arithmetic results in special cases, with direct analogues for Selmer groups of elliptic curves and for Tate–Shafarevich groups. Generalizations include the Poonen–Rains distribution for Selmer groups (Delaunay et al., 2013, Landesman, 2021).

Conclusion

The Cohen–Lenstra–Gerth heuristics, and their refinements via automorphism counts, symplectic matrix ensembles, Schur multiplier corrections, and random polynomial or matrix models, form a robust conceptual and predictive framework for the statistics of class groups and generalizations across arithmetic and topological contexts. The philosophy is that intricate arithmetic structures behave, in the large, as universality classes governed by symmetries and random matrix phenomena, subject to arithmetic corrections for roots of unity, units, and Galois symmetries (Adam et al., 2014, Rubinstein-Salzedo, 2012, Cheong et al., 2018, Koymans et al., 26 Dec 2025, Randal-Williams, 2019). Results from both number fields and function fields, verified unconditionally in select settings and supported by extensive numerical evidence in others, consistently point toward the efficacy and ubiquity of these heuristics in modeling the "randomness" of class group and related Galois structures.