Non-abelian CLM Heuristics

• Non-abelian Cohen–Lenstra–Martinet heuristics are a framework that extends classical principles to predict the distribution of arithmetic invariants in number and function fields.
• The approach uses surjection moments, random group models, and corrections for ramification and roots of unity to refine traditional class group predictions.
• It links statistical models with algebraic invariants, offering practical insights into unramified Galois groups and class group behavior in diverse field families.

Non-abelian Cohen–Lenstra–Martinet heuristics are a collection of conjectural and partially proved principles about how unramified arithmetic invariants should be distributed in families of number fields and global function fields when the relevant structure is no longer exhausted by an abelian class group. In the literature, this includes at least three closely connected regimes: distributions of maximal unramified pro-$p$ or prime-to-bad-parts Galois groups, moment formulas for counts of unramified $G$-extensions of quadratic fields, and Cohen–Lenstra–Martinet-type distributions for class groups viewed as modules in families with non-abelian Galois closure. This suggests that the subject is best understood not as a single conjecture, but as a framework organized by surjection moments, ambient Galois action, roots of unity, ramification, and the choice of ordering on the family (Boston et al., 2011).

1. Classical CLM benchmark and the non-abelian problem

For a finite group $\Gamma$ with a chosen decomposition group at infinity $\Gamma_\infty$, the standard Cohen–Lenstra–Martinet benchmark at a “good prime” $p\nmid |\Gamma|$ is a random finite $\mathbb Z_{(p)}[\Gamma]$-module $X$ satisfying

$$\mathbb{P}(X\cong M)=\frac{c}{|\operatorname{Aut}_\Gamma(M)|\,|M^{\Gamma_\infty}|}$$

for each finite $\mathbb Z_{(p)}[\Gamma]$-module $M$, together with the moment identity

$G$0

Equivalently, bounded-rank events should have positive probability and moments such as

$G$1

should be finite for every finite abelian $G$2-group $G$3 (Wang, 2022).

A major theme of the subject is that this good-prime random-module picture is only the benchmark. One line of development asks for genuinely non-abelian replacements of $G$4, such as $G$5-class tower groups or maximal unramified prime-to-bad-parts Galois groups. Another line keeps class groups abelian but treats them as modules over noncommutative orders or Hecke algebras attached to non-Galois ambient symmetry. This suggests that “non-abelian” refers both to the target Galois group and to the ambient symmetry controlling the family (Wang et al., 2019).

The classical CLM weights also admit a structural reinterpretation. For Galois fields, the relevant probabilities are inversely proportional to the number of automorphisms of “structures slightly larger than the class groups,” and, in a corrected formulation, Arakelov class groups provide the natural object whose inverse-automorphism mass recovers the usual Cohen–Lenstra–Martinet weight on torsion (Wang et al., 2019).

2. Imaginary quadratic fields, Schur $G$6-groups, and random pro-$G$7 presentations

The earliest systematic non-abelian refinement in this direction is the Boston–Bush–Hajir program for imaginary quadratic fields. For an imaginary quadratic field $G$8 and odd prime $G$9, one studies

$\Gamma$0

the Galois group of the maximal unramified pro-$\Gamma$1 extension. Complex conjugation yields a generator-inverting involution, and Koch–Venkov showed that the $\Gamma$2-class tower group of an imaginary quadratic number field is a Schur $\Gamma$3-group: a finitely generated pro-$\Gamma$4 group $\Gamma$5 with finite abelianization, $\Gamma$6, and a GI-automorphism (Boston et al., 2011).

The random model is presentation-theoretic. If $\Gamma$7 is the free pro-$\Gamma$8 group on $\Gamma$9 generators with standard GI-automorphism $\Gamma_\infty$0, and

$\Gamma_\infty$1

then one chooses $\Gamma_\infty$2 random relations from $\Gamma_\infty$3, takes the quotient by their closed normal closure, and mixes over $\Gamma_\infty$4 according to the Cohen–Lenstra distribution of the abelianization rank. The induced finite-level mass formula depends explicitly on $\Gamma_\infty$5, the generator rank, and the relator invariant $\Gamma_\infty$6, and for finite Schur $\Gamma_\infty$7-groups the stable mass is the exact non-abelian analogue of the classical Cohen–Lenstra weight (Boston et al., 2011).

The decisive invariant is the non-abelian moment. If $\Gamma_\infty$8 is a finite $\Gamma_\infty$9-group with GI-automorphism and $p\nmid |\Gamma|$0 denotes continuous $p\nmid |\Gamma|$1-equivariant surjections, then the Boston–Bush–Hajir measure $p\nmid |\Gamma|$2 satisfies

$p\nmid |\Gamma|$3

Boston and Matchett Wood proved that these moments characterize the measure, and in the function field analogue of imaginary quadratic extensions of $p\nmid |\Gamma|$4 they proved that the corresponding averages are $p\nmid |\Gamma|$5 as $p\nmid |\Gamma|$6, under the conditions $p\nmid |\Gamma|$7 and $p\nmid |\Gamma|$8 (Boston et al., 2016).

This moment formalism is foundational. It identifies the correct non-abelian analogue of the classical surjection moments, shows that abelianization pushes $p\nmid |\Gamma|$9 forward to the Cohen–Lenstra measure, and provides a setting in which moment identities imply an underlying distribution rather than merely a collection of asymptotic averages (Boston et al., 2016).

3. Quadratic fields, closure types, GI-extensions, and reduced Schur multipliers

A complementary formulation counts unramified $\mathbb Z_{(p)}[\Gamma]$0-extensions of quadratic fields that are Galois over $\mathbb Z_{(p)}[\Gamma]$1. Here the relevant object is not only $\mathbb Z_{(p)}[\Gamma]$2, but also the index-$\mathbb Z_{(p)}[\Gamma]$3 closure type

$\mathbb Z_{(p)}[\Gamma]$4

Melanie Matchett Wood formulates the average

$\mathbb Z_{(p)}[\Gamma]$5

for good $\mathbb Z_{(p)}[\Gamma]$6, and predicts $\mathbb Z_{(p)}[\Gamma]$7 when $\mathbb Z_{(p)}[\Gamma]$8 is not good. Here $\mathbb Z_{(p)}[\Gamma]$9 is the set of order-$X$0 elements outside the kernel of $X$1, “good” means that this set is a single conjugacy class, and $X$2 is the reduced Schur multiplier attached to $X$3 and $X$4 (Wood et al., 2017).

This conjecture recovers the classical abelian odd-order moments when $X$5 is abelian. It also isolates a genuinely non-abelian correction: even over $X$6, the factor $X$7 behaves as a roots-of-unity correction that is invisible in the classical odd-order abelian setting. The function field analogue confirms the same pattern: Frobenius-fixed components of the relevant Hurwitz spaces are counted by $X$8, and this is exactly the geometric source of the correction (Wood et al., 2017).

Brandon Alberts supplies a complementary admissibility criterion through GI-extensions. If $X$9 has index $$\mathbb{P}(X\cong M)=\frac{c}{|\operatorname{Aut}_\Gamma(M)|\,|M^{\Gamma_\infty}|}$$0, then $$\mathbb{P}(X\cong M)=\frac{c}{|\operatorname{Aut}_\Gamma(M)|\,|M^{\Gamma_\infty}|}$$1 is a GI-extension of $$\mathbb{P}(X\cong M)=\frac{c}{|\operatorname{Aut}_\Gamma(M)|\,|M^{\Gamma_\infty}|}$$2 if it is generated by involutions outside $$\mathbb{P}(X\cong M)=\frac{c}{|\operatorname{Aut}_\Gamma(M)|\,|M^{\Gamma_\infty}|}$$3, equivalently if

$$\mathbb{P}(X\cong M)=\frac{c}{|\operatorname{Aut}_\Gamma(M)|\,|M^{\Gamma_\infty}|}$$4

for an automorphism $$\mathbb{P}(X\cong M)=\frac{c}{|\operatorname{Aut}_\Gamma(M)|\,|M^{\Gamma_\infty}|}$$5 such that $$\mathbb{P}(X\cong M)=\frac{c}{|\operatorname{Aut}_\Gamma(M)|\,|M^{\Gamma_\infty}|}$$6 is generated by $$\mathbb{P}(X\cong M)=\frac{c}{|\operatorname{Aut}_\Gamma(M)|\,|M^{\Gamma_\infty}|}$$7. This immediately yields vanishing results: if $$\mathbb{P}(X\cong M)=\frac{c}{|\operatorname{Aut}_\Gamma(M)|\,|M^{\Gamma_\infty}|}$$8 has no GI-extension, then the corresponding averages $$\mathbb{P}(X\cong M)=\frac{c}{|\operatorname{Aut}_\Gamma(M)|\,|M^{\Gamma_\infty}|}$$9 are $\mathbb Z_{(p)}[\Gamma]$0. In particular, for primes $\mathbb Z_{(p)}[\Gamma]$1, almost all finite $\mathbb Z_{(p)}[\Gamma]$2-groups do not have a GI-extension, hence $\mathbb Z_{(p)}[\Gamma]$3 for almost all such groups (Alberts, 2016).

The same framework also produces divergence. Alberts proves

$\mathbb Z_{(p)}[\Gamma]$4

and more generally argues that for groups generated by involutions, Malle-type lower bounds suggest $\mathbb Z_{(p)}[\Gamma]$5 (Alberts, 2016). The resulting picture is sharply non-abelian: finite, zero, and infinite averages all occur, and the closure type $\mathbb Z_{(p)}[\Gamma]$6 is an essential part of the problem rather than auxiliary data.

4. Imaginary $\mathbb Z_{(p)}[\Gamma]$7-extensions and random admissible $\mathbb Z_{(p)}[\Gamma]$8-groups

A broader non-abelian Cohen–Lenstra–Martinet theory appears in the study of imaginary $\mathbb Z_{(p)}[\Gamma]$9-extensions of $M$0 or $M$1. For such a field $M$2, one considers the maximal unramified extension split completely at $M$3, removes the degrees divisible by the bad primes, and obtains the prime-to-$M$4 quotient

$M$5

By Schur–Zassenhaus, this is naturally a $M$6-group, and it is the non-abelian replacement of the class group in this setting (Liu et al., 29 Jul 2025).

The central structural input is a presentation theorem. For suitable $M$7, and for sufficiently large $M$8,

$M$9

for some $G$00 with

$G$01

The extra $G$02-fixed relation is the algebraic shadow of ramification at $G$03 and is the feature distinguishing the imaginary theory from the totally real case (Liu et al., 29 Jul 2025).

This presentation yields a random-group model. One takes a free admissible $G$04-group, imposes $G$05 random $G$06-relation orbits, and chooses the last relation from the $G$07-fixed locus. The resulting measure $G$08 has explicit masses and moments

$G$09

In the function field case these moments are proved, first in an iterated large-$G$10, large-discriminant limit and then for each sufficiently large $G$11, by counting points on Hurwitz stacks (Liu et al., 29 Jul 2025).

The logical role of moments is reinforced by Sawin’s uniqueness theorem. For measures on finite level-$G$12 $G$13-groups, convergence of all $G$14-equivariant surjection moments, together with polynomial growth bounds, forces convergence of point masses. In particular, the level-$G$15 measures arising in the Liu–Wood–Zureick-Brown setting are determined by their moments (Sawin, 2020). This makes moment identities a genuine mechanism for defining non-abelian distributions rather than a merely heuristic diagnostic.

5. Roots of unity, bad primes, and family-sensitive corrections

The presence of roots of unity forces a further refinement. For a $G$16-extension $G$17 split completely at $G$18, Liu defines an invariant

$G$19

Its descended version $G$20 determines the Weil pairing of the corresponding curve and agrees with the prime-to-$G$21-torsion part of the EVW–Wood lifting invariant on Hurwitz schemes. The refined function field moment theorem then counts only those $G$22-equivariant surjections with prescribed pushforward of $G$23, and the average is $G$24 when $G$25, and $G$26 otherwise (Liu, 2022).

This leads to a roots-of-unity correction in the conjectural number field moments. If $G$27 is the maximal integer such that $G$28, the predicted average number of surjections acquires the factor

$G$29

together with the Martinet-type unit correction $G$30 (Liu, 2022). In this form, roots of unity are not an auxiliary perturbation but part of the primary invariant.

A different correction arises from bad primes. Wang shows that when $G$31, the good-prime CLM benchmark can fail in a qualitative manner. Genus theory and the Roquette–Zassenhaus invariant subgroup

$G$32

produce lower bounds such as

$G$33

and, under natural counting conjectures, one gets

$G$34

for every $G$35 whenever $G$36 is non-random for the permutation action (Wang, 2022). In $G$37-cubic fields and $G$38-quartic fields, this produces explicit unconditional or conditional failures of naive CLM behavior, and the paper stresses that the phenomenon is ordering-sensitive (Wang, 2022).

Hecke reciprocity introduces yet another family-sensitive correction. In the pure cubic family $G$39, ordered by $G$40, the average size of $G$41 is $G$42 in the tame family and $G$43 in the wild family, a dichotomy not accounted for by previously available heuristics. The underlying mechanism is a reciprocity law for $G$44 in odd-degree extensions, and the resulting random model yields

$G$45

in the Hecke-ramified case, but

$G$46

in aberrant Hecke-unramified subfamilies (Shnidman et al., 16 Jun 2025). This shows that even after fixing the abstract Galois closure group, finer $G$47-structure of the family can change the moment law.

6. Ambient non-abelian symmetry, Arakelov corrections, and proved families

A separate but closely related strand keeps class groups abelian while treating them in non-abelian ambient Galois settings. Bartel and Lenstra show that the classical CLM heuristics fail in two distinct ways: hidden module-theoretic restrictions in imaginary abelian extensions, and a mismatch between discriminant ordering and subfield distributions. Their corrected conjecture requires $G$48 finite, orders fields by

$G$49

the norm of the product of ramified primes, and replaces $G$50 by $G$51, where $G$52 is the Pontryagin dual of the Arakelov class group (Bartel et al., 2018). The exact sequence

$G$53

explains why the usual CLM weight on finite torsion modules is the shadow of a natural inverse-automorphism measure on lattices of fixed rational type (Bartel et al., 2018).

A more refined equivariant version is given by Bartel, Lenstra, and de Valence through Arakelov class groups of random number fields. Chinburg’s $G$54 conjecture implies restrictions on the class of $G$55 in $G$56, and this leads to new counterexamples to naive CLM with non-abelian Galois group $G$57. In these families the original random-module support is too large: the corrected heuristic must restrict to modules in a fixed Grothendieck class (Bartel et al., 2020).

Wood’s module-theoretic reinterpretation of CLM for non-Galois fields moves in the same direction. A new theorem on the capitulation kernel shows that, for primes good for the relevant idempotent,

$G$58

so class groups of non-Galois fields are controlled by Hecke-module structures coming from the non-abelian Galois closure. The resulting distribution is again of inverse-automorphism type, but now on modules over an integral Hecke algebra rather than merely on abstract finite abelian groups (Wang et al., 2019).

There are also exact non-abelian average results in the good-prime regime. For any transitive permutation $G$59-group $G$60 containing a transposition, and any number field $G$61, there exists a constant $G$62 such that

$G$63

More sharply, if $G$64 is the unique index-$G$65 subfield and $G$66, then

$G$67

which is exactly the Cohen–Martinet prediction for the relative class group statistic (Oliver et al., 2021). The same paper emphasizes, however, that the full average of $G$68 can differ from the naive CM prediction because discriminant ordering biases the distribution of the resolvent subfield (Oliver et al., 2021).

Taken together, these results isolate the recurrent obstructions any future non-abelian Cohen–Lenstra–Martinet heuristic must absorb from the outset: closure type, reduced Schur multipliers, roots of unity, ramification-forced bad primes, Hecke reciprocity, Grothendieck-class constraints, and the ordering of the family. This suggests that no single random law on finite groups or finite modules, applied uniformly across all primes and all orderings, can capture the observed behavior without conditioning on these structural invariants.