Presentations of Galois groups of unramified extensions of global fields and its predicted distribution

Abstract: Motivated by the work of Liu, we study certain canonical quotients of $G_{\emptyset}<sup>T(K)$ -- the Galois group of the maximal unramified extension of a global field $K$ that is split completely at a finite nonempty set of places in $T$ -- for $Γ$-extensions $K/Q$, and prove they have presentations of a particular form. This presentation leads us to the construction of a new random group model as in the work of Liu, Wood, and Zureick-Brown that predicts the distribution of $G_{\emptyset}<sup>T(K)$ as we vary among $Γ$-extensions $K/Q$ with prescribed local conditions at places in $T$, giving a generalization of the non-abelian Cohen-Lenstra-Martinet Heuristics. The key generalization is that $Q$ can be an arbitrary global field, while this comes at a cost of introducing a prime-to-$|\text{Cl}_T(Q)|$ condition in addition to avoiding roots of unity, $|Γ|$, and the characteristic if $Q$ is a function field.

• The paper presents a comprehensive framework for explicit presentations of Galois groups from unramified extensions over arbitrary global fields with prescribed local splitting conditions.
• It introduces a new random group model that predicts the distribution and key moment statistics of these groups, generalizing earlier abelian heuristics.
• The work unifies classical Cohen-Lenstra heuristics with non-abelian generalizations, clarifying the role of roots of unity and cyclotomic exceptions in the theory.

Presentations and Predicted Distribution of Galois Groups of Unramified Extensions of Global Fields

Introduction and Motivations

This work establishes a comprehensive framework for the presentation and distribution of Galois groups of maximal unramified extensions of global fields, incorporating prescribed local splitting conditions and extending beyond earlier abelian and number field settings. By generalizing the techniques of Liu, Wood, and Zureick-Brown, and integrating modern cohomological methods and random group models, the paper provides new structural and statistical results that connect to and extend the non-abelian Cohen-Lenstra-Martinet heuristics. Unlike prior work restricted to the base field $Q = \mathbb{Q}$ (and its analogues in function fields), this work considers arbitrary global fields $Q$, yielding a more robust and general theory, albeit with necessary diophantine and group-theoretic restrictions related to roots of unity and class group torsion.

Structural Results: Presentations of Galois Groups

One primary achievement is the explicit presentation of the pro-$\mathcal{C}$ completions of Galois groups $G_{\mathcal{O}^T}(K)$, where $K/Q$ is a finite Galois (typically, $\Gamma$-) extension, and the extension is unramified and split at a set $T$ of places. Building on prior results for totally real number fields and pro-$\ell$ quotients, the paper proves that, under reasonable coprimality and finiteness hypotheses, the group $G_{\mathcal{O}^T}(K)^{\mathcal{C}}$ admits, for sufficiently large $n$,

$Q$0

for certain elements $Q$1 related to the decomposition subgroups at the places in $Q$2. Here, $Q$3 is the free admissible pro-$Q$4 $Q$5-group on $Q$6 generators, and the specific placement of the relators reflects both global and local cohomological constraints (see Theorem 1, Main Text).

When $Q$7 is specialized to the class of abelian groups, this presentation recovers (in the prime-to-$Q$8 part) the known random matrix model presentations for class groups underlying the classical Cohen-Lenstra heuristics, as well as their non-abelian generalizations. The analysis carefully distinguishes the appearance of "augmentation ideals" and the importance of the choice and structure of local decomposition groups.

Random Group Models and Heuristic Distributions

The second core contribution is the construction and analysis of a new random group model motivated by these group presentations, modeled on and extending the "random Schur group" and "random admissible group" heuristics previously developed for number fields and function fields [see "A predicted distribution for Galois groups of maximal unramified extensions," Invent. Math.; also [LWZB], [BBH], [BBH2]]. The new model applies to arbitrary global fields and prescribes moments (averages of surjective $Q$9-equivariant homomorphisms to a fixed finite group $\mathcal{C}$0) for the distribution of possible Galois groups, incorporating local conditions via the decomposition subgroups $\mathcal{C}$1.

The key conjecture (Conjecture 1.2) asserts that, after removing a thin exceptional set (extensions entangled with cyclotomic extensions or picking up extra roots of unity), the average number of $\mathcal{C}$2-equivariant surjections from the Galois group $\mathcal{C}$3 to a given admissible finite $\mathcal{C}$4-group $\mathcal{C}$5 equals

$\mathcal{C}$6

where the probability model is built from random relators sampled according to Haar measure on the relevant admissible groups.

Integration with and Extension of Heuristics

This model recovers both the classical Cohen-Lenstra (-Martinet) heuristics for class groups and their non-abelian analogues as conjectures in suitable settings, including function fields, provided all constraints on roots of unity and decomposition groups are imposed. These predictions are shown to interpolate between several cases:

• The abelian case (recovering probabilities for class groups as in Cohen-Lenstra),
• The non-abelian case of the maximal unramified pro-$\mathcal{C}$7 or prime-to-$\mathcal{C}$8 Galois groups (as considered in [BBH, BBH2]),
• Relative settings (varying over $\mathcal{C}$9-extensions with specified local type),
• Both number fields and function fields, with the distinctions controlled via the analysis of stable cohomology and the Frobenius-fixed point counts in Hurwitz spaces.

Additionally, the heuristic is rigorously justified in the abelian case via analytic approaches and the proven independence of exceptional extensions (Proposition 3.7), and is refined in the non-abelian setting by removing the "big fiber" and roots of unity obstructions made explicit in recent counterexamples to unconditional versions of Malle's conjecture for Galois extension counts.

Mathematical and Heuristic Rigor

The entire framework is underpinned by a detailed cohomological calculation and probabilistic analysis of random walks on admissible group presentations, together with precise classification and analysis of the impact of roots of unity and local conditions. The measure on the space of possible Galois groups is well-defined (Theorem 5.10) and countably additive, and the associated random group models admit a computation of all finite moments. The H-moments associated with this measure are shown (Theorem 6.2) to equal the proposed ratios, solidifying the conjectural identification between statistical averages over extensions and the random group model.

The theoretical implication is a unifying and predictive framework for the statistics of Galois groups of maximal unramified extensions over arbitrary global fields with prescribed splitting, provided natural obstructions (roots of unity, cyclotomic subfields) are explicitly excised. The model is robust under varying the base field, the group $G_{\mathcal{O}^T}(K)$0, and the local data, and is shown to be compatible with, and in some cases implies, the conjectures on distributions made by Sawin and Wood [SW2].

Implications and Future Directions

Theoretical Implications

• The paper's generalization to arbitrary global fields and arbitrary finite Galois groups substantially extends the range and applicability of non-abelian Cohen-Lenstra-Martinet heuristics.
• Establishing these presentations as generic/random group models connects the asymptotic statistics of extension counting to concrete group-theoretical objects, reminiscent of the function field analogy and Hurwitz space enumeration.
• The formulation and analysis of exceptional (thin) subsets (related to roots of unity and cyclotomic subfields) clarify the precise domain of validity for such heuristics, explaining and reconciling known counterexamples in the literature on Galois extension growth.

Practical Directions

• The explicit group-theoretic presentations and accompanying random models support experimental verification in concrete Galois-theoretic datasets and arithmetic statistics.
• The formalism can serve as a blueprint for extending probabilistic and topological methods (e.g., stable cohomology, random walks on arithmetic groups) to analyze Galois or étale fundamental groups of arithmetic schemes and stacks.
• The approach sets the stage for deeper exploration of random models in arithmetic statistics, particularly for studying the rate of escape from "bad fibers" and the impact of roots of unity, both in number fields and function fields.

Potential for Future Research

• Proving the full non-abelian moment formula over function fields, possibly via integral theory or stably fixed point counts on moduli stacks of covers.
• Extending the model to cases where the base field has roots of unity divisible by $G_{\mathcal{O}^T}(K)$1 and incorporating the additional intricacies predicted in the conjectures of [SW2].
• Further direct computation and verification of the H-moment and probability conjectures for small groups, especially as new computational data becomes available.
• Application to related heuristics for fundamental groups of higher-dimensional varieties and their unramified extensions, leveraging the foundational group-theoretic presentations established herein.

Conclusion

The paper advances the understanding of the structure and distribution of Galois groups of maximal unramified extensions by providing explicit presentations over arbitrary global fields with prescribed local conditions and by formulating a predictive random group model integrating and extending prior heuristics in the area. Both structural theorems and statistical conjectures are rigorously developed, with substantial connections to current areas of active research in arithmetic statistics, arithmetic geometry, and group theory. The work provides a solid platform for further theoretical developments and experimental investigation in the arithmetic of unramified extensions.

Citation: "Presentations of Galois groups of unramified extensions of global fields and its predicted distribution" (2605.14158).