[Paper Review] Moments and interpretations of the Cohen-Lenstra-Martinet heuristics

Weitong Wang, Melanie Matchett Wood|arXiv (Cornell University)|Jul 25, 2019

Algebraic Geometry and Number Theory41 references4 citations

TL;DR

This paper refines and clarifies the Cohen-Lenstra-Martinet heuristics for class group distributions in number fields by showing that probabilities are inversely proportional to the number of automorphisms of a slightly enlarged structure—the class triple (class group with decomposition group at infinity). It proves that the conjectured moments of these distributions are exactly $|H^{ ilde{ u}}|^{-1}$, and that these moments uniquely determine the full distribution, providing a deeper theoretical foundation for the heuristics and extending their applicability to non-Galois fields via Hecke module actions and capitulation kernel analysis.

ABSTRACT

The goal of this paper is to prove theorems that elucidate the Cohen-Lenstra-Martinet conjectures for the distributions of class groups of number fields, and further the understanding of their implications. We start by giving a simpler statement of the conjectures. We show that the probabilities that arise are inversely proportional the to number of automorphisms of structures slightly larger than the class groups. We find the moments of the Cohen-Lenstra-Martinet distributions and prove that the distributions are determined by their moments. In order to apply these conjectures to class groups of non-Galois fields, we prove a new theorem on the capitulation kernel (of ideal classes that become trivial in a larger field) to relate the class groups of non-Galois fields to the class groups of Galois fields. We then construct an integral model of the Hecke algebra of a finite group, show that it acts naturally on class groups of non-Galois fields, and prove that the Cohen-Lenstra-Martinet conjectures predict a distribution for class groups of non-Galois fields that involves the inverse of the number of automorphisms of the class group as a Hecke-module.

Motivation & Objective

• To restate and clarify the Cohen-Lenstra-Martinet conjectures in a more accessible probabilistic framework.
• To explain why the predicted probabilities involve both $|H^{ ilde{ u}}|$ and $| ext{Aut}_\Gamma(H)|$, by introducing the concept of a 'class triple'.
• To compute and prove the exact form of the $H$-moments of the Cohen-Lenstra-Martinet distributions, showing they equal $|H^{ ilde{ u}}|^{-1}$.
• To establish that these moments uniquely determine the full distribution, enhancing the predictive power of the conjectures.
• To extend the heuristics to non-Galois number fields by constructing an integral Hecke algebra action and analyzing the capitulation kernel.

Proposed method

• Introduce the notion of a 'class triple'—the class group of a Galois number field equipped with its decomposition group at infinity—to unify the automorphism count in the probability formula.
• Prove that the number of automorphisms of a class triple is exactly $|H^{ ilde{ u}}| \cdot |\text{Aut}_\Gamma(H)|$, explaining the structure of the predicted probabilities.
• Define and compute the $H$-moment of the Cohen-Lenstra-Martinet distribution as $\mathbb{E}[|\text{Sur}_\Gamma(X,H)|] = |H^{\tilde{\nu}}|^{-1}$ for all finite $\Gamma$-modules $H$ with $H^\Gamma = 1$.
• Construct an integral model of the Hecke algebra of a finite group $\Gamma$, and show it acts naturally on class groups of non-Galois fields via the capitulation kernel.
• Use the capitulation kernel to relate class groups of non-Galois fields to Galois fields, enabling the application of the conjectures beyond Galois extensions.
• Prove that the Cohen-Lenstra-Martinet conjectures predict a distribution for non-Galois fields where the probability is inversely proportional to $|\text{Aut}_\Gamma(H)|$ when $H$ is viewed as a Hecke module.

Experimental results

Research questions

• RQ1 Why do the Cohen-Lenstra-Martinet probabilities include a factor of $|H^{\tilde{\nu}}|$ in addition to $|\text{Aut}_\Gamma(H)|$?
• RQ2 What is the precise form of the $H$-moments predicted by the Cohen-Lenstra-Martinet heuristics for class groups of $\Gamma$-fields?
• RQ3 Can the moments of the Cohen-Lenstra-Martinet distributions uniquely determine the full distribution?
• RQ4 How can the Cohen-Lenstra-Martinet conjectures be extended to non-Galois number fields?
• RQ5 What role does the capitulation kernel play in connecting class groups of non-Galois fields to Galois fields?

Key findings

• The probabilities in the Cohen-Lenstra-Martinet conjectures are inversely proportional to the number of automorphisms of the class triple, which combines the class group and decomposition group at infinity.
• The $H$-moment of the Cohen-Lenstra-Martinet distribution is exactly $|H^{\tilde{\nu}}|^{-1}$, where $H$ is a finite $\Gamma$-module with $H^\Gamma = 1$.
• The moments uniquely determine the Cohen-Lenstra-Martinet distribution, establishing a strong theoretical foundation for the conjectures.
• The paper constructs an integral Hecke algebra action on class groups of non-Galois fields, enabling the extension of the heuristics beyond Galois extensions.
• The distribution for class groups of non-Galois fields predicted by the conjectures involves the inverse of the number of automorphisms of the class group as a Hecke module.
• The rank of the capitulation kernel is shown to be computable via the Galois closure, allowing consistent distributional predictions across field extensions.