Random matrices, the Cohen-Lenstra heuristics, and roots of unity

TL;DR

This paper explores the distribution of ideal class groups in number field extensions, proposing a new heuristic based on random matrices that accounts for roots of unity, improving upon previous conjectures.

Contribution

It introduces a random matrix heuristic from function fields that refines the Cohen-Lenstra-Martinet heuristics, especially in cases with roots of unity.

Findings

Malle's numerical evidence suggests previous heuristics are inaccurate with roots of unity.

A new heuristic from function fields aligns better with observed data.

The paper proposes a generalized conjecture extending Malle's work.

Abstract

The Cohen-Lenstra-Martinet heuristics predict the frequency with which a fixed finite abelian group appears as an ideal class group of an extension of number fields, for certain sets of extensions of a base field. Recently, Malle found numerical evidence suggesting that their proposed frequency is incorrect when there are unexpected roots of unity in the base field of these extensions. Moreover, Malle proposed a new frequency, which is a much better match for his data. We present a random matrix heuristic (coming from function fields) that leads to a function field version of Malle's conjecture (as well as generalizations of it).