On class groups of random number fields

TL;DR

This paper challenges existing heuristics on class groups of number fields, providing counterexamples, reformulations, and a new conjecture to better understand their distribution.

Contribution

It discredits the Cohen--Lenstra--Martinet heuristics, offers corrected versions, and reformulates them using Arakelov class groups.

Findings

Disproof of the Cohen--Lenstra--Martinet heuristics

Reformulation using Arakelov class groups

Proposal of a new, rigorous conjecture

Abstract

The main aim of the present paper is to disprove the Cohen--Lenstra--Martinet heuristics in two different ways and to offer possible corrections. We also recast the heuristics in terms of Arakelov class groups, giving an explanation for the probability weights appearing in the general form of the heuristics. We conclude by proposing a rigorously formulated Cohen--Lenstra--Martinet conjecture.