The Global Cohen-Lenstra Heuristic

TL;DR

This paper extends the Cohen-Lenstra heuristic by defining a probability measure on all finite abelian groups, not just p-groups, using properties generated by uniform conditions, resolving a long-standing open problem.

Contribution

It introduces a new probability measure on all finite abelian groups based on uniform properties, broadening the scope of the Cohen-Lenstra heuristic.

Findings

Defined a probability measure on all finite abelian groups

Extended the heuristic beyond p-groups to all finite abelian groups

Solved an open problem from 1984 regarding the measure's domain

Abstract

The Cohen-Lenstra heuristic is a universal principle that assigns to each group a probability that tells how often this group should occur "in nature". The most important, but not the only, applications are sequences of class groups, which behave like random sequences of groups with respect to the so-called Cohen-Lenstra probability measure. So far, it was only possible to define this probability measure for finite abelian $p$-groups. We prove that it is also possible to define an analogous probability measure on the set of \emph{all} finite abelian groups when restricting to the $\Sigma$-algebra on the set of all finite abelian groups that is generated by uniform properties, thereby solving a problem that was open since 1984.