Cohen-Lenstra heuristics and local conditions

TL;DR

This paper proves function field theorems that support Cohen-Lenstra heuristics for real quadratic fields, explores local conditions' effects on class group distributions, and refines these heuristics with new conjectures and partial proofs.

Contribution

It extends Cohen-Lenstra heuristics to include local conditions and the Picard group, and proves related theorems using homological stability results.

Findings

Support for Cohen-Lenstra heuristics in function fields

Evidence that local conditions do not affect class group distributions

Partial proofs of refined conjectures in number fields

Abstract

We prove function field theorems supporting the Cohen-Lenstra heuristics for real quadratic fields, and natural strengthenings of these analogs from the affine class group to the Picard group of the associated curve. Our function field theorems also support a conjecture of Bhargava on how local conditions on the quadratic field do not affect the distribution of class groups. Our results lead us to make further conjectures refining the Cohen-Lenstra heuristics, including on the distribution of certain elements in class groups. We prove instances of these conjectures in the number field case. Our function field theorems use a homological stability result of Ellenberg, Venkatesh, and Westerland.