The second moment of the size of the $2$-class group of monogenized cubic fields

TL;DR

This paper establishes upper bounds on the second moments of the 2-class group sizes in monogenized cubic fields, supporting conjectures about monogenicity's influence on class group distributions.

Contribution

It provides the first bounds on the second moments of 2-class groups for monogenized cubic fields, confirming conjectures about their distribution.

Findings

Second moment of 2-class group size is at most 3 for totally real fields.

Second moment of 2-class group size is at most 6 for complex fields.

Second moment of narrow 2-class group size is at most 9 for totally real fields.

Abstract

We prove that when totally real (resp., complex) monogenized cubic number fields are ordered by height, the second moment of the size of the $2$-class group is at most $3$ (resp., at most $6$). In the totally real case, we further prove that the second moment of the size of the narrow $2$-class group is at most $9$. This result gives further evidence in support of the general observation, first made in work of Bhargava--Hanke--Shankar and recently formalized into a set of heuristics in work of Siad--Venkatesh, that monogenicity has an altering effect on class group distributions. All of the upper bounds we obtain are tight, conditional on tail estimates.