On the average $2$-torsion in class groups and narrow class groups of cubic orders with prescribed shape

TL;DR

This paper investigates the average size of 2-torsion in class groups and narrow class groups of cubic fields and orders with specific geometric shape conditions, extending previous methods to new settings.

Contribution

It refines Bhargava and Varma's methods to analyze 2-torsion in class groups of cubic orders with prescribed shape, providing explicit average sizes.

Findings

Average 2-torsion size in class groups: 5/4 for totally real, 3/2 for complex cubic fields.

Average 2-torsion size in narrow class groups: 2 for totally real cubic fields.

Results extend to cubic orders with local conditions at all primes.

Abstract

We study the distribution of $2$-torsion in class groups and narrow class groups of cubic fields and cubic orders subject to prescribed shape conditions. The \emph{shape} of a cubic order in a number field is a natural geometric invariant taking values in the modular surface $\mathbb{H}/\operatorname{GL}_2(\mathbb{Z})$. Fix a subset $W$ of the modular surface with positive hyperbolic measure and boundary of measure zero. Refining the methods of Bhargava and Varma, we prove that among cubic fields with shape in $W$, the average size of the $2$-torsion subgroup of the class group is $5/4$ for totally real fields and $3/2$ for complex fields, while the average size of the $2$-torsion subgroup of the narrow class group for totally real cubic fields is $2$. We also obtain analogous results for cubic orders satisfying prescribed local conditions at all primes.