Asymptotics of $p$-torsion subgroup sizes in class groups of monogenized cubic fields

TL;DR

This paper provides computational evidence and new conjectures regarding the asymptotic average sizes of p-torsion subgroups in class groups of monogenized cubic fields, extending previous results for p=2.

Contribution

It introduces novel conjectures predicting p-torsion sizes for all primes p in monogenized cubic fields, building on recent asymptotic results.

Findings

Computational evidence supports existing asymptotic results for p=2.

New conjectures proposed for asymptotic p-torsion sizes for all primes p.

Enhanced understanding of class group torsion structures in cubic fields.

Abstract

Bhargava, Hanke, and Shankar have recently shown that the asymptotic average $2$-torsion subgroup size of the family of class groups of monogenized cubic fields with positive and negative discriminants is $3/2$ and $2$, respectively. In this paper, we provide strong computational evidence for these asymptotes. We then develop a pair of novel conjectures that predicts, for $p$ prime, the asymptotic average $p$-torsion subgroup size in class groups of monogenized cubic fields.