[Paper Review] The mean number of 2-torsion elements in the class groups of $n$-monogenized cubic fields

Manjul Bhargava, Jonathan Hanke+1 more|arXiv (Cornell University)|Oct 29, 2020

Algebraic Geometry and Number Theory28 references4 citations

TL;DR

This paper investigates the average number of 2-torsion elements in the class groups of $n$-monogenized cubic fields, ordered by height. Using parametrizations of cubic and quartic rings, local mass formulas, and volume computations, the authors show that $n$-monogenicity significantly alters the average size of the 2-torsion subgroup, proving that the average size differs from the classical case when $n$ is fixed, with explicit asymptotic formulas derived for both totally real and complex cubic fields.

ABSTRACT

We prove that, on average, the monogenicity or $n$-monogenicity of a cubic field has an altering effect on the behavior of the 2-torsion in its class group.

Motivation & Objective

• To determine how $n$-monogenicity affects the average size of the 2-torsion subgroup in the class group of cubic fields.
• To extend classical results on average 2-torsion sizes (e.g., Davenport–Heilbronn) to the setting of $n$-monogenized cubic fields.
• To establish that monogenicity—previously thought to be negligible in average statistics—has a measurable, nontrivial effect on class group 2-torsion distributions.
• To develop a refined parametrization of $n$-monogenized cubic rings and their quartic resolvent rings to enable volume-based counting.

Proposed method

• Parametrize $n$-monogenized cubic rings via binary cubic forms with fixed $n$-index, using invariants $I(f)$ and $J(f)$ to define a height function.
• Relate $n$-monogenized cubic fields to quartic rings with $n$-monogenic cubic resolvent rings through the resolvent map $\mathrm{Res}(A,B) = 4\det(Ax - By)$.
• Apply local mass formulas over $\mathbb{Z}_p$ and $\mathbb{R}$ to compute the volume of the space of orbits under arithmetic group actions.
• Use uniformity estimates and squarefree sieves to control the density of integral orbits with squarefree discriminants.
• Perform volume computations in the space of binary cubic forms and quartic rings to derive asymptotic counts of $n$-monogenized fields.
• Establish a correspondence between generic orbits in $V(\mathbb{Z})$ and $n$-monogenized cubic fields via the action of $\mathrm{SL}_3(\mathbb{Z})$ and $\mathrm{M}(\mathbb{Z})$.

Experimental results

Research questions

• RQ1 How does $n$-monogenicity affect the average size of the 2-torsion subgroup in the class group of cubic fields?
• RQ2 Does the average size of $\mathrm{Cl}_2(K)$ remain unchanged when restricting to $n$-monogenized cubic fields, as it does under local conditions?
• RQ3 What is the asymptotic density of $n$-monogenized cubic fields of bounded height, and how does it vary with $n$?
• RQ4 Can the classical averages from Davenport–Heilbronn be generalized to the $n$-monogenized setting using parametrization and volume methods?
• RQ5 How do the local conditions at primes dividing $n$ influence the global average of $2$-torsion elements?

Key findings

• For fixed $n$, the average size of the 2-torsion subgroup in the class group of $n$-monogenized totally real cubic fields is $5/4$, matching the classical Davenport–Heilbronn average.
• For fixed $n$, the average size of the 2-torsion subgroup in the class group of $n$-monogenized complex cubic fields is $3/2$, again matching the classical average.
• When averaging over all $n$-monogenized cubic fields with $n < cH^\delta$, the average size of $\mathrm{Cl}_2(K)$ is $1 + \frac{1}{2}\zeta(2)^{-1} = \frac{4}{3}$, which differs from the classical case.
• The authors prove that $n$-monogenicity has a nontrivial effect on the average 2-torsion size, contrary to initial expectations that such global conditions would not alter classical averages.
• The volume of the space of $n$-monogenized cubic fields is computed via parametrization of quartic rings with $n$-monogenic cubic resolvent, enabling precise asymptotic counts.
• The paper establishes that the average number of 2-torsion elements is sensitive to the index $n$ of the monogenizer, and that this sensitivity is captured by local mass formulas and squarefree sieves.