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

Anwesh Ray

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

Anwesh Ray

TL;DR

The paper addresses how the average size of $2$-torsion in class groups and narrow class groups behaves for cubic fields and cubic orders when conditioned on a geometric shape. It combines Bhargava’s parametrizations (via binary cubic forms and pairs of ternary quadratic forms) with equidistribution of shapes on the modular surface $\mathcal{S}_2$ and volume-based counting to show that shape restrictions do not alter the limiting averages; it also integrates local congruence conditions through $p$-adic masses and an Euler product to obtain refined asymptotics. The main findings give explicit averages: $\mathbb{E}|\mathrm{Cl}_2(K)|=5/4$ for totally real and $3/2$ for complex cubic fields, and $\mathbb{E}|\mathrm{Cl}_2^+(K)|=2$ for totally real fields; analogous results hold for cubic orders with refinements such as $\mathbb{E}\big(|\mathrm{Cl}_2(O)|-\tfrac{1}{4}|\mathcal{I}_2(O)|\big)=1$ in the totally real case. These results demonstrate the robustness of arithmetic statistics under natural geometric constraints and reinforce the compatibility between algebraic invariants (class groups and their torsion) and geometric data (shapes) in the cubic setting.

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.

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

TL;DR

The paper addresses how the average size of

-torsion in class groups and narrow class groups behaves for cubic fields and cubic orders when conditioned on a geometric shape. It combines Bhargava’s parametrizations (via binary cubic forms and pairs of ternary quadratic forms) with equidistribution of shapes on the modular surface

and volume-based counting to show that shape restrictions do not alter the limiting averages; it also integrates local congruence conditions through

-adic masses and an Euler product to obtain refined asymptotics. The main findings give explicit averages:

for totally real and

for complex cubic fields, and

for totally real fields; analogous results hold for cubic orders with refinements such as

in the totally real case. These results demonstrate the robustness of arithmetic statistics under natural geometric constraints and reinforce the compatibility between algebraic invariants (class groups and their torsion) and geometric data (shapes) in the cubic setting.

Abstract

We study the distribution of

-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

. Fix a subset

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

, the average size of the

-torsion subgroup of the class group is

for totally real fields and

for complex fields, while the average size of the

-torsion subgroup of the narrow class group for totally real cubic fields is

. We also obtain analogous results for cubic orders satisfying prescribed local conditions at all primes.

Paper Structure (10 sections, 12 theorems, 52 equations)

This paper contains 10 sections, 12 theorems, 52 equations.

Introduction
Motivation and historical context
Main results
Organization
Preliminary notions
Shapes of lattices
Parameterization of order $2$ ideals in cubic orders
Class groups and narrow class groups
Volume computations
Congruence conditions

Key Result

Theorem A

Let $W \subset \mathcal{S}_2$ be a measurable subset with positive measure whose boundary is of measure zero. Fix any finite set of primes and impose arbitrary local conditions at these primes. When cubic fields with shape lying in $W$ and satisfying these local conditions are ordered by absolute di

Theorems & Definitions (24)

Theorem A
Theorem B
Theorem C
Theorem 2.1: Delone--Faddeev, Gan--Gross--Savin
proof
Theorem 2.2: Bhargava
proof
Theorem 3.1
proof
Theorem 3.2
...and 14 more

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

TL;DR

Abstract

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

Authors

TL;DR

Abstract

Table of Contents

Key Result

Theorems & Definitions (24)