Lower order terms in the shape of cubic fields

TL;DR

This paper proves the equidistribution of lattice shapes of cubic fields ordered by discriminant, providing a detailed spectral analysis and addressing a known barrier to equidistribution.

Contribution

It offers a new spectral estimate with a lower order main term and distinguishes contributions of reducible and irreducible forms using Shintani's method.

Findings

Establishes equidistribution of cubic field shapes in a spectral framework

Provides a precise geometric and spectral description of the equidistribution barrier

Includes a detailed analysis of reducible and irreducible binary cubic forms

Abstract

We demonstrate equidistribution of the lattice shape of cubic fields when ordered by discriminant, giving an estimate in the Eisenstein series spectrum with a lower order main term. The analysis gives a separate discussion of the contributions of reducible and irreducible binary cubic forms, following a method of Shintani. Our work answers a question posed at the American Institute of Math by giving a precise geometric and spectral description of an evident barrier to equidistribution in the lattice shape.