# Equidistribution of shapes of complex cubic fields of fixed quadratic resolvent

## TL;DR

This paper proves that the shapes of complex cubic fields become uniformly distributed along certain geodesics on the modular surface as their discriminant increases, and that these shapes uniquely identify the fields.

## Contribution

It establishes the equidistribution of shapes of complex cubic fields along geodesics and shows that the shape is a complete invariant for these fields.

## Key findings

- Shapes lie on geodesics defined by the trace-zero form.
- Shapes become equidistributed with respect to hyperbolic measure as discriminant grows.
- Shape uniquely determines the complex cubic field within the family.

## Abstract

We show that the shape of a complex cubic field lies on the geodesic of the modular surface defined by the field's trace-zero form. We also prove a general such statement for all orders in \'etale Q-algebras. Applying a method of Manjul Bhargava and Piper H to results of Bhargava and Ariel Shnidman, we prove that the shapes lying on a fixed geodesic become equidistributed with respect to the hyperbolic measure as the discriminant of the complex cubic field goes to infinity. We also show that the shape of a complex cubic field is a complete invariant (within the family of all cubic fields).